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Gravitational effects on chemically reacting laminar boundary layer flows over a horizontal flat plate
GRAVITATIONAL EFFECTS ON CHEMICALLY REACTING LAMINAR BOUNDARY LAYER FLOWS OVER A HORIZONTAL FLAT PLATE MOSHE LAVID Guggenheim Laboratories, Princeton University AND A. L. BERLAD State University of New York at Stony Brook
This is a theoretical study of the chemically reacting laminar boundary layer flow over a horizontal flat plate with gravitationally induced buoyant force. A diffusion flame sheet model was invoked to deseribe the combustion process. It was found that the effects of gravity on the purely force convection flow are characterized by a dimensionless coordinate quantity G r J R e , 5/2. The governing equations were obtained by expanding the dependent variables into series in terms of this coordinate quantity. The problem inherently is a coordinate perturbation and it is treated as such. A numerical solution of the zeroth and first order governing equations subject to the appropriate physical boundary conditions was obtained for various external flow conditions such as wall temperature, freestream oxygen concentration and aiding and opposing flows. It was shown that the cross stream buoyancy induced body force acts effectively to produce a streamwise pressure gradient in the fluid adjacent to the plate surface. The pressure gradient is favorable in aiding flows (plate facing upwards) and is adverse in opposing flows (plate facing downwards). Hence, the local boundary layer flow is accelerated or decelerated relative to the corresponding gravity-flee forced eonveetion flow. Correspondingly, there is an increase or decrease in the local skin friction and heat transfer rates, and consequently a decrease or increase in the flame "stand-off" distance, depending upon whether aiding or opposing flow occurs, respectively. Based on the present study we conclude that buoyancy plays an important role in boundary layer diffusion flames, and by retaining the buoyant force the current model is capable of explaining phenomena that have been observed experimentally but have not been predicted by classical boundary layer theory. Such previously unaccounted for effects include the acceleration of the boundary layer flow (sometimes a velocity overshoot is exhibited), and a decrease in the flame "stand-off" distance.
I. I n t r o d u c t i o n
Chemically reacting b o u n d a r y layer flows have been investigated theoretically and experimentally over the last forty years. The present theoretical research is concerned mainly with the gravitational effects on such laminar b o u n d a r y layer flows over a horizontal flat plate. Generally speaking, the aim is to investigate the interaction of c o m b i n e d forced and free convection, the former due to the external flow a n d the latter due to the b u o y a n t force. Literature abounds with studies on t o m -
b i n e d convection in inert b o u n d a r y layer flows, dealing mainly with the fluid mechanical a n d heat transfer aspects, and with studies on "chemically reacting b o u n d a r y layer flows neglecting gravitational effects. However, no attempt has been reported yet to develop a physical and mathematical model designed to elucidate the structure of a diffusion flame w i t h i n a forced convection b o u n d a r y layer i n f l u e n c e d by gravity such as is exhibited in flows over a wedge. A review of the existing literature reveals that b u o y a n c y effects due to gravity are generally neglected with the exception of some
1557
1558
MATHEMATICAL MODELING
works dealing with b u r n i n g of droplets, 1'2 a n d axisymmetric jet d i f f u s i o n flames, a,4 and with experiments of b u r n i n g of fuel surfaces. 5,6 However, in these works either the gravitational force is in the direction of the m a i n flow (i.e. vertical wall, falling droplets) or it is a purely free convection flow. In many practical applications involving combustion, ranging from h y b r i d c o m b u s t i o n to fire prevention, the fuel and the oxidant are initially not premixed, and the c o m b u s t i o n exhibits a structure of a diffusion flame. Inasmuch as these a p p l i c a t i o n s are usually not in a zero-gravity e n v i r o n m e n t it is of importance to analyze the effects of gravity on the diffusion flame, and only after a more complete understanding of the c o m b u s t i o n process with b u o y a n c y is achieved a n d the importance of gravity is d e t e r m i n e d will it be p o s s i b l e to predict the structure of diffusion flames u n d e r various gravity conditions. This is particularly true in cases where experimental data is sparse, such as in spacecraft environments of low (zero)--gravity level a n d d u r i n g high-G maneuvers of aircrafts. II. Physical M o d e l A sufficiently general model has been selected and it is illustrated i n Fig. 1. T h e combustion is w i t h i n the b o u n d a r y layer, developed b y the flow of main gas stream containing oxidant a n d inerts over a horizontal flat plate. The plate can be considered as presenting three general cases: (1) a c o n d e n s e d phase fuel that can evaporate (or sublimate); (2) a solid fuel that pyrolyzes with gas generation rate that can be d e s c r i b e d by a 'surface' pyrolytic law; (3) a porous plate from the surface of which l i q u i d fuel is injected. T h e gaseous fuel is carried b y molecular diffusion a n d convection from the solid surface into the b o u n d a r y layer where it can react with the oxidant. The oxidant is also transported to the
BOUNDARY (~
Ue
Iue,
FLAME S H E E T \ y
Te ~
x
.
Yo
O .
_
~
L .
LAYER EDGE
/ _ _. _ _ _ . . DIFFUSION ZONE
T.7_~ IYF,w
Flc. 1. Physical model with flame sheet approximation over a horizontal flat plate.
flame b y diffusion a n d convection from the main stream. Heat released by the chemical reaction is transmitted to the surface by diffusion and convection, causing continuous gaseous fuel generation. Thus, steady state can be m a i n t a i n e d and the process is self-sustaining. This physical m o d e l has been treated previously with infinite kinetics b y Chen a n d Toong v a n d with finite kinetics b y W a l d m a n , s Krier a n d Kerzner, 9 a n d Krishnamurthy a n d Williams. 1~ However, the effects of gravity have not been s t u d i e d yet, as the b o d y force was deleted from the governing equations. A general d i m e n s i o n l e s s analysis of the two m o m e n t u m equations with b u o y a n t forces for b o u n d a r y layer flows over a wedge was carried out elsewhere, ]~ a n d it was shown that the structure of the b o u n d a r y layer is d e t e r m i n e d b y a mixed convection, dimensionless quantity {, which is essentially a ratio of the b u o y a n c y force to the inertia force. In general, { < 1 indicates a p r e d o m i n a n t l y forced convection flow perturbed b y natural convection, ~_ > 1 suggests a p r e d o m i n a n t l y free convection flow perturbed b y forced convection, and { of order unity corresponds to a flow in which both effects are comparable. This research is confined to study the effects of free convection due to non-zero gravity field on the p r e d o m i nantly forced convection flow, i.e. { < 1. The two distinct coordinate quantities w h i c h characterize the gravitational effects are; Gr x
{~ = sin c~ - Re x2
(1)
Gr x
{.
= cos
~, -
-
(2)
RexS/e The former is derived from the longitudinal m o m e n t u m equation a n d is characteristic for wedge and vertical wall, and the latter is derived from the transverse m o m e n t u m equation and is characteristic for horizontal flat plate and extremely small inclination. It is most important to note that both quantities, {~ a n d u, are not constant parameters but functions of the streamwise coordinate. {x varies directly proportional to x and { u varies directly proportional to the square root of x. The eoordinate d e p e n d e n c e of { suggests that at near region it is quite possible that { is less than unity and as the flow proceeds downstream, it increases with x and m a y exceed unity in far region. This naturally results in a ehange in the structure of the b o u n d a r y layer adjacent to the wedge. It consists of a forced convection d o m i n a t e d flow near region and a free convec-
EFFECTS ON FLOWS OVER A FLAT PLATE tion dominated flow far region connected by an intermediate region in which forced and free convections are of comparable magnitude. Inasmuch as we have confined the present work to the case ~ < 1, the flow under investigation remains forced convection dominated. A flame sheet approximation was invoked to describe the diffusion controlled combustion process within the boundary layer. The presence of the discontinuous flame sheet introduces discontinuities in the gradients of the temperature and of the mass concentration across it. This difficulty was resolved by Shvab-Zeldovich 12 formulation which defines temperature-oxidizer and fuel-oxidizer coupling variables, the profiles of which become continuous throughout the boundary layer.
IlL Governing Equations
The governing equations for the steady, chemically reacting laminar boundary layer flows over a wedge with gravitational force are derived in detail in Ref. 11. These equations do not admit similar solutions, because of the nature of the coupling between the momentum and energy (species) equations and owing to the dependence of the buoyant force on the strealnwise coordinate, and approximation methods are to be invoked. Two methods, local similarity and perturbation, have been considered and the latter is utilized in the present research. The coordinate dependence of { indicates that the problem is inherently a coordinate expansion rather than a regular parameter expansion. In view of that a coordinate perturbation of the governing equations about the forced convection flow has been employed resulting in the equations for the zeroth and higher order solutions. The equations for the zeroth and first order solutions are given below for the case of a horizontal flat plate. (For reason of space the equations for a wedge flow and the equations for second order solution are not repeated here and they can be found in Ref. 11.) M o m e n t u m Equations:
i
I
+2f:f 1 -2~,of91 +7~ofo; {/-direction
(5) (6)
The plus (+) and minus ( - ) signs in Eq. (6) refer to aiding and opposing flows, respectively. We designate as aiding flows, those flows in which buoyancy force has a positive component in the direction of the main flow', and as opposing flows, those flows in which the buoyancy force has a component opposite to the freestream velocity. Aiding flows are obtained for the tipper half of the wedge (plate facing upwards), if the natural direction of the gravity field is toward the tipper side of the surface. Opposing flow are obtained if the direction of the gravity field is reversed, or if the lower half of the wedge is considered (plate facing downwards).
