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A synthesized windfield model for tornado missile transport
Nuclear Engineeringand Design 52 (1979) 135-144 North-HollandP u b l i ~ Company
A S~IZED
WINDFIELD MODEL FOR TORNADO MISSILE TRANSPORT
W.L. DUNN * and L.A. TWISDALE ** Carolina Power& Light Company, Raleigh, NC 27602, USA
Received 14 August 1978
A tornado windfleld model has been developed for use in a probabillstic assessment of the tornado miaile hazard to nuclear power plants. Tornado flow characteristicshave been identified which are significant in terms of mit~je_transport phenomena. In ordez to account for both modeling uncertainty and the mtural variabLtltyobserved among tornadoes, several random variables are specifiedin the model, in¢]Ddin~" tornado intensity, path width, translational speed, radius to maximum tangential velocity, ratio of radial-to-tangentialwind speeds, vertical variation of core size, and boundmy layer thickness. Comider~ the lack of agreement regardingdetailed tornado dynamics as well as the difficulty in establlshtn8a pr/oH conservativeflow characteristicsfor missile transport, the windfleid model was synthesized from theoretical, obser~tional, and probabilistic considerations.A slsniftcant aspect of the model is that the parameters can be adjusted to make the intensity, size and velocityvariables consistent with the tornado lmth width boundary specification.The modell~ considerations are discussed, the windfield model and caleulatlonalprocedure presented, sample wln~__f~_Idcomponent velocity profiles illustrated, and missile velocity statistics given for a simulation case study involvingseveralthousand m_Ite3ehistories. tex models have been suggested in the literature. The empirical models have relied heavily on the observation and analysis of the 1957 Dallas tornado by Hoecker [4,5]. Bates and Swanson [6] have utilized Hoecker's results, as have Paddelford [7], Iotti [8], Lee [9] and others. The major limitations of these models are the primary reliance on the analysis of a single tornado, as Morton [10] notes, and extrapola. tion of the data to higher windspeeds. The theoretical vortex models generally have assumed axisymme. tric flows in incompresm'ble fluids. Booth and Hobbs [11], Dergarabedian and FendeU [12], Simiu and Cortes [13] and others have utilized a Rankine vortex type model. In a tornado missile investigation, Bhattacharyya et al. [14], used a boundary layer model which was derived from the work of Wen [15] and some intermediate results of Kuo [16]. The model developed by Redmann etal. [17] utilized Carrier's work [18] and yielded a wind definition derived from theoretical considerations and observational data. Sun et al. [19] constructed a simplified engineering vortex model of which the Rankine and Hoecker vortices are special cases. Kun [16] developed a velocity distribution for a tornado windfield with an axisymmetric flow in the boundary layer of a maintained vortex. Shanahan [20] utilized the cyclostrophic
1. Introduction As a natural hazard experienced in several global regions, tornadoes pose a potential threat to the safety of nuclear power plants sited in these areas. Their physical characterization is thus an important requirement in the analysis and design of safety-related structures. Although cons/derable progress has been made recently in the theoretical and experimental model. ling of tornadoes, there is stiff disagreement regarding the nature of detailed tornado dynamics, for instance see Kessler [I], Lewellen [2],and Davies-Jones [3]. Fundamental difficulties which tend to impede rapid progress in this area include the extreme variability in observed tornado events, the complexity of contributing physical mechanisms, and the three.dimensional character of the flow. For the purposes of nuclear facility structural design, a tornado windfield descrip. t~on to a height of several hundred feet above the ground surface is needed. Numerous empirical and theoretical tornado vor* Present add~eas: Department of Nuclear En~eering, North CaroAna State Untverity, Raleigh, NC 27650, USA. ** Present address: Ope~tious Antlysis ~ a , Research Triangle Institute, Research T r i ~ e Park, NC 27709, USA. 135
136
W.L. Dunn and L.A. Twisdale / Windfield model for tornado missile transpom
equation, vertical momentum, and continuity considerations to develop a model for engineering design purposes. Numerical models have also been discussed, for instance by Davies-Jones and Vickers [21], by Chi [22] and by Schlichting [23]. None of these individual models explicitly describes both the possible variation of the windspeed components in a given tornado and the natural variation of windfield characteristics among tornadoes of different sizes and intensities. The progress that has been achieved in tornado modelling has naturally been aimed more at deducing basic flow characteristics (determinsticalty): than at assessing those flow features which may be important to the transport of missiles. The tornado windfield model developed herein departs from the conventional approaches in two important aspects: (1) it is a synthesized model which relies on probabilistic methodology to structure the fundamental and modelling uncertainties associated with a scientific description of the phenomena; and (2) it contains variable flow features which are significant in missile trajectory prediction and are consistent with tornado path geometry statistics. The major sources of uncertainty and important flow considerations include turbulent flow characteristics, suction vortex phenomena, meridional flow pattern, vertical variation of the radius of maximum tangential velocity, and the ratio of the radial to tangential velocity components. In the following, the input random variables are characterized and the windfield model is presented. A calculational procedure is described which permits solution of the windfield equations consistent with tornado path width constraints. The model behavior and variability are demonstrated in plots of the windfield velocity components, and statistical results of tornado missile transport simulations using the model are included.
