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Chapter 3 Basic Principles of Open Channel Hydraulics
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CHAPTER 3 B A S I C PRINCIPLES O F OPEN CHANNEL HYDRAULICS 3.1
INTRODUCTION
I n examining h y d r a u l i c p r o c e s s e s on a l l u v i a l f a n s , it i s n e c e s s a r y f o r t h e non-engineer t o become f a m i l i a r w i t h some of t h e b a s i c p r i n c i p l e s o f open-channel flow: and f o r t h e e n g i n e e r w i t h a background i n f l u i d m e c h a n i c s a n d h y d r a u l i c e n g i n e e r i n g t o re-examine and r e - c o n s i d e r some o f t h e b a s i c p r i n c i p l e s o f t h e s e t e c h n i c a l a r e a s from t h e v i e w p o i n t of a l l u v i a l f a n s . I t i s t h e p u r p o s e of t h i s chaptertoprovideabriefand rudimentarytreatmentofseveral o f t h e b a s i c p r i n c i p l e s of open-channel h y d r a u l i c s . I n p a r t i c u l a r , t h i s chapterwillconsiderspecificenergy:normaloruniformflowonmild, c r i t i c a l , and s t e e p s l o p e s ; t h e s t a b i l i t y of a l l u v i a l c h a n n e l s ; and t h e mechanics of d e b r i s flows. Although s p e c i f i c r e f e r e n c e s w i l l be mentioned t h r o u g h o u t t h i s c h a p t e r , t h e r e a r e a number of b a s i c r e f e r e n c e s which p r o v i d e a comprehensive d i s c u s s i o n of t h e s e s u b j e c t s ; see f o r example, Chow ( 1 9 5 9 ) , Henderson ( 1 9 6 6 ) , French ( 1 9 8 5 ) , and Graf ( 1 9 7 1 ) . 3.2
S P E C I F I C ENERGY
By d e f i n i t i o n , t h e s p e c i f i c energy of a one d i m e n s i o n a l openchannel flow r e l a t i v e t o t h e bottom of t h e c h a n n e l is (3.2.1)
w h e r e E = s p e c i f i c e n e r g y ( m ) ; y = d e p t h o f flow ( m ) ; a = k i n e t i c e n e r g y c o r r e c t i o n f a c t o r which c o r r e c t s f o r t h e non-uniformity of t h e v e l o c i t y p r o f i l e ( d i m e n s i o n l e s s ) ; u = Q/A = a v e r a g e v e l o c i t y of flow (rn/s); Q = channel v o l u m e t r i c flow r a t e ( m 3 / s ) ; A = flow a r e a ( m 2 ) ; and g = a c c e l e r a t i o n of g r a v i t y ( m / s 2 ) . Equation ( 3 . 2 . 1 ) d e r i v e s d i r e c t l y from t h e B e r n o u l l i energy e q u a t i o n (law of c o n s e r v a t i o n of energy) and i s s u b j e c t t o a number of assumptions. Primary among t h e s e a s s u m p t i o n s i s t h a t c o s 8 Il o r t h a t 8 < 1 0 ' where 8 = s l o p e a n g l e of t h e c h a n n e l . For a more complete d e r i v a t i o n of Equation ( 3 . 2 . 1 ) , t h e r e a d e r i s referred t o French (1985) o r Streeter and Wylie ( 1 9 7 5 ) . Examination o f Equation ( 3 . 2 . 1 ) f o r t h e c a s e t h a t a = 1, Q i s known, a n d t h e c h a n n e l g e o m e t r y i s d e f i n e d d e m o n s t r a t e s t h a t E i s o n l y afunctionofthedepthofflow, y , F i g s . 3 . 2 . 1 . For a s p e c i f i e d Q a n d
83
Y
yz
FIG.
3.2.1
S p e c i f i c energy c u r v e .
a d e f i n e d channel geometry, i f y is p l o t t e d a s a f u n c t i o n of E, F i g . 3.2.lb, t h e r e s u l t i s a hyperbolawithbranches i n t h e f i r s t andthird q u a d r a n t s . T h e b r a n c h o f t h e h y p e r b o l a i n t h e t h i r d q u a d r a n t i s of no i n t e r e s t s i n c e it r e p r e s e n t s n e g a t i v e v a l u e s of b o t h E and y , which have no p h y s i c a l meaning. The branch o f t h e hyperbola i n t h e f i r s t q u a d r a n t h a s two limbs. The limb AC of t h i s branch i s asymptotic t o On t h e t h e E a x i s , and t h e limb AB is a s y m p t o t i c t o t h e l i n e y = E . curveshown i n F i g . 3 . 2 . l b , t h e p o i n t l a b e l e d A r e p r e s e n t s t h e m i n i m u m s p e c i f i c energy r e q u i r e d t o p a s s t h e flow Q through t h e d e f i n e d c h a n n e l . The c o - o r d i n a t e s of t h e p o i n t l a b e l e d A can b e found by t a k i n g t h e f i r s t d e r i v a t i v e of Equation ( 3 . 2 . 1 ) w i t h r e s p e c t t o t h e d e p t h o f flow and s e t t i n g t h e r e s u l t e q u a l t o z e r o o r Q2
E = y + 2gA2 dE
dA
- 0
(3.2.2)
With r e f e r e n c e t o F i g . 3 . 2 . l a , t h e d i f f e r e n t i a l flow a r e a d A n e a r t h e f r e e s u r f a c e can be approximated by dA
I
Tdy
84
or
dA _ -- =
T
dY With t h i s a p p r o x i m a t i o n f o r dA/dy E q u a t i o n ( 3 . 2 . 2 ) becomes (3.2.3) where D = A/T = h y d r a u l i c d e p t h .
Rearrangement o f E q u a t i o n ( 3 . 2 . 3 )
yields (3.2.4) o r d e f i n i n g t h e Froude number
(3.2.5) where F = Froude number and E q u a t i o n ( 3 . 2 . 5 ) d e f i n e s what i s termed
criticalflow;thatis,theminimumamountofspecificenergyrequired
t o p a s s t h e flow
Q t h r o u g h t h e c h a n n e l geometry s p e c i f i e d .
With
r e g a r d t o t h e above development and F i g s . 3 . 2 . 1 , t h e f o l l o w i n g s h o u l d be n o t e d . F i r s t , i n a c h a n n e l of l a r g e s l o p e a n g l e 6 and a f 1, it can beeasilydemonstratedthat t h e c r i t e r i o n forminimum s p e c i f i c e n e r g y o r c r i t i c a l flow is (3.2.6)
Second, i n F i g . 3 . 2 . l b , t h e E-y curves f o r f l o w r a t e s g r e a t e r t h a n Q
l i e t o t h e r i g h t o f t h e curve BAC; and E-y curves f o r f l o w r a t e s l e s s t h a n Q l i e t o t h e l e f t o f c u r v e BAC. T h i r d , by d e f i n i t i o n , f l o w s f o r which F > 1 a r e termed s u p e r c r i t i c a l and l i e on l i m b AC o f t h e E-y
c u r v e ; f l o w s f o r which F
= 1 a r e termed c r i t i c a l and o c c u r a t p o i n t A;
and f l o w s f o r w h i c h F < l a r e t e r m e d s u b c r i t i c a l and l i e o n b r a n c h ABof t h e E-y c u r v e .
Fourth, with t h e exception of P o i n t A, t h e c r i t i c a l
point, everyvalueofE suchthatE > E ,
( c r i t i c a l s p e c i f i c energy) has
a s s o c i a t e d w i t h it t w o p o s s i b l e d e p t h s o f f l o w . 3.2.lb,
i f t h e s p e c i f i c energy of
Forexample, i n F i g .
flow i s E l ,
then t h e l i n e DF
85
demonstrates that there are two possible depths of flow; y1 and y2. FLoodplaln/Ovarbank
Floodplaln/Overbank
7
7
Sectlon
Sectlon
"/'\
Primary
' - ! ! Channel - - , I
FIG. 3.2.2
Schematic of a channel of compound section.
The depth of flow y 1 is associated with a supercritical flow having a specific energy E l ; and the depth of flow y, is associated with a subcritical flow which also has a specific energy E,. The possible depths of flow y1 and y, are known as the alternate depths of flow. Fifth, in channels of compound section, Fig. 3.2.2, the problem of specific energy and the computation of the correct Froude number becomes complex. Although a discussion of this area of study is beyond the scope of this treatment, the interested reader may find a summary in French (1985) and specific discussions in Blalock (1980), Blalock and Sturm (1981), Petryk and Grant (1975), and Konemann
.
(1982)
The comment in the foregoing paragraph regarding alternate depths of flow leads directly to a consideration of the problem of accessability and controls. Through the terminology of accessability, it is meant to indicate that with Q and the chann'el geometry given a priori a means of deciding which of the alternate depths on the E-y curve is accessable must be found. The result is the identificationoftheactual downstreamdepthofflowthat is possible with the specified upstream conditions. Accessability arguments appeal to logic rather than mathematical proof. As an example of such arguments, consider a rectangular channel of constantwidthb which conveys a steady flowQ. In the otherwise horizontal bed of the channel, there is a smooth upward step of height 42, Fig. 3.2.3a. Given this situation, an E-y curve can be constructed, Fig. 3.2.3b. In this figure, assume that the flowatsection1inFig. 3.2.3ais represented by point Aon theEy curve. Note, the point A' has the same specific energy as point A, and the choice of point A as the starting point derives from prior or given knowledge of the Froude number at section 1. The choice of
86
p o i n t A i n d i c a t e s t h a t F < 1; and hence, t h e flow a t s e c t i o n 1 is s u b c r i t i c a l . T h e l o c a t i o n of t h e s p e c i f i c energy on t h i s curve r e p r e s e n t i n g s e c t i o n 2 can be determined by t h e a p p l i c a t i o n of t h e Bernoulli energy equation between s e c t i o n s 1 and 2 o r - 2
- 2
u2
U1
2g + y 1
= -
2g
+
y2 +
AZ
E l = E2 + A Z E2 = El
-
AZ
where i t i s a s s u m e d t h a t t h e e n e r g y d i s s i p a t i o n b e t w e e n s e c t i o n s l a n d 2 isnegligible. Havingmathematically d e t e r m i n e d t h e v a l u e o f E 2 w e must now determine t h e c o r r e c t depth of flow a t s e c t i o n 2 . T h e w i d t h of t h e channel, by d e f i n i t i o n , does n o t change: and t h e r e f o r e , t h e l l f l o w p o i n t l v c a n o n l y m o v e o n t h e c u r v e d e f i n e d i n F i g .3.2.333. Ifthe w i d t h of t h e channel v a r i e d , t h e n flow p e r u n i t w i d t h would vary and t h e flow p o i n t could move o f f t h e curve shown i n Fig. 3.2.3b and onto t h e foregoing o t h e r E-y curves. With regard t o F i g . 3.2.3b, statement means t h a t t h e flow p o i n t cannot Ivjurnp1* from p o i n t B t o B 1 Thus, i f p o i n t B' is t o b e a n a c c e s s a b l e s o l u t i o n o f t h i s problem, t h e flow p o i n t must p a s s through p o i n t C. Movement of t h e flow p o i n t t o pointC i s o n l y p o s s i b l e i f t h e increase i n c h a n n e l b o t t o m e l e v a t i o n i s g r e a t e r t h a n t h a t s p e c i f i e d . This h y p o t h e t i c a l s i t u a t i o n i s r e p r e s e n t e d by a dashed l i n e i n Fig. 3 . 2 . 3 a . Therefore, t h e conclusion i s t h a t of t h e p o s s i b l e d e p t h s of flow a t s e c t i o n 2 , only t h e depth of flow a s s o c i a t e d with p o i n t B i s accessable from p o i n t A with t h e upstream flow c o n d i t i o n s a s s p e c i f i e d . The foregoing d i s c u s s i o n t a c i t l y assumes t h a t t h e r e i s a s o l u t i o n t o t h e problem with t h e upstream c o n d i t i o n s a s s p e c i f i e d . I n f a c t , it is q u i t e easy t o s p e c i f y a s t e p s i z e such t h a t a s o l u t i o n with t h e given upstream c o n d i t i o n s does n o t e x i s t . For example, i n Fig. 3.2.3a, i f t h e s t e p s i z e e x c e e d s ~ z , ,t h e n t h e r e i s n o s o l u t i o n t o t h e problem. T h i s s i t u a t i o n is p h y s i c a l l y explained by n o t i n g t h a t when t h e s t e p s i z e exceeds A Z , t h e channel h a s been s u f f i c i e n t l y o b s t r u c t e d by t h e s t e p so t h a t t h e upstream s p e c i f i c energy i s not adequate t o p a s s t h e flow. That is, t h e flow has been choked by t h e s t e p , I n s u c h a s i t u a t i o n , t h e flowupstreamofthestepmust increase indepthuntilthespecificenergyavailableupstreamissufficientto p a s s t h e flow over t h e o b s t r u c t i o n .
.
87
The following comments regardingspecificenergyshouldbemade. First , in the vicinity of the critical point , point C in Figs. 3.2.3 , Y
E
FIGS. 3.2.3
(a) (b) Schematic definition of the accessability problem.
small changes in specific energy can result in large changes in the depth of flow. Second, as noted previously in this section, in channels of compound section, the problemsof specificenergy andthe locationofthecriticalpointorpointscanbecomesignificantlymore complex and difficult. Third, the foregoing paragraphs have provided only a synoptic discussion of specific energy. For a more detailed discussion of this topic, the reader is referred to either French (1985) or Henderson (1966). 3.3
UNIFORM/NORMAL FLOW By definition, uniform flow occurs when 1. 2.
The depth, flow area, and velocity at every channel cross section are constant: and the energy grade line, water surface, and channel bottom are all parallel.
Although uniform flow is a theoretical ideal which can only occur in very long, straight channels of a fixed geometry, this theoretical concept can be used with great effect in the solution of very applied problems. One of the equations that describes uniform flow is the Manning equation or
88
(3.3.1) where Q = flow rate (m3/s): A = flow area (m2); R = A/P = hydraulic radius (m): P = wetted perimeter (m): S = slope of the energy grade line, water surface, or channel bottom: n = Manning roughness coefficient; and $ = a coefficient used to account for the system of units used. Ifthe SI system ofunits specifiedabove isused, then $ If both = 1; if the English system of units is used, then $ = 1.49. sides of Equation (3.3.1) are divided by the flow area, then a second form of the Manning equation results or (3.3.2) where
u
= average velocity of flow (m/s if $ = 1.0) InEquations (3.3.1) and (3.3.2)thecoefficientncharacterizes the roughness of the channel boundary and is a well established coefficient for estimating boundary friction effects. There are a number of methods for estimating values of n in perennial channels: see for example, Barnes (1967), Chow (1959), French (1985), Henderson Although some of the methods used for (1966), and Urquhart (1975). perennial channels are applicable to ephemeral channels on alluvial fans, it is worth noting several semi-empirical methods which have b e e n d e v e l o p e d t o e s t i m a t e v a l u e s o f n from amechanical sizeanalysis of the materials which compose the channel boundary. Perhaps the best known semi-empirical method of estimating n is t h a t p r o p o s e d b y S t r i c k l e r i n 1 9 2 3 ; see for exampleSimons andsenturk (1976), which hypothesized that
n = 0.047d'/6
(3.3.3)
where d = diameter in millimeters of the uniform sand pasted to the sides and bottom of an experimental flume used by Strickler. While Equation (3.3.3) has limited validity in natural channels, it specifies a basic functional relationship between n and d. Among the results functionally similar to Equation (3.3.3) are: 1/6
n = 0.013d
(3.3.3a) 65
where d6 = diameter in millimeters such that 65% of the channel
89
boundary material, by weight, is smaller, Raudkivi (1976). 1/6
n = 0.039d
(3.3.3b) 30
whered,, =diameter infeetsuchthat50% ofthechannelboundary material by weight is smaller, Garde and Raju (1978). If the SI system of units is used, then Equation (3.3.335) becomes 1/6
n = 0.047d
(3.3.3c) 50
where d is in meters, Subramanya (1982). 1/6
n = 0.038d
(3.3.3d) 90
where d,, = diameter in meters such that 90% of the channel boundary material by weight is smaller, Meyer-Peter and Muller (1948). Equation (3.3.34) wasdeveloped for channels withbeds formed of mixed materials with a significant proportion of coarse grained sizes. 1/6
n = 0.026d
(3.3.3e) 75
where d,, = diameter in inches such that 75% of the channel boundarymaterialbyweight issmaller, Laneand Carlson (1953). Equation (3.3.3e) was derived from experiments in which the channels were paved with cobbles. Of the above equations for estimating a value of Manning's n, in the case of alluvial fans Equation (3.3.34) is perhaps the most appropriate. Jarrett (1984) recognized that the available guidelines for estimating resistance coefficients for high-gradient; that is, channelswithslopesgreaterthan0.002, arebased onlimited dataand are handicapped by a lack of easily applied methods for evaluating changes of boundary resistance with the depth of flow. Jarrett (1984) exa.mined 21 high-gradient streams in the Rocky Mountains with stable beds andminimallyvegetatedbanks and developed an empirical equation for n or
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n =
0.3g~0.38~-0*16
(3.3.4)
w h e r e R = h y d r a u l i c r a d i u s i n feet. W h i l e t h e s t r e a m s u s e d b y J a r r e t t (1984) t o develop Equation ( 3 . 3 . 4 ) had s l o p e s s i m i l a r t o t h o s e found on a l l u v i a l f a n s , t h e channel beds were composed of large c o b b l e s a n d boulders whichmay ormay n o t b e t y p i c a l of channels found o n a l l u v i a l f a n s . F u r t h e r , flows i n t h e channels s t u d i e d by t h i s i n v e s t i g a t o r were subcritical while flows on a l l u v i a l f a n s a r e g e n e r a l l y b e l i e v e d t o be c r i t i c a l o r s u p e r c r i t i c a l . I n Equation (3.3.1) , t h e parameter AR2 l 3 is known as t h e s e c t i o n f a c t o r , and f o r a channel of s p e c i f i e d geometry where A R 2 I 3 always i n c r e a s e s w i t h i n c r e a s i n g d e p t h s o f flow, each f l o w r a t e o r d i s c h a r g e hasacorrespondinguniquedepthof f l o w a t whichuniform f l o w o c c u r s . Examination of Equation ( 3 . 3 . 1 ) demonstrates t h a t f o r a uniform flow t h e flow r a t e i s a f u n c t i o n of 1) t h e geometric shape of t h e channel, 2 ) a boundary r e s i s t a n c e c o e f f i c i e n t such a s Manning's n, 3 ) t h e l o n g i t u d i n a l s l o p e of t h e channel, and 4 ) t h e depth of flow o r
Q = f(r,n,s,y,,)
(3.3.5)
where 'I = shape f a c t o r and Y,, = normal depth of flow. I f f o u r of t h e f i v e v a r i a b l e s i n Equation (3.3.5) a r e known, t h e n t h e v a l u e of t h e f i f t h v a r i a b l e can be estimated. Although a d e t a i l e d d i s c u s s i o n of t h e computation of uniform flow i s beyond t h e scope of t h i s book, t h e i n t e r e s t e d r e a d e r is r e f e r r e d t o French (1985) f o r a complete d i s c u s s i o n of t h i s t o p i c . I f Q , n, y,, a n d t h e channelgeometry a r e s p e c i f i e d , thenEquation ( 3 . 3 . 1 ) can be solved e x p l i c i t l y f o r t h e s l o p e of t h e channel which a l l o w s t h e f l o w t o o c c u r a t anormal depth. B y d e f i n i t i o n , t h i s s l o p e wouldbethenormal slope. I f t h e s l o p e o f t h e c h a n n e l i s v a r i e d w h i l e Q and n are h e l d c o n s t a n t , t h e n it is p o s s i b l e t o determine a s l o p e value s u c h t h a t normal f l o w w i l l o c c u r w i t h F = l ; t h a t i s , normal depth w i l l a l s o be c r i t i c a l depth. T h i s slopewould betermed t h e l i m i t i n g c r i t i c a l s l o p e . Slopes g r e a t e r t h a n t h e l i m i t i n g c r i t i c a l s l o p e a r e termed s t e e p s l o p e s and s u s t a i n s u p e r c r i t i c a l flow. 3.4
ALLUVIAL CHANNEL STABILITY
Before c o n s i d e r i n g t h e p r o c e s s e s a f f e c t i n g channel s t a b i l i t y and sediment t r a n s p o r t i n an a l l u v i a l channel e x p l i c i t l y , l e t us c o n s i d e r t h e t r a c t i v e f o r c e method of d e s i g n i n g unlined channels. Scour o r e r o s i o n on t h e perimeter of a channel occurs when t h e particleswhichcomposethe channel p e r i m e t e r a r e s u b j e c t e d t o f o r c e s
91
which are of s u f f i c i e n t magnitude t o cause movement.
The p a r t i c l e s
r e s t i n g o n t h e b o t t o m o f t h e channel a r e s u b j e c t t o o n l y flowgenerated f o r c e s while t h e material on t h e s l o p i n g s i d e s of t h e channel is s u b j e c t e d t o both flow generated f o r c e s and a g r a v i t a t i o n a l f o r c e which a c t s t o r o l l t h e p a r t i c l e s down t h e s l o p i n g channel s i d e . I n uniform flow, it can be e a s i l y shown; see f o r example, French (1985), t h a t t h e t r a c t i v e f o r c e on p a r t i c l e s due t o flow can be approximated a s F, = YALS
(3.4.1)
where F, = f l o w g e n e r a t e d t r a c t i v e f o r c e , Y = f l u i d s p e c i f i c weight, A = flow a r e a , L = l o n g i t u d i n a l l e n g t h considered, a n d S = l o n g i t u d i n a l s l o p e of t h e channel. I f Equation (3.4.1) i s accepted, t h e n t h e u n i t t r a c t i v e f o r c e is YALS To=--
PL
-
YRS
(3.4.2)
where r 0 = average v a l u e of t h e t r a c t i v e f o r c e ( t h i s i s a c t u a l l y a stress r a t h e r t h a n a f o r c e ) p e r u n i t of wetted a r e a and P = w e t t e d perimeter. I n wide channels, y w I R , and Equation (3.4.2) becomes To =
vy,s
(3.4.3)
A t t h i s p o i n t , it s h o u l d b e n o t e d t h a t inchannelsoftrapezoidalcross
s e c t i o n t h e maximum t r a c t i v e f o r c e on t h e bed has been found t o be approximately yy,S and on t h e sides 0 . 7 6 ~ ~ ~ s . When p a r t i c l e s on t h e perimeter of t h e channel a r e i n a s t a t e of impending motion, t h e f o r c e s a c t i n g t o cause t h e motion a r e i n e q u i l i b r i u m with t h e f o r c e s opposing motion. I n t h e c a s e of p a r t i c l e s on t h e bottom of t h e channel, t h e f o r c e causing motion i s A e r L where r L = u n i t t r a c t i v e f o r c e on a l e v e l s u r f a c e and A, = e f f e c t i v e a r e a on which r L o p e r a t e s . The f o r c e r e s i s t i n g motion i s t h e g r a v i t a t i o n a l forceorW,tanawhereW, =submergedparticleweight and a = t h e angle of repose of t h e p a r t i c l e . Then, when motion i s incipient A e r L = W,tana
or
(3.4.4)
92
T~
=
ws
- tana
(3.4.5)
*e
On the sloping sides of the channel, particles are subject to both a flow generated tractive force T ~ A , and a downslope gravitational component W,sinr where 7 s = tractive force on side slopes and r = side slope angle. The resultant forceactingtocause motion is
i +
(w,tanr)'
(A,T,)'
The force resisting motion is W,cosrtana, and setting these forces equal for the case of incipient motion W,(cosr) (tana) =
4 (W,tanr) ' +
or
zS =
WE
- (cosr)(tana) Ae
1
1
-
tan2r
tan'a
Equations (3.4.5) and ( 3 . 4 . 7 ) tractive force ratio K, or
7L
sin'a
(Ae7,)
(3.4.6)
(3.4.7)
are usually combined to form the
(3.4.8)
The tractive force ratio is a function of both the side slope angle and the angle of repose of the material which composes the perimeter. Laboratory data are available for bothcohesive andnoncohesive materials, Lane (1955) or French (1985). F r o m t h e c o n c e p t o f t r a c t i v e forcechanneldesign,theconcept of a stable hydraulic section is derived. In the design of a channel section, usually trapezoidal in shape, the tractive force is equal to its permissablevalue ononly aportion ofthe perimeter- usuallythe sides. It seems rational toattempt todefine achannel sectionsuch that incipient particle motion prevails at all points on the channel perimeter. Glover and Florey (1951) did this for a channel carrying clear water through non-cohesive materials. The specific
93
Schematic, d e f i n i t i o n of v a r i a b l e s f o r a s t a b l e h y d r a u l i c channel i n non-cohesive materials. F I G . 3.4.,1
assumptions made by t h e s e i n v e s t i g a t o r s i n t h e i r a n a l y s i s were: 1.