Energy Equations: i
Zeroth order:
- - 0 o + foO[~ = 0
pr
(7)
1
- - 0 " 1 + foO'l - f'o01
First order:
pr
+9~f1 % = 0
(s)
Species Equations: 1 --go Sc
Zeroth order:
+ f0g0 = 0
(7')
1
--Sc gl" -[- ( 0 g ; - f o g l
First order:
+ 2flgo=O
(8')
Here prime denotes differentiation with respect to the conventional nondimensional boundary layer coordinate "q,
P~Ue n-
(2s)l/2
f
fu0 (P/P~)dy
p,,IXeu~dx
(9) (10)
0
x-direction
fg' + fofo = 0
y-direction
I~ = 0
(3)
The variable f is the nondimensional stream function defined such that f ' = u / %
(4)
O(x,y) = (2sp/2 f(s,n) First order: x-direction
=0
- ( 1 / 8 ) 1/2 fi~ - (P0 - 1) = 0
s=
Zeroth order:
1559
0+
f'[' + fof; - f'ofl
pu = - -
Oy
pv -
0q, Ox
(11) (12)
1560
MATHEMATICAL MODELING
f(sm) =
{f,
)Co +
+ '~2f2 + ...
(13)
where ~ for a h o r i z o n t a l flat plate yields, = G r ~ / R e ~ 3/2
(14)
Equation of State:
T h e a d d i t i o n a l fifth e q u a t i o n r e q u i r e d to solve the five d e p e n d e n t variables, f, ~, /5, 0, g is d e r i v e d from t h e e q u a t i o n of state, w h i c h c a n b e written in a g e n e r a l form,
a n d the local G r a s h o f a n d R e y n o l d s n u m b e r s are, Cry-
gxav 2 -Gr~
Re ~ =
(Pp~_/p~)
UeX
(15)
(16)
1)
T h e v a r i a b l e s / 3 a n d ~ are the n o n d i m e n s i o n a l pressure a n d d e n s i t y , respectively.
U p o n s u b s t i t u t i n g the d e p e n d e n t variables, p, p a n d T b y the n o n d i m e n s i o n a l variables /3, 15 a n d 0 (or g), this e q u a t i o n b e c o m e s very difficult to deal with. A detailed d e r i v a t i o n can b e f o u n d i n Ref. 11, a n d the f o l l o w i n g e q u a t i o n s have b e e n o b t a i n e d ; Zeroth order: Po = (B4 - D o B s ) O o
~3 = ;3o + ~731 + ~z~52 + ...
(17)
fi = P') q- ~151 -~- ~2~)2
(18)
1 3 T - - 13T, W
13 Y - 1 3 ~',w
(27)
D o = O f o r ~ 1 = O - - ~ l f , Oo~ 0r First order:
(19) PO + ~ f i l = (B4 - DiBs)(Oo +1~0,)
13T, e -- 13T, W
g -
+ B6 - D o B 7
where
T h e n o r m a l i z e d S h v a b - Z e l d o v i c h variables 0 a n d 13 are d e f i n e d as,
o-
(26)
p = pRT
+ B 6 - DIB 7
(20)
(28)
where Where 13T a n d 13v are the t e m p e r a t u r e - o x i d i z e r a n d fnel-oxidizer c o u p l i n g f u n c t i o n s , respectively. ~3r = ~xr - a 0
(21)
13r = a F - a o
(22)
a n d a t , a 0 a n d a F are the d i m e n s i o n l e s s temperatnre, o x y g e n mass fraction a n d fuel mass fraction, respectively.
D~=Oforll=O--*~f,O
o+~01~Of
D1 = l f o r x I =- xlf--->'qe, O0 + ~0~ > 0 f All B's are c o n s t a n t s d e f i n e d b y the followi n g expressions,
T~ B1-
T~(1 + B 2) QYo,e
B2aW o Cp T e
f T C,dT TO Os
YF, w B3= - r
(23)
N
B 4 = (1 -- B1)(1 + B 2) B 5=B2(1-B3)
i=1
%-
Yo,e
y,
B 6 = B1(1 + B 2)
i = (O,F)
(24)
w,(,, 7 - .,,;) w h e r e C , is the specific heat of the m i x t u r e N
c,, = ~ i=1
B 7 = B2B 3
where Q is the heat released b y the c h e m i c a l reaction N
Y, c,,,,
(25)
Q=~h, i=1
o
W
, /wtt, - - i )
EFFECTS ON FLOWS OVER A FLAT PLATE go, w = 0
~b is the stoichiometric mass ratio
d,-
~Wo
at
1561
(fuel-oxidizer at the wall)
(31')
~--+ ac
bW~ f0,e = 1
1 Of = 1 + Yo,~/cbYF, w
and
(freestream velocity)
(32)
0o,~ = 1 (freestream temperature-oxidizer)
T h e five functions fo, 00, go, 130 and P0 are associated with the pure forced convection flow while f l , 0i, gl,/31 a n d Pl give the first order deviations clue to the gravitational effects. It is worth noting that b y retaining the b u o y a n t force in the m o m e n t u m equations the pressure field is perturbed. H o w e v e r the pressure perturbation has different effects on each of the governing equations. It is of first order in the m o m e n t u m equation, of second order in the equation of state a n d there is no effect at all on the energy a n d species equations. Nevertheless all the d e p e n d e n t variables are effeeted b y b u o y a n c y through the first order perturbation since the g o v e r n i n g equations a n d the b o u n d a r y conditions are coupled.
go,,, = 1
(freestream fuel-oxidizer)
(33) (33')
PO,e ~ 0
(pressure merges to freestream value)
(34)
First Order: At
~q = 0 f'l.w = 1
(freestream velocity)
(35)
B fl,w
--O;,w 2Pr
(36)
0~,w = 0
(37)
gl,w = 0
(37')
IV. Boundary Conditions and Assumptions
f;,,, = o
(as)
E i g h t b o u n d a r y conditions for each solution are required to specify c o m p l e t e l y the problem, three for the stream function, two for the temperature-oxidizer, two for the fuel-oxidizer, a n d one for the pressure. (The equation of state is algebraic and does not need any b o u n d a r y conditions).
0~,e = 0
(39)
g~,e = 0
(39')
~3~,~= 0
(40)
at
"q ---~ ~
where 13 is the mass transfer driving force, referred to throughout the present work as the heat ratio,
Y Cv(TeB =
C v,,(T w-
Tw) + 0
. . . aW~ T s.~) + L + (Cv, s - C v) T w
. (41)
Zeroth Order:
Assumptions:
At
In addition to the conventional assumptions introduced by the b o u n d a r y layer theory, the following ones have been e m p l o y e d in deducing the governing equations and b o u n d a r y conditions;
lq = 0 fo,w= 0
(no slip)
(29)
B fo, w = - - - O ' o . w Pr (energy b a l a n c e at the wall)
(30)
0,,,w = 0 (temperature-oxidizer at the wall)
(31)
(i) Lewis n u m b e r is unity, Pr = Sc of order unity. (2) the products pu and p'2D are constant (3) no thermal or pressure gradient diffusion, (4) no radiative energy transfer,
1562
MATHEMATICAL MODELING
(5) equal binary diffusion coefficients, Fick's law is applicable. (6) one step, gas phase, irreversible chemical reaction (7) infinite chemical kinetics, flame sheet model (8) no surface reaction, gas emerging from surface is entirely fuel. (9) C,, (for the mixture), T w and Yo,e are constant. (10) ~ < 1 (ratio of buoyant to inertia force). V. Solution
A comparison of the energy equations (Eqs. (7), (8)) and the species equations (Eqs. (7') (8')), and the corresponding boundary conditions (Eqs. (31) and (31'), (33) and (33'), (37) and (37'), (39) and (39')), reveals that by employing Le = 1, Prandtl and Schmidt numbers are equal resulting that 0 and g are identical functions. Thus, it is sufficient to solve for any one of them. Moreover, the zeroth order momentum equation in the transverse direction, Eq. (4), subject to the boundary condition described by Eq. (34) yields a trivial solution for the nondimensional pressure, /30 = 0
(42)
namely, no pressure changes across the boundary layer in the case of pure convection flow. Since every nth order ordinary differential equation can be replaced by an equivalent system of n first order equations, we get a system of eleven equations as follows; 5 equations from the zeroth order solution (3 from the momentum equation and 2 from the energy (or species) equation and 6 equations from the first order solution (4 from the momentum equations and 2 from the energy (or species) equation). The equation of state is an algebraic equation and therefore does not alter the number of ordinary differential equations in our system. Examining the present boundary conditions we find that a boundary value problem with split boundary conditions is exhibited. In order to have an initial value problem with all boundary conditions specified at one point (wall) we need the following five auxiliary boundary conditions;
folw, .Oo,w, f'~',w, Oi,w, ~l,W instead of the five boundary conditions already obtained at the outer edge (see Eqs. (32), (33), (38), (39), (40)). The method of Smith et a133