2. Modelling considerations and random variable specification In terms of missile transport, a primary influence on the synthesis of a tornado wind model is that for given maximum windspeed, conservative flow characteristics cannot generally be established a priori. The potential for variations in the initial conditions and for diverse aerodynamic properties of different missile types inhibits the use of introspection and sensi-
tivity analyses to specify deterministically the tornado flow characteristics which maximize a set of missile transport variables. Because of the resulting limitations of any attempt to specify a "conservative" deterministic wind definition, a windfield model with probabifistic variables which specify tile basic flow characteristics is developed. In order to maintain model simplicity, the random effects associated with turbulence, suction vortices, and "disorganized" three dimensional flows are indirectly considered in the probabilistic missile transport model discussed by Twisdale, Dunn, and Davis [24]. The following discussion reviews the assumed tornado flow characteristics and identifes the random variables used in the wind model. Observational and theoretical intormation indicates that the general flow in a tornado vortex is characterized by a tangential velocity component which rises rapidly as the radial distance from the center of the tornado increases, reaches a maximum at the edge of the core, and beyond that decays approximately inversely with radius. Vertically, the tangential velocity increases rapidly from the ground surface of the top of the sub4ayer and more slowly from there to the top of the boundary layer. This increase witl~in the boundary layer may be monotonic or slightly oscillatory. Theoretical considerations suggest that the radial velocity is large and directed inward near the center of the tornado. This inflow increases rapidly with height to a maximum and then decreases, vanishing at the top of the boundary layer. Although some theoretical models indicate radial outflows at some elevations within the core, all exhibit inflows near the ground in the region of missile transport. Previous vertical velocity profiles support the concept of an updraft inside the core, but there is uncertainty regarding its detailed form and attenuation outside the core. For example, some modellers suggest a two-cell vortex in which the updraft encloses a central downdraft in the inner cell. In general, however, the vertical velocity windfield component is expected to decrease with radius (approaching zero outside the core) and increase with height to account for meridional flow continuity with the radial inflow. These flow features are illustrated in fig. 1 for a single cyclonic vortex. A wind field model incorporating the basic flow features discussed above is synthesized as the energy
I¢.L. Dunn and L.A. Twisdale/ ~¢indfleld model for tornado misstTetransport source in the missile transport model. Velocity component profiles of several deterministic models have been auessed to estimate enveloping characteristics of the flow parameters. The elements of uncertainty are structured into the model through probabilistic characterization of the input variables. These random variables are: the magnitude of the ratio factor of radial to tangential components (7), the radius (PmO)to the maximum tangential velocity at a specified height (Zo), the linear variation of this radius with height as specified by the slope (3'), the ground roughness as specified by the reference boundary layer thickness (8o), the reference rotational velocity (UrO,O), the translational speed (Ut), and the tornado width (IV). The sampling distributions for the variables Uro,o, W, and Ut can be developed from the available tornado data record (e.g., Twisdale et al. [25]). The height at which the refer-
137
ence intensity is specified, Zo, is taken as 20 feet above the ground plane. This is a repretentative mean value of the height of structures and trees associated with the damage descriptions of the tornado intensity classification system. The variable Uro,o is the assumed tornado rotational velocity at the edge of the core and at a height z 0, and is not necessarily the maximum rotational velocity in the tornado. The remaining random variables control the internal flow characteristics of the tornado and are characterized from available data sources. Values of 7 reported in the literature range from effectively zero (i.e., negligible radial velocity) to about 0.6. The tornadoes discussed by Simiu and Cortex [13] have values ranging from 0 to 0.54; the tornado considered by Redmann et al. [ 17 ] has a value 7 = 0,5 at z = 0. To provide for a representative range of radial velocities, 7 is taken as uniformly
I I i i I
ToP oF
',.---"
--
BOUNDARY._.~" LAYER ~...
"''',~
"~EDGE
OF CORE
"---I ..... 41~- " " II /
I
~}C~
iiiIll
MERIOIONAL FLOW STREAM LINE II
,p Fig. 1. Cl=~cterlsfic=of a =k~lecyclonicvortex.
13 8
W.L. Dunn and L.A. Twisdale / Windfield model fi)r tornado missile transpo,~i
distributed within the interval (0.1,0.6). This provides an expected value of 0.35 and also permits relative extremes corresponding to small and large inflows. The special case 3' = 0 is not considered to be critical since missiles would tend to be more quickly centrifuged from the cylindrical vortex, e.g., see Simiu and Cortes [13]. The radial distance to the maximum tangential windspeed near the ground surface, Pmo, has generally been considered to be in the range of 40 to 150 feet [4, 7,19, 26]. The tornado model given by Redmann et al. [17] has a 528-foot core and is a notable exception. It is noted that specification ofpm 0 cannot be made independently of the width and translational velocity of the tornado. For the case Ut = 0,Pmo must be less than I4//2, or the core would exceed the tornado wind boundary. The effect of translational velocity is a significant loss of windfield symmetry. Also, since tornado widths ranging over orders of magnitudes (e.g., from 168 feet to 16 368 feet in the FPP data system [29]) have occurred, a specification which covers this range is needed. Based upon the above geometrical considerations and the tangential velocity profile, the following scheme is utilized to specify Pmo. First, a value is sampled from a linearly decreasing frequency density function in the range {(,Om0)L , C o m o ) u } , where the bounds are defined by: j'O.1 W*x (,Omo)Z = rain qO0 ft s ,
(1)
(Pm0)U = min
(2)
If the sampled value, P m 0 , lS not compatible with the tornado width specification, W*, it is reduced in increments as required for a feasible model solution, as discussed in section 4. This scheme provides for a probabilistic characterization of #m0 consistent with suggested values and avoids the situation in which Pmo is so large that the windfield cannot assume the specified windspeed at the boundaries. Variation of the core radius with height has been observed by Hoecker [4] and is potentially significant in missile transport analysis, as noted by Costello [28] and Simiu and Cortes [ 12]. Values of the slope (S) of the radius to the maximum tangential velocity have been suggested in the range 0 to 0.45 [6,16,17,27 ]. Bates and Swanson [6] refer to a value for the Dallas tornado of S = 0.45 in the lowest 200 feet. Reference
to Hoecker's work [4] indicates that a value of 0.1 above 150 feet is appropriate. Several investigators have used a cylindrical model above 200 feet and a Hoecker type model below 200 teet. To include the potential for both cylindrical and conical type t o t nadoes S is assumed to be uniformly distributed from 0 to 0.3 and to apply to all heights within the boundary layer. There is limited guidance on tile proper choices for the boundary layer thicknesses 60 and a m = 6(Pmo ). Redmann et al. [17] use an effective boundary layer of 1000 feet; Kuo [16] shows a boundary layer which increases with radial distance out to 2.5 Pmo, then slowly decreases, indicating that 6 m ~< aoo The majority of the missile transport events are expected to occur at relatively low elevations (less than 200 feet) which will be considered to be entirely within the boundary layer. Considering variations in ground roughness and the effects of 6 o on the vertical velocity component, the value of 6rn has been selected as 400 feet and 60 is considered to be uniformly distributed between 400 and 500 feet.