B o u n d a r y p a r t i c l e s a r e h e l d i n p l a c e b y t h e submergedweight component of t h e p a r t i c l e s a c t i n g i n a d i r e c t i o n normal t o t h e bed of t h e channel.
2.
A t andabovethewater surface, t h e s i d e s l o p e o f thechannel
is a t t h e a n g l e of repose of t h e material composing t h e channel perimeter. 3.
A t t h e c e n t e r l i n e of t h e channel s e c t i o n , t h e channel s i d e
s l o p e is z e r o , and t h e t r a c t i v e f o r c e generated by t h e flow is a l o n e s u f f i c i e n t t o maintain a s t a t e of i n c i p i e n t p a r t i c l e motion. 4.
Atpointsbetweenthecenterlineofthechanneland i t s edge, p a r t i c l e s a r e i n a s t a t e of i n c i p i e n t motion.
5.
The t r a c t i v e f o r c e a c t i n g on an a r e a of t h e channel is equal t o t h e component of t h e weight of t h e water above t h e a r e a a c t i n g i n t h e d i r e c t i o n o f flow: t h a t i s , t h e r e is n o l a t e r a l t r a n s f e r of t r a c t i v e f o r c e .
Giventheaboveassumptions a n d t h e s c h e m a t i c d e f i n i t i o n o f v a r i a b l e s i n Fig. 3 . 4 . 1 , it can be shownthat t h e g e o m e t r i c s h a p e o f t h e c h a n n e l of s t a b l e h y d r a u l i c s e c t i o n is given by
94
y = y w cos
[1'
(3.4.9)
the flow area by A=-
2TYN
(3.4.10)
I(
and the wetted perimeter by
2YN
E(sin
p = -
sin
(3.4.11)
a)
a
where the symbolism E(sin a) designates a complete elliptic integral of the second kind. The discharge of this stable channel is then estimated directly from the Manning equation or 8/3
Q -
2.98 y N n(tan
a)
(cOS
,)'I3
[E(sin
&
a)]'13
(3.4.12)
It is worth noting that the tractive force methodology represents onlyone techniquewhich canbe usedto designor analyzea channel in non-cohesive alluvial materials. Other methodologies have equal validity and in some cases have particular advantages. For example, there isthethresholdofmovementhypothesisof Shields; see for example, Henderson (1966) or French (1985). This hypothesis is stated in terms of two dimensionless variables R, =
u*d -
(3.4.13)
V
and (3.4.14)
where R, = a Reynolds number based on the shear velocity of the flow, the size of the particles composing the perimeter, and the kinematic viscosity of the fluid;
95
I -
U* I:
4 gRS
(3.4.15)
=shearvelocity, v = fluid kinematicviscosity; S, = specificgravity ofthe soil particles composingthe channel perimeter (S, = 2.65 inthe general case) ; and d = diameter of the soil particles composing the channel perimeter. d is usually taken as the diameter such that 25% of the particles, measured by weight, are larger. Shields' results are usually summarized in a graphical form, Fig. 3.4.2. In this figure, the curve denotes and defines the threshold of particle movement.
-i
SHIELD'S DIAGRAM
Particle Movement
No Particle Movement
FIG. 3.4.2 number.
Threshold of movement as a function of particle Reynolds
A second alternative to the tractive force methodology is the regime theory. Although this theory has beenvigorously attackedby modern engineers because of its lack of a rational and vigorous derivation, it is essentially the technique recommendedby theU.S. Federal Emergency Management Agency (FEMA) for identifying flood hazard zones on alluvial fans; see for example, Anon (1983), Dawdy (1979), and Anon (1985). The regime theory has also been termed the Indian approach because the original field studies performed in supportofthistheoryweredone inwhat isnow IndiaandPakistan. In the modern hydrologic and geologic literature, the regime theory is often termed the theory of hydraulic geometry because it is used to
96
e s t i m a t e t h e channel widthand d e p t h a n d t h e v e l o c i t y of flow. Given t h a t FEMA has adapted t h e regime t h e o r y ( o r t h e o r y of h y d r a u l i c geometry) a s i t s p r e f e r r e d technique of channel a n a l y s i s f o r f l o o d flows on a l l u v i a l f a n s , t h e apparent e a s e with which r e s u l t s can be obtained using t h i s t h e o r y , and t h e w i d e d i s c u s s i o n t h i s t h e o r y has received i n t h e hydrologic, g e o l o g i c , and h y d r a u l i c engineering l i t e r a t u r e , it i s w o r t h w h i l e t o c o n s i d e r t h e o r i g i n , l i m i t a t i o n s , and r e s u l t s available. A s n o t e d b y Graf ( 1 9 7 1 ) , i n a n o p e n - c h a n n e l w i t h r i g i d b o u n d a r i e s and uniform flow, t h e r e is only one degree of freedom - t h e depth of flow which can be estimated fromtheManning equation. Uniform flow i n an open-channel w i t h non-cohesive and moveable boundaries has a minimum of t h r e e degrees of freedom - width, depth, and l o n g i t u d i n a l s l o p e . The t h r e e degrees of freedomderive f r o m t h e h y p o t h e s i s t h a t i n a channel with e r o d i b l e boundaries a depth of flow w i l l be e s t a b l i s h e d which depends on t h e l o n g i t u d i n a l s l o p e of t h e channel, t h e width of t h e channel, and t h e d i s c h a r g e . Blench ( 1 9 6 1 ) introduced a f o u r t h degree of freedomby a s s e r t i n g t h a t a r t i f i c i a l l y s t r a i g h t channel s e c t i o n s are u n s t a b l e b e c a u s e e r o s i o n o f t h e channel banks w i l l e v e n t u a l l y r e s u l t i n channel meanders. The regime t h e o r y w a s a p p a r e n t l y f i r s t enunciated by Kennedy i n 1895. Since t h a t t i m e , a rather l a r g e body of d a t a and information regarding t h i s t h e o r y h a s and c o n t i n u e s t o evolve even i n t h e f a c e of vigorous a t t a c k s on t h e v a l i d i t y of t h e theory: see f o r example, E i n s t e i n a n d C h i e n (1956). Theterminologyofhydraulicgeometrywas a p p a r e n t l y f i r s t i n t r o d u c e d b y Leopoldand Maddock (1953) t o d e s c r i b e t h e observed v a r i a t i o n s of width, v e l o c i t y , and depth of flow a s a f u n c t i o n of flow r a t e , Figs. 3 . 4 . 3 . Because average d a t a w e r e used, Figs. 3 . 4 . 3 is somewhat misleading regarding t h e v a l i d i t y of t h e t h e o r y and f o r t h i s reason s i m i l a r d a t a a r e p l o t t e d i n F i g s . 3 . 4 . 3 f o r comparison with t h e d a t a i n Figs. 3 . 4 . 4 , P h i l l i p s and H a r l i n ( 1 9 8 4 ) . With r e f e r e n c e t o Figs. 3 . 4 . 3 and 3 . 4 . 4 , i n t h e i r most g e n e r a l form, t h e regime o r h y d r a u l i c geometry e q u a t i o n s f o r channel width, depth, and v e l o c i t y of flow are T = C,Qb
(3.4.16)
Y = C2Qf
(3.4.17)
and
-
u = C,Qm
(3.4.18)
97
TOP WIDTH, T b t e r s l
10
1
10
n o n RATE. 0
[cubic metere per second)
loo
98 VELOCITY OF ROn, u h a t e r s per eecond) 10
t
H1
ROW UTE.Q (cubic meters per second) (C)
FIGS. 3.4.3 Hydraulic geometry forthe RioPuerco at Rio Puerco, New Mexico, average data from Leopold and Maddock (1953). whereC1, C,, C,,b, f, andm areundetermined empiricalcoefficients. It is generally assumed that Equations (3.4.16) (3.4.18) must satisfy the continuity equation or if the channel cross section is
-
appproximately rectangular ~ y =i C,Q~C,Q~C,Q'
(3.4.19)
If Equation (3.4.19) is to be satisfied, then the undetermined exponents must satisfy b + f + m = l and
(3.4.20)
the coefficients
c,c,c,
= 1
(3.4.21)
In many cases, Equations (3.4.20) and (3.4.21) are not satisfied. Among the reasons for this deviation are 1) data error, 2) the method used to fit the regression line to the data, 3) validity of the power function relationship, and 4) the absence of physical adjustment of the channel to some discharges, Rhodes (1977). In considering the applicability of the regime or hydraulic
g e o m e t r y t h e o r y t o h y d r a u l i c processeson alluvial fans, thecriteria for their use noted by Blench (1957) should be considered. These criteria are:
99
0
I
0.1
0.01
FLOW RATE. 0
(cubic metere per eecond)
FLOW R A E , 0
(cubic meters per second)
DEPTH OF FLOW, y betere)
'
0.01 0.01
0.1
(b)
1
100
V M C I l Y OF R O W .
1
u
(#tors
par recondl
°
i
0O
0.1 0. 0.01
o=
0.1
ROW RATE. B (cubic w t e r r per rscondl
FIGS. 3 . p . 4 , Hydraulicgeometr OftheHuerfanoRiver, Colorado, data from Phillips and Harlin (198i). 1.
The channel is straight.
2.
The channel sides behave as if they were hydraulically smooth. Note, boundary surfaces are classified as being either hydraulically smooth or rough by comparing the laminar sublayer thickness and the roughness height: see for example, French (1985) or Schlichting (1968). If the boundary roughness is such that theroughness elements are covered by the laminar sublayer, then by definition the boundary is hydraulically smooth.
3.
The bed or bottom width of the channel must be greater than three times the depth of flow.
4.
The side slopes of the channel must approximate those for cohesive materials in nature.
5.
The water discharge of the channel is steady.
6.
The sediment discharge of the channel is steady.
101
7.
Channels that move a non-cohesive load of sediment along the bed in dune formation.
8.
The Froudenumber ofthe flowmust be less thanone; that is the flow is subcritical (see for example Section 2 of this chapter). Note, Chang (1980) mentioned that canal designers using the regime theory have often used a Froude number criterion for establishing the range of applicability of regime theory designs. According to Chang (1980), the Froude number of the flow should be kept a t a v a l u e of approximately 0.2 and neverallowed toexceed a value of 0.3.
9.
Thesizeofmaterial composingthe channelboundary mustbe small relative to the depth of flow.
10.
The channel flow rate and the rate of sediment transport must be equilibrium; that is, a stable hydraulic section has been achieved.
Althoughmany investigatorshave assertedthat the coefficients C,, C,, C, and the exponents b, f, and m are universal constants, sufficient evidence now existsto demonstrate that this assertion is without merit. Some o f t h e results regardingthese coefficientsand exponents are summarized in Table 3.4.1. Studies of the regime or hydraulicgeometryequationshaveprimarilyexaminedthevariationof the coefficients either at-a-stationwith time or with longitudinal channel distance. The results in Table 3.4.ldemonstrate that there is a significant variation in coefficient values with geographical area, climate, time, and distance along the channel. At this point, it is appropriate to note that when dealing empiricalorsemi-empiricalequationsthatthesystemofunitsusedto derive the exponents and coefficients associated with these are equations is an important consideration. For example, consider the situation in which it is asserted that T = C,Qb
(3.4.22)
where the English system of units has been used to estimate values of C, and b. Further assume that in Equation (3.4.22)
102
TABLE 3.4.1 Summary of regime/hydraulic geometry coefficients and exponents from various literature sources
1
I I I I
I IData obtained from U.S. Geological I lsurvey gaging stations on perenI lnial rivers. I I I 1 b=0.26 1 f=0.40 I m=0.34 [at-a-station (primarily for arid I I and semi/arid regions) I I I I I I I b=0.50 I f=0.40 1 m=O.10 Idownstream (rivers throughout the IUnited States) I I I ........................................................................... Carlston I I I IData from Leopold and Maddock l(1953). Regression lines fitted (1969) I I I
Leopold and Maddock (1953)
1
I
I I I I I b-0.461 I f-0.383 I I I b=0.499 I f=0.320
I
I
II C1=8.36 III
Iby principle of least squares. I I I I m=0.155 [downstream (10 U.S. river basins) I I I m=0.180 [downstream (Yellowstone River
I IBasin: arid and semi-arid) I I C2=0.27 1 C3=0.45 IC values in SI units
I I I C,=4.63 I C2=0.28 1 c3=o.78 IC values in English units
________________________________________---------------------------------Dawdy I b=0.40 I fz0.40 I I I (1979) I I 1 I c1=12.
I c2=o.09 I
Ic values in SI units
I C,=9.5
I C2=0.07 I
IC values in English units
I
Nixon (1959)
I
I
I
I I
I
IData from 11 rivers in England land Wales. Equations apply to lrivers with sediment loads so lsmall as to have little effect on I lchannel shape. Discharges used I I I I b=0.500 I f=0.333 I m=0.167 Irepresent bankfull discharges.
I
I
I
I
I
I I
I I
Cl=l.65
I
I
I I
I
I
IRivers construct their own geoImetries. Profiles of rivers in lhumid climates tend toward equal [power per unit area and per unit [length (minimum work).
C2=o.5451 c3=1.1121c values in SI units
I Cl=0.9111 C2=0.5461 C3=2.01 IC values in English units ........................................................................... Langbein (1964)
I I
I
I
I I
I I
I
I I b=0.23 I I b-0.53
I I
I
I f=0.42
I
I f=0.37
I
I
I I
I
I m=O.35 I
I
I
lat-a-station
I
I m=0.10
(downstream
I
I I I I I
IAt-a-station data from 7 stations Ion Sungei Kinta, Perak, Western IMalaysia. Downstream data from 9 [stations.
I
I
I
........................................................................... Ee (1970)
I I I I
I I I
I
I I
I
b=0.09
I f=0.61
I m=0.31
I
lat-a-station
I b=0.29 I f=0.44 I m=0.28 Idownstream ...........................................................................
103
Knighton (1974) and
I
/Data from the Bolin and Dean River /Great Britain. Five reaches of 1500 m were chosen to represent a lwide range of channel sizes and Idischarge conditions. Within each [reach, 3 measurement stations were [chosen. The drainage area was )primarily covered by varied jglacial deposits. Bankfull discharges ranged from 1 m3/s (up;stream) to 10 m3/s (downstream) I I 1.261f~ 10.24~m< lat-a-station 0.63 I 0.71 I I I f=O.40 I m=O.48 1 I I
I I
I
10.01
I 0.33 I I b=0.11 I I h=
~
0.61
0.54 f= 0.46
I
{
I
0.311 0:23Im= 0 161
1
I
0 081 2%
0:23)15% 0.38150%
Downstream variation at per cent of time discharge is equaled or exceeded
I I ________________________________________--------------------------------Park (1977)
I I
I
I I
I
I I I
I I
I
I I O
I
10.06
I I I
IAnalysis of at-a-station geometry lof 139 streams and the downstream [geometry of 72 streams representling several different environments I (primarily perrennial streams with [some ephemeral streams)
/0.07
lat-a-station
I
0.71 I I I 0 . 0 3 5 b ~ 10.091fs I-0.51
1
I C1=4.5 8
I
c2=o.132 1c3=1.37
I
Williams (1978)
i
1
O
I I I I I I I I
IChanges in variables are such that lthe total effect of action, work, lor adjustment is a minimum. All lvariables strive to resist any Iimposed change with the net result lbeing that all of them change [equally. In checking this assump[tion only streams with loose parlticles on the bed were considered.
io.031ms I 0.81
iat-a-station I
I
io.io
I
0.78
I c values in SI units
I
........................................................................... [Basic principle - minimum rate of Yaw, I I I lenergy dissipation which states Song, and I I I lthat a system is in an equilibWoldenbergl I I lrium condition when its rate of I (1981) I I I I
I
lb=0.41
I I I
lf=0.41
I I
lenergy dissipation is at its lminimum value.
Im=O.l8
I
I
I
104
c,
= 1.00
and b has a v a l u e of a. I f T h a s u n i t s of [ f t ] and Q h a s u n i t s of [ f t 3 / s ] , then C , must have t h e following u n i t s t o p r e s e r v e t h e dimensional homogeneity of t h e equation
c,
[ft]
T
= - =
Qa
[ft3/s]"
-
[ft-seca]
ft3U
or C, = [ f t ' 1-3a)-seca~
The a p p r o p r i a t e conversion f o r C , given i n English u n i t s t o u n i t s would be C, [ S I ] = C,
[English]
(0.3048) 1ft1-3a
C,
in SI
1
Thus, i n c o n s i d e r i n g c o e f f i c i e n t s i n v o l v e d i n t h e regime o r h y d r a u l i c geometry t h e o r i e s it i s necessary t o transform them from t h e English s y s t e n ' o f u n i t s t o t h e S I system t h a t t h e v a l u e of t h e exponents be known. I n Equation ( 3 . 4 . 2 2 ) , C , and b are u s u a l l y determined from a l e a s t s q u a r e s r e g r e s s i o n a n a l y s i s o f t h e equation obtained by t a k i n g thenaturallogarithmsofbothsidesoftheequation, C a r l s t o n (1969), or log(T)
=
lOg(C,) + b lOg(Q)
where l o g ( C , ) i s t h e y a x i s i n t e r c e p t of t h e l i n e and b i s t h e s l o p e .
Therefore, b is independent of t h e system of u n i t s used and no conversion i s r e q u i r e d . I n t h e p a s t , many i n v e s t i g a t o r s have been remiss i n s p e c i f y i n g t h e u n i t s a s s o c i a t e d with t h e c o e f f i c i e n t s C , , C , , and C, and t h e r e a d e r i s cautioned r e g a r d i n g t h i s p o t e n t i a l problem. Theprimaryadvantageoftheregime o r h y d r a u l i c geometrytheory i s t h a t i t p r o v i d e s a v e r y s i m p l e summaryofthecomplicatedandpoorly understood r e l a t i o n s h i p s t h a t e x i s t among t h e channel and flow c h a r a c t e r i s t i c s i n a l l u v i a l c h a n n e l s . However, t h i s t h e o r y a l s o h a s a number of inadequacies and disadvantages. F i r s t , as noted p r e v i o u s l y i n t h i s s e c t i o n , t h e s e t h e o r i e s are e m p i r i c a l and have no theoretical justification. Second, t h e v a l i d i t y o f simplepower f u n c t i o n s a s d e s c r i p t o r s o f channel c h a r a c t e r i s t i c s d u r i n g p e r i o d s of v a r y i n g d i s c h a r g e h a s b e e n
105
questioned by a number of investigators: for example, Wolman (1955) and Richards (1973, 1976). Richards (1973) noted among other objections that there are discontinuities in the depth-discharge relationship following the transition fromthe lower to upper flow
\ O
/
0 b
1,
f
O
0,8 082
0.6
0.4
0.2
0,4
0,6
0.8
1.
0 1.0
F I G . 3.4.5 Chan es of at-a-stream hydraulic geometry exponents during a single Flood event, data from Knighton (1975).
regime: see also Dawdy (1961). Richards (1973) hypothesized that these discontinuities could be the result of non-linear changes in channelroughnesswiththedepthof flowandsuggestedthatnon-linear
power relationships may yield a better description of the system. Currently, there is not sufficient evidence to substantiate the suggested modification. Third, Rhodes (1977) studied 315 sets of hydraulic geometry equations with the aid of a trilateral diagram: see for example Fig. 3.4.5. Given the scatter of the m, f, and b values in his figure, he concluded that the average values of at-a-station hydraulicgeometry relationships may have but little meaning and may give erroneous predictions of actual channel responses to changing discharge. Fourth, there i s e v i d e n c e t o s u g g e s t t h a t d u r i n g e v e n s m a l l flood events significant changes in channel morphology may occur which cannot be predicted or explained by these hypotheses. For example,
106
Knighton (1975) mentioned that during one study of at-a-station hydraulic geometry a flood event with a return period of 2.5 years significantly modified the hydraulic exponents at one station, Fig. 3.4.5. Knighton (1975) c o n c l u d e d t h a t a t - a - s t a t i o n r e l a t i o n s canbe drastically changed by a single flow event if the channel cross section is in an unstable condition. Fifth, these theories fail to recognizethe important influence ofthesediment loadonchannel width, depth, andcapacity, Simonsand Albertson (1960) Sixth, regime and hydraulic geometry theory was developed to describe steady flow, steady sediment transport, and equilibrium channel conditions. The extension of these theoriesto the analysis of highly transient condition in an alluvial channel may be both unrealistic and improper.
.
DEBRIS F M W MECHANICS The terminology debris flow has been imprecisely used to describe the movement of a wide variety of soil and water mixtures. For example, there isnot aclear andprecise definitionaldifference among the movement of soil andwatermixturesvariouslytermeddebris flows, mud flows, creep, and landslides. For this reason, the foregoing terms may, and probably do, convey different images to geologists and geomorphologists than they do to engineers. To some degree, the lack of precise definitions results from the fact that mass movements of rock, earth, and water can vary greatly in speed, water content, c h a r a c t e r i s t i c s o f t h e s o l i d m a t e r i a l t r a n s p o r t e d , and the distribution of the solid materials within the flow. Some authors have attempted to differentiate between debris and mud flows on the basis of the basis of particle size, for example Schuster and Jkrizek (1978). In contrast to definitions based on the size of materials transported, the definition of debris flows proposed by Takahashi (1980) is to be preferred from the viewpoint of fluid mechanics. This definition states (in paraphrase): 3.5
Adebris f l o w i s a m i x t u r e o f a l l s i z e s o f s e d i m e n t . Boulders accumulate andtumble at the front ofthe debriswave and form a lobe, behind which follows the finer grained more fluidic debris, Takahashi (1980, p. 381). In comparison with this definition, Sharp and Nobles (1953, pp. 551552) offeredthe followingdescriptionofa llmudflowllthatoccurredin
107
1941 at Wrightwood, California: "A bouldery embankment formed at the front of more viscous
surges, a n d t h e b o u l d e r s t h e r e i n r o l l e d , twisted, andshifted about but for themost partdid notappear t o b e rolledunder. Instead, theywerepushedalongbythe finermore fluiddebris impounded behind the bouldery dam and swept along by the mud leaking through it. The similarity between the Takahashi definition and the Sharp and Nobles description is unmistakable. The advantage of the Takahashi definition is that it focuses on the fluid and flow characteristics that distinguish debris flows from other fluid flows. The density andviscosity of debris flows are important fluid characteristics by whichdebris flowscanbedistinguished fromother typesof flowssuch a s l l p u r e w a t e r l g o r w a t e r t r a n s p o r t i n g s e d i m e n t . Tobeclassifiedasa debris flow, a flow should have a density approximatelytwice that of normal water (at 20 degrees Centigrade, the density ofwater is998.2 kg/m3) and a viscosity significantly larger than that of water [at 20 degrees Centigrade, the absolute or dynamic viscosity of water is 1.005 x kg/(m-s)]. Because of their relatively large viscosities, debris flows are generally assumed to be laminar flows, and this assumed characteristic accounts for the fact that separate layers of the flow do not mix except at the leading edge of the flow. Note, turbulent flow situations are the most common in hydraulic engineering practice, and in such flows, the flow paths are highly irregular; that is, there is significant mixing between the layers, and flow losses vary as the square of the velocity. In laminar flow, the fluid particles move along smooth flow paths with each layer gliding smoothly over the adjacent layers, and flow losses vary directly with the velocity. Laminar and turbulent Newtonian flows are usually differentiated between on the basis of the ratio of the inertial to the viscous forces in the flow; a ratio commonly known as the Reynolds number, R, which is by definition R=-
UL p
(3.5.1)
U
where
u = average velocity of flow, L = a characteristic length which
in open-channel flow is usually taken as the hydraulic radius R, p = density ofthe fluid flowing, and u =dynamic orabsolute viscosityof
108
0.3
0.01
I
Pierson (1981)
l500-300C
I I
2.3
____
I
Segerstroem (1950)
Blackwelder (1928)
___
----
I
I 1 I
I I I Sharp and1 Nobles I (1953) I I I
___ ----
---
.urry
5-7 1 4 0 (bylviscous, laminar 1veight)ldebris flow, I Islurry
:igantic lasses o ieteroleneous iaterial ;hot fro1 barrow :anyons nto ope1 alleys
I I I
1740 0.2 (measure (bulk density) at surface
several tens of m/s to several cm/s
I
I I I
5-7
0.6
I I 1 I I I I I I
I
,arent tonian cosity
-
----
I
I
I I
I
I 1.2-3.0 I (average I velocity
I
1
I
I I I I I I
lo5
--
4-8 I
I
I I I I I I I I
--
_-
1
.8-
I I 1
-6
-
I
4
.5
I I 2x104 I 6X1O4
I I I
___
I
I
I
I
1
I I____
I
I
5-7 I
0.2
----
I
I
(measure at surface
I
I I 1 I I---Pack (1923) I I I I
:e moto oil
5.0
____
I Takahashil---(1980) I
I 4 0 (bylTurbulent muddy Iweight)lstreamflow be[tween surges
1730 1.0 (measure (bulk density) at surface
6
I
I
I
I
I I I I I I I
I I
/Higher velocity lviscous debris [flow with relnewed turbulence
22
----
----
-----
---
I
IFlowage of a [mixture of all (sediment sizes I
I I
I
JFollowing the \first impulse /were tremendous I quantities of lrock waste rangling in size from Jsmall to very (large boulders
I I
I I
I
I I I I
I I
JFlood transportled large quantilties of washed lgravel, sand, Iclay, and small I boulders I I 25-30 /Highly fluidic, 1 (by Islimy, cement(weight [like mud conltaining abundant I I stones 1 I I
Reynolds numbers above 12,500 are consideredto beturbulent. Openchannel flows with Reynoldsnumbers between500 and12,500 aretermed transitional flows. The critical Reynolds number is by definition the Reynolds number at which a laminar flow becomes a turbulent flow. In general, the Ilcritical Reynolds numberwwis actually a range of Reynolds numbers as indicated above. Finally, the assertion that debris flows are laminar has also been used to account for the observation that boulders apparently ride on the top of debris flows
109
for long distances.