was utilized for a search of these required boundary conditions at the wall which produce the solution satisfying the outer boundary conditions. The eleven ordinary differential equations subject to the appropriate boundary conditions at the wall were numerically integrated using Runge-Kutta method on the SUNY at Stony Brook IBM 360/370 computer system. The integration is started with the zeroth order solution by assigning initial values to f~'w and 00 w- These initial values are iterated t~ll the freestream conditions are met. After the zeroth order solution is obtained the integration is continued with both zeroth and first order equations by assigning initial values at the wall to the first order boundary conditions (the zeroth order solution is independent of the first order, and hence is not effected), and iterating these values till the functions at the outer edge merge smoothly into the freestream values. The numerical computations have been applied to a horizontal flat plate covered with fuel. The fuel is ethyl alcohol burning in air at the stoichiometrie ratio and at atmosphereic pressure. Four cases representing various external conditions such as wall temperature, freestream oxygen concentration and direction of the gravity field, were analyzed (the direction of the gravity field determines whether the flow is aiding or opposing). Each case was solved with three different heat ratio values (the conventional mass transfer number), B = 0.5, 1 and 2. For each heat ratio value three solutions corresponding to the relative importance of the buoyancy effects were obtained. First solution is for no buoyancy effects at all, ~ = 0. Second solution is for a particular perturbation quantity ~ describing the relative contribution of buoyancy to the basic forced convection flow. The third solution is for 26 which means that the gravitational effects are doubled. Thus, altogether 36 solutions have been obtained as listed in Table I. Flow conditions, the corresponding perturbation quantities, Reynolds numbers and Grashof numbers are given in Table II. The representative value that has been used throughout the work to describe the gravitational effects is, = 1.0 x 10 -a Before applying the computer program to the present work it was tested by letting it to reproduce the following three well-known published solutions: Emmons and Leigh 14 solution for Blasius function with blowing and suction: Sparrow and Minkowycz 15 solution
EFFECTS ON FLOWS OVER A FLAT PLATE
1563
TABLE I External flow conditions for the 36 various solutions obtained Perturbation quantity ~
0
0.5 x 10-a
1.0 x 10-a
External Conditions
B
B
B
Case 1
T w = 288~ Yo,e = 0.232 aiding flow
0.5 1.0 2.0
0.5 1.0 2.0
0.5 1.0 2.0
Case 2
T w = 576~ Yo.e 0.232 aiding flow
0.5 1.0 2.0
0.5 1.0 2.0
0.5 1.0 2.0
Case 3
T w = 288~ Yo,r = 0.500 aiding flow
0.5 1.0 2.0
0.5 1.0 2.0
0.5 1.0 2.0
Case 4
T w = 288~ Y0,~= 0.232 opposing flow
0.5 1.0 2.0
0.5 1.0 2.0
0.5 1.0 2.0
=
for b u o y a n c y effects on horizontal boundary layer; and C h e n and T o o n g v solution for laminar boundary layer w e d g e flow with diffusion flame. T h e first r e p r o d u c e d solution is c h e c k i n g the m o m e n t u m e q u a t i o n with blowing boundary condition at the wall. T h e second solution is testing if the present model is capable of describing the b u o y a n c y effects for c h e m i c a l l y inert flow by shutting off the combustion. T h e last solution is e x a m i n i n g if the current model describes the structure of a diffusion flame over a w e d g e w i t h o u t buoyancy effects by setting ~ = 0. All reproduced results were compared to the original p u b l i s h e d studies and showed excellent agreement. Thus, we positively c o n c l u d e that if the model successfully describes on
the one hand the b u o y a n c y effects without c o m b u s t i o n and on the other hand the diffusion flame w i t h o u t buoyancy effects it is also capable of describing the mutual interaction b e t w e e n both phenomena, namely the buoyancy effects on chemically reacting flows.
VI. Results and Discussion
V e l o c i t y P r o f i l e s (Fig. 2):
T h e velocity profiles are presented in Fig. 2 for aiding and o p p o s i n g flows u n d er the same external conditions. E x a m i n i n g tl~ese profiles we find that with no gravitational effects the velocity profile is similar to that
TABLE II Flow conditions for a horizontal flat plate, L[m]--length of the plate, u e [m/secJ--freestream velocity.
Ue[m]
2.0
3.0
2.5
Re x
Gr x
~ x
L [m]
10-a
10-6
10 a
Re x 10 -a
Gr x 10 -6
~x 10 a
Be x 10 -a
Gr x 10 -6
~x 10 a
0.03 0.04 0.05
3.82 5.09 6.37
1.07 2.54 4.97
1.19 1.37 1.53
4.78 6.37 7.96
1.07 2.54 4.97
0.68 0.79 0.88
5.73 7.64 9.55
1.07 2.54 4.97
0.43 0.50 0.56
1564
MATHEMATICAL MODELING 1.2
I0.0
1.0
/
/
AIDING 3W
~"
8.0
/
.8
ZERO OPPOSING _GRAVITY
i.-
/
n.- 6.0 23
-~L--FLOW
E)
/ ii /L zGR.0AVITY / ill
o
_1
w >
r~ hi
~; 4.0
hi I"-
OPPOSING
.4
/--:
/ .2 /
2.0
FLOW
/
/
/
/
1.0
//
0
/
FIG. 3. Temperature profiles, T w = 288~ = 0.232, B = 1.
0 ~
0 2.0 4.0 6.0 BOUNDARY LAYER THICKNESS,'/'/
F[c. 2. Velocity profiles, T w = 228~ = 0.232, B = 2.
|
210 4.0 6.0 0 BOUNDARY LAYER THICKNESS,'r]
Yo,e
Yo,e
obtained b y E m m o n s a n d Leigh t4 for Blasius function with b l o w i n g . Buoyaney effects accelerate a i d i n g flows a n d decelerate o p p o s i n g flows. This can b e p h y s i c a l l y explained b y the existence of a horizontal pressure gradient which is favorable for the former flow a n d adverse for the latter. It is also o b s e r v e d that b y increasing either the heat ratio, or the wall temperature or the freestream oxygen concentration the f l u i d in a i d i n g flows is more accelerated d u e to t h e increase in the favorable pressure gradient (see Fig. 5), a n d exceeds sometimes the freestream velocity. Such velocity overshoots were experimentally measured b y H i r a n o et al. x6av on horizontal flat plate with diffusion flame. W h e n c o m b u s t i o n occurs inside the b o u n d a r y layer the temperature near the flame increases, the d e n s i t y decreases a n d the flame is accelerated to a velocity greater than the freestream velocity due to the i n d u c e d (by buoyancy) pressure gradient in spite of viscous retardation. Because of the acceleration of the flow the velocity gradient at the wall becomes larger increasing the wall shear stress a n d hence the transition Reynolds number. For o p p o s i n g flows the b u o y a n c y forces retard the fluid w i t h i n the b o u n d a r y layer so that these act like an adverse pressure gradient causing a decrease in the velocity.
Temperature Profiles (Fig. 3): The temperature increases from the wall temperature to the flame temperature a n d then decreases to the a m b i e n t temperature. T h e m a x i m u m temperature occurs at the flame sheet due to the fact that infinite reaction rate is assumed. The m a x i m u m temperature increases with increasing, B, T w and Yo,e" Buoyance effects do not change the m a x i m u m temperature a n d the only effect is inereasing temperature gradients at the wall by m o v i n g the flame closer to the surface for a i d i n g flow. In case of o p p o s i n g flows the flame moves away from the surface resulting in a decrease in temperature gradients at the wall. It is also observed that the temperature gradients at the flame are d i s c o n t i n u o u s but this has been taken care of b y i n t r o d u c i n g Shvab-Zeldovich transformation.
Mass Fraction Profiles (Fig. 4): As a c o n s e q u e n c e of the flame sheet approximation the flame front is located at a p o i n t where the nmss fractions of both fuel a n d oxidizer are zero, a n d the mass fraction gradient is discontinuous. Fuel mass fraction Y~ is decreasing from its initial value at the wall YFW. ( Y v w d e p e n d s only on B and Yoe) to zero at th'e flame. Oxidizer mass fraction is decreasing from its freestream value Yo,e, as it is transported t o w a r d the flame, to zero at
EFFECTS ON FLOWS OVER A FLAT PLATE 2.0
1565
~o--_60.0 E
,\
w
_J
.~~ 1.5
>- -50.0
-
-
rr
z_0
S
g
A'D'NG FLOW
Tw *k Y0,e 288 0.500 576 0.232
288
Z
0.232
,AIDING FLOWS
F-
~ -40.0 m W T p-
'~ .5 <
YF
'\~'f/' YO O ~vvy 0 2.0 4.0 6.0 BOUNDARY LAYER THICKNESS, 77
FIG. 4. Mass fraction profiles, T w = 288~ = 0.232, B = i.
-30.0
0 n,"
TY
LIJ
-20.0
:E
W rr (..)
Yo, e
_z - I 0 . 0 W Z:) 03 CO ',' n~
the flame. It is observed that increasing heat ratio and decreasing freestream oxidizer concentration move the flame away from the surface. Buoyancy effects tend to move the flame back to the surface in a i d i n g flows. This t e n d e n c y is a u g m e n t e d in eases of high wall temperature and high freestream oxidizer concentration. In o p p o s i n g flows the t e n d e n c y is the reverse.
q0 I~~ZERO GRAVITY~
I
0 2.0 4.0 6.0 BOUNDARY LAYER THICKNESS,~7
FIG. 5. Pressure profiles, 13 = l.
creases the pressure effects within the boundary layer due to buoyancy.