3. Windfield model The radial and vertical coordinates in the tornado frame are represented by # and z, respectively. The tornado frame origin is located at the vortex center at zero elevation and translates at constant speed Ut, as sampled from the specified probability distribution. The radius to maximum tangential velocity is a random variable which can vary with height, as previously discussed, according to Pm = Pm0 + S*(z - z 0 ) ,
z ~ 6(p) ,
Pm =pm0 + S * [ 6 ( / 9 ) - z 0 ] , Z > 6 ( p ) ,
(3)
where S* is the sampled value of the slope for a given tornado, the point (Pm0, z0) specifies the position at which the reference rotational velocity, Uro,o applies, and the boundary layer thickness, 6, varies as 6(p) = 1(56m -- 26;) + 2p(6~ -- am)/3Pmo , 0 < p ~< 2.5 Pmo , 6(p) = 6 o exp[-O.Ol(p/pm
(4) 0 - 2.5)] , p > 2.5 P m o ,
Here ~ m = ~(,Omo) and 60 is the sampled value of the
W.L. Dunn and L.A. Twisdale/ Windfieldmodel for tornado missiletmnRTort maximum boundary layer thickness, which occurs at p = 2.5 Pmo. This form is utilized because of its relative simplicity (linear near the center, exponential far away) and because it exhibits the expected general behavior of minimum thickness at p = 0 and slow asymptotic variation. A typical boundary layer profde is given in section 5. Radial, tangential, vertical, and rotational velocity components are represented by Ur, Ue, U= and Ure, respectively, where
u,e = ( ~ + u~) ~r' .
(5)
The random variable 3'is taken to be the magnitude of the ratio of radial to tangential components at the reference position; thus, the sampled value relates Ur and Uo according to 3'* = IUr(Pmo, Zo)/Uo(Pmo, z0)[.
(6)
The models for tangential and radial velocity utilize the work of Kuo [16]. The tangential velocity model is synthesized as
Ue(p, z) = Uo [m(p)/p] G(p, z)
(7)
139
and
3"= 3'*G(Pmo, Zo)/F(P~,o, Zo).
(13)
It is noted that the magnitudes of the radial and tangential components increase rapidly from zero at p = 0 to a maximum at Prn and vary as 1/p for larger p. The parameter a determines the asymptotic behavior: for a < 0 the vortex vanishes at the finite dis. tance, Po = -b/a; for a = 0 the vortex approaches zero asymptotically, and for a > 0 the vortex approaches a f'mite positive value. The parameter ~', representing an effective sublayer thickness, determines the values of Us and Ur at the "ground" (z = 0). The functions G and F specify the vertical variations, Ue generally increasing with height up to the top of the boundary layer, Ur generally decreasing. The parameter a determines the rate of vertical variation. Values of a = 10 and [" = 20 feet have been used in the results reported later in the paper. The vertical velocity model is generated using the continuity condition applied to an imaginary cylinder of radius • and height z, i.e.,
where U o is a normalizirtg variable,
- Oz
u=(p, z) = a
1
~p Ur(P, z) + r Uv(P, z).
(14)
re(p) =Pm [ 1 -- exp [-- 1.25643Colp m)2 ] , 0 < p
pm~p
re(p)= 0 ,
P > Po,
Po = -b/a,
a < 0,
po "*°°
a ~ O,
,
G6~, z) = I - exp {-~
The fo~m, obtained by integration,is
(s)
u=(o, z)= Uo3' ~ ( p ) , nLo, ~ z) + m(p)
,(15)
where
(9)
H(p, z) = 6a~--~)e-a~lS(P) { l + 6a~)
z < ~6o),
c,~, z) = 1 - exp {--a[~(p) + f]l~(p) ), z> ~(p),
(1o) and the parameters a and b are determined for a specific tornado to match path width (W'), as discussed in section 4. The radial velocity is given by
Although no additional random variables have been utilized in the specification of Uz, variations in Pmo, 3', ~'o, and'S provide for significant variations in the vertical velocity profdes.
t
u,(p, z) = Uo'r [m(p)/p] e'~, z), where
4. Calculation pcocedurc
~%o, z) = (z + f)/~(p)exp[-a(z + ~/6(p)], z<8(p),
Using the windfield model, a ~ c i f t c tornado is defined by the parameters ~ , o , U~, W',Pmo, S', 7", and 6~. The dependent varisbles 7' and Uo are then calculated, the former by eq. (13) and the ktter according
~Tp, z)=O
z > ~(p), (12)
140
W.L. Dunn and L.A. Twisdale / Windfield model for tornado missile transport
to
along an axis defined by
U 0 = Uro,o/moG(Pmo, Zo)(1 + 72) I/2 ,
(l 7)
where m o is the constant mo = m(Pm0)/Pmo = 1 - e x p ( - 1.25643) = 0.71533. (18) Determination of the parameters a and b in eqs. (8) and (9) which provide model compatibility with the tornado intensity and width variables follows from velocity vector geometry considerations. Tornado path width (W*) is defined as the distance between constant horizontal velocities (Uh = [U2to + U2 ] 1/2) corresponding to gale intensity winds (vb = 73 mph) at elevation z o such that Uh ~< Ub outside the tornado path. This wind boundary definition follows from the FPP tornado classification system [29] which utilizes damage caused by near-ground windspeeds to estimate W for each tornado. From fig. 2, it is noted that the effect of translational velocity is a loss of windfield symmetry such that the distance from the stationary vortex center (point O) to the left boundary (Rt) is always less than the distance to the right boundary (Rr). It can be shown that the new position of zero horizontal velocity (O') is always shifted to the second quadrant for cyclonic flow and
R~ - - - w ~ , . . ~
~= tan-lfUr(Pr,Zo)/Uo(Or, Zo)r ~ tan--l~/.