Rnte o f Angular Deformation FIG. 3.5.1 Schematic of shear stress as a fupction of the rate of angular deformation for various types of fluids. Table 3.5.1 summarizes some of the historic field and laboratory data available regarding debris flows. With regard to these data, the following observations can be made. First, the Reynoldsnumbers ofthese flows, Column ( Z ) , arelow, a n d t h i s t e n d s t o s u b s t a n t i a t e t h e assumption that debris flows are laminar. Second, the velocity of these flows, Column (4), are reasonably large. Third, both the density, Column (5), and the viscosity, Column (6), of these flows is large relative to those of pure water. Fourth, the slopes on which debris flows occur, Column ( 7 ) , are relatively large. Fifth, the observed water content of these flows is relatively small. 3.5.1
Fluid Types Of all the fluid properties, from the viewpoint of fluid mechanics, viscosity is the one which deserves the most serious consideration. By definition, viscosity is the fluid propertywhich characterizes the resistance of the fluid to shear: and for this reason, fluids m a y b e classifiedbythe functional relationship that exists between the shear stress applied and the rate of angular deformation of the fluid. The rate of angular deformation of the fluid is usually measured by the velocity gradient within the fluid.
110
Four t y p e s of f l u i d s a r e and may i n t h e f u t u r e be important t o t h e d i s c u s s i o n of d e b r i s flow mechanics. F i r s t , Newtonian f l u i d s , such as water, e x h i b i t a l i n e a r r e l a t i o n s h i p between t h e s h e a r stress a p p l i e d , T , and t h e r a t e . o f angulardeformationwhich i s measuredby t h e v e l o c i t y g r a d i e n t d u / d y . I n t h i s type of f l u i d , T
=
{z]
(3.5.2)
where u = a b s o l u t e o r dynamic v i s c o s i t y of t h e f l u i d , F i g . 3.5.1. Second, a Bingham p l a s t i c i s a f l u i d i n which t h e s h e a r stress a p p l i e d must exceed a y i e l d stress v a l u e b e f o r e a s h e a r r a t e can be d e f i n e d o r flow occurs. A f t e r t h e y i e l d stress, r Y , i s exceeded, t h e r e i s a l i n e a r r e l a t i o n s h i p between t h e a p p l i e d s h e a r s t r e s s a n d t h e r a t e o f a n g u l a r s h e a r . Q u a n t i t a t i v e l y , t h e r e l a t i o n s h i p between t h e a p p l i e d s h e a r stress and t h e r a t e of angular deformation f o r a Bingham p l a s t i c i s (3.5.3)
where uB = Bingham p l a s t i c v i s c o s i t y , Fig. 3.5.1. Third, apseudo-plastic f l u i d h a s n e i t h e r a y i e l d s t r e s s n o r d o e s it e x h i b i t a l i n e a r r e l a t i o n s h i p betweenthe s h e a r stress a p p l i e d a n d t h e r a t e of a n g u l a r deformation of t h e f l u i d . Rather, a pseudop l a s t i c f l u i d is characterized b y a progressivelydecreasing slopeof t h e a p p l i e d s h e a r s t r e s s v e r s u s t h e r a t e of a n g u l a r deformation, Fig. 3.5.1. Fourth, a d i l a t a n t f l u i d a l s o h a s n e i t h e r a y i e l d s t r e s s n o r d o e s it e x h i b i t a l i n e a r r e l a t i o n s h i p betweenthe a p p l i e d s h e a r stress and t h e r a t e of a n g u l a r deformation of t h e f l u i d . I n c o n t r a s t t o a pseudo-plastic f l u i d , a d i l a t a n t f l u i d is c h a r a c t e r i z e d by a p r o g r e s s i v e l y i n c r e a s i n g s l o p e of t h e a p p l i e d s h e a r stress v e r s u s t h e rate of a n g u l a r deformation, Fig. 3.5.1. Mathematical formulations r e l a t i n g t h e s h e a r stress t o t h e r a t e of a n g u l a r deformation of t h e f l u i d s i m i l a r t o Equations ( 3 . 5 . 2 ) o r ( 3 . 5 . 3 ) are n o t a v a i l a b l e f o r pseudo-plastic and d i l a t a n t f l u i d s due t o t h e f a c t t h a t du/dy i s a f u n c t i o n of t h e a p p l i e d s h e a r stress. A number of e m p i r i c a l r e l a t i o n s h i p s between T and du/dy have been proposed f o r t h e s e types of f l u i d s ; and among t h e s i m p l e s t is t h a t
111
c i t e d by Hughes and Brighton ( 1 9 6 7 ) o r 7
(3.5.4)
=
where k = a c o n s t a n t r e l a t e d t o t h e consistency of t h e f l u i d and n = a
constant w h i c h i s ameasure ofhow t h e f l u i d d e v i a t e s from aNewtonian f l u i d w i t h n < 1 f o r a pseudo-plastic f l u i d and n > 1 f o r a d i l a t a n t f l u i d . W i t h t h e e m p i r i c a l r e l a t i o n s h i p d e f i n e d b y Equation ( 3 . 5 . 4 ) , t h e a p p a r e n t v i s c o s i t y , ua, o f t h e pseudo-plasticand d i l a t a n t f l u i d s is d e f i n e d by (3.5.5) A t t h i s p o i n t , it is r e l e v a n t t o n o t e t h a t a c h o i c e o f a f l u i d t y p e ( o r model) f o r a d e b r i s flow is n o t an i n s i g n i f i c a n t d e c i s i o n . F o r example, i n t h e foregoing m a t e r i a l laminar and t u r b u l e n t flows have been d i f f e r e n t i a t e d between on t h e b a s i s of a c r i t i c a l Reynolds number. The concept of a c r i t i c a l Reynolds number d e r i v e s from a c o n s i d e r a t i o n o f t h e p a r t i a l d i f f e r e n t i a l equationswhich g o v e r n t h e flowof Newtonianfluids. I n t h e c a s e o f Binghamplastic f l u i d s , flow s t a b i l i t y ; and hence t h e c l a s s i f i c a t i o n of a flow as laminar o r t u r b u l e n t , h a s been shown t o depend on t h e Binghamnumber r a t h e r t h a n t h e Reynolds number o r (3.5.6) UBU
u
where = average v e l o c i t y of flow, y = depth of flow, and B = Bingham number; see f o r example, Enos ( 1 9 7 7 ) o r Hampton ( 1 9 7 2 ) . Hedstrom ( 1 9 5 2 ) furtherdemonstratedthatinthecaseof Binghamfluids flowing i n p i p e s t h a t t h e c r i t i c a l Reynolds number i s only a f u n c t i o n of t h e Bingham number, Fig. 3.5.2. Note, i n p i p e flow, t h e c r i t i c a l Reynolds number is u s u a l l y t a k e n t o b e approximately2,OOO. Hampton (1972) showedthatthecurvederivedbyHedstromrelatingthecritica1 Reynolds number and t h e Bingham number was i n g e n e r a l agreement with
t h e experimental d a t a of B a b b i t t and C a l d w e l l ( 1 9 4 0 ) and Gregory ( 1 9 2 7 ) r e g a r d i n g t h e flow of c l a y s l u r r i e s i n p i p e s , Table 3 . 5 . 2 and Fig. 3 . 5 . 2 . With r e g a r d t o t h e d a t a i n T a b l e 3 . 5 . 2 a n d p l o t t e d i n F i g . 3 . 5 . 2 , t h e following o b s e r v a t i o n s can made. F i r s t , t h e Babbitt and Caldwell ( 1 9 4 0 ) and Gregory ( 1 9 2 7 ) d a t a c o n s i s t e n t l y p l o t above t h e
112
70.2
42.9
73.5
28.6
75.0
21.0
76.7
13.8
79.7
0.91
83.9
0.39
1 I I I I I I I I I I I I I 1 I I I I I I I I I
I
1,210
2.23
1,200
1.63
1,180
1.49
1,160
1.19
1,150
0.89
1,120
I I 1 I 1 I I I I I I I I I I 1 I I I 1 I 1
I I I I
I
I
I
- -- - - - - 86.4
2.23
81.4
0.9581
76.7
2.68
70.9
5.60
I
I I I 1 I
0.743 0.743 0.743 0.743
I I I I I I I I
1,170 1,130 1,175 1,225
I I I I I I I I
7,370 13,500 19,600
5.34 4.89 4.73
0.0127 0.0254 0.0508 0.0762
4.33 4.48 4.12 4.21
0.0127 0.0254 0.0508 0.0762
3.66 3.94 3.36 3.51
I
3,360 7,250 12,400 19,400
0.0254 0.0508 0.0762
3.36 2.75 2.90
I
6,660 10,900 17,200
0.0127 0.0254 0.0508 0.0762
2.35 2.41 2.32 2.35
I 1 I I
2,880 5,910 11,400 17,300
0.0127 0.0254 0.0508 0.0762
1.65 1.62 1.55 1.55
I I I I
2,640 5,180 9,920 14,900
0.0102
I 1 I
I
I
2,960 6,140 12,400 17,300
I 1 I I I I 1 I I I I I I
I I 1 1 I I 1 I I I I I I I
9.16 20.0 31.0 3.76 7.25 15.8 23.1 4.49 8.32 19.5 28.0 7.0 17.1 24.4 4.14 8.07 16.8 24.8 3.37 6.88 14.4 21.6
I I I I 1 I I 1 I I I I I
I 1 I I I I I 1 1 I 1 I I I
138,000 248,000 360,000 55,000 114,000 209,000 321,000 46,500 100,000 171,000 268,000 85,400 140,000 221,000 29,800 61,200 118,000 17,900 21,000 41,100 78,700 118,000
____________________----_-___-------
Gregory Data from Hampton 0.5271
I 1 I I I I 1 I I I 1 I
0.0254 0.0508 0.0762
(1972)
0.5541
I
8.910
0.0102
0.6711
10,400
0.0102
1.09
I
17,600
0.0102
1.83
I
I 1
I
30,800
I I I I I I I
I
13.1 19.6 33.8 42.1
I I 1 I I I I
I
565,000 68,500 111,000
187,000
1 7.56 I 0.743 I 1 , 2 2 5 I 0 . 0 1 0 2 1 2.14 I 3 7 , 1 0 0 I 48.6 I 218,000 I I I I I I I I 64.8 I 10.3 I 0.743 I 1 , 2 5 8 1 0.0102 1 2.59 I 4 4 , 6 0 0 I 5 4 . 6 I 264,000 .______------------______________________--------------.................................................................................... 67.5
curve defined by Hedstrom (1952). However, thetwo datasets andthe curve suggest the same general relationship between the critical Reynolds number and the Bingham number for clay slurry pipe flow. Second, although similar data for open-channel Bingham plastic fluid flows are not available, it can be safely assumed that except for a scale factor an analogous relationship betweenthe critical Reynolds number and the Bingham number would exist. In Fig. 3.5.3 critical Reynolds numbers forNewtonianopen-channel flows andBinghamnumbers are plotted as a function of the velocity and depth of flow for a material with a yield stress of r y = 10 N/m2 (0.21 lb/ft2) and a Bingham fluidviscosityofu, =lkg/m-s (0.02lsl/ft-s). Enos (1977) claimed that these values of r y and u s adequately described the fluid
113
loo. ow
Turbulent
f
?
--
BAmm Am
CWwaL DATA
10.000
1
FIG. 3.5.2 number.
!
loo loo0 B I N W MLBER Critical Reynolds number as a function of the Bingham 10
VELOCITY
-
OF R O W . ./I
A
wo.
B
0.001
-
0.01 -
B
o.ooo1
0.001
0.01
DEPTH OF R O Y .
0.1 I
1
---- -lo
B
0.10
B
1.0
B
10.
B
100.
- 1000. -
B
FIG. 3.5.3 Critical Reynolds numbers for Newtonian open-channel flows and Bingham numbers as a function of the velocity and depth of flow for a material with a yield stress of T , = 10 N/m and a Bingham fluid viscosity of u B = 1-kg/m-s. properties of the material commonly found in debris flows. Third, Fig. 3.5.2 i n d i c a t e s t h a t a s t h e B i n g h a m n u m b e r increasesthecritical Reynolds number also increases. The implication is that eventhough a Bingham plastic fluid flow may have a Reynolds number that would indicate the flow was turbulent, if the fluid was water, Column (9), Table 3.5.2; the non-Newtonian behavior of the fluid must be considered. Further, t h i s o b s e r v a t i o n s u g g e s t s t h a t r e l a t i v e l y h i g h energy levels, relative to pure water, are necessary for a Bingham
114
plastic fluid flowtobeturbulent. Fourth, the foregoing discussion has only considered the question of flow classification from the viewpoint of clay slurries in pipes and has not considered the effect of granular materials such as those found in natural debris flows. However, Bagnold (1956) demonstrated that granularmaterials tendto make the critical Reynolds number of a flow larger rather than smaller. Thus, there i s c u r r e n t l y n o r e a s o n t o b e l i e v e t h a t t h e a b o v e discusssion is not applicable to fluids which behave as Bingham plastic fluids even though they contain not only clay but also granular materials. Finally, there is the question of how viscosity or apparent viscosity should be measured i n a debris flow. For example, should the viscosity o f t h e flow as awhole bemeasured oronlytheviscosity of the dispersion medium, a clay slurry in most debris flows? There is currently no definitive answer to this question. Analytic Results for Debris Flows DeLeonandJeppson (1982) notedthat todate onlytwo closed form or analytic solutions for the vertical distribution of the longitudinal velocity in a debris flowhave beendeveloped. Johnson (1970) assumed that a debris flow behaved as a Bingham plastic fluid and obtained a form of the Poisson equation; see for example, Schlichting (1968), that described the movement of debris flows in a rectangular channel. In contrast, Takahashi (1978 and 1980) assumed a dilatant fluid model and was ableto defineboth themovement ofthe fluid-soil mixture, the steady state depth of flow, and a resistance coefficient which has subsequently shown to be related to the Chezy resistance coefficient for uniform flow. The original work of Takahashi was extended by DeLeon and Jeppson (1982) and Jeppson and Rodriguez (1983) to gradually varied and unsteady debris flows, respectively. Because of these extensions of the original work by Takahashi to areas of applied interest, the dilatant fluid model of debris flows will be considered first and then the Bingham plastic fluid model second. Takahashi (1980) in deriving the vertical distribution of longitudinal velocity in a debris flow occurring in a rectangular channel began with the equation of volumetric continuity or 3.5.2
aY
a (UY)
- +-= at
ax
0
and the unsteady equation of conservation of momentum or
(3.5.7)
115
-
-
au - au -++-=gsine-gcos at ax
(3.5.8)
8
u
= cross-sectional mean where x = longitudinal distance, t = time, velocityof flow, y = debris flowdepthof flow, 0 =slope angleofthe debris channel bed, and k = a resistance or frictional coefficient. Fromthisstartingpoint, Takahashiproceededtoderive anexpression for the resistance coefficient k or
(3.5.9)
and an expression for the steady state depth of the debris flow or k<
Y=-
(3.5.10)
g sine
w h e r e a = a n e m p i r i c a l l y e s t a b l i s h e d v a l u e w h i c h d e p e n d s o n t h e flow, a = friction angle of the moving grains, cd = grain concentration by volume of the debris flow, p = density of water, C , = grain concentration by volume in the stationary bed of the flow, and d = grain diameter. Although Equations (3.5.9) and (3.5.10) are analytically and experimentally useful, from an applied viewpoint these equations are not useful since they require the quantification of variables not usually known, a priori, to the investigator. DeLeon and Jeppson (1982) usedaone-dimensionalhydraulicengineeringapproachandwere abletoanalytically demonstrate that the resistance coefficient k i n Equations (3.5.9) and (3.5.10) was related to the Chezy C a standard resistance coefficient for open-channel flow: see for example, French (1985), or
-
(3.5.11) Substitution of Equation (3.5.11) in Equation (3.5.9) yields
116
(3.5.12)
cd
pm
= (u-pm)
tan
(tan a
8
- tan
8)
I
provided that the value of Cd is less than C,; otherwise,
cd
= O.gc*
and
density of the debris flow mixture. The valueofthechezy C, which is a l s o r e l a t e d t o t h e M a n n i n g n ; see for example, French (1985), is usually choosen on the basis of experience and a field inspection of the channel. In the case of debris flows, standard hydraulic engineering practice may not be applicable to the selection of an appropriate value of C. The Chezy coefficient derives from the semi-empirical Chezy equation for uniform flow in open channels which states pm =
(3.5.13)
where R = hydraulic radius and S = in the general case, the friction slope of the flow. In Equation (3.5.13) , the units associated with the resistance coefficient C are (length) '/*/(time). It can be easily shown that the Chezy resistance coefficient is equivalent to Manning n's or (3.5.14) In Equation (3.5.14) , $ has a value of 1.49 if English units are used and a value of 1.00 if SI units are used. The Chezy coefficient is also related tothe Darcy-Weisbach frictional coefficient f for pipe flow or
c
=
J8g/f
(3.5.15)
see for example, French (1985). . In the case of laminar flow, the
117
Darcy-Weisbach friction factor f is an inverse function of the Reynolds number or 64 f E -
R
(3.5.16)
Combining Equations (3.5.15) and (3.5.16) yields (3.5.17) For SI units, Equation (3.5.17) becomes C = 1.1076
(3.5.18a)
and in the case of English units
c
= 2 . 0 0 6 5
(3.5.18b)
DeLeon and Jeppson (1982) used the limited field and laboratory data currently available for debris flows to estimate values of C. TheresultsofthisanalysisaresummarizedinTable3.5.3. Thereis, see Columns (8), (9), and (10) of Table 3.5.3, reasonably good agreement among the Chezy coefficients estimated by the three techniques used by DeLeon and Jeppson. Given the very limited data available for actual debris flows, DeLeon and Jeppson (1982) asserted that there were sufficient similarities between debris flows and sludge flows to allow the data summarized in Table 3.5.3 to be combined with the sludge flow data of Babbitt and Caldwell (1939) to yield a data set large enough for statistical analysis. Fromthis combined data set, a least squares analysis yielded the following best fit relationship for C and R. C = 1.02 R o S 5 '
(3.5.19a)
for SI units and C = 1.85 R o a S 2
(3.5.19b)
forEnglishunits. Acomparison ofthe coefficientsand exponents in Equations (3.5.19) and those in Equations (3.5.18) show excellent agreement.
118
I
I
Sharp and1 Nobles 1 (1953) I
6
1
1.20
Pierson
5
I I I
5.00
(1981)
I I I
I
Takahashil (1980) I
18
I
I
Takahashil (1980)
I I
I
I
I
1 I
1.00
18
I
I
I
I 1 I
20,000
1,000
1,940
I
I
I I
I 1 I I I 1
1.00
I
I
I I
I 1 0.762 I I I 1 0.4817 I I 1 0.044 I I
1,410
I 1 0.041 I
I
I 1
I I I I
I I I I I I
I
I I 1 4.24 I 3.65 I I I I I I 500 1 24.35 I 24.75 I I I I 8.25 1 8.36 1 3.18 10.97
I
I
I 10.39 I to I
I 9.08 I
to
80.9321
I 1 1o2.Oe2l
I
I
to
9.96 3.57
to
I 11.18
........................................................................
I Value results from slight modification of the coefficients. Values assumed by comparison with data given by Babbitt and Caldwell (1939).
At this point, only one problem remains to be solved before debris flows can be analyzed by the techniques of traditional onedimensional open-channel hydraulics. Up to this point methods of estimating the density and viscosity of a debris flow, a priori, have not been developed; and hence, neither R nor C can be estimated. DeLeonandJeppson (1982) a s s e r t e d t h a t b o t h t h e d e n s i t y a n d v i s c o s i t y of a debris flow vary as a function of the depth and velocity of flow. These investigators, with some justification, offered the following approximation forthe ratioofthe densityto thedynamic viscosity in a debris flow
P
-
u
E
-
k
R*
(3.5.20)
where p = debris flow bulk density, k = an empirical constant, and a = an empirical exponent whose value is assumed to be one and R = hydraulic radius. Usingthe dataof Sharpand Nobles (1953), avalue for k was estimated; and hence, P
10
- I -
u
R
(3.5.21a)
119
f o r S I u n i t s and P 3 - = -
u
(3.5.21b)
R
f o r English u n i t s . With t h e foregoing hypotheses, t h e Chezy equation f o r uniform flowinanopenchannelcanbeusedtoestimatedebris flowdepth. For example, b q i n n i n g w i t h Equation ( 3 . 5 . 1 3 )
or Q2
A3S
- C2
=
(3.5.22)
P
C 2 is e s t i m a t e d by combining Equations (3.5.19a) and (3.5.21a) f o r
computations i n t h e S I system of u n i t s c2
=
[
1.02
R0.52]
*
=
'1
(,.,, [? P u
0 . 5 2
)
2
(3.5.23)
S u b s t i t u t i n g Equation ( 3 . 5 . 2 3 ) i n Equation ( 3 . 5 . 2 2 ) and rearranging
Q2P
-
11.41 A''96
Q 1 * 0 4 S= 0
(3.5.24)
which i s an i m p l i c i t equation f o r t h e d e b r i s flow depth y. Equation ( 3 . 5 . 2 4 ) can be solved f o r y by t r i a l and e r r o r o r by numerical methods; f o r example, a Newton i t e r a t i v e scheme, Conte ( 1 9 6 5 ) . DeLeon and Jeppson ( 1 9 8 2 ) and Jeppson and Rodriguez ( 1 9 8 3 ) used t h e r e s u l t s d i s c u s s e d above and t h e techniques of t r a d i t i o n a l o p e n channel h y d r a u l i c s t o develop numerical s o l u t i o n s f o r g r a d u a l l y v a r i e d b u t s t e a d y and u n s t e a d y d e b r i s flows, r e s p e c t i v e l y . A n o t e o f c a u t i o n , some c a r e should be e x e r c i s e d i n u s i n g t h i s approach p r i m a r i l y because t h e l a c k of d a t a t h a t i s needed t o v e r i f y t h e assumptions i n h e r e n t i n t h i s development. For example, it has not b e e n p r o v e n t h a t d e b r i s f l o w s a r e d i l a t a n t f l u i d s nor c a n t h e v a l i d i t y of Equation ( 3 . 5 . 2 1 ) be b l i n d l y accepted. F u r t h e r , i s t h e Reynolds number an a p p r o p r i a t e parameter f o r d i f f e r e n t i a t i n g between laminar
120
3.5.4 S c h e m a t i c * d e f i n i t i o n of v a r i a b l e s f o r Bingham p l a s t i c f l u i d model of a d e b r i s flow.