Flame Height (Fig. 6): Pressure Profiles (Fig. 5): T h e pressure profiles in a i d i n g flow are shown in Fig. 5 for three different external conditions. The pressure is increasing across the b o u n d a r y layer from its initial value at the surface to the edge where it is merging smoothly with the freestream pressure. Rotem and Claassen, TM and Pera a n d Gebhart xv.2~ reported the same p h e n o m e n a for natural convection flows adjacent to horizontal and slightly inclined surfaces. T h e physical explanation for the pressure gradient across the b o u n d a r y layer with gravitational force is that the only force available to balance the b u o y a n t force in the transverse direction is a pressure force acting in an opposite direction to the b u o y a n t force. Therefore, in aiding flows, where the b u o y a n t force is acting in the positive direction of the y-axis, the pressure force is acting d o w n w a r d resulting in a positive pressnre gradient across the b o u n d a r y layer. For o p p o s i n g flows the behavior is the reverse and hence pressure is decreasing with increasing distance from the surface. It is also f o u n d that increasing either B, or Tw, or Y.... in-
The flame height is presented in Fig. 6, and it is observed that the flame sheet moves away from the surface as the heat ratio increases. The effect of gravity is to b r i n g the flame closer to the surface in a i d i n g flows, and to move it away from the surface in o p p o s i n g flow. T h e prediction of the present study that fl'ame " s t a n d o f f " distances are lower under gravitational effects is in good agreement with some experimental works. Krishnamurthy a n d Williams 2~ for a laminar c o m b u s t i o n of P M M A on a flat plate c o n c l u d e d that "measured values of flame standoff distances are lower than the value predicted b y classical theory b y more than a factor of 2." We may attribute these differences to the role that b u o y a n c y plays in the flow. We found that the differences of flame heights with and without b u o y a n c y can be of the order of two d e p e n d i n g on the flow conditions. Moreover, it was observed in experiments c o n d u c t e d in zero gravity either b y using a drop tower or test flights flying Keplerian parabola that "flames are approximately 50% longer and w i d e r in zero gravity (no buoyancy) than in normal gravity. ''3 Un-
1566
MATHEMATICAL
MODELING
:3.5 J
f
J
J J f f
30
g--
///~TY /// J I"''! / / / ~ ' ~ . I D I N G FLOW
~2.5 I.s.I 1"
:~2O W
/
_1 I.i..
1.5
,.%
I
I
.5
I
t
1.0 1.5 HEAT RATIO, B
Fie. 6. Flame height, T w = 288~
2.0
2.5
Yo.e = 0.232.
.5
.4
~.
AIDING FLOW
z0 ~ . 3 I--0 or la_
Z x-" o3
.2
\ \ \
0
I
0
.5
\ OPPOSING FLOW
I
I
I
1.0 1.5 HEAT RATIO, B
FIG. 7. Skin friction, T w = 288~
2.0
Yo, e = 0.232.
2.5
EFFECTS ON FLOWS OVER A FLAT PLATE fortunately, all these experiments were concerned with jet diffusion a,4 or free convection 5,~or droplet burning, 2 and not with forced convective flow over a flat plate. In any event the trend is the same.
Skin Friction (Fig. 7): The local skin friction is presented in Fig. 7. It is observed that b y increasing the heat ratio the shear stress at the surface is decreased. This can be explained b y recognizing that an increase in heat ratio corresponds to an increase in the b l o w i n g rate, which in turn, reduces the shear stress in the vicinity of the surface. 14 The gravitational effects tend to increase the shear stress in a i d i n g flows thus stabilizing the flows, and to reduce the shear stress in o p p o s i n g flows hastening separation. This is physically explained b y the favorable pressure gradient whieh accelerates the flow, causing steeper velocity gradients at the surface.
VII. Conclusions Since chemically reacting b o u n d a r y l a y e r flows are associated with high temperature and low density, steep temperature and density gradients prevail across the thin b o u n d a r y layer, a u g m e n t i n g the b u o y a n t force. This research has shown that the cross stream b u o y a n c y - i n d n e e d b o d y force acts effectively to p r o d u c e a streamwise pressure gradient in the fluid adjacent to the plate surface. The pressure gradient is favorable in a i d i n g flows and adverse in o p p o s i n g flows. Hence, the local b o u n d a r y layer flow is accelerated or decelerated relative to the c o r r e s p o n d i n g gravity-free forced convection flow. Correspondingly, there is an increase or decrease in the local skin friction and heat transfer rates, and consequently a decrease or increase in the flame sheet "standoff" distance, d e p e n d i n g u p o n whether aiding or o p p o s i n g flow exists, respectively. These theoretical predictions are in full agreement with observed experimental results. Based on the present study it is cone l u d e d that buoyancy plays an important role in b o u n d a r y layer diffusion flames, previous studies that deleted the b u o y a n t force, for reasons of simplification, result in underestimation of the combustion rate and the flame velocity. In addition, neglecting buoyancy yields erroneous values for such o t h e r important physical quantities as flame "standoff" distance, skin friction, heat transfer rates and b o u n d a r y layer thickness.
1567
Nomenclature a B b cf C1,
n u m b e r of moles of oxidizer heat ratio (mass transfer driving force) n u m b e r of moles of fuel local skin friction coefficient specific heat of t h e gas mixture at constant pressure f dimensionless stream function (7;r Grashof n u m b e r Gr m o d i f i e d Grashof n u m b e r g normalized fuel-oxidizer function h~ standard heat of formation per unit mass at T o L latent heat and all heat losses to the surroundings Le Lewis n u m b e r N total n u m b e r of chemical species present /3 nondimensional pressure increment across the b o u n d a r y layer Pr Prandtl n u m b e r Q heat released b y the cheinical reaction R universal gas constant Re Reynolds n u m b e r s similarity i n d e p e n d e n t variable in xdirection Sc Schmidt n u m b e r T temperature u velocity c o m p o n e n t in the x-direction v velocity c o m p o n e n t in the {/-direction W molecular weight x cartesian coordinate parallel to the surface Y mass fraction ~j cartesian coordinate normal to the surface c~i dimensionless mass fraction ~xr dimensionless temperature [3 r dimensionless temperature-oxidizer function [3 r dimensionless fuel-oxidizer functiofi "q n o n d i m e n s i o n a l b o u n d a r y layer coordinate 0 normalized temperature-oxidizer function tx absolute viscosity v kinematic viscosity v'i stoichiometric coefficient for species i a p p e a r i n g as a reactant v'i' stoichiometric coefficient for species i a p p e a r i n g as a product perturbation quantity, ratio of b u o y a n t force to inertia force p density 15 n o n d i m e n s i o n a l density
1568
MATHEMATICAL M O D E L I N G
Subscripts 0,1, e F f i 0 s x y W ~c
7. CHEN, T. N. AND TOOXG, T. Y.: Progress in
t h e order of t h e solution freestream fuel flame species oxidizer solid p h a s e in x-direction in y - d i r e c t i o n wall ambient conditions
Acknowledgment Portions of this research were supported under NASA grant NSG-3051.
REFERENCES 1. FENDELL, F. E. AND SMITH, E. B.: AIAA J. 15, 1984 (1967). 2. OKAJIMA,S. AND KlrMAGAI, S.: Fifteenth Symposium (International) on Combustion, p. 401, The Combustion Institute, 1975. 3. COCHRAN,T. H.: Experimental Investigation of Laminar Gas Jet Diffusion Flame in Zero Gravity, NASA TN D-6523, 1972. 4. EDELMAN, R. B., FORTUNE, O. F., WEILERSTEIN, G. COCHRAN, T. H. AND HAGGARD, J. B., J~.: Fourteenth Symposium (International) on Combnstion, p. 399, The Combustion Institute, 1973. 5. KIMZEY,J. H., Downs, W. R., ELDRED, C. H. AND NORMS, C. W.: Flammability in Zero Gravity Environment, NASA TR R-246, 1966. 6. DE RIS, J. ANDORLOFF, L.: Fifteenth Symposium (International) on Combustion, p, 175, The Combnstion Institute, 1975.
Astronautics and Aeronautics, Vol. 15, Heterogeneous Combnstion, p. 643, Academic Press, 1964. 8. WALDMAX,C. H.: Theoretical Studies of Diffusion Flame Structnres. Ph.D. Thesis, Princeton University, 1969; also available as AFOSR SR 69-0350 TR, 1969. 9. KmER, H. AND KERZNER, H.: AIAA J. 11, 1691
(1973). 10. KRISItNAMURTHY,L. AND WILLIAMS, F. A.: SIAM
J. Appl. Math. 20, 590 (1971). 11. LAVID, M.: Buoyancy Effects on Chemically Reacting Laminar Bonndary Layer Flows, Ph.D. thesis, State University of New York at Stony Brook, 1974. 12. WILLIAMS,F. A.: Combustion Theory, AddisonWesley, 1965. 13. SMITH, A. M. O. Ago CLUTTER,D. W.: AIAA J. 3, 639 (1965). 14. EMMOXS, H. W. AND LEIGH, D. C.: Tabulation of the Blasins Function with Blowing and Suction. Aeronautical Research Council Current Paper 157, 1954. 15. SPARROW,E. M. ANn MINKO~CZ, W. J.: Intern. J. Heat Mass Transfer 5, 505, (1962). 16. HmAxO, T., IWAI, K. AND KANNO,Y.: Astronautica Acta 17, 811 (1972). 17. HmANO,T. ANDK1NOSHITA,M.: Fifteenth Symposium (International) on Combustion, p. 379, The Combustion Institute, 1975. 18. ROTEM, Z. AND CLAASSEN,L.: J. Fluid Mech. 38, 173 (1969). 19. PERA, L. AND GEBHART, B.: Intern. J. Heat Mass Transfer 15, 269 (1972). 20. PE~A, L. AND GEBHART, B.: Inter. J. Heat Mass Transfer 16, 1131 (1973). 21. KRISHNAMURTHY,L. AND WILLIAMS, F. A.: Fourteenth Symposinm (International) on Combustion, p. 1151, The Combustion Institute, 1973.