(19)
For given distance p from the stationary vortex center, we note that on the left side the maximum Uh always occurs at point d whereas the minimum value of Uh always occurs at point b. For p > Pro, the line defined by the points c and O provides an accurate and simple reference axis to determine the left boundary Uh = Ub. Thus, R t is related to vb by {[v0(R~, z o ) - vt] 2 + u~(&, Zo)) '/2 = vb = J07 fps. (20) On the right-hand side, the maximum Uh always occurs at point e and the minimum always occurs at point a for given P and constant % The net horizontal windspeed at point e is larger than that at point f since the rotational and translational velocity vectors are parallel there. Thus, as shown in fig. 2, we treat the boundary as occurring at a radial distance Or from the tornado origin, where Rr = Or cos ~,
(21)
and the right-hand boundary condition is Ur0 (,Or, Zo) + Ut = Ob •
(22)
For 7 = 0 we note that R r = Pr ; also, depending on the variation of 7 with p, it is possible for the net horizontal velocity at some point between e and f to exceed that at R = Rr. However, an involved treatment is required to determine the exact point, and the approximation is good to place the right boundary at R = Rr. We then need only the identities
Rr -------~.
lw
W* = R t +Rr
(23)
and O~
LEFT BOUNDARY
f
I CENTER LINE
Uh=vb
RIGHT BOUNDARY
[ Uh=v b
Fig. 2. Tornado windspeed specification at path width boundary.
m(Pmo)/Pmo = m o = a + b/Pmo ,
(24)
with eq. (18) through eq. (22) to find the appropriate a and b parameters. This is accomplished by an iteratire procedure beginning with an assumed minimum value o f R t = Pmo and incrementing outward until the set of equations is satisfied. This procedure "fits" the model to the given tornado path width, forcing the windfield to assume the proper magnitude at the boundaries. For some low-intensity, high-translational velocity tornadoes, the net horizintal velocity may
W.L. Dunn and L.A. Twlsdale / g~ndfleld model for tormdo mm~ie transport ue
not reach the boundary velocity at the left edge of the core. In these cases, we take simply R~ = 0 and t h m R , = W'.
z
ool
//
\ \
• =
Some typical results are given in order to demonstrate some of the characteristics of the windfield model and its utilization in tornado missile simulations. Radial, tangential, vertical and rotational velocities are plotted in figs. 3 - 6 for the case Ur0,o = 406.27 fps, Ut = 7.33 fps, and ~o = 500 ft., with the remaining perameter values noted on the figures. The radial velocities given in fig. 3 demonstrate the variation in magnitude obtained for the stated conditions due to variation in the parameter 7. The effects of ground roughness and the extent of ground interaction are accounted for by allowing variation of the parameters 7 and 6 o. It is noted that the vertical velocity model is also directly proportional to 7, so that significant variation of both radial and vertical ue
~oo =
f
~oo
/l
":
5. Results
300
141
_ oo
5
200 I00
e (z)
o
I 100
160 rt ,
=
p
ioo
2o0
3oo
~oo
5oo
60o
~)
o
ioo
~oo
3oo
~oo
5oo
~t'ps )u~
F ~ 4. Romtion~ wlodty mad bou~=~tryt,ye~thlckneu.
magnitudes is possible, simulating gross ground interaction effects. The boundary layer thickneu is also affected by ground condRions. Smaller localboundary layer thicknesses genzrafiy remit in smaller verticalvelocities, but the effect is not large. Figs. 3 and fig. 4 indicate that radial, tangential (and hence rotational) velocities are nonzero at z = 0 and that rotational velocity is nearly constant with height above about 100 ft. The radial distances to maximum tangential velocity in fig. 3 differ for different heights as a consequence of the use of S = 0.I76 in the model. The two cases of infinite(figs.3 and 4) and finite (fig. 5) radius to zero windfield are demonstrated. Note that the vertical velocity in fig. 3 remains positive, though small, outside the core, whereas in fig. 5 it becomes negative for large p. This
lOO
o
i0o
2OO I
300 !
-ioo
~00 !
5OO 4
60o 0
7OO I
~f ' P S ~ A
z=O y=0.6
~ t , Pmv =
Ur Uz (~s) 200 loo
~
0o ft
md~l ~ .
ers
160 ft
S
~ 0.176
W
= 1668 ft
•
= .o9267
b
~ 99.6256
Y
~ c.~5
I1\
i
\
°0
o
I
Fig. 3. Tae~nfial, ~
Fsa'Ieters Omo = 30 tt
-2oo
(t~,)
Velocity
"p (et)
I
~100
s.
.: 165 t't
"
'
(~1
% = 3hO f t
and vertical velodt~sas functiom of Fig. 5. Veloc/ty profiles fat a ~
tornado of flMte extent.
142
W.L. Dunn and L.A. Twisdale / Windfield model for tornado missile transport z
Omo = 160 f% S = 0.176, V = O.L5 W
400
1668 ft
a
.09267
b
99.6256
3 ~ 30C
]
i00
Ur
(fps)
-200
-I00
200
300
400
500
(fgs)
1;0
i00
200
300
(fps )
Uz
Fig. 6. Tangential, radial and vertical velocities as functions of height for 0 = 160 ft. is determined by the values of a and b which result from the boundary fitting process (the parameter "a" is positive in one case and negative in the other), since d m / d p = a for large p. The boundary layer thickness is also plotted in fig. 4.