FIG.
and t u r b u l e n t f l u i d s when t h e f l u i d s a r e non-Newtonian? ThedevelopmentofaBinghamplastic f l u i d m o d e l o f a d e b r i s flow b e g i n s w i t h a s t e a d y flow f o r c e b a l a n c e similartothatusedtodevelop t h e Chezy uniform flow equation f o r Newtonian f l u i d s . Following Johnson ( 1 9 7 0 ) , s t e a d y d e b r i s flow i n a s e m i - c i r c u l a r c h a n n e l w i l l b e considered, Fig. 3.5.4. A f o r c e balance f o r t h e c o n t r o l volume defined i n t h i s f i g u r e y i e l d s
(3.5.25)
where F, = g r a v i t a t i o n a l f o r c e component causing flow F, = Y
nr
- AL s i n 6 2
F, = v i s c o u s f l u i d f o r c e r e s i s t i n g motion which i s equal t o a s h e a r
stress T times t h e area on which t h i s stress a c t s o r nrAL, and Y = Binghamplastic f l u i d s p e c i f i c w e i g h t . F o r s m a l l v a l u e s o f 6 , s i n 6 i s approximately equal t o t h e s l o p e o f t h e debris flow s u r f a c e s , . With t h e s e d e f i n i t i o n s and assumptions, Equation (3.5.25) becomes Ynr2
TilrAL
=
Am, 2
121
and s i m p l i f y i n g Y r T = -
2
(3.5.26)
SD
Equation (3.5.26) i s v a l i d f o r a l l and T h a s v a l u e of z e r o a t r = 0 and a maximum v a l u e a t r = R. For a Bingham p l a s t i c f l u i d , T i s given by Equation (3.5.3) and s u b s t i t u t i q n of t h i s equation i n Equation (3.5.26) y i e l d s (3.5.27)
which is o n l y v a l i d f o r T 2 7 , ; when T < throughou out the f l u i d , t h e r e i s no flow and t h e above a n a l y s i s beginning w i t h Equation (3.5.25) i s n o t v a l i d . Rearrangement of Equation (3.5.25) y i e l d s
dr
ug
or S,
-
zY
1
for
T
2
(3.5.28)
T~
where t h e n e g a t i v e s i g n i n Equation (3.5.28) d e r i v e s from t h e choice of co-ordinate systems ( v e l o c i t y d e c r e a s e s w i t h i n c r e a s i n g r ) , Johnson (1970). I n t e g r a t i o n of Equation (3.5.28) and e v a l u a t i o n of t h e c o e f f i c i e n t of i n t e g r a t i o n r e s u l t s i n a n e q u a t i o n f o r t h e v e l o c i t y d i s t r i b u t i o n i n a s e m i - c i r c u l a r channel under s t e a d y d e b r i s flow conditions o r S,
-
(R-r)Ty
1
for
T
2
T~
(3.5.29)
Asnoted intheprecedingparagraphs, t h e s h e a r stress vanishes a t r = 0. Therefore, it is c o n c l u d e d t h a t t h e r e m u s t be a v a l u e of r i n any flow where T = T~ and t h i s a r c s e p a r a t e s t h e a r e a of t h e channel where t h e v e l o c i t y of f l o w v a r i e s a c c o r d i n g t o E q u a t i o n (3.5.29) from t h e area of t h e channel where t h e f l u i d moves a t a uniform v e l o c i t y L e t R, be t h e r a d i u s ( t h i s i s o f t e n termed a plug flow) , Fig. 3.5.5. a t which T = T~ and where (du/dr) = 0 . R , can t h e n be determined from Equation (3.5.28)
122
7
FIG. 3.5.5
Definition of debris flow plug.
(3 -5.30)
T h e v e l o c i t y a t w h i c h t h e p l u g movescan bedetermined by substituting
Equation (3.5.30) in Equation (3.5.29). When R, 2 R, there is no flow. Johnson (1970) noted that these results are in general agreement with the limited laboratory results that are available; that is, debris flows increase their thickness as the flow rate increases; decrease their thickness as the flow rate decreases: and reach minimum thickness when flow ceases. Johnson (1970) also presented results for other channelshapes. For example, in infinitely wide rectangular channels, (3.5.31)
123
Yc
=
TY
-
(3.5.32)
YSD
whereH=totaldepthof flow, y = a C a r t e s i a n c o - o r d i n a t e w i t h t h e zero value being taken at the surface of the debris flow, and y c = the thickness of the debris plug. The Bingham plastic fluid model of debris flows proposed by Johnson (1970)m u s t b e c o n s i d e r e d a v i a b l e a l t e r n a t i v e t o t h e d i l a t a n t fluid model discussed previously in this section. However, in comparison with the dilatant fluidmodel, the Bingham plastic fluid model is limited f r o m t h e v i e w p o i n t o f a p p l i c a t i o n . For example, the Bingham model discussed here 1.
requiresthat the yieldstress ofthe Binghamfluid beknown but provides no method of estimating the value of this variable from data that would be generally available, and
2.
has not been extended to gradually varied or unsteady flows
It must noted that the extensions and hence the usefulness of the dilatantfluidmodelisdependentonanumberofempiricalassumptions which may or may not be justified; for example, Equation (3.5.20) and Equations (3.5.21). At the present time, there are not sufficient
field or laboratory dataavailable toeither endorseor reject either of these models. HYDRAULIC/PHYSICAL MODELS Unlike many other areas of modern engineering and scientific endeavor, important discoveries, designs, and insights in hydraulic engineering and geology continue to derive from observations and theoretical calculations made in conjunctionwith laboratory studies utilizingphysicalmodels. Theuse ofphysicalmodelsto developnew knowledge requires a comprehensiveunderstanding ofthe principles of similitude; see for example, Streeter andwylie (1975). Withregard to similarity, there are three types of similarity important to hydraulic/physical model studies: 3.6
1.
Geometric Similarity: Two objects are only similar if the ratios of all corresponding dimensions are similar. The terminology geometric similarity therefore pertains to similarity in form.
124
2.
Kinematic Similarity: Two motions are termed kinematically similar only when the paths of motion are geometrically similar and when the ratiosofthevelocities of the two motions are equal.
3.
Dynamic Similarity: Two motions are termed dynamically similar only if the rasios of the masses involved are equal and the ratios of l the forces involved are equal.
In most physical model studies, geometric and kinematic similarity can be rather easily achieved; however, complete dynamic similarity is often difficult, if not impossible, to achieve. In most complex modeling situations,themodelingprocessbeginswitha scalingofthe governing differential equations; see for example, French '(1985). From the scaling process, a number of dimensionless numbers are derived such as the Reynolds, Froude, Bingham, and Weber numbers. In the material which follows, the concepts of hydraulic/physical models relevant to the study of hydraulic processes on alluvial fans are summarized. Detailed treatments of the subject of hydraulic/physical models are available in French (1985) andsharp (1981). For specificexamples ofthe applicationof these principles to alluvial fans, the reader is referred to Anon. (1981)
.
Froude Law Models Froude law models assert that the primary force causing flow is gravity and that all other forces such as surface tension and fluid friction are small relative to the gravitational force and can be neglected. F r o u d e l a w m o d e l s c o m p r i s e t h e p r i m a r y t y p e o f model used to study traditional open-channel flows. If only Froudenumber similarityis required, then bydefinition 3.6.1
F,, = F,
(3.6.1)
where F = Froude number and the subscripts M and P indicate model and prototype Froude numbers, respectively. Substitution of the definition of the Froude number, Equation (3.2.5) , in Equation (3.6.1) yields after rearrangement an expression for the velocity ratio
125
UM
UP
u , = -ufiu,
gHLN
1 / 2
- [gpLpl
(3.6.2) =
where the subscript R indicates the ratio of model to prototype variables, U, = velocity ratio, L, = length scale ratio, and g, = gravity ratio. Since the acceleration of gravity cannot, from a practical viewpoint, be altered between the prototype and model: i.e., g, = 1, Equation (3.6.2) becomes (3.6.3) Hydraulic models must often be distorted: that is, the longitudinal and vertical scale ratios are not equal. The need for length scale distortion results from the fact that most channels are much longer than they are deep and laboratory space for hydraulic/physicalmodel studies is always limited. For example, in one physical model study cited by Anon. (1981) the vertical scale ratio, Y,, wasl/lOwhilethe longitudinal scale ratio, L,, was1/150. If a distorted model is necessary, then Equation (3.6.3) becomes
"R
=
&
(3.6.4)
and the time scale ratio can be demonstrated to be T,
=
LR
-
(3.6.5)
Other scaling ratios for a distorted Froude number model are summarized in Table 3.6.1. 3.6.2
Moveable Bed Models When the movement of the materials which compose the sides and b e d o f a c h a n n e l o r t h e m i g r a t i o n o f a c h a n n e l a c r o s s a soil surfaceare primary considerations, as they are in the study of hydraulic processes on alluvial fans, then the use of a moveable bed hydraulic modelisnecessary. Theproperdesignand useof amoveable bedmodel is much more complex and difficult thanthe designand usedof a fixed
126
TABLE 3.6.1 Scaling ratios for a distorted Froude law model
.............................................................
Variable
Scaling Ratio
.............................................................
.............................. IJ R Vertical Length ................................ YR Time (hydraulic) ............................... LR/(YR)'/' Velocity ....................................... (Y,)'/' LR(YR)=/' Flow Rate ...................................... Force .......................................... _ _ _._. _. _. _. ._._. _. _. _. ._._. _. _. _. ._._. _. _. _. ._._. _. _. _. ._._. _. _. _. ._._ _ _ _ _ _ _ _ _ ~ _ ~L,(Y,)' ~________~~~~~~~ Horizontal Length
bed model because 1.
The boundary roughness of the moveable bed model is not determined by design but is controlled by the motion of the material which comprises the model.
2.
Not only must the moveable bed model correctly simulate the movement of water but it must also correctly simulate the movement of sediment.
There are three methods by which moveable bed models can be designed. Of these, the method commonly referenced as trial and error;thatis,theadjustmentofmodelvariablesuntilthehistorical record of prototype performance is reproduced, is not appropriate to the s t u d y o f h i g h l y t r a n s i e n t a n d p o o r l y d o c u m e n t e d e p h e m e r a l f l o w s o n alluvial fans. For this reason, the trial and error technique of moveable bed hydraulicmodel designwill not be discussedhere. The remaining two techniques of design can be classified as theoretical and semi-empirical. 3.6.3
Theoretical Techniques Equations (3.4.13) and (3.4.14) , as noted in Section 3.4 , can be used to define the threshold of particle movement. It can therefore be asserted that if F, and R, are the same in the model and prototype thatthe stateofthebedofthe channelwill bethe same. Then, using the hypothesis of Einstein and Barbarossa (1956) regarding the shear stress on the bottom ofthe channel and a roughness-to-particle size
127
r e l a t i o n s h i p such a s Equation (3.3.3) it can be shown: see fa: example, French (1985) , t h a t a moveable bed model can be developed on t h e b a s i s of t h e following equations (3.6.6)
(3.6.7)
(3.6.8)
where R, = h y d r a u l i c r a d i u s r a t i o , V , = k i n e m a t i c v i s c o s i t y r a t i o , a = S, -.1, S , = sediment s p e c i f i c g r a v i t y , and d, = sediment s i z e r a t i o . With regard t o Equations (3.6.6) (3.6.8) , R, is a f u n c t i o n of both Y andL, and v R w o u l d u s u a l l y h a v e a v a l u e o f 1. Therefore, i f onemodel
-
r a t i o i s selected, t h e n t h e o t h e r model r a t i o s c a n b e determined from Equations (3.6.6) - (3.6.8). With regard t o Equations (3.6.6) - (3.6.8) , t h e following comments should be noted and considered: 1.
I f Y, = L, (an u n d i s t o r t e d model)
, t h e n a s c a l e model cannot
be b u i l t ; t h a t is, t h e model w i l l be t h e same s i z e a s t h e prototype. 2.
I f t h e s p e c i f i c g r a v i t y r a t i o , a,, i s s e l e c t e d i n a d v a n c e a n d t h e kinematic v i s c o s i t y r a t i o has a v a l u e of u n i t y , t h e n d, = a R - l I 3
(3.6.9)
Equation (3.6.9) i n d i c a t e s t h a t i f t h e moveable m a t e r i a l used i n t h e model has a s p e c i f i c g r a v i t y less t h a n t h e materialintheprototype,thenthesizeofthematerialused inthemodelmustbelargerthanthat found i n t h e p r o t o t y p e . 3.
I f R, cannot be assumed t o be equal t o Y,, t h e n t h e s c a l e r a t i o s Y, and L, must be found by t r i a l and e r r o r .
4.
The r a t i o of sediment t r a n s p o r t p e r u n i t width t o flow p e r u n i t w i d t h m u s t b e l a r g e r i n themodel t h a n i n t h e p r o t o t y p e .
128
1.0
p
0.10
0.01 1.
FIG. 3.6.1
10.
R#I
100.
1000,
RelationofbedformationtopositionintheF,-R,plane.
5.
The particle size ratio in Equation (3.6.8) characterizes the roughness of the channel while the same parameter in Equations (3.6.6) and (3.6.7) characterizes the sediment transport properties.
6.
Inthedesignof traditionalmoveable bedmodels, theFroude number model law is not considered to be an absolute rule since if the Froude number is small a slight difference between F, and F, can be tolerated. However, as will be subsequentlyshown, it is believedthat theFroude numberof flows occurring on an alluvial fan may not be small; and therefore, small difference between F, and F, may be significant.
Recall that the primary assumption of the foregoing development is that if F, and R, are the same in the model and the prototype, then an appropriate physical model will have been developed. A number of investigators have asserted; see for example, French (1985), that this requirement can and should berelaxed. Fig. 3.6.ldemonstrates the relation of bed formation to position in the F, - R, plane. In viewofthis figure, it is apparentthat toachieve similaritybetween model and prototype the values of F, and R, do not have to be exactly
129
equal but the points in the F, - R, plane representing the mode 1 and the prototype should be within the same bed form band in Fig. 3.6.1. Zwamborn (1966, 1967, 1969) has developeda differenttechnique for designing moveable bed hydraulic models. In summary, this technique requires that the following criteria be satisfied. 1.
Sioce most water' flows of practical importance are turbulent , the Reynolds number for the model should exceed 600. This was a tacit assumption in the previous discussion.
2.
Dynamic similarity between the model and the prototype is achieved when a.
The Froude number for the model and prototype are equal.
b.
The frictional forces acting in themodel andprototype are scaled correctly.
c.
Geometric scale distortion cannot be large: for example , YR
- < 1/4 LR 3.
Sincesedimentmotion i s c l o s e l y r e l a t e d t o t h e b e d formtype and bed form types are a function of R*, F,, and u*/Usr the value of R, for the model should fall within the range determined by F and u*/U,. French (1985) provided an appropriate figure for doing this.
4.
Given the foregoing criteria, the following scale factors can be determined Flow Rate Q R = LRY,1'5
Hydraulic Time
(3.6.10)
130
T,
=z
LR -
(3.6.11)
UR
Sedimentological Time (3.6.12) where T, = sediment time scale, A = relative submerged sediment density, and q s = sediment bed load per unit width. If ( q s ) Ris not accurately known, T, cannot be calculated, but Zwamborn (1966, 1967, 1969) suggested that (Ts) R
I :
loT,
(3.6.13)
With regard to the discussion above, hydraulic time is a time scale based on the speed of water wave propagation while sedimentological time is, at least conceptually, based on the speed of sediment wave propagation. Finally, themethodologyusedbyAnon. (1981)todesign moveable bed hydraulic models of alluvial fans is similiar to that discussed above. 3.6.4
Empirical The regime or hydraulic geometry theory can also be used to develop scaling ratios for moveable bed hydraulicmodels. If it is assumed that T Y
-
S
.-
-
h
(3.6.14)
Q113
(3.6.15)
Q-1Is
(3.6.16)
then it can be shown by induction that (3.6.17) (3.6.18) and (3.6.19) Y,
=
LR213
Q R = L,YR312
(3.6.20) (3.6.21)
131
-
(3.6.21), the following With regard to Equations (3.6.14) observations can be made. First, the model described by these Second, the equations i s a d i s t o r t e d s c a l e m o d e 1 , E q u a t i o n (3.6.20). flow rate scaling, Equation (3.6.21) agrees with the Froude law scaling, Table 3.6.1. Third, t h i s t e c h n i q u e t a c i t l y a s s u m e s t h a t t h e model bed material is the same as that found in the prototype. 3.7
CONCLUSION The foregoing sections of this chapter have attempted to introduce the reader to a limited number of the basic and traditional principles of open-channel hydraulics which the author believes are important to understanding hydraulic processes on alluvial fans. The coverage of the various principles discussed in this chapter is, by intention, not comprehensive. However, there are sufficient references given forthe interested reader to further investigate at his leisure all of the topics. This chapter has also briefly discussed the mechanics of debris flows which is a rapidly developing area of technology. There is currently significant interest in the mechanisms which initiate debris flows, Croft (1967), Campbell (1975), and MacArthur et a1 (1986); the mechanics of debris flows, Chen (1983, 1984, 1985, 1986a, 1986b, 1 9 8 6 ~ ) ;the identification of areas susceptible to debris flows, Mears (1977), James et a1 (1986), and Kumar (1986); and the frequency of debris flows, Campbell (1975) and Mears (1977). Finally, it should be noted that post-event analysis often mistakes debris flow floods for fluvial floods, Costa (1983).
REFERENCES Anon., 1981. Flood plain management tools for alluvial fans study docvmentation. Pre ared b Anderson-Nichols Inc. , Pal4 Alto, California. For: Federal gkergency Management Agency, Washington. Anon., 1983. Appendix G: Alluvial Fan Studies. In: Flood Insurance Guidelines and Specifications for Stud Contractors .St%d&-37 July 1983 , Federal Emergency Managemenf Agency, Washington. i6-1:A6-10. Anon., 1985. Alluvial fan flooding metqodolo y: an analysis. Prepared by: DMA Consulting Engineers , Marina dey Rey California. For: Federal Emergency Management Agency, Washingto;. Babbitt H.E. aqd Caldwell, D.H. , 1939. Laminar flow of slud es in iipes with s ecial reference to sewage sludge. Bulletin Ilyinois ngineering Experiment Station, 319. Babbitt, H.E. andCaldwel1, D.H., 1940. Turbulent flowof sludges in
132
pipes. Bulletin Illinois Engineering Experiment Station, 323. Bagnold R.A., 1956. The flow of cohesionless rains in fluids. Transactions of the Royal Society of London, Serqes A, 249 (964): 235-297. Barnes H.H., Jr., 1967. Roughness characteristics of natural channeis. U. S. Geologipal Survey Water Supply Paper 1849 , U. S. , Geological Survey, Washington. Blackw lder, E. 1928. Mudflow as a qeologic a ent iv semi-arid mountafns. Geoiogical Society of Amer ca Bullet%, 39. 465-583. E. , I I 1980. Minimum specific energy i 0 en hannels Blalock M. of com 6und secttori Thesis presented to the Geor ?a Fnstltute of Technotogy Atlanth Geor ia in ar ial ful&,lment of ,the requirements for the degree p% Mdater oPScXence n Clvil Engineering. Blalock M. E. and Sturm, T. W. 1981. Minimum s p e d ic energy in com oun6. open channel. ASCE, fournal of the Hydraulfcs Division, 1077HY6) 699-717. Blench T., 1957. e ime Behpviour of Canals and Rivers. ButteAorths Scientif fc %ublications, London. Blench T. 1961. Hydraulics pf Canals and Rivers of Mobile h: Butterworths Civil Engineering Reference Book, 2nd BoundaGy Edition. Butterworths, London. Campbell R.H.., 1975. Soil slips qebris flows and raingtorms in the Santa)Monica Mountains and vicinity, SoutherfiCalifornia. U.S. Geological Survey Professional Paper 851, Washington. Carlston, C.W., 1969. Downstream variations in the h draglic geometry of streams: s ecial em hasis on mean velocity. Lerican ournal of Science, 2 6 8 499-508.
. .
Chan H.H., 1980. Stable alluvia: canal design. ASCE, Journal of the 8Ldraulics Division, 106 (HY5). 873-891. Chen, C., 1983. On frontier between rheolo and-mudflowmechanic?. In: Proqeedin s of the Conference on K o n t i e r s in H draulic En ineering. L e r i c a n Society of Civil Engineers, New Yo&: 113113. Chen, C. , 1984. Hydraulic ,concepts in debris .flow simulation. In: D.S. Bowles (editor), Delineation of Landslide, Flash Flood, and Debris Flow Hazards in Utah. Utah WaterResearch Laboratory, Logan, Utah: 236-259. Chen, C. , 1985. Present status of research in debris flow modeling. In: Proceedin s of the Speciality Conference Hydraulics and Hydrology in t%e Small Com uter Age. American Society of Civil Engineers, New York: 733-791. Chen C., 1986a. Bingham pJastic or Bagnold's diJatant fluid aF a rheoiogical model of a debris flow? In: Proceedings of the Third Inteyna$ional osium on River Sedimentation. Jackson, Mississippi: 1 6 % ? ? 6 3 5 . Chen, C., 1986b. Chinese concepts of modeling hyperconcentrated streamflgw and debris flow. In.: Proceedings of the Third International osium on River Sedimentation. Jackson, Mississippi: 16%??657. Chen, C. 1986c. Visco lastic fluidmodel fro debris flow routing. In: M. Aaramouz, G.R. gaumli and W,J. Brick (editors), Water Forum '86: World Water Issues in hvolution. American Society of Civil Engineers, New York: 10-18. Chow, V.T., 1959. Company, New York.
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Hydraulics.
McGraw-Hill
Book
Conte S.D., 1965. Elementary Numerical Analysis. McGraw-Hill Book bompany, New York. Costa, J.E., 1983. Paleohydraulic reconstruction of flash flood eaks from boulder deposit? in the Colorado Front Range. Geological Eociety of America Bulletin, 17.: 986-1004. Croft, A.R., 1967. Rainstow debris flows: a problem in public welfare, Agricultural Experiment Station, University of Arizona,
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Tucson.
-
2.
Dawdy, D.R., 1961. Depth-dischar e relations in alluvial streams discontinuous rating curves. U. Geological Survey Water Supply Paper 1498-C, Washington. Dawdy D.R., 1979. Flood frequency estimates on alluvial fans. ASCE, 'Journal of the Hydraulics Division, 105 (HY11): 1407-1413. DeLeon A.A. and Jeppson, R.W. , 1982,. Hydraulics and numerical soluti6ns of steady-state but s atially varied debris flows. UWRL/H-82/03 , Utah Water Research La%oratory, Utah State University, Logan, Utah. Ee C.S. 1970. The width de th and velocity of the Sungei Kinta, Pecak. keographica, 6: f2-8f. Einstein H.A. and Barbarossa N.p. , 1956. River Fhannel roughness. Transactions of the American 'Society of Civil Engineers, 177: 440457. Einstein, H.A. and Chien N. 1926. Einqtein-Chiqn,on di$torted models. Transactions of 'the h e r i c a n Society of Civil Engineers , 121: 459-462. Enos P 1977. Flow regimes in debris flow. Sedimentology, 24: 133-142: French, R.H., 1985. Open-Channel Hydraulics. McGraw-Hill Book Company, Inc., New York, 705 pp. Garde R.J. and Ran a Rgju K.G., 1978. Mechanics of Sediment Trangbortation and A?luvial' Stream Problems. Wiley Eastern, New Delhi. Glover R.E. and Florey .L., 1951. Stable channel profiles. Hyd 325. U.S. Bureau of Rec amation, Hydraulic Laboratory Repoft Washington. Graf W.H., 1971. Hydraulics of Sediment Transport. McGraw-Hill Book'Company, New York. Gre ory G. 1927. Pumping cla slurry through a four-inch pipe. Mec%ani&al kngineering, 49 : 608-616. Hampton M.A. 1972. The role of subaqueous debris flow in eneratin turbidity currents. Journal of Sedimentary Petrology, 82: 775-783.
80.