This is a theoretical study of the chemically reacting laminar boundary layer flow over a horizontal flat plate with gravitationally induced buoyant force. A diffusion flame sheet model was invoked to deseribe the combustion process. It was found that the effects of gravity on the purely force convection flow are characterized by a dimensionless coordinate quantity G r J R e , 5/2. The governing equations were obtained by expanding the dependent variables into series in terms of this coordinate quantity. The problem inherently is a coordinate perturbation and it is treated as such. A numerical solution of the zeroth and first order governing equations subject to the appropriate physical boundary conditions was obtained for various external flow conditions such as wall temperature, freestream oxygen concentration and aiding and opposing flows. It was shown that the cross stream buoyancy induced body force acts effectively to produce a streamwise pressure gradient in the fluid adjacent to the plate surface. The pressure gradient is favorable in aiding flows (plate facing upwards) and is adverse in opposing flows (plate facing downwards). Hence, the local boundary layer flow is accelerated or decelerated relative to the corresponding gravity-flee forced eonveetion flow. Correspondingly, there is an increase or decrease in the local skin friction and heat transfer rates, and consequently a decrease or increase in the flame "stand-off" distance, depending upon whether aiding or opposing flow occurs, respectively. Based on the present study we conclude that buoyancy plays an important role in boundary layer diffusion flames, and by retaining the buoyant force the current model is capable of explaining phenomena that have been observed experimentally but have not been predicted by classical boundary layer theory. Such previously unaccounted for effects include the acceleration of the boundary layer flow (sometimes a velocity overshoot is exhibited), and a decrease in the flame "stand-off" distance.
I. I n t r o d u c t i o n
Chemically reacting b o u n d a r y layer flows have been investigated theoretically and experimentally over the last forty years. The present theoretical research is concerned mainly with the gravitational effects on such laminar b o u n d a r y layer flows over a horizontal flat plate. Generally speaking, the aim is to investigate the interaction of c o m b i n e d forced and free convection, the former due to the external flow a n d the latter due to the b u o y a n t force. Literature abounds with studies on t o m -
b i n e d convection in inert b o u n d a r y layer flows, dealing mainly with the fluid mechanical a n d heat transfer aspects, and with studies on "chemically reacting b o u n d a r y layer flows neglecting gravitational effects. However, no attempt has been reported yet to develop a physical and mathematical model designed to elucidate the structure of a diffusion flame w i t h i n a forced convection b o u n d a r y layer i n f l u e n c e d by gravity such as is exhibited in flows over a wedge. A review of the existing literature reveals that b u o y a n c y effects due to gravity are generally neglected with the exception of some
1557
1558
MATHEMATICAL MODELING
works dealing with b u r n i n g of droplets, 1'2 a n d axisymmetric jet d i f f u s i o n flames, a,4 and with experiments of b u r n i n g of fuel surfaces. 5,6 However, in these works either the gravitational force is in the direction of the m a i n flow (i.e. vertical wall, falling droplets) or it is a purely free convection flow. In many practical applications involving combustion, ranging from h y b r i d c o m b u s t i o n to fire prevention, the fuel and the oxidant are initially not premixed, and the c o m b u s t i o n exhibits a structure of a diffusion flame. Inasmuch as these a p p l i c a t i o n s are usually not in a zero-gravity e n v i r o n m e n t it is of importance to analyze the effects of gravity on the diffusion flame, and only after a more complete understanding of the c o m b u s t i o n process with b u o y a n c y is achieved a n d the importance of gravity is d e t e r m i n e d will it be p o s s i b l e to predict the structure of diffusion flames u n d e r various gravity conditions. This is particularly true in cases where experimental data is sparse, such as in spacecraft environments of low (zero)--gravity level a n d d u r i n g high-G maneuvers of aircrafts. II. Physical M o d e l A sufficiently general model has been selected and it is illustrated i n Fig. 1. T h e combustion is w i t h i n the b o u n d a r y layer, developed b y the flow of main gas stream containing oxidant a n d inerts over a horizontal flat plate. The plate can be considered as presenting three general cases: (1) a c o n d e n s e d phase fuel that can evaporate (or sublimate); (2) a solid fuel that pyrolyzes with gas generation rate that can be d e s c r i b e d by a 'surface' pyrolytic law; (3) a porous plate from the surface of which l i q u i d fuel is injected. T h e gaseous fuel is carried b y molecular diffusion a n d convection from the solid surface into the b o u n d a r y layer where it can react with the oxidant. The oxidant is also transported to the
BOUNDARY (~
Ue
Iue,
FLAME S H E E T \ y
Te ~
x
.
Yo
O .
_
~
L .
LAYER EDGE
/ _ _. _ _ _ . . DIFFUSION ZONE
T.7_~ IYF,w
Flc. 1. Physical model with flame sheet approximation over a horizontal flat plate.
flame b y diffusion a n d convection from the main stream. Heat released by the chemical reaction is transmitted to the surface by diffusion and convection, causing continuous gaseous fuel generation. Thus, steady state can be m a i n t a i n e d and the process is self-sustaining. This physical m o d e l has been treated previously with infinite kinetics b y Chen a n d Toong v a n d with finite kinetics b y W a l d m a n , s Krier a n d Kerzner, 9 a n d Krishnamurthy a n d Williams. 1~ However, the effects of gravity have not been s t u d i e d yet, as the b o d y force was deleted from the governing equations. A general d i m e n s i o n l e s s analysis of the two m o m e n t u m equations with b u o y a n t forces for b o u n d a r y layer flows over a wedge was carried out elsewhere, ]~ a n d it was shown that the structure of the b o u n d a r y layer is d e t e r m i n e d b y a mixed convection, dimensionless quantity {, which is essentially a ratio of the b u o y a n c y force to the inertia force. In general, { < 1 indicates a p r e d o m i n a n t l y forced convection flow perturbed b y natural convection, ~_ > 1 suggests a p r e d o m i n a n t l y free convection flow perturbed b y forced convection, and { of order unity corresponds to a flow in which both effects are comparable. This research is confined to study the effects of free convection due to non-zero gravity field on the p r e d o m i nantly forced convection flow, i.e. { < 1. The two distinct coordinate quantities w h i c h characterize the gravitational effects are; Gr x
{~ = sin c~ - Re x2
(1)
Gr x
{.
= cos
~, -
-
(2)
RexS/e The former is derived from the longitudinal m o m e n t u m equation a n d is characteristic for wedge and vertical wall, and the latter is derived from the transverse m o m e n t u m equation and is characteristic for horizontal flat plate and extremely small inclination. It is most important to note that both quantities, {~ a n d u, are not constant parameters but functions of the streamwise coordinate. {x varies directly proportional to x and { u varies directly proportional to the square root of x. The eoordinate d e p e n d e n c e of { suggests that at near region it is quite possible that { is less than unity and as the flow proceeds downstream, it increases with x and m a y exceed unity in far region. This naturally results in a ehange in the structure of the b o u n d a r y layer adjacent to the wedge. It consists of a forced convection d o m i n a t e d flow near region and a free convec-
EFFECTS ON FLOWS OVER A FLAT PLATE tion dominated flow far region connected by an intermediate region in which forced and free convections are of comparable magnitude. Inasmuch as we have confined the present work to the case ~ < 1, the flow under investigation remains forced convection dominated. A flame sheet approximation was invoked to describe the diffusion controlled combustion process within the boundary layer. The presence of the discontinuous flame sheet introduces discontinuities in the gradients of the temperature and of the mass concentration across it. This difficulty was resolved by Shvab-Zeldovich 12 formulation which defines temperature-oxidizer and fuel-oxidizer coupling variables, the profiles of which become continuous throughout the boundary layer.
IlL Governing Equations
The governing equations for the steady, chemically reacting laminar boundary layer flows over a wedge with gravitational force are derived in detail in Ref. 11. These equations do not admit similar solutions, because of the nature of the coupling between the momentum and energy (species) equations and owing to the dependence of the buoyant force on the strealnwise coordinate, and approximation methods are to be invoked. Two methods, local similarity and perturbation, have been considered and the latter is utilized in the present research. The coordinate dependence of { indicates that the problem is inherently a coordinate expansion rather than a regular parameter expansion. In view of that a coordinate perturbation of the governing equations about the forced convection flow has been employed resulting in the equations for the zeroth and higher order solutions. The equations for the zeroth and first order solutions are given below for the case of a horizontal flat plate. (For reason of space the equations for a wedge flow and the equations for second order solution are not repeated here and they can be found in Ref. 11.) M o m e n t u m Equations:
i
I
+2f:f 1 -2~,of91 +7~ofo; {/-direction
(5) (6)
The plus (+) and minus ( - ) signs in Eq. (6) refer to aiding and opposing flows, respectively. We designate as aiding flows, those flows in which buoyancy force has a positive component in the direction of the main flow', and as opposing flows, those flows in which the buoyancy force has a component opposite to the freestream velocity. Aiding flows are obtained for the tipper half of the wedge (plate facing upwards), if the natural direction of the gravity field is toward the tipper side of the surface. Opposing flow are obtained if the direction of the gravity field is reversed, or if the lower half of the wedge is considered (plate facing downwards).