Typical vertical variations of the three wind components are shown in fig. 6. It is noted that U0 is not monotonically increasing with height for constant p, as a consequence of the fact that this specific tornado is non-cyclindrical (S = 0.176); m fact, Uo does monotonically increase with height for constant P / P m , All three components are constant above the boundary layer. For a narrow, high intensity tornado, as shown in fig. 6, the vertical velocity is quite high inside the small core since all the radial inflow becomes a concentrated vertical updraft distributed over a small area. Velocity characteristics of the six standard missiles postulated by the NRC [30] have been assessed using this windfield model and tile probabilistic missile transport model given by Twisdale, Dunn, and Davis [24]. Twenty tornadoes were simulated using the random variable probability density functions described herein. The tornado intensity, translational velocity, and path width variables were characterized
Table 1 Missile velocity parameter estimators Missile
NRC region
No. of observations Vm
Vi
Impact o, Vi (if/s) o
#
o
~
o
Postulated design velocity [30] (ft/s)
Max. o, V m (ft/s)
Equiv. impact o, V~ (ft/s)
Wood beam
I II III
1661 1642 1640
199 210 254
73.40 83.75 75.91
33.91 37.55 33.02
96.07 104.32 100.84
58.74 59.33 36.47
53.34 65.52 53.99
41.23 54.13 28.32
272.6 229.9 190.5
6" Pipe
I II III
1423 1173 1014
160 106 75
91.93 83.90 74.14
30.84 30.41 20.07
97.87 68.38 67.58
38.47 29.60 18.89
49.75 33.90 28.31
28.95 21.95 14.76
170.8 137.9 32.8
1" Steel rod
I II III
1425 1257 1173
190 118 101
95.85 76.02 70.17
34.35 29.34 20.00
98.83 78.82 64.88
47.75 30.95 33.32
54.07 36.81 25.92
37.42 26.55 18.00
167.5 131.4 26.3
Utility pole
I II III
1441 1315 1181
159 121 93
95.32 80.37 72.75
37.55 30.07 18.71
110.46 76.67 62.16
51.87 28.12 29.98
51.74 37.72 35.28
38.73 25.26 20.64
180.6 157.6 85.4
12" Pipe
I
II III
1217 894 673
108 58 27
86.86 71.00 67.14
31.26 19.36 24.27
92.57 57.86 66.67
34.93 20.17 25.40
45.95 22.95 39.53
23.24 15.87 12.57
154.3 91.9 23.0
I II III
1274 1033 799
71 32 27
54.25 46.36 38.07
19.95 15.46 14.83
56.66 54.11 40.27
33.17 18.76 11.31
* * *
Vehicle
o = velocity.
* * *
193.8 170.8 134.6
If.L. Dunn and L.A. Twisdale / P/indfleld model for tornado missile transport
from available data records [25]. A total of 10 000 missile histories (500 per tornado) were simulated for the maximum F'-scale intensity range [25] in each of the three NRC tornado regions [25]. A hypothetical layout of six safety-related structures (targets) was postulated with the missiles uniformly distributed within a rectangular zone centered on the plant. The missile initial elevations were uniformly distributed between five and fifty feet above the ground, except the vehicle missile which was injected between five and ten feet. The maximum velocity attained by each missile during its trajectory was determined; for those missiles striking one of the targets, impact velocity and effective normal-collinear impact veocity were calculated. The means and standard deviations of these quantities are given in table 1 ; also listed is the reference velocity postulated by the NRC [30] for a normal-collinear impact of each missile type in each region. These results suggest that the currently acceptable design velocities are very conservative when compared to an actual plant case study using the windfield model described herein. 6. Discussion A probabilistic tornado windfield model has been synthesized which includes the variable flow characteristics that appear to be important in missile transport analysis. The inherent random behavior of tornadoes and the fundamental modelling uncertainty resulting from incomplete knowledge of tornado dynamics have been treated through the characterization of several windfield parameters as random variables. The approach assumes that a prior/reasoning cannot be utilized exclusively to specify conservative flow characteristics for a variety of missile shapes and initial conditions. By introducing random variables to account for the major sources of uncertainty and significant flow variations, and synthesizing functions consistent with theoretical considerations and observational information, a model was developed which potentially envelopes the expected variety of tornado flow characteristics. The resulting model exhibits the following significant characteristics: (a) The asymptotic behavior of windfield components with radial distance can be adjusted to match the right and left boundaries to the tornado intensity, path width, and translational velocity variables,
143
as specified from their respective probability density functions. Positions for tornado path width wind boundary velocity comparisons have been identified for a translating vortex and a calculational procedure developed to insure tornado wind field compatibility with path width constraints. @) The shape of the envelope of the core region can by cylindrical or conical, with the radius to maximum tangential windspeed, Pro, a random variable dependent on tornado path width. (c) The vertical updraft within the core satisfies continuity with the radial inflow and depends on the magnitude of the radial velocity, the core size, and the boundary layer thickness. This variability allows the possibility of large updrafts, particularly for narrow tornadoes. Tornado missile ILmulations indi. care that such updrafts for high-intensity tornadoes may be capable of lifting certain missiles to altitudes of several hundred feet. (d) The model applies to the observed range of intensity, width, and translational speeds and the dependent tornado parameters are evaluated consistent with these specified tornado characteristics. Also, the vortex windfield components behave reasonably at limiting values o f p and z. At p = 0, radial and tangential velocities vanish but a vertical updraft exists; for large p, all velocities approach zero or some small finite value; above the boundary layer the windfield is a Rankine-like vortex; and at z = 0, radial and tangential velocities are conservatively taken as non-zero (in effect considering the phne z = 0 as the top of a sublayer) whereas vertical velocity is zero. Utilizing this model, wind velocity components have been presented, as functions of radial and vertical distances, for selected values of the input param. eters. In addition, missile velocity statistics were generated by sampling from a variety of high-inten. sity tornadoes and simulating missile trajectories. Average impact velocities are generally lower than those postulated by the NRC (see table 1) and the mean effective normal collinear impact velocities are significantly below NRC postulated values. In addition, the model has been used in an extensive simulation study of missile transport and damage probability assessment, as described by Twisdale, Dunn and Chu [25], for which existing deterministic models were not well suited.
144
W.L. Dunn and L.A. Twisdale
Windfield model for tornado missile transpor~
Acknowledgements The assistance of Dr. J. Hsu in reviewing the existing tornado models and suggesting Kuo's formulation as a starting point for this analysis is appreciated. This work was sponsored by the Electric Power Research Institute as Research Project 616-1, with Dr. B.B. Chu serving as project manager.