Hedstrolp B.O.A., 1952. Flow of plastic materials in pipes. Industrik and Engineering Chemistry, 44: 651-656. Henderson, F.M., 1966. Openchannel Flow. MacmillanCo., NewYork, 522 pp. Hughes, W.F. and Brighton, J.A., 1967. Fluid Dynamics, McGraw-Hill Book Company, New York. James, L.D., Pitcher D.O.,Heefner, S.,Hall, B.R., Paxman, S.W., and Weston, A. , 1986. Flood risk below stee mountain slopes. In: M. Karamouz G.R. Baumli, and W.J. ,Brick ?editors) Water Forum 186: World WAter Issues in Evolution. American kocietv of Civil * Engineers, New York: 203-210. Jarrett, R.D. 1984. Hydraulics of hi h- radient streams. ASCE, Journal of Hyhraulic Engineering, 110 7117: 1519-1539. Jep son R.W. and Rodri S.A., 1983. Hydraulics of solving unsteadi debris flow. aLFfiL83 03 Utah Water Research Laboratory, Utah State University, Logan, &ah: Johnson A.M., 1970. PhysJcal Processes in Geology. Freeman, Cooper & Company, San Francisco. Knighton A.D., 1974. variation inthewidth-discharge relationand some imp'lications for h draulic geometry. Geologic Society of America Bulletin, 85: lo&-1076. Knighton, geometry.
A.D. 1975. Variations in at-a-station American Journal of Science, 275: 186-218.
Konemann, N,
1982.
Discussion of
"Minimum specific
hydraulic energy in
134
Compound Open Channel". ASCE, Journal of the Hydraulics Division, 108 (HY3): 462-464. Kumar. S.. 1986. Enaineerins methodoloav for delineatins debris flow hazards in Los Afigeles Co'unt In: 7 4 . Karamouz G.R.-Bauml$, and W.J. Brick (vditors) Water $ & p m '8q: World Waker Issues in Evolution. American Society of Civil Engineers, New York: 19-26. Lane E.W., 1955. Design of stable channels. Transactions of the American Society of Civil Engineers, 120: 1234-1279. Lane F.W. and Carlson E.J., $953. Some factors affqcting the stability of canals condtructed in coarse ranular materials. In: Proceedings of the Minnesota Internationa? Hydraulics Convention. Langbein W.B., 1964. Geomet of riverchannels. ASCE, Journal of the Hydr6ulics Division, 90 3 Y 2 ) : 301-312. Leo old, L.B. and Langbein W.B. 1962. The concept of-entro y in langscape ?valuation. U. d. Geoiogical Survey Professional gaper 500A, Washington. Leopold, L.B. and Maddock, T. , Jr. , 1953. The hydraulic geomet stream channels and some hysio ra hic jmplications. Geological Survey Professional Saper Washington. Leopold, L.B., Wolman, M.G., and Miller, J.P., 1964. Fluvial FroceSses in geomorphology. Freeman, Cooper and Company, San rancisco.
?d,
7.;:
Mears A.I., 1977. Debris-fl,owhazard analysis andmitigation: an exampie from Glenwood Springs , Colorado. Colorado Geological Survey, Denver, Colorado. Meyer-Peter, E. and Mu ler R., 1948. Formqlas for bed-load trans ort In: Proceedings, of the 3rd Meeting of the IAHR, StockRolm; 39-64. Nixon, M., 1959. Astudyofbank-fulJdischar esofrivers inEngland and Wales. Proceedings of the Institution 0% Civil Engineers, 12: 157-174. Pack F.J., 1923. Torrential potential of desert waters. PanAmerican Geologist, 40: 349-356. Park C.C., 1977. World-wide variations in hydraulic georpetry expohents of stream channels: an analysis and some observations. Journal of Hydrology, 33: 133-146. Petyyk, S. andGrant E.U., 1975. Critical, fJoy inriverswithflood lains. ASCE, Jourdal of the Hydraulics Division, 101 (HY7): 933846. Philli $ P.J. and Harlin J.M., 1984. Spatial dependenc of hydrau? 16 geometr onents in a subalpine stream. Journa5 of Hydrology, 71: 277-%8. Pierson T.C. 1981. Dominant artic,le 8 ppo tmechanisms in ebris flows ad Mt. Thomas, New Zealan$ and ImplYcatIfons for flowmobhity. Sedimentology, 28: 49-60. Raudkivi, A.J., 1976. Loose Boundary Hydraulics. Pergamon Press, New York. m o d e s D.D. 1977. The b-f-m diagram: gr phical representation and inte retation of at-a-station hydraul ?c geometry. American Journal science, 277: 73-96. Richa ds X.S. 1973. Hydraulic geometry and channel .rou hness a non-lfne6r system. American Journal of Science, 273. 897-896. R'chards K . S 1976. Complexw dth-discharge relatio 8 in natural river segtionk: Geological Soctety of America Bulletfn, 87: 199206. Schlichting, H., 1968. Boundary-Layer Theory. McGraw-Hill Book
3
-
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Company, New York. Schuster, R.L. and Jkrizek, R., 1978. Land slides anal sis and control. TranSpOrtatiQn Research Board National Research gouncil National Academy of Science, Washingtoi. Segerstroem, K., 1950. Erosion studies in Paricutin, State of Michgacan Mexico. Bulletin 965-A, U.S. Geological Survey, WashingtoA. Sharp, J.J., 1981. Hydraulic Modelling, Butterworth, London. Shar R.P. and Nqbles, L.H., 1953. Mudflow of 19.41 at Wri Qtwood, Southrn California. Geological Society of America Bullezin, 64 : 547-560. Simons D.p. and AJbertson,.M.L., 1960. Uniform water conveyance cQannqds in a1luv:al material. ASCE, Journal of the Hydraulics Division, 86 (HY5). 33-71. Simons D.B. and Senturk ,F., 1976. Sediment Transport Technology. Water fiesources Publications, Fort Collins, Co. Sfreeter V. L. and Wylie E. B., 1975. Fluid Mechanics. McGrawHill Book Company, New Ydrk. Subraman a K., 1982. Flow in 0 en Channels, Vol. 1. TataMcGrawHill Pubxikhing Company, New Deyhi. Takahashi T., 1978. Mechan'cal characteristics of. debris flow. ASCE, J o u h a l of the Hydraulks Division, 104 (HY8). 1153-1169. Takahashi T., 1980. Debris,flow,ona rismaticopenchannel. ASCE, Journal of the Hydraulics Division, PO6 (HY3): 381-396. art W.J., 1975. Hydyaulics. In: En ineerin Field Manyal. Debartment of Agriculture , Soil %onserva%ion Service, Washington. Williams G.P., 1978. Hydraulic geometry pf river cross sections theory ok minimup variance. U. S Geological Survey Professional Paper 1029, Washington. Wolman M.G 1955. The qatural channel of Brandywine Creek Pennsyivanii.' U. S. Geological Survey Professional Paper 271: Washington. 1981. Hydraulic Yang C.T., Sopg, C.C.S., and Wolden erg M,J Water Resources Tom& and minimum rate of energy d?ssibatioi: esearz, 17 (4): 1014-1018 Zwamborn J.A., 1966. Reproducibi ity in .hydraulic models of prototype river morphology. La H o u h l e Blanche, 3: 291-298. Zwamborn J.A. 1967. Solution of river problems with moveable-bed hydraulih models. MEG 597. Council for Scientific and Industrial Research, Pretoria, South Africa. Zwamborn J.A., 1969. Hydraulic models. MEG 795. Council for Scientific and Industrial Research, Pretoria, South Africa.
.
8?!?
.
CHAPTER 3 B A S I C PRINCIPLES O F OPEN CHANNEL HYDRAULICS 3.1
INTRODUCTION
I n examining h y d r a u l i c p r o c e s s e s on a l l u v i a l f a n s , it i s n e c e s s a r y f o r t h e non-engineer t o become f a m i l i a r w i t h some of t h e b a s i c p r i n c i p l e s o f open-channel flow: and f o r t h e e n g i n e e r w i t h a background i n f l u i d m e c h a n i c s a n d h y d r a u l i c e n g i n e e r i n g t o re-examine and r e - c o n s i d e r some o f t h e b a s i c p r i n c i p l e s o f t h e s e t e c h n i c a l a r e a s from t h e v i e w p o i n t of a l l u v i a l f a n s . I t i s t h e p u r p o s e of t h i s chaptertoprovideabriefand rudimentarytreatmentofseveral o f t h e b a s i c p r i n c i p l e s of open-channel h y d r a u l i c s . I n p a r t i c u l a r , t h i s chapterwillconsiderspecificenergy:normaloruniformflowonmild, c r i t i c a l , and s t e e p s l o p e s ; t h e s t a b i l i t y of a l l u v i a l c h a n n e l s ; and t h e mechanics of d e b r i s flows. Although s p e c i f i c r e f e r e n c e s w i l l be mentioned t h r o u g h o u t t h i s c h a p t e r , t h e r e a r e a number of b a s i c r e f e r e n c e s which p r o v i d e a comprehensive d i s c u s s i o n of t h e s e s u b j e c t s ; see f o r example, Chow ( 1 9 5 9 ) , Henderson ( 1 9 6 6 ) , French ( 1 9 8 5 ) , and Graf ( 1 9 7 1 ) . 3.2
S P E C I F I C ENERGY
By d e f i n i t i o n , t h e s p e c i f i c energy of a one d i m e n s i o n a l openchannel flow r e l a t i v e t o t h e bottom of t h e c h a n n e l is (3.2.1)
w h e r e E = s p e c i f i c e n e r g y ( m ) ; y = d e p t h o f flow ( m ) ; a = k i n e t i c e n e r g y c o r r e c t i o n f a c t o r which c o r r e c t s f o r t h e non-uniformity of t h e v e l o c i t y p r o f i l e ( d i m e n s i o n l e s s ) ; u = Q/A = a v e r a g e v e l o c i t y of flow (rn/s); Q = channel v o l u m e t r i c flow r a t e ( m 3 / s ) ; A = flow a r e a ( m 2 ) ; and g = a c c e l e r a t i o n of g r a v i t y ( m / s 2 ) . Equation ( 3 . 2 . 1 ) d e r i v e s d i r e c t l y from t h e B e r n o u l l i energy e q u a t i o n (law of c o n s e r v a t i o n of energy) and i s s u b j e c t t o a number of assumptions. Primary among t h e s e a s s u m p t i o n s i s t h a t c o s 8 Il o r t h a t 8 < 1 0 ' where 8 = s l o p e a n g l e of t h e c h a n n e l . For a more complete d e r i v a t i o n of Equation ( 3 . 2 . 1 ) , t h e r e a d e r i s referred t o French (1985) o r Streeter and Wylie ( 1 9 7 5 ) . Examination o f Equation ( 3 . 2 . 1 ) f o r t h e c a s e t h a t a = 1, Q i s known, a n d t h e c h a n n e l g e o m e t r y i s d e f i n e d d e m o n s t r a t e s t h a t E i s o n l y afunctionofthedepthofflow, y , F i g s . 3 . 2 . 1 . For a s p e c i f i e d Q a n d
83
Y
yz
FIG.
3.2.1
S p e c i f i c energy c u r v e .
a d e f i n e d channel geometry, i f y is p l o t t e d a s a f u n c t i o n of E, F i g . 3.2.lb, t h e r e s u l t i s a hyperbolawithbranches i n t h e f i r s t andthird q u a d r a n t s . T h e b r a n c h o f t h e h y p e r b o l a i n t h e t h i r d q u a d r a n t i s of no i n t e r e s t s i n c e it r e p r e s e n t s n e g a t i v e v a l u e s of b o t h E and y , which have no p h y s i c a l meaning. The branch o f t h e hyperbola i n t h e f i r s t q u a d r a n t h a s two limbs. The limb AC of t h i s branch i s asymptotic t o On t h e t h e E a x i s , and t h e limb AB is a s y m p t o t i c t o t h e l i n e y = E . curveshown i n F i g . 3 . 2 . l b , t h e p o i n t l a b e l e d A r e p r e s e n t s t h e m i n i m u m s p e c i f i c energy r e q u i r e d t o p a s s t h e flow Q through t h e d e f i n e d c h a n n e l . The c o - o r d i n a t e s of t h e p o i n t l a b e l e d A can b e found by t a k i n g t h e f i r s t d e r i v a t i v e of Equation ( 3 . 2 . 1 ) w i t h r e s p e c t t o t h e d e p t h o f flow and s e t t i n g t h e r e s u l t e q u a l t o z e r o o r Q2
E = y + 2gA2 dE
dA
- 0
(3.2.2)
With r e f e r e n c e t o F i g . 3 . 2 . l a , t h e d i f f e r e n t i a l flow a r e a d A n e a r t h e f r e e s u r f a c e can be approximated by dA
I
Tdy
84
or
dA _ -- =
T
dY With t h i s a p p r o x i m a t i o n f o r dA/dy E q u a t i o n ( 3 . 2 . 2 ) becomes (3.2.3) where D = A/T = h y d r a u l i c d e p t h .
Rearrangement o f E q u a t i o n ( 3 . 2 . 3 )
yields (3.2.4) o r d e f i n i n g t h e Froude number
(3.2.5) where F = Froude number and E q u a t i o n ( 3 . 2 . 5 ) d e f i n e s what i s termed
criticalflow;thatis,theminimumamountofspecificenergyrequired
t o p a s s t h e flow
Q t h r o u g h t h e c h a n n e l geometry s p e c i f i e d .
With
r e g a r d t o t h e above development and F i g s . 3 . 2 . 1 , t h e f o l l o w i n g s h o u l d be n o t e d . F i r s t , i n a c h a n n e l of l a r g e s l o p e a n g l e 6 and a f 1, it can beeasilydemonstratedthat t h e c r i t e r i o n forminimum s p e c i f i c e n e r g y o r c r i t i c a l flow is (3.2.6)
Second, i n F i g . 3 . 2 . l b , t h e E-y curves f o r f l o w r a t e s g r e a t e r t h a n Q
l i e t o t h e r i g h t o f t h e curve BAC; and E-y curves f o r f l o w r a t e s l e s s t h a n Q l i e t o t h e l e f t o f c u r v e BAC. T h i r d , by d e f i n i t i o n , f l o w s f o r which F > 1 a r e termed s u p e r c r i t i c a l and l i e on l i m b AC o f t h e E-y
c u r v e ; f l o w s f o r which F
= 1 a r e termed c r i t i c a l and o c c u r a t p o i n t A;
and f l o w s f o r w h i c h F < l a r e t e r m e d s u b c r i t i c a l and l i e o n b r a n c h ABof t h e E-y c u r v e .
Fourth, with t h e exception of P o i n t A, t h e c r i t i c a l
point, everyvalueofE suchthatE > E ,
( c r i t i c a l s p e c i f i c energy) has
a s s o c i a t e d w i t h it t w o p o s s i b l e d e p t h s o f f l o w . 3.2.lb,
i f t h e s p e c i f i c energy of
Forexample, i n F i g .
flow i s E l ,
then t h e l i n e DF
85
demonstrates that there are two possible depths of flow; y1 and y2. FLoodplaln/Ovarbank
Floodplaln/Overbank
7
7
Sectlon
Sectlon
"/'\
Primary
' - ! ! Channel - - , I
FIG. 3.2.2
Schematic of a channel of compound section.
The depth of flow y 1 is associated with a supercritical flow having a specific energy E l ; and the depth of flow y, is associated with a subcritical flow which also has a specific energy E,. The possible depths of flow y1 and y, are known as the alternate depths of flow. Fifth, in channels of compound section, Fig. 3.2.2, the problem of specific energy and the computation of the correct Froude number becomes complex. Although a discussion of this area of study is beyond the scope of this treatment, the interested reader may find a summary in French (1985) and specific discussions in Blalock (1980), Blalock and Sturm (1981), Petryk and Grant (1975), and Konemann
.
(1982)
The comment in the foregoing paragraph regarding alternate depths of flow leads directly to a consideration of the problem of accessability and controls. Through the terminology of accessability, it is meant to indicate that with Q and the chann'el geometry given a priori a means of deciding which of the alternate depths on the E-y curve is accessable must be found. The result is the identificationoftheactual downstreamdepthofflowthat is possible with the specified upstream conditions. Accessability arguments appeal to logic rather than mathematical proof. As an example of such arguments, consider a rectangular channel of constantwidthb which conveys a steady flowQ. In the otherwise horizontal bed of the channel, there is a smooth upward step of height 42, Fig. 3.2.3a. Given this situation, an E-y curve can be constructed, Fig. 3.2.3b. In this figure, assume that the flowatsection1inFig. 3.2.3ais represented by point Aon theEy curve. Note, the point A' has the same specific energy as point A, and the choice of point A as the starting point derives from prior or given knowledge of the Froude number at section 1. The choice of
86
p o i n t A i n d i c a t e s t h a t F < 1; and hence, t h e flow a t s e c t i o n 1 is s u b c r i t i c a l . T h e l o c a t i o n of t h e s p e c i f i c energy on t h i s curve r e p r e s e n t i n g s e c t i o n 2 can be determined by t h e a p p l i c a t i o n of t h e Bernoulli energy equation between s e c t i o n s 1 and 2 o r - 2
- 2
u2
U1
2g + y 1
= -
2g
+
y2 +
AZ
E l = E2 + A Z E2 = El
-
AZ
where i t i s a s s u m e d t h a t t h e e n e r g y d i s s i p a t i o n b e t w e e n s e c t i o n s l a n d 2 isnegligible. Havingmathematically d e t e r m i n e d t h e v a l u e o f E 2 w e must now determine t h e c o r r e c t depth of flow a t s e c t i o n 2 . T h e w i d t h of t h e channel, by d e f i n i t i o n , does n o t change: and t h e r e f o r e , t h e l l f l o w p o i n t l v c a n o n l y m o v e o n t h e c u r v e d e f i n e d i n F i g .3.2.333. Ifthe w i d t h of t h e channel v a r i e d , t h e n flow p e r u n i t w i d t h would vary and t h e flow p o i n t could move o f f t h e curve shown i n Fig. 3.2.3b and onto t h e foregoing o t h e r E-y curves. With regard t o F i g . 3.2.3b, statement means t h a t t h e flow p o i n t cannot Ivjurnp1* from p o i n t B t o B 1 Thus, i f p o i n t B' is t o b e a n a c c e s s a b l e s o l u t i o n o f t h i s problem, t h e flow p o i n t must p a s s through p o i n t C. Movement of t h e flow p o i n t t o pointC i s o n l y p o s s i b l e i f t h e increase i n c h a n n e l b o t t o m e l e v a t i o n i s g r e a t e r t h a n t h a t s p e c i f i e d . This h y p o t h e t i c a l s i t u a t i o n i s r e p r e s e n t e d by a dashed l i n e i n Fig. 3 . 2 . 3 a . Therefore, t h e conclusion i s t h a t of t h e p o s s i b l e d e p t h s of flow a t s e c t i o n 2 , only t h e depth of flow a s s o c i a t e d with p o i n t B i s accessable from p o i n t A with t h e upstream flow c o n d i t i o n s a s s p e c i f i e d . The foregoing d i s c u s s i o n t a c i t l y assumes t h a t t h e r e i s a s o l u t i o n t o t h e problem with t h e upstream c o n d i t i o n s a s s p e c i f i e d . I n f a c t , it is q u i t e easy t o s p e c i f y a s t e p s i z e such t h a t a s o l u t i o n with t h e given upstream c o n d i t i o n s does n o t e x i s t . For example, i n Fig. 3.2.3a, i f t h e s t e p s i z e e x c e e d s ~ z , ,t h e n t h e r e i s n o s o l u t i o n t o t h e problem. T h i s s i t u a t i o n is p h y s i c a l l y explained by n o t i n g t h a t when t h e s t e p s i z e exceeds A Z , t h e channel h a s been s u f f i c i e n t l y o b s t r u c t e d by t h e s t e p so t h a t t h e upstream s p e c i f i c energy i s not adequate t o p a s s t h e flow. That is, t h e flow has been choked by t h e s t e p , I n s u c h a s i t u a t i o n , t h e flowupstreamofthestepmust increase indepthuntilthespecificenergyavailableupstreamissufficientto p a s s t h e flow over t h e o b s t r u c t i o n .
.
87
The following comments regardingspecificenergyshouldbemade. First , in the vicinity of the critical point , point C in Figs. 3.2.3 , Y
E
FIGS. 3.2.3
(a) (b) Schematic definition of the accessability problem.
small changes in specific energy can result in large changes in the depth of flow. Second, as noted previously in this section, in channels of compound section, the problemsof specificenergy andthe locationofthecriticalpointorpointscanbecomesignificantlymore complex and difficult. Third, the foregoing paragraphs have provided only a synoptic discussion of specific energy. For a more detailed discussion of this topic, the reader is referred to either French (1985) or Henderson (1966). 3.3
UNIFORM/NORMAL FLOW By definition, uniform flow occurs when 1. 2.
The depth, flow area, and velocity at every channel cross section are constant: and the energy grade line, water surface, and channel bottom are all parallel.
Although uniform flow is a theoretical ideal which can only occur in very long, straight channels of a fixed geometry, this theoretical concept can be used with great effect in the solution of very applied problems. One of the equations that describes uniform flow is the Manning equation or
88
(3.3.1) where Q = flow rate (m3/s): A = flow area (m2); R = A/P = hydraulic radius (m): P = wetted perimeter (m): S = slope of the energy grade line, water surface, or channel bottom: n = Manning roughness coefficient; and $ = a coefficient used to account for the system of units used. Ifthe SI system ofunits specifiedabove isused, then $ If both = 1; if the English system of units is used, then $ = 1.49. sides of Equation (3.3.1) are divided by the flow area, then a second form of the Manning equation results or (3.3.2) where
u
= average velocity of flow (m/s if $ = 1.0) InEquations (3.3.1) and (3.3.2)thecoefficientncharacterizes the roughness of the channel boundary and is a well established coefficient for estimating boundary friction effects. There are a number of methods for estimating values of n in perennial channels: see for example, Barnes (1967), Chow (1959), French (1985), Henderson Although some of the methods used for (1966), and Urquhart (1975). perennial channels are applicable to ephemeral channels on alluvial fans, it is worth noting several semi-empirical methods which have b e e n d e v e l o p e d t o e s t i m a t e v a l u e s o f n from amechanical sizeanalysis of the materials which compose the channel boundary. Perhaps the best known semi-empirical method of estimating n is t h a t p r o p o s e d b y S t r i c k l e r i n 1 9 2 3 ; see for exampleSimons andsenturk (1976), which hypothesized that
n = 0.047d'/6
(3.3.3)
where d = diameter in millimeters of the uniform sand pasted to the sides and bottom of an experimental flume used by Strickler. While Equation (3.3.3) has limited validity in natural channels, it specifies a basic functional relationship between n and d. Among the results functionally similar to Equation (3.3.3) are: 1/6
n = 0.013d
(3.3.3a) 65
where d6 = diameter in millimeters such that 65% of the channel
89
boundary material, by weight, is smaller, Raudkivi (1976). 1/6
n = 0.039d
(3.3.3b) 30
whered,, =diameter infeetsuchthat50% ofthechannelboundary material by weight is smaller, Garde and Raju (1978). If the SI system of units is used, then Equation (3.3.335) becomes 1/6
n = 0.047d
(3.3.3c) 50
where d is in meters, Subramanya (1982). 1/6
n = 0.038d
(3.3.3d) 90
where d,, = diameter in meters such that 90% of the channel boundary material by weight is smaller, Meyer-Peter and Muller (1948). Equation (3.3.34) wasdeveloped for channels withbeds formed of mixed materials with a significant proportion of coarse grained sizes. 1/6
n = 0.026d
(3.3.3e) 75
where d,, = diameter in inches such that 75% of the channel boundarymaterialbyweight issmaller, Laneand Carlson (1953). Equation (3.3.3e) was derived from experiments in which the channels were paved with cobbles. Of the above equations for estimating a value of Manning's n, in the case of alluvial fans Equation (3.3.34) is perhaps the most appropriate. Jarrett (1984) recognized that the available guidelines for estimating resistance coefficients for high-gradient; that is, channelswithslopesgreaterthan0.002, arebased onlimited dataand are handicapped by a lack of easily applied methods for evaluating changes of boundary resistance with the depth of flow. Jarrett (1984) exa.mined 21 high-gradient streams in the Rocky Mountains with stable beds andminimallyvegetatedbanks and developed an empirical equation for n or
90
n =
0.3g~0.38~-0*16
(3.3.4)
w h e r e R = h y d r a u l i c r a d i u s i n feet. W h i l e t h e s t r e a m s u s e d b y J a r r e t t (1984) t o develop Equation ( 3 . 3 . 4 ) had s l o p e s s i m i l a r t o t h o s e found on a l l u v i a l f a n s , t h e channel beds were composed of large c o b b l e s a n d boulders whichmay ormay n o t b e t y p i c a l of channels found o n a l l u v i a l f a n s . F u r t h e r , flows i n t h e channels s t u d i e d by t h i s i n v e s t i g a t o r were subcritical while flows on a l l u v i a l f a n s a r e g e n e r a l l y b e l i e v e d t o be c r i t i c a l o r s u p e r c r i t i c a l . I n Equation (3.3.1) , t h e parameter AR2 l 3 is known as t h e s e c t i o n f a c t o r , and f o r a channel of s p e c i f i e d geometry where A R 2 I 3 always i n c r e a s e s w i t h i n c r e a s i n g d e p t h s o f flow, each f l o w r a t e o r d i s c h a r g e hasacorrespondinguniquedepthof f l o w a t whichuniform f l o w o c c u r s . Examination of Equation ( 3 . 3 . 1 ) demonstrates t h a t f o r a uniform flow t h e flow r a t e i s a f u n c t i o n of 1) t h e geometric shape of t h e channel, 2 ) a boundary r e s i s t a n c e c o e f f i c i e n t such a s Manning's n, 3 ) t h e l o n g i t u d i n a l s l o p e of t h e channel, and 4 ) t h e depth of flow o r
Q = f(r,n,s,y,,)
(3.3.5)
where 'I = shape f a c t o r and Y,, = normal depth of flow. I f f o u r of t h e f i v e v a r i a b l e s i n Equation (3.3.5) a r e known, t h e n t h e v a l u e of t h e f i f t h v a r i a b l e can be estimated. Although a d e t a i l e d d i s c u s s i o n of t h e computation of uniform flow i s beyond t h e scope of t h i s book, t h e i n t e r e s t e d r e a d e r is r e f e r r e d t o French (1985) f o r a complete d i s c u s s i o n of t h i s t o p i c . I f Q , n, y,, a n d t h e channelgeometry a r e s p e c i f i e d , thenEquation ( 3 . 3 . 1 ) can be solved e x p l i c i t l y f o r t h e s l o p e of t h e channel which a l l o w s t h e f l o w t o o c c u r a t anormal depth. B y d e f i n i t i o n , t h i s s l o p e wouldbethenormal slope. I f t h e s l o p e o f t h e c h a n n e l i s v a r i e d w h i l e Q and n are h e l d c o n s t a n t , t h e n it is p o s s i b l e t o determine a s l o p e value s u c h t h a t normal f l o w w i l l o c c u r w i t h F = l ; t h a t i s , normal depth w i l l a l s o be c r i t i c a l depth. T h i s slopewould betermed t h e l i m i t i n g c r i t i c a l s l o p e . Slopes g r e a t e r t h a n t h e l i m i t i n g c r i t i c a l s l o p e a r e termed s t e e p s l o p e s and s u s t a i n s u p e r c r i t i c a l flow. 3.4
ALLUVIAL CHANNEL STABILITY
Before c o n s i d e r i n g t h e p r o c e s s e s a f f e c t i n g channel s t a b i l i t y and sediment t r a n s p o r t i n an a l l u v i a l channel e x p l i c i t l y , l e t us c o n s i d e r t h e t r a c t i v e f o r c e method of d e s i g n i n g unlined channels. Scour o r e r o s i o n on t h e perimeter of a channel occurs when t h e particleswhichcomposethe channel p e r i m e t e r a r e s u b j e c t e d t o f o r c e s
91
which are of s u f f i c i e n t magnitude t o cause movement.