Energy Equations: i
Zeroth order:
- - 0 o + foO[~ = 0
pr
(7)
1
- - 0 " 1 + foO'l - f'o01
First order:
pr
+9~f1 % = 0
(s)
Species Equations: 1 --go Sc
Zeroth order:
+ f0g0 = 0
(7')
1
--Sc gl" -[- ( 0 g ; - f o g l
First order:
+ 2flgo=O
(8')
Here prime denotes differentiation with respect to the conventional nondimensional boundary layer coordinate "q,
P~Ue n-
(2s)l/2
f
fu0 (P/P~)dy
p,,IXeu~dx
(9) (10)
0
x-direction
fg' + fofo = 0
y-direction
I~ = 0
(3)
The variable f is the nondimensional stream function defined such that f ' = u / %
(4)
O(x,y) = (2sp/2 f(s,n) First order: x-direction
=0
- ( 1 / 8 ) 1/2 fi~ - (P0 - 1) = 0
s=
Zeroth order:
1559
0+
f'[' + fof; - f'ofl
pu = - -
Oy
pv -
0q, Ox
(11) (12)
1560
MATHEMATICAL MODELING
f(sm) =
{f,
)Co +
+ '~2f2 + ...
(13)
where ~ for a h o r i z o n t a l flat plate yields, = G r ~ / R e ~ 3/2
(14)
Equation of State:
T h e a d d i t i o n a l fifth e q u a t i o n r e q u i r e d to solve the five d e p e n d e n t variables, f, ~, /5, 0, g is d e r i v e d from t h e e q u a t i o n of state, w h i c h c a n b e written in a g e n e r a l form,
a n d the local G r a s h o f a n d R e y n o l d s n u m b e r s are, Cry-
gxav 2 -Gr~
Re ~ =
(Pp~_/p~)
UeX
(15)
(16)
1)
T h e v a r i a b l e s / 3 a n d ~ are the n o n d i m e n s i o n a l pressure a n d d e n s i t y , respectively.
U p o n s u b s t i t u t i n g the d e p e n d e n t variables, p, p a n d T b y the n o n d i m e n s i o n a l variables /3, 15 a n d 0 (or g), this e q u a t i o n b e c o m e s very difficult to deal with. A detailed d e r i v a t i o n can b e f o u n d i n Ref. 11, a n d the f o l l o w i n g e q u a t i o n s have b e e n o b t a i n e d ; Zeroth order: Po = (B4 - D o B s ) O o
~3 = ;3o + ~731 + ~z~52 + ...
(17)
fi = P') q- ~151 -~- ~2~)2
(18)
1 3 T - - 13T, W
13 Y - 1 3 ~',w
(27)
D o = O f o r ~ 1 = O - - ~ l f , Oo~
(19) PO + ~ f i l = (B4 - DiBs)(Oo +1~0,)
13T, e -- 13T, W
g -
+ B6 - D o B 7
where
T h e n o r m a l i z e d S h v a b - Z e l d o v i c h variables 0 a n d 13 are d e f i n e d as,
o-
(26)
p = pRT
+ B 6 - DIB 7
(20)
(28)
where Where 13T a n d 13v are the t e m p e r a t u r e - o x i d i z e r a n d fnel-oxidizer c o u p l i n g f u n c t i o n s , respectively. ~3r = ~xr - a 0
(21)
13r = a F - a o
(22)
a n d a t , a 0 a n d a F are the d i m e n s i o n l e s s temperatnre, o x y g e n mass fraction a n d fuel mass fraction, respectively.
D~=Oforll=O--*~f,O
o+~01~Of
D1 = l f o r x I =- xlf--->'qe, O0 + ~0~ > 0 f All B's are c o n s t a n t s d e f i n e d b y the followi n g expressions,
T~ B1-
T~(1 + B 2) QYo,e
B2aW o Cp T e
f T C,dT TO Os
YF, w B3= - r
(23)
N
B 4 = (1 -- B1)(1 + B 2) B 5=B2(1-B3)
i=1
%-
Yo,e
y,
B 6 = B1(1 + B 2)
i = (O,F)
(24)
w,(,, 7 - .,,;) w h e r e C , is the specific heat of the m i x t u r e N
c,, = ~ i=1
B 7 = B2B 3
where Q is the heat released b y the c h e m i c a l reaction N
Y, c,,,,
(25)
Q=~h, i=1
o
W
, /wtt, - - i )
EFFECTS ON FLOWS OVER A FLAT PLATE go, w = 0
~b is the stoichiometric mass ratio
d,-
~Wo
at
1561
(fuel-oxidizer at the wall)
(31')
~--+ ac
bW~ f0,e = 1
1 Of = 1 + Yo,~/cbYF, w
and
(freestream velocity)
(32)
0o,~ = 1 (freestream temperature-oxidizer)
T h e five functions fo, 00, go, 130 and P0 are associated with the pure forced convection flow while f l , 0i, gl,/31 a n d Pl give the first order deviations clue to the gravitational effects. It is worth noting that b y retaining the b u o y a n t force in the m o m e n t u m equations the pressure field is perturbed. H o w e v e r the pressure perturbation has different effects on each of the governing equations. It is of first order in the m o m e n t u m equation, of second order in the equation of state a n d there is no effect at all on the energy a n d species equations. Nevertheless all the d e p e n d e n t variables are effeeted b y b u o y a n c y through the first order perturbation since the g o v e r n i n g equations a n d the b o u n d a r y conditions are coupled.
go,,, = 1
(freestream fuel-oxidizer)
(33) (33')
PO,e ~ 0
(pressure merges to freestream value)
(34)
First Order: At
~q = 0 f'l.w = 1
(freestream velocity)
(35)
B fl,w
--O;,w 2Pr
(36)
0~,w = 0
(37)
gl,w = 0
(37')
IV. Boundary Conditions and Assumptions
f;,,, = o
(as)
E i g h t b o u n d a r y conditions for each solution are required to specify c o m p l e t e l y the problem, three for the stream function, two for the temperature-oxidizer, two for the fuel-oxidizer, a n d one for the pressure. (The equation of state is algebraic and does not need any b o u n d a r y conditions).
0~,e = 0
(39)
g~,e = 0
(39')
~3~,~= 0
(40)
at
"q ---~ ~
where 13 is the mass transfer driving force, referred to throughout the present work as the heat ratio,
Y Cv(TeB =
C v,,(T w-
Tw) + 0
. . . aW~ T s.~) + L + (Cv, s - C v) T w
. (41)
Zeroth Order:
Assumptions:
At
In addition to the conventional assumptions introduced by the b o u n d a r y layer theory, the following ones have been e m p l o y e d in deducing the governing equations and b o u n d a r y conditions;
lq = 0 fo,w= 0
(no slip)
(29)
B fo, w = - - - O ' o . w Pr (energy b a l a n c e at the wall)
(30)
0,,,w = 0 (temperature-oxidizer at the wall)
(31)
(i) Lewis n u m b e r is unity, Pr = Sc of order unity. (2) the products pu and p'2D are constant (3) no thermal or pressure gradient diffusion, (4) no radiative energy transfer,
1562
MATHEMATICAL MODELING
(5) equal binary diffusion coefficients, Fick's law is applicable. (6) one step, gas phase, irreversible chemical reaction (7) infinite chemical kinetics, flame sheet model (8) no surface reaction, gas emerging from surface is entirely fuel. (9) C,, (for the mixture), T w and Yo,e are constant. (10) ~ < 1 (ratio of buoyant to inertia force). V. Solution
A comparison of the energy equations (Eqs. (7), (8)) and the species equations (Eqs. (7') (8')), and the corresponding boundary conditions (Eqs. (31) and (31'), (33) and (33'), (37) and (37'), (39) and (39')), reveals that by employing Le = 1, Prandtl and Schmidt numbers are equal resulting that 0 and g are identical functions. Thus, it is sufficient to solve for any one of them. Moreover, the zeroth order momentum equation in the transverse direction, Eq. (4), subject to the boundary condition described by Eq. (34) yields a trivial solution for the nondimensional pressure, /30 = 0
(42)
namely, no pressure changes across the boundary layer in the case of pure convection flow. Since every nth order ordinary differential equation can be replaced by an equivalent system of n first order equations, we get a system of eleven equations as follows; 5 equations from the zeroth order solution (3 from the momentum equation and 2 from the energy (or species) equation and 6 equations from the first order solution (4 from the momentum equations and 2 from the energy (or species) equation). The equation of state is an algebraic equation and therefore does not alter the number of ordinary differential equations in our system. Examining the present boundary conditions we find that a boundary value problem with split boundary conditions is exhibited. In order to have an initial value problem with all boundary conditions specified at one point (wall) we need the following five auxiliary boundary conditions;
folw, .Oo,w, f'~',w, Oi,w, ~l,W instead of the five boundary conditions already obtained at the outer edge (see Eqs. (32), (33), (38), (39), (40)). The method of Smith et a133