References [1] E. Kessler, Proc. Symp. on Tornadoes, Texas Tech. Univ. June (1976) p. 431. [2] W.S. LeweUen, Proc. Symp. on Tornadoes, Texas Tech. Univ., June 1976 p. 107. [3] R.P. Davies-Jones, Proc. Symp. on Tornadoes, Texas Tech. Univ., June, 1976 p. 151. [4] W.H. Hoecker, Jr., Monthly Weather Rev. 88 (1960) 167. [5 ] W.H. Hoecker, Jr., Monthly Weather Rev. 89(12) (1961) 533. [6] F.C. Bates and A.E. Swanson, Res. Rep. Black & Veach, (1967). [7] D.F. Paddleford, Westinghouse Elec. Corp., WCAP-7897, April 1969. [8] R.C. Iotti, Velocities of Tornado Generated Missiles, Ebasco Sere., ETR-1003, February 1975. [9] A.J.H. Lee, ASCE Speciality Conf. on Structural Design of Nuclear Plant Facilities, Chicago, IL, December 17-18, 1973. [10] B.R. Morton, Geophysical Vortices, Progress in Aeronautical Sciences, Vol. 7, (Pergamon Press, Oxford, 1966). [ 11 ] D.R. Beeth and S.H. Hobbs, Topical Rept. B&R-001, Brown & Root, Houston, TX (May 1975). [12] P. Dergarabedian and F. FendeU, J. Atmospheric Sei. XXI, (1973) 26. [13] Emil Simiu and M. Cortes, Tornado Borne Missile Speeds, Nat. Bur. Stds., NBSIR 76-10-50, April 1976.
[ 14] A.K. Bhattacharyya, R.C. Boritz, and P.K. Niyogi, 2nd ASCE Specialty Conf. on Structural Design of Nuclear Power Plant Facilities, New Orleans, LA, December 8 - 1 0 1975 p. 562. [151 Y.K. Wen, J. Struct. Div., Proc. ASCE, Vol. 101, (1975) 169. [16] H.L. Kuo, Axisymmetric Flows in the Boundary Layer of a Maintained Vortex, J. Atmospheric Sci., 23, (1971) 20. [ 17 ] G.H. Redmann et al., EPRI 308, Te chn. Rep. 1 (February 1976). [18J G.F. Carrier, J. Fluid Mech., 49 (1971) 133. [19] G.N. Sun, E.G. Burdestle, and R.O. Barnett, Nucl. Eng. Des. 44 (1977) 407. [20] J.A. Shanahan, Proc. Syrup. on Tornadoes, Texas Tech. Univ., June 1976 p. 251. [21] R.P. Davies-Jones and G.T. Vickers, NOAA Techn. Mem. ERL NSSL-57, November 1971. [221 S.W. Chi, Tellus, XXVI, (1974) 444. [23] H. Schlichting, Boundary Layer Theory (McGraw-Hill, New York, 1965). [241 LA. Twisdale, W.L. Dunn, and T.L. Davis, Nucl. Eng. Des. 51 (1978) 295. [25 ] L.A. Twisdale, W.L. Dunn, and J. Chu et al., EPRI NP-768 and NP-769, Elec. Power Res. Insti., Palo Alto, CA, May 1978. [26] Nuclear Regulatory Commission, Design Basis Tornado for Nuclear Power Plants, Regulatory Guide 1.76, April 1974. [27] J.R. Eagleman, V.U. Muinhead, and N. Willems, Thunderstorms, Tornadoes, and Building Damage (D.C. Heath, 1975). [28] J.F. Costello, Proc. on Tornadoes, Lubbock, TX June, 1976, p. 349. [29] T.T. Fujita and A.D. Pearson, 8th Conf. on Severe Local Storms, October 1973. [30] Nuclear Regulatory Commission, Section 3.5.1.4, Standard Rev. Plan, June 1975.
A S~IZED
WINDFIELD MODEL FOR TORNADO MISSILE TRANSPORT
W.L. DUNN * and L.A. TWISDALE ** Carolina Power& Light Company, Raleigh, NC 27602, USA
Received 14 August 1978
A tornado windfleld model has been developed for use in a probabillstic assessment of the tornado miaile hazard to nuclear power plants. Tornado flow characteristicshave been identified which are significant in terms of mit~je_transport phenomena. In ordez to account for both modeling uncertainty and the mtural variabLtltyobserved among tornadoes, several random variables are specifiedin the model, in¢]Ddin~" tornado intensity, path width, translational speed, radius to maximum tangential velocity, ratio of radial-to-tangentialwind speeds, vertical variation of core size, and boundmy layer thickness. Comider~ the lack of agreement regardingdetailed tornado dynamics as well as the difficulty in establlshtn8a pr/oH conservativeflow characteristicsfor missile transport, the windfleid model was synthesized from theoretical, obser~tional, and probabilistic considerations.A slsniftcant aspect of the model is that the parameters can be adjusted to make the intensity, size and velocityvariables consistent with the tornado lmth width boundary specification.The modell~ considerations are discussed, the windfield model and caleulatlonalprocedure presented, sample wln~__f~_Idcomponent velocity profiles illustrated, and missile velocity statistics given for a simulation case study involvingseveralthousand m_Ite3ehistories. tex models have been suggested in the literature. The empirical models have relied heavily on the observation and analysis of the 1957 Dallas tornado by Hoecker [4,5]. Bates and Swanson [6] have utilized Hoecker's results, as have Paddelford [7], Iotti [8], Lee [9] and others. The major limitations of these models are the primary reliance on the analysis of a single tornado, as Morton [10] notes, and extrapola. tion of the data to higher windspeeds. The theoretical vortex models generally have assumed axisymme. tric flows in incompresm'ble fluids. Booth and Hobbs [11], Dergarabedian and FendeU [12], Simiu and Cortes [13] and others have utilized a Rankine vortex type model. In a tornado missile investigation, Bhattacharyya et al. [14], used a boundary layer model which was derived from the work of Wen [15] and some intermediate results of Kuo [16]. The model developed by Redmann etal. [17] utilized Carrier's work [18] and yielded a wind definition derived from theoretical considerations and observational data. Sun et al. [19] constructed a simplified engineering vortex model of which the Rankine and Hoecker vortices are special cases. Kun [16] developed a velocity distribution for a tornado windfield with an axisymmetric flow in the boundary layer of a maintained vortex. Shanahan [20] utilized the cyclostrophic
1. Introduction As a natural hazard experienced in several global regions, tornadoes pose a potential threat to the safety of nuclear power plants sited in these areas. Their physical characterization is thus an important requirement in the analysis and design of safety-related structures. Although cons/derable progress has been made recently in the theoretical and experimental model. ling of tornadoes, there is stiff disagreement regarding the nature of detailed tornado dynamics, for instance see Kessler [I], Lewellen [2],and Davies-Jones [3]. Fundamental difficulties which tend to impede rapid progress in this area include the extreme variability in observed tornado events, the complexity of contributing physical mechanisms, and the three.dimensional character of the flow. For the purposes of nuclear facility structural design, a tornado windfield descrip. t~on to a height of several hundred feet above the ground surface is needed. Numerous empirical and theoretical tornado vor* Present add~eas: Department of Nuclear En~eering, North CaroAna State Untverity, Raleigh, NC 27650, USA. ** Present address: Ope~tious Antlysis ~ a , Research Triangle Institute, Research T r i ~ e Park, NC 27709, USA. 135
136
W.L. Dunn and L.A. Twisdale / Windfield model for tornado missile transpom