The p a r t i c l e s
r e s t i n g o n t h e b o t t o m o f t h e channel a r e s u b j e c t t o o n l y flowgenerated f o r c e s while t h e material on t h e s l o p i n g s i d e s of t h e channel is s u b j e c t e d t o both flow generated f o r c e s and a g r a v i t a t i o n a l f o r c e which a c t s t o r o l l t h e p a r t i c l e s down t h e s l o p i n g channel s i d e . I n uniform flow, it can be e a s i l y shown; see f o r example, French (1985), t h a t t h e t r a c t i v e f o r c e on p a r t i c l e s due t o flow can be approximated a s F, = YALS
(3.4.1)
where F, = f l o w g e n e r a t e d t r a c t i v e f o r c e , Y = f l u i d s p e c i f i c weight, A = flow a r e a , L = l o n g i t u d i n a l l e n g t h considered, a n d S = l o n g i t u d i n a l s l o p e of t h e channel. I f Equation (3.4.1) i s accepted, t h e n t h e u n i t t r a c t i v e f o r c e is YALS To=--
PL
-
YRS
(3.4.2)
where r 0 = average v a l u e of t h e t r a c t i v e f o r c e ( t h i s i s a c t u a l l y a stress r a t h e r t h a n a f o r c e ) p e r u n i t of wetted a r e a and P = w e t t e d perimeter. I n wide channels, y w I R , and Equation (3.4.2) becomes To =
vy,s
(3.4.3)
A t t h i s p o i n t , it s h o u l d b e n o t e d t h a t inchannelsoftrapezoidalcross
s e c t i o n t h e maximum t r a c t i v e f o r c e on t h e bed has been found t o be approximately yy,S and on t h e sides 0 . 7 6 ~ ~ ~ s . When p a r t i c l e s on t h e perimeter of t h e channel a r e i n a s t a t e of impending motion, t h e f o r c e s a c t i n g t o cause t h e motion a r e i n e q u i l i b r i u m with t h e f o r c e s opposing motion. I n t h e c a s e of p a r t i c l e s on t h e bottom of t h e channel, t h e f o r c e causing motion i s A e r L where r L = u n i t t r a c t i v e f o r c e on a l e v e l s u r f a c e and A, = e f f e c t i v e a r e a on which r L o p e r a t e s . The f o r c e r e s i s t i n g motion i s t h e g r a v i t a t i o n a l forceorW,tanawhereW, =submergedparticleweight and a = t h e angle of repose of t h e p a r t i c l e . Then, when motion i s incipient A e r L = W,tana
or
(3.4.4)
92
T~
=
ws
- tana
(3.4.5)
*e
On the sloping sides of the channel, particles are subject to both a flow generated tractive force T ~ A , and a downslope gravitational component W,sinr where 7 s = tractive force on side slopes and r = side slope angle. The resultant forceactingtocause motion is
i +
(w,tanr)'
(A,T,)'
The force resisting motion is W,cosrtana, and setting these forces equal for the case of incipient motion W,(cosr) (tana) =
4 (W,tanr) ' +
or
zS =
WE
- (cosr)(tana) Ae
1
1
-
tan2r
tan'a
Equations (3.4.5) and ( 3 . 4 . 7 ) tractive force ratio K, or
7L
sin'a
(Ae7,)
(3.4.6)
(3.4.7)
are usually combined to form the
(3.4.8)
The tractive force ratio is a function of both the side slope angle and the angle of repose of the material which composes the perimeter. Laboratory data are available for bothcohesive andnoncohesive materials, Lane (1955) or French (1985). F r o m t h e c o n c e p t o f t r a c t i v e forcechanneldesign,theconcept of a stable hydraulic section is derived. In the design of a channel section, usually trapezoidal in shape, the tractive force is equal to its permissablevalue ononly aportion ofthe perimeter- usuallythe sides. It seems rational toattempt todefine achannel sectionsuch that incipient particle motion prevails at all points on the channel perimeter. Glover and Florey (1951) did this for a channel carrying clear water through non-cohesive materials. The specific
93
Schematic, d e f i n i t i o n of v a r i a b l e s f o r a s t a b l e h y d r a u l i c channel i n non-cohesive materials. F I G . 3.4.,1
assumptions made by t h e s e i n v e s t i g a t o r s i n t h e i r a n a l y s i s were: 1.
B o u n d a r y p a r t i c l e s a r e h e l d i n p l a c e b y t h e submergedweight component of t h e p a r t i c l e s a c t i n g i n a d i r e c t i o n normal t o t h e bed of t h e channel.
2.
A t andabovethewater surface, t h e s i d e s l o p e o f thechannel
is a t t h e a n g l e of repose of t h e material composing t h e channel perimeter. 3.
A t t h e c e n t e r l i n e of t h e channel s e c t i o n , t h e channel s i d e
s l o p e is z e r o , and t h e t r a c t i v e f o r c e generated by t h e flow is a l o n e s u f f i c i e n t t o maintain a s t a t e of i n c i p i e n t p a r t i c l e motion. 4.
Atpointsbetweenthecenterlineofthechanneland i t s edge, p a r t i c l e s a r e i n a s t a t e of i n c i p i e n t motion.
5.
The t r a c t i v e f o r c e a c t i n g on an a r e a of t h e channel is equal t o t h e component of t h e weight of t h e water above t h e a r e a a c t i n g i n t h e d i r e c t i o n o f flow: t h a t i s , t h e r e is n o l a t e r a l t r a n s f e r of t r a c t i v e f o r c e .
Giventheaboveassumptions a n d t h e s c h e m a t i c d e f i n i t i o n o f v a r i a b l e s i n Fig. 3 . 4 . 1 , it can be shownthat t h e g e o m e t r i c s h a p e o f t h e c h a n n e l of s t a b l e h y d r a u l i c s e c t i o n is given by
94
y = y w cos
[1'
(3.4.9)
the flow area by A=-
2TYN
(3.4.10)
I(
and the wetted perimeter by
2YN
E(sin
p = -
sin
(3.4.11)
a)
a
where the symbolism E(sin a) designates a complete elliptic integral of the second kind. The discharge of this stable channel is then estimated directly from the Manning equation or 8/3
Q -
2.98 y N n(tan
a)
(cOS
,)'I3
[E(sin
&
a)]'13
(3.4.12)
It is worth noting that the tractive force methodology represents onlyone techniquewhich canbe usedto designor analyzea channel in non-cohesive alluvial materials. Other methodologies have equal validity and in some cases have particular advantages. For example, there isthethresholdofmovementhypothesisof Shields; see for example, Henderson (1966) or French (1985). This hypothesis is stated in terms of two dimensionless variables R, =
u*d -
(3.4.13)
V
and (3.4.14)
where R, = a Reynolds number based on the shear velocity of the flow, the size of the particles composing the perimeter, and the kinematic viscosity of the fluid;
95
I -
U* I:
4 gRS
(3.4.15)
=shearvelocity, v = fluid kinematicviscosity; S, = specificgravity ofthe soil particles composingthe channel perimeter (S, = 2.65 inthe general case) ; and d = diameter of the soil particles composing the channel perimeter. d is usually taken as the diameter such that 25% of the particles, measured by weight, are larger. Shields' results are usually summarized in a graphical form, Fig. 3.4.2. In this figure, the curve denotes and defines the threshold of particle movement.
-i
SHIELD'S DIAGRAM
Particle Movement
No Particle Movement
FIG. 3.4.2 number.
Threshold of movement as a function of particle Reynolds
A second alternative to the tractive force methodology is the regime theory. Although this theory has beenvigorously attackedby modern engineers because of its lack of a rational and vigorous derivation, it is essentially the technique recommendedby theU.S. Federal Emergency Management Agency (FEMA) for identifying flood hazard zones on alluvial fans; see for example, Anon (1983), Dawdy (1979), and Anon (1985). The regime theory has also been termed the Indian approach because the original field studies performed in supportofthistheoryweredone inwhat isnow IndiaandPakistan. In the modern hydrologic and geologic literature, the regime theory is often termed the theory of hydraulic geometry because it is used to
96
e s t i m a t e t h e channel widthand d e p t h a n d t h e v e l o c i t y of flow. Given t h a t FEMA has adapted t h e regime t h e o r y ( o r t h e o r y of h y d r a u l i c geometry) a s i t s p r e f e r r e d technique of channel a n a l y s i s f o r f l o o d flows on a l l u v i a l f a n s , t h e apparent e a s e with which r e s u l t s can be obtained using t h i s t h e o r y , and t h e w i d e d i s c u s s i o n t h i s t h e o r y has received i n t h e hydrologic, g e o l o g i c , and h y d r a u l i c engineering l i t e r a t u r e , it i s w o r t h w h i l e t o c o n s i d e r t h e o r i g i n , l i m i t a t i o n s , and r e s u l t s available. A s n o t e d b y Graf ( 1 9 7 1 ) , i n a n o p e n - c h a n n e l w i t h r i g i d b o u n d a r i e s and uniform flow, t h e r e is only one degree of freedom - t h e depth of flow which can be estimated fromtheManning equation. Uniform flow i n an open-channel w i t h non-cohesive and moveable boundaries has a minimum of t h r e e degrees of freedom - width, depth, and l o n g i t u d i n a l s l o p e . The t h r e e degrees of freedomderive f r o m t h e h y p o t h e s i s t h a t i n a channel with e r o d i b l e boundaries a depth of flow w i l l be e s t a b l i s h e d which depends on t h e l o n g i t u d i n a l s l o p e of t h e channel, t h e width of t h e channel, and t h e d i s c h a r g e . Blench ( 1 9 6 1 ) introduced a f o u r t h degree of freedomby a s s e r t i n g t h a t a r t i f i c i a l l y s t r a i g h t channel s e c t i o n s are u n s t a b l e b e c a u s e e r o s i o n o f t h e channel banks w i l l e v e n t u a l l y r e s u l t i n channel meanders. The regime t h e o r y w a s a p p a r e n t l y f i r s t enunciated by Kennedy i n 1895. Since t h a t t i m e , a rather l a r g e body of d a t a and information regarding t h i s t h e o r y h a s and c o n t i n u e s t o evolve even i n t h e f a c e of vigorous a t t a c k s on t h e v a l i d i t y of t h e theory: see f o r example, E i n s t e i n a n d C h i e n (1956). Theterminologyofhydraulicgeometrywas a p p a r e n t l y f i r s t i n t r o d u c e d b y Leopoldand Maddock (1953) t o d e s c r i b e t h e observed v a r i a t i o n s of width, v e l o c i t y , and depth of flow a s a f u n c t i o n of flow r a t e , Figs. 3 . 4 . 3 . Because average d a t a w e r e used, Figs. 3 . 4 . 3 is somewhat misleading regarding t h e v a l i d i t y of t h e t h e o r y and f o r t h i s reason s i m i l a r d a t a a r e p l o t t e d i n F i g s . 3 . 4 . 3 f o r comparison with t h e d a t a i n Figs. 3 . 4 . 4 , P h i l l i p s and H a r l i n ( 1 9 8 4 ) . With r e f e r e n c e t o Figs. 3 . 4 . 3 and 3 . 4 . 4 , i n t h e i r most g e n e r a l form, t h e regime o r h y d r a u l i c geometry e q u a t i o n s f o r channel width, depth, and v e l o c i t y of flow are T = C,Qb
(3.4.16)
Y = C2Qf
(3.4.17)
and
-
u = C,Qm
(3.4.18)
97
TOP WIDTH, T b t e r s l
10
1
10
n o n RATE. 0
[cubic metere per second)
loo
98 VELOCITY OF ROn, u h a t e r s per eecond) 10
t
H1
ROW UTE.Q (cubic meters per second) (C)
FIGS. 3.4.3 Hydraulic geometry forthe RioPuerco at Rio Puerco, New Mexico, average data from Leopold and Maddock (1953). whereC1, C,, C,,b, f, andm areundetermined empiricalcoefficients. It is generally assumed that Equations (3.4.16) (3.4.18) must satisfy the continuity equation or if the channel cross section is
-
appproximately rectangular ~ y =i C,Q~C,Q~C,Q'
(3.4.19)
If Equation (3.4.19) is to be satisfied, then the undetermined exponents must satisfy b + f + m = l and
(3.4.20)
the coefficients
c,c,c,
= 1
(3.4.21)
In many cases, Equations (3.4.20) and (3.4.21) are not satisfied. Among the reasons for this deviation are 1) data error, 2) the method used to fit the regression line to the data, 3) validity of the power function relationship, and 4) the absence of physical adjustment of the channel to some discharges, Rhodes (1977). In considering the applicability of the regime or hydraulic
g e o m e t r y t h e o r y t o h y d r a u l i c processeson alluvial fans, thecriteria for their use noted by Blench (1957) should be considered. These criteria are:
99
0
I
0.1
0.01
FLOW RATE. 0
(cubic metere per eecond)
FLOW R A E , 0
(cubic meters per second)
DEPTH OF FLOW, y betere)
'
0.01 0.01
0.1
(b)
1
100
V M C I l Y OF R O W .
1
u
(#tors
par recondl
°
i
0O
0.1 0. 0.01
o=
0.1
ROW RATE. B (cubic w t e r r per rscondl
FIGS. 3 . p . 4 , Hydraulicgeometr OftheHuerfanoRiver, Colorado, data from Phillips and Harlin (198i). 1.
The channel is straight.
2.
The channel sides behave as if they were hydraulically smooth. Note, boundary surfaces are classified as being either hydraulically smooth or rough by comparing the laminar sublayer thickness and the roughness height: see for example, French (1985) or Schlichting (1968). If the boundary roughness is such that theroughness elements are covered by the laminar sublayer, then by definition the boundary is hydraulically smooth.
3.
The bed or bottom width of the channel must be greater than three times the depth of flow.
4.
The side slopes of the channel must approximate those for cohesive materials in nature.
5.
The water discharge of the channel is steady.
6.
The sediment discharge of the channel is steady.
101
7.
Channels that move a non-cohesive load of sediment along the bed in dune formation.
8.
The Froudenumber ofthe flowmust be less thanone; that is the flow is subcritical (see for example Section 2 of this chapter). Note, Chang (1980) mentioned that canal designers using the regime theory have often used a Froude number criterion for establishing the range of applicability of regime theory designs. According to Chang (1980), the Froude number of the flow should be kept a t a v a l u e of approximately 0.2 and neverallowed toexceed a value of 0.3.
9.
Thesizeofmaterial composingthe channelboundary mustbe small relative to the depth of flow.
10.
The channel flow rate and the rate of sediment transport must be equilibrium; that is, a stable hydraulic section has been achieved.
Althoughmany investigatorshave assertedthat the coefficients C,, C,, C, and the exponents b, f, and m are universal constants, sufficient evidence now existsto demonstrate that this assertion is without merit. Some o f t h e results regardingthese coefficientsand exponents are summarized in Table 3.4.1. Studies of the regime or hydraulicgeometryequationshaveprimarilyexaminedthevariationof the coefficients either at-a-stationwith time or with longitudinal channel distance. The results in Table 3.4.ldemonstrate that there is a significant variation in coefficient values with geographical area, climate, time, and distance along the channel. At this point, it is appropriate to note that when dealing empiricalorsemi-empiricalequationsthatthesystemofunitsusedto derive the exponents and coefficients associated with these are equations is an important consideration. For example, consider the situation in which it is asserted that T = C,Qb
(3.4.22)
where the English system of units has been used to estimate values of C, and b. Further assume that in Equation (3.4.22)
102
TABLE 3.4.1 Summary of regime/hydraulic geometry coefficients and exponents from various literature sources
1
I I I I
I IData obtained from U.S. Geological I lsurvey gaging stations on perenI lnial rivers. I I I 1 b=0.26 1 f=0.40 I m=0.34 [at-a-station (primarily for arid I I and semi/arid regions) I I I I I I I b=0.50 I f=0.40 1 m=O.10 Idownstream (rivers throughout the IUnited States) I I I ........................................................................... Carlston I I I IData from Leopold and Maddock l(1953). Regression lines fitted (1969) I I I
Leopold and Maddock (1953)
1
I
I I I I I b-0.461 I f-0.383 I I I b=0.499 I f=0.320
I
I
II C1=8.36 III
Iby principle of least squares. I I I I m=0.155 [downstream (10 U.S. river basins) I I I m=0.180 [downstream (Yellowstone River
I IBasin: arid and semi-arid) I I C2=0.27 1 C3=0.45 IC values in SI units
I I I C,=4.63 I C2=0.28 1 c3=o.78 IC values in English units
________________________________________---------------------------------Dawdy I b=0.40 I fz0.40 I I I (1979) I I 1 I c1=12.
I c2=o.09 I
Ic values in SI units
I C,=9.5
I C2=0.07 I
IC values in English units
I
Nixon (1959)
I
I
I
I I
I
IData from 11 rivers in England land Wales. Equations apply to lrivers with sediment loads so lsmall as to have little effect on I lchannel shape. Discharges used I I I I b=0.500 I f=0.333 I m=0.167 Irepresent bankfull discharges.
I
I
I
I
I
I I
I I
Cl=l.65
I
I
I I
I
I
IRivers construct their own geoImetries. Profiles of rivers in lhumid climates tend toward equal [power per unit area and per unit [length (minimum work).
C2=o.5451 c3=1.1121c values in SI units
I Cl=0.9111 C2=0.5461 C3=2.01 IC values in English units ........................................................................... Langbein (1964)
I I
I
I
I I
I I
I
I I b=0.23 I I b-0.53
I I
I
I f=0.42
I
I f=0.37
I
I
I I
I
I m=O.35 I
I
I
lat-a-station
I
I m=0.10
(downstream
I
I I I I I
IAt-a-station data from 7 stations Ion Sungei Kinta, Perak, Western IMalaysia. Downstream data from 9 [stations.
I
I
I
........................................................................... Ee (1970)
I I I I
I I I
I
I I
I
b=0.09
I f=0.61
I m=0.31
I
lat-a-station
I b=0.29 I f=0.44 I m=0.28 Idownstream ...........................................................................
103
Knighton (1974) and
I
/Data from the Bolin and Dean River /Great Britain. Five reaches of 1500 m were chosen to represent a lwide range of channel sizes and Idischarge conditions. Within each [reach, 3 measurement stations were [chosen. The drainage area was )primarily covered by varied jglacial deposits. Bankfull discharges ranged from 1 m3/s (up;stream) to 10 m3/s (downstream) I I 1.261f~ 10.24~m< lat-a-station 0.63 I 0.71 I I I f=O.40 I m=O.48 1 I I
I I
I
10.01
I 0.33 I I b=0.11 I I h=
~
0.61
0.54 f= 0.46
I
{
I
0.311 0:23Im= 0 161
1
I
0 081 2%
0:23)15% 0.38150%
Downstream variation at per cent of time discharge is equaled or exceeded
I I ________________________________________--------------------------------Park (1977)
I I
I
I I
I
I I I
I I
I
I I O
I
10.06
I I I
IAnalysis of at-a-station geometry lof 139 streams and the downstream [geometry of 72 streams representling several different environments I (primarily perrennial streams with [some ephemeral streams)
/0.07
lat-a-station
I
0.71 I I I 0 . 0 3 5 b ~ 10.091fs I-0.51
1
I C1=4.5 8
I
c2=o.132 1c3=1.37
I
Williams (1978)
i
1
O
I I I I I I I I
IChanges in variables are such that lthe total effect of action, work, lor adjustment is a minimum. All lvariables strive to resist any Iimposed change with the net result lbeing that all of them change [equally. In checking this assump[tion only streams with loose parlticles on the bed were considered.
io.031ms I 0.81
iat-a-station I
I
io.io
I
0.78
I c values in SI units
I
........................................................................... [Basic principle - minimum rate of Yaw, I I I lenergy dissipation which states Song, and I I I lthat a system is in an equilibWoldenbergl I I lrium condition when its rate of I (1981) I I I I
I
lb=0.41
I I I
lf=0.41
I I
lenergy dissipation is at its lminimum value.
Im=O.l8
I
I
I
104
c,
= 1.00
and b has a v a l u e of a. I f T h a s u n i t s of [ f t ] and Q h a s u n i t s of [ f t 3 / s ] , then C , must have t h e following u n i t s t o p r e s e r v e t h e dimensional homogeneity of t h e equation
c,
[ft]
T
= - =
Qa
[ft3/s]"
-
[ft-seca]
ft3U
or C, = [ f t ' 1-3a)-seca~
The a p p r o p r i a t e conversion f o r C , given i n English u n i t s t o u n i t s would be C, [ S I ] = C,
[English]
(0.3048) 1ft1-3a
C,
in SI
1
Thus, i n c o n s i d e r i n g c o e f f i c i e n t s i n v o l v e d i n t h e regime o r h y d r a u l i c geometry t h e o r i e s it i s necessary t o transform them from t h e English s y s t e n ' o f u n i t s t o t h e S I system t h a t t h e v a l u e of t h e exponents be known. I n Equation ( 3 . 4 . 2 2 ) , C , and b are u s u a l l y determined from a l e a s t s q u a r e s r e g r e s s i o n a n a l y s i s o f t h e equation obtained by t a k i n g thenaturallogarithmsofbothsidesoftheequation, C a r l s t o n (1969), or log(T)
=
lOg(C,) + b lOg(Q)
where l o g ( C , ) i s t h e y a x i s i n t e r c e p t of t h e l i n e and b i s t h e s l o p e .