was utilized for a search of these required boundary conditions at the wall which produce the solution satisfying the outer boundary conditions. The eleven ordinary differential equations subject to the appropriate boundary conditions at the wall were numerically integrated using Runge-Kutta method on the SUNY at Stony Brook IBM 360/370 computer system. The integration is started with the zeroth order solution by assigning initial values to f~'w and 00 w- These initial values are iterated t~ll the freestream conditions are met. After the zeroth order solution is obtained the integration is continued with both zeroth and first order equations by assigning initial values at the wall to the first order boundary conditions (the zeroth order solution is independent of the first order, and hence is not effected), and iterating these values till the functions at the outer edge merge smoothly into the freestream values. The numerical computations have been applied to a horizontal flat plate covered with fuel. The fuel is ethyl alcohol burning in air at the stoichiometrie ratio and at atmosphereic pressure. Four cases representing various external conditions such as wall temperature, freestream oxygen concentration and direction of the gravity field, were analyzed (the direction of the gravity field determines whether the flow is aiding or opposing). Each case was solved with three different heat ratio values (the conventional mass transfer number), B = 0.5, 1 and 2. For each heat ratio value three solutions corresponding to the relative importance of the buoyancy effects were obtained. First solution is for no buoyancy effects at all, ~ = 0. Second solution is for a particular perturbation quantity ~ describing the relative contribution of buoyancy to the basic forced convection flow. The third solution is for 26 which means that the gravitational effects are doubled. Thus, altogether 36 solutions have been obtained as listed in Table I. Flow conditions, the corresponding perturbation quantities, Reynolds numbers and Grashof numbers are given in Table II. The representative value that has been used throughout the work to describe the gravitational effects is, = 1.0 x 10 -a Before applying the computer program to the present work it was tested by letting it to reproduce the following three well-known published solutions: Emmons and Leigh 14 solution for Blasius function with blowing and suction: Sparrow and Minkowycz 15 solution
EFFECTS ON FLOWS OVER A FLAT PLATE
1563
TABLE I External flow conditions for the 36 various solutions obtained Perturbation quantity ~
0
0.5 x 10-a
1.0 x 10-a
External Conditions
B
B
B
Case 1
T w = 288~ Yo,e = 0.232 aiding flow
0.5 1.0 2.0
0.5 1.0 2.0
0.5 1.0 2.0
Case 2
T w = 576~ Yo.e 0.232 aiding flow
0.5 1.0 2.0
0.5 1.0 2.0
0.5 1.0 2.0
Case 3
T w = 288~ Yo,r = 0.500 aiding flow
0.5 1.0 2.0
0.5 1.0 2.0
0.5 1.0 2.0
Case 4
T w = 288~ Y0,~= 0.232 opposing flow
0.5 1.0 2.0
0.5 1.0 2.0
0.5 1.0 2.0
=
for b u o y a n c y effects on horizontal boundary layer; and C h e n and T o o n g v solution for laminar boundary layer w e d g e flow with diffusion flame. T h e first r e p r o d u c e d solution is c h e c k i n g the m o m e n t u m e q u a t i o n with blowing boundary condition at the wall. T h e second solution is testing if the present model is capable of describing the b u o y a n c y effects for c h e m i c a l l y inert flow by shutting off the combustion. T h e last solution is e x a m i n i n g if the current model describes the structure of a diffusion flame over a w e d g e w i t h o u t buoyancy effects by setting ~ = 0. All reproduced results were compared to the original p u b l i s h e d studies and showed excellent agreement. Thus, we positively c o n c l u d e that if the model successfully describes on
the one hand the b u o y a n c y effects without c o m b u s t i o n and on the other hand the diffusion flame w i t h o u t buoyancy effects it is also capable of describing the mutual interaction b e t w e e n both phenomena, namely the buoyancy effects on chemically reacting flows.
VI. Results and Discussion
V e l o c i t y P r o f i l e s (Fig. 2):
T h e velocity profiles are presented in Fig. 2 for aiding and o p p o s i n g flows u n d er the same external conditions. E x a m i n i n g tl~ese profiles we find that with no gravitational effects the velocity profile is similar to that
TABLE II Flow conditions for a horizontal flat plate, L[m]--length of the plate, u e [m/secJ--freestream velocity.
Ue[m]
2.0
3.0
2.5
Re x
Gr x
~ x
L [m]
10-a
10-6
10 a
Re x 10 -a
Gr x 10 -6
~x 10 a
Be x 10 -a
Gr x 10 -6
~x 10 a
0.03 0.04 0.05
3.82 5.09 6.37
1.07 2.54 4.97
1.19 1.37 1.53
4.78 6.37 7.96
1.07 2.54 4.97
0.68 0.79 0.88
5.73 7.64 9.55
1.07 2.54 4.97
0.43 0.50 0.56
1564
MATHEMATICAL MODELING 1.2
I0.0
1.0
/
/
AIDING 3W
~"
8.0
/
.8
ZERO OPPOSING _GRAVITY
i.-
/
n.- 6.0 23
-~L--FLOW
E)
/ ii /L zGR.0AVITY / ill
o
_1
w >
r~ hi
~; 4.0
hi I"-
OPPOSING
.4
/--:
/ .2 /
2.0
FLOW
/
/
/
/
1.0
//
0
/
FIG. 3. Temperature profiles, T w = 288~ = 0.232, B = 1.
0 ~
0 2.0 4.0 6.0 BOUNDARY LAYER THICKNESS,'/'/
F[c. 2. Velocity profiles, T w = 228~ = 0.232, B = 2.
|
210 4.0 6.0 0 BOUNDARY LAYER THICKNESS,'r]
Yo,e
Yo,e
obtained b y E m m o n s a n d Leigh t4 for Blasius function with b l o w i n g . Buoyaney effects accelerate a i d i n g flows a n d decelerate o p p o s i n g flows. This can b e p h y s i c a l l y explained b y the existence of a horizontal pressure gradient which is favorable for the former flow a n d adverse for the latter. It is also o b s e r v e d that b y increasing either the heat ratio, or the wall temperature or the freestream oxygen concentration the f l u i d in a i d i n g flows is more accelerated d u e to t h e increase in the favorable pressure gradient (see Fig. 5), a n d exceeds sometimes the freestream velocity. Such velocity overshoots were experimentally measured b y H i r a n o et al. x6av on horizontal flat plate with diffusion flame. W h e n c o m b u s t i o n occurs inside the b o u n d a r y layer the temperature near the flame increases, the d e n s i t y decreases a n d the flame is accelerated to a velocity greater than the freestream velocity due to the i n d u c e d (by buoyancy) pressure gradient in spite of viscous retardation. Because of the acceleration of the flow the velocity gradient at the wall becomes larger increasing the wall shear stress a n d hence the transition Reynolds number. For o p p o s i n g flows the b u o y a n c y forces retard the fluid w i t h i n the b o u n d a r y layer so that these act like an adverse pressure gradient causing a decrease in the velocity.
Temperature Profiles (Fig. 3): The temperature increases from the wall temperature to the flame temperature a n d then decreases to the a m b i e n t temperature. T h e m a x i m u m temperature occurs at the flame sheet due to the fact that infinite reaction rate is assumed. The m a x i m u m temperature increases with increasing, B, T w and Yo,e" Buoyance effects do not change the m a x i m u m temperature a n d the only effect is inereasing temperature gradients at the wall by m o v i n g the flame closer to the surface for a i d i n g flow. In case of o p p o s i n g flows the flame moves away from the surface resulting in a decrease in temperature gradients at the wall. It is also observed that the temperature gradients at the flame are d i s c o n t i n u o u s but this has been taken care of b y i n t r o d u c i n g Shvab-Zeldovich transformation.
Mass Fraction Profiles (Fig. 4): As a c o n s e q u e n c e of the flame sheet approximation the flame front is located at a p o i n t where the nmss fractions of both fuel a n d oxidizer are zero, a n d the mass fraction gradient is discontinuous. Fuel mass fraction Y~ is decreasing from its initial value at the wall YFW. ( Y v w d e p e n d s only on B and Yoe) to zero at th'e flame. Oxidizer mass fraction is decreasing from its freestream value Yo,e, as it is transported t o w a r d the flame, to zero at
EFFECTS ON FLOWS OVER A FLAT PLATE 2.0
1565
~o--_60.0 E
,\
w
_J
.~~ 1.5
>- -50.0
-
-
rr
z_0
S
g
A'D'NG FLOW
Tw *k Y0,e 288 0.500 576 0.232
288
Z
0.232
,AIDING FLOWS
F-
~ -40.0 m W T p-
'~ .5 <
YF
'\~'f/' YO O ~vvy 0 2.0 4.0 6.0 BOUNDARY LAYER THICKNESS, 77
FIG. 4. Mass fraction profiles, T w = 288~ = 0.232, B = i.
-30.0
0 n,"
TY
LIJ
-20.0
:E
W rr (..)
Yo, e
_z - I 0 . 0 W Z:) 03 CO ',' n~
the flame. It is observed that increasing heat ratio and decreasing freestream oxidizer concentration move the flame away from the surface. Buoyancy effects tend to move the flame back to the surface in a i d i n g flows. This t e n d e n c y is a u g m e n t e d in eases of high wall temperature and high freestream oxidizer concentration. In o p p o s i n g flows the t e n d e n c y is the reverse.
q0 I~~ZERO GRAVITY~
I
0 2.0 4.0 6.0 BOUNDARY LAYER THICKNESS,~7
FIG. 5. Pressure profiles, 13 = l.
creases the pressure effects within the boundary layer due to buoyancy.