equation, vertical momentum, and continuity considerations to develop a model for engineering design purposes. Numerical models have also been discussed, for instance by Davies-Jones and Vickers [21], by Chi [22] and by Schlichting [23]. None of these individual models explicitly describes both the possible variation of the windspeed components in a given tornado and the natural variation of windfield characteristics among tornadoes of different sizes and intensities. The progress that has been achieved in tornado modelling has naturally been aimed more at deducing basic flow characteristics (determinsticalty): than at assessing those flow features which may be important to the transport of missiles. The tornado windfield model developed herein departs from the conventional approaches in two important aspects: (1) it is a synthesized model which relies on probabilistic methodology to structure the fundamental and modelling uncertainties associated with a scientific description of the phenomena; and (2) it contains variable flow features which are significant in missile trajectory prediction and are consistent with tornado path geometry statistics. The major sources of uncertainty and important flow considerations include turbulent flow characteristics, suction vortex phenomena, meridional flow pattern, vertical variation of the radius of maximum tangential velocity, and the ratio of the radial to tangential velocity components. In the following, the input random variables are characterized and the windfield model is presented. A calculational procedure is described which permits solution of the windfield equations consistent with tornado path width constraints. The model behavior and variability are demonstrated in plots of the windfield velocity components, and statistical results of tornado missile transport simulations using the model are included.
2. Modelling considerations and random variable specification In terms of missile transport, a primary influence on the synthesis of a tornado wind model is that for given maximum windspeed, conservative flow characteristics cannot generally be established a priori. The potential for variations in the initial conditions and for diverse aerodynamic properties of different missile types inhibits the use of introspection and sensi-
tivity analyses to specify deterministically the tornado flow characteristics which maximize a set of missile transport variables. Because of the resulting limitations of any attempt to specify a "conservative" deterministic wind definition, a windfield model with probabifistic variables which specify tile basic flow characteristics is developed. In order to maintain model simplicity, the random effects associated with turbulence, suction vortices, and "disorganized" three dimensional flows are indirectly considered in the probabilistic missile transport model discussed by Twisdale, Dunn, and Davis [24]. The following discussion reviews the assumed tornado flow characteristics and identifes the random variables used in the wind model. Observational and theoretical intormation indicates that the general flow in a tornado vortex is characterized by a tangential velocity component which rises rapidly as the radial distance from the center of the tornado increases, reaches a maximum at the edge of the core, and beyond that decays approximately inversely with radius. Vertically, the tangential velocity increases rapidly from the ground surface of the top of the sub4ayer and more slowly from there to the top of the boundary layer. This increase witl~in the boundary layer may be monotonic or slightly oscillatory. Theoretical considerations suggest that the radial velocity is large and directed inward near the center of the tornado. This inflow increases rapidly with height to a maximum and then decreases, vanishing at the top of the boundary layer. Although some theoretical models indicate radial outflows at some elevations within the core, all exhibit inflows near the ground in the region of missile transport. Previous vertical velocity profiles support the concept of an updraft inside the core, but there is uncertainty regarding its detailed form and attenuation outside the core. For example, some modellers suggest a two-cell vortex in which the updraft encloses a central downdraft in the inner cell. In general, however, the vertical velocity windfield component is expected to decrease with radius (approaching zero outside the core) and increase with height to account for meridional flow continuity with the radial inflow. These flow features are illustrated in fig. 1 for a single cyclonic vortex. A wind field model incorporating the basic flow features discussed above is synthesized as the energy
I¢.L. Dunn and L.A. Twisdale/ ~¢indfleld model for tornado misstTetransport source in the missile transport model. Velocity component profiles of several deterministic models have been auessed to estimate enveloping characteristics of the flow parameters. The elements of uncertainty are structured into the model through probabilistic characterization of the input variables. These random variables are: the magnitude of the ratio factor of radial to tangential components (7), the radius (PmO)to the maximum tangential velocity at a specified height (Zo), the linear variation of this radius with height as specified by the slope (3'), the ground roughness as specified by the reference boundary layer thickness (8o), the reference rotational velocity (UrO,O), the translational speed (Ut), and the tornado width (IV). The sampling distributions for the variables Uro,o, W, and Ut can be developed from the available tornado data record (e.g., Twisdale et al. [25]). The height at which the refer-
137
ence intensity is specified, Zo, is taken as 20 feet above the ground plane. This is a repretentative mean value of the height of structures and trees associated with the damage descriptions of the tornado intensity classification system. The variable Uro,o is the assumed tornado rotational velocity at the edge of the core and at a height z 0, and is not necessarily the maximum rotational velocity in the tornado. The remaining random variables control the internal flow characteristics of the tornado and are characterized from available data sources. Values of 7 reported in the literature range from effectively zero (i.e., negligible radial velocity) to about 0.6. The tornadoes discussed by Simiu and Cortex [13] have values ranging from 0 to 0.54; the tornado considered by Redmann et al. [ 17 ] has a value 7 = 0,5 at z = 0. To provide for a representative range of radial velocities, 7 is taken as uniformly
I I i i I
ToP oF
',.---"
--
BOUNDARY._.~" LAYER ~...
"''',~
"~EDGE
OF CORE
"---I ..... 41~- " " II /
I
~}C~
iiiIll
MERIOIONAL FLOW STREAM LINE II
,p Fig. 1. Cl=~cterlsfic=of a =k~lecyclonicvortex.