Therefore, b is independent of t h e system of u n i t s used and no conversion i s r e q u i r e d . I n t h e p a s t , many i n v e s t i g a t o r s have been remiss i n s p e c i f y i n g t h e u n i t s a s s o c i a t e d with t h e c o e f f i c i e n t s C , , C , , and C, and t h e r e a d e r i s cautioned r e g a r d i n g t h i s p o t e n t i a l problem. Theprimaryadvantageoftheregime o r h y d r a u l i c geometrytheory i s t h a t i t p r o v i d e s a v e r y s i m p l e summaryofthecomplicatedandpoorly understood r e l a t i o n s h i p s t h a t e x i s t among t h e channel and flow c h a r a c t e r i s t i c s i n a l l u v i a l c h a n n e l s . However, t h i s t h e o r y a l s o h a s a number of inadequacies and disadvantages. F i r s t , as noted p r e v i o u s l y i n t h i s s e c t i o n , t h e s e t h e o r i e s are e m p i r i c a l and have no theoretical justification. Second, t h e v a l i d i t y o f simplepower f u n c t i o n s a s d e s c r i p t o r s o f channel c h a r a c t e r i s t i c s d u r i n g p e r i o d s of v a r y i n g d i s c h a r g e h a s b e e n
105
questioned by a number of investigators: for example, Wolman (1955) and Richards (1973, 1976). Richards (1973) noted among other objections that there are discontinuities in the depth-discharge relationship following the transition fromthe lower to upper flow
\ O
/
0 b
1,
f
O
0,8 082
0.6
0.4
0.2
0,4
0,6
0.8
1.
0 1.0
F I G . 3.4.5 Chan es of at-a-stream hydraulic geometry exponents during a single Flood event, data from Knighton (1975).
regime: see also Dawdy (1961). Richards (1973) hypothesized that these discontinuities could be the result of non-linear changes in channelroughnesswiththedepthof flowandsuggestedthatnon-linear
power relationships may yield a better description of the system. Currently, there is not sufficient evidence to substantiate the suggested modification. Third, Rhodes (1977) studied 315 sets of hydraulic geometry equations with the aid of a trilateral diagram: see for example Fig. 3.4.5. Given the scatter of the m, f, and b values in his figure, he concluded that the average values of at-a-station hydraulicgeometry relationships may have but little meaning and may give erroneous predictions of actual channel responses to changing discharge. Fourth, there i s e v i d e n c e t o s u g g e s t t h a t d u r i n g e v e n s m a l l flood events significant changes in channel morphology may occur which cannot be predicted or explained by these hypotheses. For example,
106
Knighton (1975) mentioned that during one study of at-a-station hydraulic geometry a flood event with a return period of 2.5 years significantly modified the hydraulic exponents at one station, Fig. 3.4.5. Knighton (1975) c o n c l u d e d t h a t a t - a - s t a t i o n r e l a t i o n s canbe drastically changed by a single flow event if the channel cross section is in an unstable condition. Fifth, these theories fail to recognizethe important influence ofthesediment loadonchannel width, depth, andcapacity, Simonsand Albertson (1960) Sixth, regime and hydraulic geometry theory was developed to describe steady flow, steady sediment transport, and equilibrium channel conditions. The extension of these theoriesto the analysis of highly transient condition in an alluvial channel may be both unrealistic and improper.
.
DEBRIS F M W MECHANICS The terminology debris flow has been imprecisely used to describe the movement of a wide variety of soil and water mixtures. For example, there isnot aclear andprecise definitionaldifference among the movement of soil andwatermixturesvariouslytermeddebris flows, mud flows, creep, and landslides. For this reason, the foregoing terms may, and probably do, convey different images to geologists and geomorphologists than they do to engineers. To some degree, the lack of precise definitions results from the fact that mass movements of rock, earth, and water can vary greatly in speed, water content, c h a r a c t e r i s t i c s o f t h e s o l i d m a t e r i a l t r a n s p o r t e d , and the distribution of the solid materials within the flow. Some authors have attempted to differentiate between debris and mud flows on the basis of the basis of particle size, for example Schuster and Jkrizek (1978). In contrast to definitions based on the size of materials transported, the definition of debris flows proposed by Takahashi (1980) is to be preferred from the viewpoint of fluid mechanics. This definition states (in paraphrase): 3.5
Adebris f l o w i s a m i x t u r e o f a l l s i z e s o f s e d i m e n t . Boulders accumulate andtumble at the front ofthe debriswave and form a lobe, behind which follows the finer grained more fluidic debris, Takahashi (1980, p. 381). In comparison with this definition, Sharp and Nobles (1953, pp. 551552) offeredthe followingdescriptionofa llmudflowllthatoccurredin
107
1941 at Wrightwood, California: "A bouldery embankment formed at the front of more viscous
surges, a n d t h e b o u l d e r s t h e r e i n r o l l e d , twisted, andshifted about but for themost partdid notappear t o b e rolledunder. Instead, theywerepushedalongbythe finermore fluiddebris impounded behind the bouldery dam and swept along by the mud leaking through it. The similarity between the Takahashi definition and the Sharp and Nobles description is unmistakable. The advantage of the Takahashi definition is that it focuses on the fluid and flow characteristics that distinguish debris flows from other fluid flows. The density andviscosity of debris flows are important fluid characteristics by whichdebris flowscanbedistinguished fromother typesof flowssuch a s l l p u r e w a t e r l g o r w a t e r t r a n s p o r t i n g s e d i m e n t . Tobeclassifiedasa debris flow, a flow should have a density approximatelytwice that of normal water (at 20 degrees Centigrade, the density ofwater is998.2 kg/m3) and a viscosity significantly larger than that of water [at 20 degrees Centigrade, the absolute or dynamic viscosity of water is 1.005 x kg/(m-s)]. Because of their relatively large viscosities, debris flows are generally assumed to be laminar flows, and this assumed characteristic accounts for the fact that separate layers of the flow do not mix except at the leading edge of the flow. Note, turbulent flow situations are the most common in hydraulic engineering practice, and in such flows, the flow paths are highly irregular; that is, there is significant mixing between the layers, and flow losses vary as the square of the velocity. In laminar flow, the fluid particles move along smooth flow paths with each layer gliding smoothly over the adjacent layers, and flow losses vary directly with the velocity. Laminar and turbulent Newtonian flows are usually differentiated between on the basis of the ratio of the inertial to the viscous forces in the flow; a ratio commonly known as the Reynolds number, R, which is by definition R=-
UL p
(3.5.1)
U
where
u = average velocity of flow, L = a characteristic length which
in open-channel flow is usually taken as the hydraulic radius R, p = density ofthe fluid flowing, and u =dynamic orabsolute viscosityof
108
0.3
0.01
I
Pierson (1981)
l500-300C
I I
2.3
____
I
Segerstroem (1950)
Blackwelder (1928)
___
----
I
I 1 I
I I I Sharp and1 Nobles I (1953) I I I
___ ----
---
.urry
5-7 1 4 0 (bylviscous, laminar 1veight)ldebris flow, I Islurry
:igantic lasses o ieteroleneous iaterial ;hot fro1 barrow :anyons nto ope1 alleys
I I I
1740 0.2 (measure (bulk density) at surface
several tens of m/s to several cm/s
I
I I I
5-7
0.6
I I 1 I I I I I I
I
,arent tonian cosity
-
----
I
I
I I
I
I 1.2-3.0 I (average I velocity
I
1
I
I I I I I I
lo5
--
4-8 I
I
I I I I I I I I
--
_-
1
.8-
I I 1
-6
-
I
4
.5
I I 2x104 I 6X1O4
I I I
___
I
I
I
I
1
I I____
I
I
5-7 I
0.2
----
I
I
(measure at surface
I
I I 1 I I---Pack (1923) I I I I
:e moto oil
5.0
____
I Takahashil---(1980) I
I 4 0 (bylTurbulent muddy Iweight)lstreamflow be[tween surges
1730 1.0 (measure (bulk density) at surface
6
I
I
I
I
I I I I I I I
I I
/Higher velocity lviscous debris [flow with relnewed turbulence
22
----
----
-----
---
I
IFlowage of a [mixture of all (sediment sizes I
I I
I
JFollowing the \first impulse /were tremendous I quantities of lrock waste rangling in size from Jsmall to very (large boulders
I I
I I
I
I I I I
I I
JFlood transportled large quantilties of washed lgravel, sand, Iclay, and small I boulders I I 25-30 /Highly fluidic, 1 (by Islimy, cement(weight [like mud conltaining abundant I I stones 1 I I
Reynolds numbers above 12,500 are consideredto beturbulent. Openchannel flows with Reynoldsnumbers between500 and12,500 aretermed transitional flows. The critical Reynolds number is by definition the Reynolds number at which a laminar flow becomes a turbulent flow. In general, the Ilcritical Reynolds numberwwis actually a range of Reynolds numbers as indicated above. Finally, the assertion that debris flows are laminar has also been used to account for the observation that boulders apparently ride on the top of debris flows
109
for long distances.
Rnte o f Angular Deformation FIG. 3.5.1 Schematic of shear stress as a fupction of the rate of angular deformation for various types of fluids. Table 3.5.1 summarizes some of the historic field and laboratory data available regarding debris flows. With regard to these data, the following observations can be made. First, the Reynoldsnumbers ofthese flows, Column ( Z ) , arelow, a n d t h i s t e n d s t o s u b s t a n t i a t e t h e assumption that debris flows are laminar. Second, the velocity of these flows, Column (4), are reasonably large. Third, both the density, Column (5), and the viscosity, Column (6), of these flows is large relative to those of pure water. Fourth, the slopes on which debris flows occur, Column ( 7 ) , are relatively large. Fifth, the observed water content of these flows is relatively small. 3.5.1
Fluid Types Of all the fluid properties, from the viewpoint of fluid mechanics, viscosity is the one which deserves the most serious consideration. By definition, viscosity is the fluid propertywhich characterizes the resistance of the fluid to shear: and for this reason, fluids m a y b e classifiedbythe functional relationship that exists between the shear stress applied and the rate of angular deformation of the fluid. The rate of angular deformation of the fluid is usually measured by the velocity gradient within the fluid.
110
Four t y p e s of f l u i d s a r e and may i n t h e f u t u r e be important t o t h e d i s c u s s i o n of d e b r i s flow mechanics. F i r s t , Newtonian f l u i d s , such as water, e x h i b i t a l i n e a r r e l a t i o n s h i p between t h e s h e a r stress a p p l i e d , T , and t h e r a t e . o f angulardeformationwhich i s measuredby t h e v e l o c i t y g r a d i e n t d u / d y . I n t h i s type of f l u i d , T
=
{z]
(3.5.2)
where u = a b s o l u t e o r dynamic v i s c o s i t y of t h e f l u i d , F i g . 3.5.1. Second, a Bingham p l a s t i c i s a f l u i d i n which t h e s h e a r stress a p p l i e d must exceed a y i e l d stress v a l u e b e f o r e a s h e a r r a t e can be d e f i n e d o r flow occurs. A f t e r t h e y i e l d stress, r Y , i s exceeded, t h e r e i s a l i n e a r r e l a t i o n s h i p between t h e a p p l i e d s h e a r s t r e s s a n d t h e r a t e o f a n g u l a r s h e a r . Q u a n t i t a t i v e l y , t h e r e l a t i o n s h i p between t h e a p p l i e d s h e a r stress and t h e r a t e of angular deformation f o r a Bingham p l a s t i c i s (3.5.3)
where uB = Bingham p l a s t i c v i s c o s i t y , Fig. 3.5.1. Third, apseudo-plastic f l u i d h a s n e i t h e r a y i e l d s t r e s s n o r d o e s it e x h i b i t a l i n e a r r e l a t i o n s h i p betweenthe s h e a r stress a p p l i e d a n d t h e r a t e of a n g u l a r deformation of t h e f l u i d . Rather, a pseudop l a s t i c f l u i d is characterized b y a progressivelydecreasing slopeof t h e a p p l i e d s h e a r s t r e s s v e r s u s t h e r a t e of a n g u l a r deformation, Fig. 3.5.1. Fourth, a d i l a t a n t f l u i d a l s o h a s n e i t h e r a y i e l d s t r e s s n o r d o e s it e x h i b i t a l i n e a r r e l a t i o n s h i p betweenthe a p p l i e d s h e a r stress and t h e r a t e of a n g u l a r deformation of t h e f l u i d . I n c o n t r a s t t o a pseudo-plastic f l u i d , a d i l a t a n t f l u i d is c h a r a c t e r i z e d by a p r o g r e s s i v e l y i n c r e a s i n g s l o p e of t h e a p p l i e d s h e a r stress v e r s u s t h e rate of a n g u l a r deformation, Fig. 3.5.1. Mathematical formulations r e l a t i n g t h e s h e a r stress t o t h e r a t e of a n g u l a r deformation of t h e f l u i d s i m i l a r t o Equations ( 3 . 5 . 2 ) o r ( 3 . 5 . 3 ) are n o t a v a i l a b l e f o r pseudo-plastic and d i l a t a n t f l u i d s due t o t h e f a c t t h a t du/dy i s a f u n c t i o n of t h e a p p l i e d s h e a r stress. A number of e m p i r i c a l r e l a t i o n s h i p s between T and du/dy have been proposed f o r t h e s e types of f l u i d s ; and among t h e s i m p l e s t is t h a t
111
c i t e d by Hughes and Brighton ( 1 9 6 7 ) o r 7
(3.5.4)
=
where k = a c o n s t a n t r e l a t e d t o t h e consistency of t h e f l u i d and n = a
constant w h i c h i s ameasure ofhow t h e f l u i d d e v i a t e s from aNewtonian f l u i d w i t h n < 1 f o r a pseudo-plastic f l u i d and n > 1 f o r a d i l a t a n t f l u i d . W i t h t h e e m p i r i c a l r e l a t i o n s h i p d e f i n e d b y Equation ( 3 . 5 . 4 ) , t h e a p p a r e n t v i s c o s i t y , ua, o f t h e pseudo-plasticand d i l a t a n t f l u i d s is d e f i n e d by (3.5.5) A t t h i s p o i n t , it is r e l e v a n t t o n o t e t h a t a c h o i c e o f a f l u i d t y p e ( o r model) f o r a d e b r i s flow is n o t an i n s i g n i f i c a n t d e c i s i o n . F o r example, i n t h e foregoing m a t e r i a l laminar and t u r b u l e n t flows have been d i f f e r e n t i a t e d between on t h e b a s i s of a c r i t i c a l Reynolds number. The concept of a c r i t i c a l Reynolds number d e r i v e s from a c o n s i d e r a t i o n o f t h e p a r t i a l d i f f e r e n t i a l equationswhich g o v e r n t h e flowof Newtonianfluids. I n t h e c a s e o f Binghamplastic f l u i d s , flow s t a b i l i t y ; and hence t h e c l a s s i f i c a t i o n of a flow as laminar o r t u r b u l e n t , h a s been shown t o depend on t h e Binghamnumber r a t h e r t h a n t h e Reynolds number o r (3.5.6) UBU
u
where = average v e l o c i t y of flow, y = depth of flow, and B = Bingham number; see f o r example, Enos ( 1 9 7 7 ) o r Hampton ( 1 9 7 2 ) . Hedstrom ( 1 9 5 2 ) furtherdemonstratedthatinthecaseof Binghamfluids flowing i n p i p e s t h a t t h e c r i t i c a l Reynolds number i s only a f u n c t i o n of t h e Bingham number, Fig. 3.5.2. Note, i n p i p e flow, t h e c r i t i c a l Reynolds number is u s u a l l y t a k e n t o b e approximately2,OOO. Hampton (1972) showedthatthecurvederivedbyHedstromrelatingthecritica1 Reynolds number and t h e Bingham number was i n g e n e r a l agreement with
t h e experimental d a t a of B a b b i t t and C a l d w e l l ( 1 9 4 0 ) and Gregory ( 1 9 2 7 ) r e g a r d i n g t h e flow of c l a y s l u r r i e s i n p i p e s , Table 3 . 5 . 2 and Fig. 3 . 5 . 2 . With r e g a r d t o t h e d a t a i n T a b l e 3 . 5 . 2 a n d p l o t t e d i n F i g . 3 . 5 . 2 , t h e following o b s e r v a t i o n s can made. F i r s t , t h e Babbitt and Caldwell ( 1 9 4 0 ) and Gregory ( 1 9 2 7 ) d a t a c o n s i s t e n t l y p l o t above t h e
112
70.2
42.9
73.5
28.6
75.0
21.0
76.7
13.8
79.7
0.91
83.9
0.39
1 I I I I I I I I I I I I I 1 I I I I I I I I I
I
1,210
2.23
1,200
1.63
1,180
1.49
1,160
1.19
1,150
0.89
1,120
I I 1 I 1 I I I I I I I I I I 1 I I I 1 I 1
I I I I
I
I
I
- -- - - - - 86.4
2.23
81.4
0.9581
76.7
2.68
70.9
5.60
I
I I I 1 I
0.743 0.743 0.743 0.743
I I I I I I I I
1,170 1,130 1,175 1,225
I I I I I I I I
7,370 13,500 19,600
5.34 4.89 4.73
0.0127 0.0254 0.0508 0.0762
4.33 4.48 4.12 4.21
0.0127 0.0254 0.0508 0.0762
3.66 3.94 3.36 3.51
I
3,360 7,250 12,400 19,400
0.0254 0.0508 0.0762
3.36 2.75 2.90
I
6,660 10,900 17,200
0.0127 0.0254 0.0508 0.0762
2.35 2.41 2.32 2.35
I 1 I I
2,880 5,910 11,400 17,300
0.0127 0.0254 0.0508 0.0762
1.65 1.62 1.55 1.55
I I I I
2,640 5,180 9,920 14,900
0.0102
I 1 I
I
I
2,960 6,140 12,400 17,300
I 1 I I I I 1 I I I I I I
I I 1 1 I I 1 I I I I I I I
9.16 20.0 31.0 3.76 7.25 15.8 23.1 4.49 8.32 19.5 28.0 7.0 17.1 24.4 4.14 8.07 16.8 24.8 3.37 6.88 14.4 21.6
I I I I 1 I I 1 I I I I I
I 1 I I I I I 1 1 I 1 I I I
138,000 248,000 360,000 55,000 114,000 209,000 321,000 46,500 100,000 171,000 268,000 85,400 140,000 221,000 29,800 61,200 118,000 17,900 21,000 41,100 78,700 118,000
____________________----_-___-------
Gregory Data from Hampton 0.5271
I 1 I I I I 1 I I I 1 I
0.0254 0.0508 0.0762
(1972)
0.5541
I
8.910
0.0102
0.6711
10,400
0.0102
1.09
I
17,600
0.0102
1.83
I
I 1
I
30,800
I I I I I I I
I
13.1 19.6 33.8 42.1
I I 1 I I I I
I
565,000 68,500 111,000
187,000
1 7.56 I 0.743 I 1 , 2 2 5 I 0 . 0 1 0 2 1 2.14 I 3 7 , 1 0 0 I 48.6 I 218,000 I I I I I I I I 64.8 I 10.3 I 0.743 I 1 , 2 5 8 1 0.0102 1 2.59 I 4 4 , 6 0 0 I 5 4 . 6 I 264,000 .______------------______________________--------------.................................................................................... 67.5
curve defined by Hedstrom (1952). However, thetwo datasets andthe curve suggest the same general relationship between the critical Reynolds number and the Bingham number for clay slurry pipe flow. Second, although similar data for open-channel Bingham plastic fluid flows are not available, it can be safely assumed that except for a scale factor an analogous relationship betweenthe critical Reynolds number and the Bingham number would exist. In Fig. 3.5.3 critical Reynolds numbers forNewtonianopen-channel flows andBinghamnumbers are plotted as a function of the velocity and depth of flow for a material with a yield stress of r y = 10 N/m2 (0.21 lb/ft2) and a Bingham fluidviscosityofu, =lkg/m-s (0.02lsl/ft-s). Enos (1977) claimed that these values of r y and u s adequately described the fluid
113
loo. ow
Turbulent
f
?
--
BAmm Am
CWwaL DATA
10.000
1
FIG. 3.5.2 number.
!
loo loo0 B I N W MLBER Critical Reynolds number as a function of the Bingham 10
VELOCITY
-
OF R O W . ./I
A
wo.
B
0.001
-
0.01 -
B
o.ooo1
0.001
0.01
DEPTH OF R O Y .
0.1 I
1
---- -lo
B
0.10
B
1.0
B
10.
B
100.
- 1000. -
B
FIG. 3.5.3 Critical Reynolds numbers for Newtonian open-channel flows and Bingham numbers as a function of the velocity and depth of flow for a material with a yield stress of T , = 10 N/m and a Bingham fluid viscosity of u B = 1-kg/m-s. properties of the material commonly found in debris flows. Third, Fig. 3.5.2 i n d i c a t e s t h a t a s t h e B i n g h a m n u m b e r increasesthecritical Reynolds number also increases. The implication is that eventhough a Bingham plastic fluid flow may have a Reynolds number that would indicate the flow was turbulent, if the fluid was water, Column (9), Table 3.5.2; the non-Newtonian behavior of the fluid must be considered. Further, t h i s o b s e r v a t i o n s u g g e s t s t h a t r e l a t i v e l y h i g h energy levels, relative to pure water, are necessary for a Bingham
114
plastic fluid flowtobeturbulent. Fourth, the foregoing discussion has only considered the question of flow classification from the viewpoint of clay slurries in pipes and has not considered the effect of granular materials such as those found in natural debris flows. However, Bagnold (1956) demonstrated that granularmaterials tendto make the critical Reynolds number of a flow larger rather than smaller. Thus, there i s c u r r e n t l y n o r e a s o n t o b e l i e v e t h a t t h e a b o v e discusssion is not applicable to fluids which behave as Bingham plastic fluids even though they contain not only clay but also granular materials. Finally, there is the question of how viscosity or apparent viscosity should be measured i n a debris flow. For example, should the viscosity o f t h e flow as awhole bemeasured oronlytheviscosity of the dispersion medium, a clay slurry in most debris flows? There is currently no definitive answer to this question. Analytic Results for Debris Flows DeLeonandJeppson (1982) notedthat todate onlytwo closed form or analytic solutions for the vertical distribution of the longitudinal velocity in a debris flowhave beendeveloped. Johnson (1970) assumed that a debris flow behaved as a Bingham plastic fluid and obtained a form of the Poisson equation; see for example, Schlichting (1968), that described the movement of debris flows in a rectangular channel. In contrast, Takahashi (1978 and 1980) assumed a dilatant fluid model and was ableto defineboth themovement ofthe fluid-soil mixture, the steady state depth of flow, and a resistance coefficient which has subsequently shown to be related to the Chezy resistance coefficient for uniform flow. The original work of Takahashi was extended by DeLeon and Jeppson (1982) and Jeppson and Rodriguez (1983) to gradually varied and unsteady debris flows, respectively. Because of these extensions of the original work by Takahashi to areas of applied interest, the dilatant fluid model of debris flows will be considered first and then the Bingham plastic fluid model second. Takahashi (1980) in deriving the vertical distribution of longitudinal velocity in a debris flow occurring in a rectangular channel began with the equation of volumetric continuity or 3.5.2
aY
a (UY)
- +-= at
ax
0
and the unsteady equation of conservation of momentum or
(3.5.7)
115
-
-
au - au -++-=gsine-gcos at ax
(3.5.8)
8
u
= cross-sectional mean where x = longitudinal distance, t = time, velocityof flow, y = debris flowdepthof flow, 0 =slope angleofthe debris channel bed, and k = a resistance or frictional coefficient. Fromthisstartingpoint, Takahashiproceededtoderive anexpression for the resistance coefficient k or
(3.5.9)
and an expression for the steady state depth of the debris flow or k<
Y=-
(3.5.10)
g sine
w h e r e a = a n e m p i r i c a l l y e s t a b l i s h e d v a l u e w h i c h d e p e n d s o n t h e flow, a = friction angle of the moving grains, cd = grain concentration by volume of the debris flow, p = density of water, C , = grain concentration by volume in the stationary bed of the flow, and d = grain diameter. Although Equations (3.5.9) and (3.5.10) are analytically and experimentally useful, from an applied viewpoint these equations are not useful since they require the quantification of variables not usually known, a priori, to the investigator. DeLeon and Jeppson (1982) usedaone-dimensionalhydraulicengineeringapproachandwere abletoanalytically demonstrate that the resistance coefficient k i n Equations (3.5.9) and (3.5.10) was related to the Chezy C a standard resistance coefficient for open-channel flow: see for example, French (1985), or
-
(3.5.11) Substitution of Equation (3.5.11) in Equation (3.5.9) yields
116
(3.5.12)
cd
pm
= (u-pm)
tan
(tan a
8
- tan
8)
I
provided that the value of Cd is less than C,; otherwise,
cd
= O.gc*
and
density of the debris flow mixture. The valueofthechezy C, which is a l s o r e l a t e d t o t h e M a n n i n g n ; see for example, French (1985), is usually choosen on the basis of experience and a field inspection of the channel. In the case of debris flows, standard hydraulic engineering practice may not be applicable to the selection of an appropriate value of C. The Chezy coefficient derives from the semi-empirical Chezy equation for uniform flow in open channels which states pm =
(3.5.13)
where R = hydraulic radius and S = in the general case, the friction slope of the flow. In Equation (3.5.13) , the units associated with the resistance coefficient C are (length) '/*/(time). It can be easily shown that the Chezy resistance coefficient is equivalent to Manning n's or (3.5.14) In Equation (3.5.14) , $ has a value of 1.49 if English units are used and a value of 1.00 if SI units are used. The Chezy coefficient is also related tothe Darcy-Weisbach frictional coefficient f for pipe flow or
c
=
J8g/f
(3.5.15)
see for example, French (1985). . In the case of laminar flow, the
117
Darcy-Weisbach friction factor f is an inverse function of the Reynolds number or 64 f E -
R
(3.5.16)
Combining Equations (3.5.15) and (3.5.16) yields (3.5.17) For SI units, Equation (3.5.17) becomes C = 1.1076
(3.5.18a)
and in the case of English units
c
= 2 . 0 0 6 5
(3.5.18b)
DeLeon and Jeppson (1982) used the limited field and laboratory data currently available for debris flows to estimate values of C. TheresultsofthisanalysisaresummarizedinTable3.5.3. Thereis, see Columns (8), (9), and (10) of Table 3.5.3, reasonably good agreement among the Chezy coefficients estimated by the three techniques used by DeLeon and Jeppson. Given the very limited data available for actual debris flows, DeLeon and Jeppson (1982) asserted that there were sufficient similarities between debris flows and sludge flows to allow the data summarized in Table 3.5.3 to be combined with the sludge flow data of Babbitt and Caldwell (1939) to yield a data set large enough for statistical analysis. Fromthis combined data set, a least squares analysis yielded the following best fit relationship for C and R. C = 1.02 R o S 5 '
(3.5.19a)
for SI units and C = 1.85 R o a S 2
(3.5.19b)
forEnglishunits. Acomparison ofthe coefficientsand exponents in Equations (3.5.19) and those in Equations (3.5.18) show excellent agreement.