Flame Height (Fig. 6): Pressure Profiles (Fig. 5): T h e pressure profiles in a i d i n g flow are shown in Fig. 5 for three different external conditions. The pressure is increasing across the b o u n d a r y layer from its initial value at the surface to the edge where it is merging smoothly with the freestream pressure. Rotem and Claassen, TM and Pera a n d Gebhart xv.2~ reported the same p h e n o m e n a for natural convection flows adjacent to horizontal and slightly inclined surfaces. T h e physical explanation for the pressure gradient across the b o u n d a r y layer with gravitational force is that the only force available to balance the b u o y a n t force in the transverse direction is a pressure force acting in an opposite direction to the b u o y a n t force. Therefore, in aiding flows, where the b u o y a n t force is acting in the positive direction of the y-axis, the pressure force is acting d o w n w a r d resulting in a positive pressnre gradient across the b o u n d a r y layer. For o p p o s i n g flows the behavior is the reverse and hence pressure is decreasing with increasing distance from the surface. It is also f o u n d that increasing either B, or Tw, or Y.... in-
The flame height is presented in Fig. 6, and it is observed that the flame sheet moves away from the surface as the heat ratio increases. The effect of gravity is to b r i n g the flame closer to the surface in a i d i n g flows, and to move it away from the surface in o p p o s i n g flow. T h e prediction of the present study that fl'ame " s t a n d o f f " distances are lower under gravitational effects is in good agreement with some experimental works. Krishnamurthy a n d Williams 2~ for a laminar c o m b u s t i o n of P M M A on a flat plate c o n c l u d e d that "measured values of flame standoff distances are lower than the value predicted b y classical theory b y more than a factor of 2." We may attribute these differences to the role that b u o y a n c y plays in the flow. We found that the differences of flame heights with and without b u o y a n c y can be of the order of two d e p e n d i n g on the flow conditions. Moreover, it was observed in experiments c o n d u c t e d in zero gravity either b y using a drop tower or test flights flying Keplerian parabola that "flames are approximately 50% longer and w i d e r in zero gravity (no buoyancy) than in normal gravity. ''3 Un-
1566
MATHEMATICAL
MODELING
:3.5 J
f
J
J J f f
30
g--
///~TY /// J I"''! / / / ~ ' ~ . I D I N G FLOW
~2.5 I.s.I 1"
:~2O W
/
_1 I.i..
1.5
,.%
I
I
.5
I
t
1.0 1.5 HEAT RATIO, B
Fie. 6. Flame height, T w = 288~
2.0
2.5
Yo.e = 0.232.
.5
.4
~.
AIDING FLOW
z0 ~ . 3 I--0 or la_
Z x-" o3
.2
\ \ \
0
I
0
.5
\ OPPOSING FLOW
I
I
I
1.0 1.5 HEAT RATIO, B
FIG. 7. Skin friction, T w = 288~
2.0
Yo, e = 0.232.
2.5
EFFECTS ON FLOWS OVER A FLAT PLATE fortunately, all these experiments were concerned with jet diffusion a,4 or free convection 5,~or droplet burning, 2 and not with forced convective flow over a flat plate. In any event the trend is the same.
Skin Friction (Fig. 7): The local skin friction is presented in Fig. 7. It is observed that b y increasing the heat ratio the shear stress at the surface is decreased. This can be explained b y recognizing that an increase in heat ratio corresponds to an increase in the b l o w i n g rate, which in turn, reduces the shear stress in the vicinity of the surface. 14 The gravitational effects tend to increase the shear stress in a i d i n g flows thus stabilizing the flows, and to reduce the shear stress in o p p o s i n g flows hastening separation. This is physically explained b y the favorable pressure gradient whieh accelerates the flow, causing steeper velocity gradients at the surface.
VII. Conclusions Since chemically reacting b o u n d a r y l a y e r flows are associated with high temperature and low density, steep temperature and density gradients prevail across the thin b o u n d a r y layer, a u g m e n t i n g the b u o y a n t force. This research has shown that the cross stream b u o y a n c y - i n d n e e d b o d y force acts effectively to p r o d u c e a streamwise pressure gradient in the fluid adjacent to the plate surface. The pressure gradient is favorable in a i d i n g flows and adverse in o p p o s i n g flows. Hence, the local b o u n d a r y layer flow is accelerated or decelerated relative to the c o r r e s p o n d i n g gravity-free forced convection flow. Correspondingly, there is an increase or decrease in the local skin friction and heat transfer rates, and consequently a decrease or increase in the flame sheet "standoff" distance, d e p e n d i n g u p o n whether aiding or o p p o s i n g flow exists, respectively. These theoretical predictions are in full agreement with observed experimental results. Based on the present study it is cone l u d e d that buoyancy plays an important role in b o u n d a r y layer diffusion flames, previous studies that deleted the b u o y a n t force, for reasons of simplification, result in underestimation of the combustion rate and the flame velocity. In addition, neglecting buoyancy yields erroneous values for such o t h e r important physical quantities as flame "standoff" distance, skin friction, heat transfer rates and b o u n d a r y layer thickness.
1567
Nomenclature a B b cf C1,
n u m b e r of moles of oxidizer heat ratio (mass transfer driving force) n u m b e r of moles of fuel local skin friction coefficient specific heat of t h e gas mixture at constant pressure f dimensionless stream function (7;r Grashof n u m b e r Gr m o d i f i e d Grashof n u m b e r g normalized fuel-oxidizer function h~ standard heat of formation per unit mass at T o L latent heat and all heat losses to the surroundings Le Lewis n u m b e r N total n u m b e r of chemical species present /3 nondimensional pressure increment across the b o u n d a r y layer Pr Prandtl n u m b e r Q heat released b y the cheinical reaction R universal gas constant Re Reynolds n u m b e r s similarity i n d e p e n d e n t variable in xdirection Sc Schmidt n u m b e r T temperature u velocity c o m p o n e n t in the x-direction v velocity c o m p o n e n t in the {/-direction W molecular weight x cartesian coordinate parallel to the surface Y mass fraction ~j cartesian coordinate normal to the surface c~i dimensionless mass fraction ~xr dimensionless temperature [3 r dimensionless temperature-oxidizer function [3 r dimensionless fuel-oxidizer functiofi "q n o n d i m e n s i o n a l b o u n d a r y layer coordinate 0 normalized temperature-oxidizer function tx absolute viscosity v kinematic viscosity v'i stoichiometric coefficient for species i a p p e a r i n g as a reactant v'i' stoichiometric coefficient for species i a p p e a r i n g as a product perturbation quantity, ratio of b u o y a n t force to inertia force p density 15 n o n d i m e n s i o n a l density
1568
MATHEMATICAL M O D E L I N G
Subscripts 0,1, e F f i 0 s x y W ~c
7. CHEN, T. N. AND TOOXG, T. Y.: Progress in
t h e order of t h e solution freestream fuel flame species oxidizer solid p h a s e in x-direction in y - d i r e c t i o n wall ambient conditions
Acknowledgment Portions of this research were supported under NASA grant NSG-3051.
REFERENCES 1. FENDELL, F. E. AND SMITH, E. B.: AIAA J. 15, 1984 (1967). 2. OKAJIMA,S. AND KlrMAGAI, S.: Fifteenth Symposium (International) on Combustion, p. 401, The Combustion Institute, 1975. 3. COCHRAN,T. H.: Experimental Investigation of Laminar Gas Jet Diffusion Flame in Zero Gravity, NASA TN D-6523, 1972. 4. EDELMAN, R. B., FORTUNE, O. F., WEILERSTEIN, G. COCHRAN, T. H. AND HAGGARD, J. B., J~.: Fourteenth Symposium (International) on Combnstion, p. 399, The Combustion Institute, 1973. 5. KIMZEY,J. H., Downs, W. R., ELDRED, C. H. AND NORMS, C. W.: Flammability in Zero Gravity Environment, NASA TR R-246, 1966. 6. DE RIS, J. ANDORLOFF, L.: Fifteenth Symposium (International) on Combustion, p, 175, The Combnstion Institute, 1975.
Astronautics and Aeronautics, Vol. 15, Heterogeneous Combnstion, p. 643, Academic Press, 1964. 8. WALDMAX,C. H.: Theoretical Studies of Diffusion Flame Structnres. Ph.D. Thesis, Princeton University, 1969; also available as AFOSR SR 69-0350 TR, 1969. 9. KmER, H. AND KERZNER, H.: AIAA J. 11, 1691
(1973). 10. KRISItNAMURTHY,L. AND WILLIAMS, F. A.: SIAM
J. Appl. Math. 20, 590 (1971). 11. LAVID, M.: Buoyancy Effects on Chemically Reacting Laminar Bonndary Layer Flows, Ph.D. thesis, State University of New York at Stony Brook, 1974. 12. WILLIAMS,F. A.: Combustion Theory, AddisonWesley, 1965. 13. SMITH, A. M. O. Ago CLUTTER,D. W.: AIAA J. 3, 639 (1965). 14. EMMOXS, H. W. AND LEIGH, D. C.: Tabulation of the Blasins Function with Blowing and Suction. Aeronautical Research Council Current Paper 157, 1954. 15. SPARROW,E. M. ANn MINKO~CZ, W. J.: Intern. J. Heat Mass Transfer 5, 505, (1962). 16. HmAxO, T., IWAI, K. AND KANNO,Y.: Astronautica Acta 17, 811 (1972). 17. HmANO,T. ANDK1NOSHITA,M.: Fifteenth Symposium (International) on Combustion, p. 379, The Combustion Institute, 1975. 18. ROTEM, Z. AND CLAASSEN,L.: J. Fluid Mech. 38, 173 (1969). 19. PERA, L. AND GEBHART, B.: Intern. J. Heat Mass Transfer 15, 269 (1972). 20. PE~A, L. AND GEBHART, B.: Inter. J. Heat Mass Transfer 16, 1131 (1973). 21. KRISHNAMURTHY,L. AND WILLIAMS, F. A.: Fourteenth Symposinm (International) on Combustion, p. 1151, The Combustion Institute, 1973.