118
I
I
Sharp and1 Nobles 1 (1953) I
6
1
1.20
Pierson
5
I I I
5.00
(1981)
I I I
I
Takahashil (1980) I
18
I
I
Takahashil (1980)
I I
I
I
I
1 I
1.00
18
I
I
I
I 1 I
20,000
1,000
1,940
I
I
I I
I 1 I I I 1
1.00
I
I
I I
I 1 0.762 I I I 1 0.4817 I I 1 0.044 I I
1,410
I 1 0.041 I
I
I 1
I I I I
I I I I I I
I
I I 1 4.24 I 3.65 I I I I I I 500 1 24.35 I 24.75 I I I I 8.25 1 8.36 1 3.18 10.97
I
I
I 10.39 I to I
I 9.08 I
to
80.9321
I 1 1o2.Oe2l
I
I
to
9.96 3.57
to
I 11.18
........................................................................
I Value results from slight modification of the coefficients. Values assumed by comparison with data given by Babbitt and Caldwell (1939).
At this point, only one problem remains to be solved before debris flows can be analyzed by the techniques of traditional onedimensional open-channel hydraulics. Up to this point methods of estimating the density and viscosity of a debris flow, a priori, have not been developed; and hence, neither R nor C can be estimated. DeLeonandJeppson (1982) a s s e r t e d t h a t b o t h t h e d e n s i t y a n d v i s c o s i t y of a debris flow vary as a function of the depth and velocity of flow. These investigators, with some justification, offered the following approximation forthe ratioofthe densityto thedynamic viscosity in a debris flow
P
-
u
E
-
k
R*
(3.5.20)
where p = debris flow bulk density, k = an empirical constant, and a = an empirical exponent whose value is assumed to be one and R = hydraulic radius. Usingthe dataof Sharpand Nobles (1953), avalue for k was estimated; and hence, P
10
- I -
u
R
(3.5.21a)
119
f o r S I u n i t s and P 3 - = -
u
(3.5.21b)
R
f o r English u n i t s . With t h e foregoing hypotheses, t h e Chezy equation f o r uniform flowinanopenchannelcanbeusedtoestimatedebris flowdepth. For example, b q i n n i n g w i t h Equation ( 3 . 5 . 1 3 )
or Q2
A3S
- C2
=
(3.5.22)
P
C 2 is e s t i m a t e d by combining Equations (3.5.19a) and (3.5.21a) f o r
computations i n t h e S I system of u n i t s c2
=
[
1.02
R0.52]
*
=
'1
(,.,, [? P u
0 . 5 2
)
2
(3.5.23)
S u b s t i t u t i n g Equation ( 3 . 5 . 2 3 ) i n Equation ( 3 . 5 . 2 2 ) and rearranging
Q2P
-
11.41 A''96
Q 1 * 0 4 S= 0
(3.5.24)
which i s an i m p l i c i t equation f o r t h e d e b r i s flow depth y. Equation ( 3 . 5 . 2 4 ) can be solved f o r y by t r i a l and e r r o r o r by numerical methods; f o r example, a Newton i t e r a t i v e scheme, Conte ( 1 9 6 5 ) . DeLeon and Jeppson ( 1 9 8 2 ) and Jeppson and Rodriguez ( 1 9 8 3 ) used t h e r e s u l t s d i s c u s s e d above and t h e techniques of t r a d i t i o n a l o p e n channel h y d r a u l i c s t o develop numerical s o l u t i o n s f o r g r a d u a l l y v a r i e d b u t s t e a d y and u n s t e a d y d e b r i s flows, r e s p e c t i v e l y . A n o t e o f c a u t i o n , some c a r e should be e x e r c i s e d i n u s i n g t h i s approach p r i m a r i l y because t h e l a c k of d a t a t h a t i s needed t o v e r i f y t h e assumptions i n h e r e n t i n t h i s development. For example, it has not b e e n p r o v e n t h a t d e b r i s f l o w s a r e d i l a t a n t f l u i d s nor c a n t h e v a l i d i t y of Equation ( 3 . 5 . 2 1 ) be b l i n d l y accepted. F u r t h e r , i s t h e Reynolds number an a p p r o p r i a t e parameter f o r d i f f e r e n t i a t i n g between laminar
120
3.5.4 S c h e m a t i c * d e f i n i t i o n of v a r i a b l e s f o r Bingham p l a s t i c f l u i d model of a d e b r i s flow.
FIG.
and t u r b u l e n t f l u i d s when t h e f l u i d s a r e non-Newtonian? ThedevelopmentofaBinghamplastic f l u i d m o d e l o f a d e b r i s flow b e g i n s w i t h a s t e a d y flow f o r c e b a l a n c e similartothatusedtodevelop t h e Chezy uniform flow equation f o r Newtonian f l u i d s . Following Johnson ( 1 9 7 0 ) , s t e a d y d e b r i s flow i n a s e m i - c i r c u l a r c h a n n e l w i l l b e considered, Fig. 3.5.4. A f o r c e balance f o r t h e c o n t r o l volume defined i n t h i s f i g u r e y i e l d s
(3.5.25)
where F, = g r a v i t a t i o n a l f o r c e component causing flow F, = Y
nr
- AL s i n 6 2
F, = v i s c o u s f l u i d f o r c e r e s i s t i n g motion which i s equal t o a s h e a r
stress T times t h e area on which t h i s stress a c t s o r nrAL, and Y = Binghamplastic f l u i d s p e c i f i c w e i g h t . F o r s m a l l v a l u e s o f 6 , s i n 6 i s approximately equal t o t h e s l o p e o f t h e debris flow s u r f a c e s , . With t h e s e d e f i n i t i o n s and assumptions, Equation (3.5.25) becomes Ynr2
TilrAL
=
Am, 2
121
and s i m p l i f y i n g Y r T = -
2
(3.5.26)
SD
Equation (3.5.26) i s v a l i d f o r a l l and T h a s v a l u e of z e r o a t r = 0 and a maximum v a l u e a t r = R. For a Bingham p l a s t i c f l u i d , T i s given by Equation (3.5.3) and s u b s t i t u t i q n of t h i s equation i n Equation (3.5.26) y i e l d s (3.5.27)
which is o n l y v a l i d f o r T 2 7 , ; when T < throughou out the f l u i d , t h e r e i s no flow and t h e above a n a l y s i s beginning w i t h Equation (3.5.25) i s n o t v a l i d . Rearrangement of Equation (3.5.25) y i e l d s
dr
ug
or S,
-
zY
1
for
T
2
(3.5.28)
T~
where t h e n e g a t i v e s i g n i n Equation (3.5.28) d e r i v e s from t h e choice of co-ordinate systems ( v e l o c i t y d e c r e a s e s w i t h i n c r e a s i n g r ) , Johnson (1970). I n t e g r a t i o n of Equation (3.5.28) and e v a l u a t i o n of t h e c o e f f i c i e n t of i n t e g r a t i o n r e s u l t s i n a n e q u a t i o n f o r t h e v e l o c i t y d i s t r i b u t i o n i n a s e m i - c i r c u l a r channel under s t e a d y d e b r i s flow conditions o r S,
-
(R-r)Ty
1
for
T
2
T~
(3.5.29)
Asnoted intheprecedingparagraphs, t h e s h e a r stress vanishes a t r = 0. Therefore, it is c o n c l u d e d t h a t t h e r e m u s t be a v a l u e of r i n any flow where T = T~ and t h i s a r c s e p a r a t e s t h e a r e a of t h e channel where t h e v e l o c i t y of f l o w v a r i e s a c c o r d i n g t o E q u a t i o n (3.5.29) from t h e area of t h e channel where t h e f l u i d moves a t a uniform v e l o c i t y L e t R, be t h e r a d i u s ( t h i s i s o f t e n termed a plug flow) , Fig. 3.5.5. a t which T = T~ and where (du/dr) = 0 . R , can t h e n be determined from Equation (3.5.28)
122
7
FIG. 3.5.5
Definition of debris flow plug.
(3 -5.30)
T h e v e l o c i t y a t w h i c h t h e p l u g movescan bedetermined by substituting
Equation (3.5.30) in Equation (3.5.29). When R, 2 R, there is no flow. Johnson (1970) noted that these results are in general agreement with the limited laboratory results that are available; that is, debris flows increase their thickness as the flow rate increases; decrease their thickness as the flow rate decreases: and reach minimum thickness when flow ceases. Johnson (1970) also presented results for other channelshapes. For example, in infinitely wide rectangular channels, (3.5.31)
123
Yc
=
TY
-
(3.5.32)
YSD
whereH=totaldepthof flow, y = a C a r t e s i a n c o - o r d i n a t e w i t h t h e zero value being taken at the surface of the debris flow, and y c = the thickness of the debris plug. The Bingham plastic fluid model of debris flows proposed by Johnson (1970)m u s t b e c o n s i d e r e d a v i a b l e a l t e r n a t i v e t o t h e d i l a t a n t fluid model discussed previously in this section. However, in comparison with the dilatant fluidmodel, the Bingham plastic fluid model is limited f r o m t h e v i e w p o i n t o f a p p l i c a t i o n . For example, the Bingham model discussed here 1.
requiresthat the yieldstress ofthe Binghamfluid beknown but provides no method of estimating the value of this variable from data that would be generally available, and
2.
has not been extended to gradually varied or unsteady flows
It must noted that the extensions and hence the usefulness of the dilatantfluidmodelisdependentonanumberofempiricalassumptions which may or may not be justified; for example, Equation (3.5.20) and Equations (3.5.21). At the present time, there are not sufficient
field or laboratory dataavailable toeither endorseor reject either of these models. HYDRAULIC/PHYSICAL MODELS Unlike many other areas of modern engineering and scientific endeavor, important discoveries, designs, and insights in hydraulic engineering and geology continue to derive from observations and theoretical calculations made in conjunctionwith laboratory studies utilizingphysicalmodels. Theuse ofphysicalmodelsto developnew knowledge requires a comprehensiveunderstanding ofthe principles of similitude; see for example, Streeter andwylie (1975). Withregard to similarity, there are three types of similarity important to hydraulic/physical model studies: 3.6
1.
Geometric Similarity: Two objects are only similar if the ratios of all corresponding dimensions are similar. The terminology geometric similarity therefore pertains to similarity in form.
124
2.
Kinematic Similarity: Two motions are termed kinematically similar only when the paths of motion are geometrically similar and when the ratiosofthevelocities of the two motions are equal.
3.
Dynamic Similarity: Two motions are termed dynamically similar only if the rasios of the masses involved are equal and the ratios of l the forces involved are equal.
In most physical model studies, geometric and kinematic similarity can be rather easily achieved; however, complete dynamic similarity is often difficult, if not impossible, to achieve. In most complex modeling situations,themodelingprocessbeginswitha scalingofthe governing differential equations; see for example, French '(1985). From the scaling process, a number of dimensionless numbers are derived such as the Reynolds, Froude, Bingham, and Weber numbers. In the material which follows, the concepts of hydraulic/physical models relevant to the study of hydraulic processes on alluvial fans are summarized. Detailed treatments of the subject of hydraulic/physical models are available in French (1985) andsharp (1981). For specificexamples ofthe applicationof these principles to alluvial fans, the reader is referred to Anon. (1981)
.
Froude Law Models Froude law models assert that the primary force causing flow is gravity and that all other forces such as surface tension and fluid friction are small relative to the gravitational force and can be neglected. F r o u d e l a w m o d e l s c o m p r i s e t h e p r i m a r y t y p e o f model used to study traditional open-channel flows. If only Froudenumber similarityis required, then bydefinition 3.6.1
F,, = F,
(3.6.1)
where F = Froude number and the subscripts M and P indicate model and prototype Froude numbers, respectively. Substitution of the definition of the Froude number, Equation (3.2.5) , in Equation (3.6.1) yields after rearrangement an expression for the velocity ratio
125
UM
UP
u , = -ufiu,
gHLN
1 / 2
- [gpLpl
(3.6.2) =
where the subscript R indicates the ratio of model to prototype variables, U, = velocity ratio, L, = length scale ratio, and g, = gravity ratio. Since the acceleration of gravity cannot, from a practical viewpoint, be altered between the prototype and model: i.e., g, = 1, Equation (3.6.2) becomes (3.6.3) Hydraulic models must often be distorted: that is, the longitudinal and vertical scale ratios are not equal. The need for length scale distortion results from the fact that most channels are much longer than they are deep and laboratory space for hydraulic/physicalmodel studies is always limited. For example, in one physical model study cited by Anon. (1981) the vertical scale ratio, Y,, wasl/lOwhilethe longitudinal scale ratio, L,, was1/150. If a distorted model is necessary, then Equation (3.6.3) becomes
"R
=
&
(3.6.4)
and the time scale ratio can be demonstrated to be T,
=
LR
-
(3.6.5)
Other scaling ratios for a distorted Froude number model are summarized in Table 3.6.1. 3.6.2
Moveable Bed Models When the movement of the materials which compose the sides and b e d o f a c h a n n e l o r t h e m i g r a t i o n o f a c h a n n e l a c r o s s a soil surfaceare primary considerations, as they are in the study of hydraulic processes on alluvial fans, then the use of a moveable bed hydraulic modelisnecessary. Theproperdesignand useof amoveable bedmodel is much more complex and difficult thanthe designand usedof a fixed
126
TABLE 3.6.1 Scaling ratios for a distorted Froude law model
.............................................................
Variable
Scaling Ratio
.............................................................
.............................. IJ R Vertical Length ................................ YR Time (hydraulic) ............................... LR/(YR)'/' Velocity ....................................... (Y,)'/' LR(YR)=/' Flow Rate ...................................... Force .......................................... _ _ _._. _. _. _. ._._. _. _. _. ._._. _. _. _. ._._. _. _. _. ._._. _. _. _. ._._. _. _. _. ._._ _ _ _ _ _ _ _ _ ~ _ ~L,(Y,)' ~________~~~~~~~ Horizontal Length
bed model because 1.
The boundary roughness of the moveable bed model is not determined by design but is controlled by the motion of the material which comprises the model.
2.
Not only must the moveable bed model correctly simulate the movement of water but it must also correctly simulate the movement of sediment.
There are three methods by which moveable bed models can be designed. Of these, the method commonly referenced as trial and error;thatis,theadjustmentofmodelvariablesuntilthehistorical record of prototype performance is reproduced, is not appropriate to the s t u d y o f h i g h l y t r a n s i e n t a n d p o o r l y d o c u m e n t e d e p h e m e r a l f l o w s o n alluvial fans. For this reason, the trial and error technique of moveable bed hydraulicmodel designwill not be discussedhere. The remaining two techniques of design can be classified as theoretical and semi-empirical. 3.6.3
Theoretical Techniques Equations (3.4.13) and (3.4.14) , as noted in Section 3.4 , can be used to define the threshold of particle movement. It can therefore be asserted that if F, and R, are the same in the model and prototype thatthe stateofthebedofthe channelwill bethe same. Then, using the hypothesis of Einstein and Barbarossa (1956) regarding the shear stress on the bottom ofthe channel and a roughness-to-particle size
127
r e l a t i o n s h i p such a s Equation (3.3.3) it can be shown: see fa: example, French (1985) , t h a t a moveable bed model can be developed on t h e b a s i s of t h e following equations (3.6.6)
(3.6.7)
(3.6.8)
where R, = h y d r a u l i c r a d i u s r a t i o , V , = k i n e m a t i c v i s c o s i t y r a t i o , a = S, -.1, S , = sediment s p e c i f i c g r a v i t y , and d, = sediment s i z e r a t i o . With regard t o Equations (3.6.6) (3.6.8) , R, is a f u n c t i o n of both Y andL, and v R w o u l d u s u a l l y h a v e a v a l u e o f 1. Therefore, i f onemodel
-
r a t i o i s selected, t h e n t h e o t h e r model r a t i o s c a n b e determined from Equations (3.6.6) - (3.6.8). With regard t o Equations (3.6.6) - (3.6.8) , t h e following comments should be noted and considered: 1.
I f Y, = L, (an u n d i s t o r t e d model)
, t h e n a s c a l e model cannot
be b u i l t ; t h a t is, t h e model w i l l be t h e same s i z e a s t h e prototype. 2.
I f t h e s p e c i f i c g r a v i t y r a t i o , a,, i s s e l e c t e d i n a d v a n c e a n d t h e kinematic v i s c o s i t y r a t i o has a v a l u e of u n i t y , t h e n d, = a R - l I 3
(3.6.9)
Equation (3.6.9) i n d i c a t e s t h a t i f t h e moveable m a t e r i a l used i n t h e model has a s p e c i f i c g r a v i t y less t h a n t h e materialintheprototype,thenthesizeofthematerialused inthemodelmustbelargerthanthat found i n t h e p r o t o t y p e . 3.
I f R, cannot be assumed t o be equal t o Y,, t h e n t h e s c a l e r a t i o s Y, and L, must be found by t r i a l and e r r o r .
4.
The r a t i o of sediment t r a n s p o r t p e r u n i t width t o flow p e r u n i t w i d t h m u s t b e l a r g e r i n themodel t h a n i n t h e p r o t o t y p e .
128
1.0
p
0.10
0.01 1.
FIG. 3.6.1
10.
R#I
100.
1000,
RelationofbedformationtopositionintheF,-R,plane.
5.
The particle size ratio in Equation (3.6.8) characterizes the roughness of the channel while the same parameter in Equations (3.6.6) and (3.6.7) characterizes the sediment transport properties.
6.
Inthedesignof traditionalmoveable bedmodels, theFroude number model law is not considered to be an absolute rule since if the Froude number is small a slight difference between F, and F, can be tolerated. However, as will be subsequentlyshown, it is believedthat theFroude numberof flows occurring on an alluvial fan may not be small; and therefore, small difference between F, and F, may be significant.
Recall that the primary assumption of the foregoing development is that if F, and R, are the same in the model and the prototype, then an appropriate physical model will have been developed. A number of investigators have asserted; see for example, French (1985), that this requirement can and should berelaxed. Fig. 3.6.ldemonstrates the relation of bed formation to position in the F, - R, plane. In viewofthis figure, it is apparentthat toachieve similaritybetween model and prototype the values of F, and R, do not have to be exactly
129
equal but the points in the F, - R, plane representing the mode 1 and the prototype should be within the same bed form band in Fig. 3.6.1. Zwamborn (1966, 1967, 1969) has developeda differenttechnique for designing moveable bed hydraulic models. In summary, this technique requires that the following criteria be satisfied. 1.
Sioce most water' flows of practical importance are turbulent , the Reynolds number for the model should exceed 600. This was a tacit assumption in the previous discussion.
2.
Dynamic similarity between the model and the prototype is achieved when a.
The Froude number for the model and prototype are equal.
b.
The frictional forces acting in themodel andprototype are scaled correctly.
c.
Geometric scale distortion cannot be large: for example , YR
- < 1/4 LR 3.
Sincesedimentmotion i s c l o s e l y r e l a t e d t o t h e b e d formtype and bed form types are a function of R*, F,, and u*/Usr the value of R, for the model should fall within the range determined by F and u*/U,. French (1985) provided an appropriate figure for doing this.
4.
Given the foregoing criteria, the following scale factors can be determined Flow Rate Q R = LRY,1'5
Hydraulic Time
(3.6.10)
130
T,
=z
LR -
(3.6.11)
UR
Sedimentological Time (3.6.12) where T, = sediment time scale, A = relative submerged sediment density, and q s = sediment bed load per unit width. If ( q s ) Ris not accurately known, T, cannot be calculated, but Zwamborn (1966, 1967, 1969) suggested that (Ts) R
I :
loT,
(3.6.13)
With regard to the discussion above, hydraulic time is a time scale based on the speed of water wave propagation while sedimentological time is, at least conceptually, based on the speed of sediment wave propagation. Finally, themethodologyusedbyAnon. (1981)todesign moveable bed hydraulic models of alluvial fans is similiar to that discussed above. 3.6.4
Empirical The regime or hydraulic geometry theory can also be used to develop scaling ratios for moveable bed hydraulicmodels. If it is assumed that T Y
-
S
.-
-
h
(3.6.14)
Q113
(3.6.15)
Q-1Is
(3.6.16)
then it can be shown by induction that (3.6.17) (3.6.18) and (3.6.19) Y,
=
LR213
Q R = L,YR312
(3.6.20) (3.6.21)
131
-
(3.6.21), the following With regard to Equations (3.6.14) observations can be made. First, the model described by these Second, the equations i s a d i s t o r t e d s c a l e m o d e 1 , E q u a t i o n (3.6.20). flow rate scaling, Equation (3.6.21) agrees with the Froude law scaling, Table 3.6.1. Third, t h i s t e c h n i q u e t a c i t l y a s s u m e s t h a t t h e model bed material is the same as that found in the prototype. 3.7
CONCLUSION The foregoing sections of this chapter have attempted to introduce the reader to a limited number of the basic and traditional principles of open-channel hydraulics which the author believes are important to understanding hydraulic processes on alluvial fans. The coverage of the various principles discussed in this chapter is, by intention, not comprehensive. However, there are sufficient references given forthe interested reader to further investigate at his leisure all of the topics. This chapter has also briefly discussed the mechanics of debris flows which is a rapidly developing area of technology. There is currently significant interest in the mechanisms which initiate debris flows, Croft (1967), Campbell (1975), and MacArthur et a1 (1986); the mechanics of debris flows, Chen (1983, 1984, 1985, 1986a, 1986b, 1 9 8 6 ~ ) ;the identification of areas susceptible to debris flows, Mears (1977), James et a1 (1986), and Kumar (1986); and the frequency of debris flows, Campbell (1975) and Mears (1977). Finally, it should be noted that post-event analysis often mistakes debris flow floods for fluvial floods, Costa (1983).
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.
8?!?
.