13 8
W.L. Dunn and L.A. Twisdale / Windfield model fi)r tornado missile transpo,~i
distributed within the interval (0.1,0.6). This provides an expected value of 0.35 and also permits relative extremes corresponding to small and large inflows. The special case 3' = 0 is not considered to be critical since missiles would tend to be more quickly centrifuged from the cylindrical vortex, e.g., see Simiu and Cortes [13]. The radial distance to the maximum tangential windspeed near the ground surface, Pmo, has generally been considered to be in the range of 40 to 150 feet [4, 7,19, 26]. The tornado model given by Redmann et al. [17] has a 528-foot core and is a notable exception. It is noted that specification ofpm 0 cannot be made independently of the width and translational velocity of the tornado. For the case Ut = 0,Pmo must be less than I4//2, or the core would exceed the tornado wind boundary. The effect of translational velocity is a significant loss of windfield symmetry. Also, since tornado widths ranging over orders of magnitudes (e.g., from 168 feet to 16 368 feet in the FPP data system [29]) have occurred, a specification which covers this range is needed. Based upon the above geometrical considerations and the tangential velocity profile, the following scheme is utilized to specify Pmo. First, a value is sampled from a linearly decreasing frequency density function in the range {(,Om0)L , C o m o ) u } , where the bounds are defined by: j'O.1 W*x (,Omo)Z = rain qO0 ft s ,
(1)
(Pm0)U = min
(2)
If the sampled value, P m 0 , lS not compatible with the tornado width specification, W*, it is reduced in increments as required for a feasible model solution, as discussed in section 4. This scheme provides for a probabilistic characterization of #m0 consistent with suggested values and avoids the situation in which Pmo is so large that the windfield cannot assume the specified windspeed at the boundaries. Variation of the core radius with height has been observed by Hoecker [4] and is potentially significant in missile transport analysis, as noted by Costello [28] and Simiu and Cortes [ 12]. Values of the slope (S) of the radius to the maximum tangential velocity have been suggested in the range 0 to 0.45 [6,16,17,27 ]. Bates and Swanson [6] refer to a value for the Dallas tornado of S = 0.45 in the lowest 200 feet. Reference
to Hoecker's work [4] indicates that a value of 0.1 above 150 feet is appropriate. Several investigators have used a cylindrical model above 200 feet and a Hoecker type model below 200 teet. To include the potential for both cylindrical and conical type t o t nadoes S is assumed to be uniformly distributed from 0 to 0.3 and to apply to all heights within the boundary layer. There is limited guidance on tile proper choices for the boundary layer thicknesses 60 and a m = 6(Pmo ). Redmann et al. [17] use an effective boundary layer of 1000 feet; Kuo [16] shows a boundary layer which increases with radial distance out to 2.5 Pmo, then slowly decreases, indicating that 6 m ~< aoo The majority of the missile transport events are expected to occur at relatively low elevations (less than 200 feet) which will be considered to be entirely within the boundary layer. Considering variations in ground roughness and the effects of 6 o on the vertical velocity component, the value of 6rn has been selected as 400 feet and 60 is considered to be uniformly distributed between 400 and 500 feet.
3. Windfield model The radial and vertical coordinates in the tornado frame are represented by # and z, respectively. The tornado frame origin is located at the vortex center at zero elevation and translates at constant speed Ut, as sampled from the specified probability distribution. The radius to maximum tangential velocity is a random variable which can vary with height, as previously discussed, according to Pm = Pm0 + S*(z - z 0 ) ,
z ~ 6(p) ,
Pm =pm0 + S * [ 6 ( / 9 ) - z 0 ] , Z > 6 ( p ) ,
(3)
where S* is the sampled value of the slope for a given tornado, the point (Pm0, z0) specifies the position at which the reference rotational velocity, Uro,o applies, and the boundary layer thickness, 6, varies as 6(p) = 1(56m -- 26;) + 2p(6~ -- am)/3Pmo , 0 < p ~< 2.5 Pmo , 6(p) = 6 o exp[-O.Ol(p/pm
(4) 0 - 2.5)] , p > 2.5 P m o ,
Here ~ m = ~(,Omo) and 60 is the sampled value of the
W.L. Dunn and L.A. Twisdale/ Windfieldmodel for tornado missiletmnRTort maximum boundary layer thickness, which occurs at p = 2.5 Pmo. This form is utilized because of its relative simplicity (linear near the center, exponential far away) and because it exhibits the expected general behavior of minimum thickness at p = 0 and slow asymptotic variation. A typical boundary layer profde is given in section 5. Radial, tangential, vertical, and rotational velocity components are represented by Ur, Ue, U= and Ure, respectively, where
u,e = ( ~ + u~) ~r' .
(5)
The random variable 3'is taken to be the magnitude of the ratio of radial to tangential components at the reference position; thus, the sampled value relates Ur and Uo according to 3'* = IUr(Pmo, Zo)/Uo(Pmo, z0)[.
(6)
The models for tangential and radial velocity utilize the work of Kuo [16]. The tangential velocity model is synthesized as
Ue(p, z) = Uo [m(p)/p] G(p, z)
(7)
139
and
3"= 3'*G(Pmo, Zo)/F(P~,o, Zo).
(13)
It is noted that the magnitudes of the radial and tangential components increase rapidly from zero at p = 0 to a maximum at Prn and vary as 1/p for larger p. The parameter a determines the asymptotic behavior: for a < 0 the vortex vanishes at the finite dis. tance, Po = -b/a; for a = 0 the vortex approaches zero asymptotically, and for a > 0 the vortex approaches a f'mite positive value. The parameter ~', representing an effective sublayer thickness, determines the values of Us and Ur at the "ground" (z = 0). The functions G and F specify the vertical variations, Ue generally increasing with height up to the top of the boundary layer, Ur generally decreasing. The parameter a determines the rate of vertical variation. Values of a = 10 and [" = 20 feet have been used in the results reported later in the paper. The vertical velocity model is generated using the continuity condition applied to an imaginary cylinder of radius • and height z, i.e.,
where U o is a normalizirtg variable,
- Oz
u=(p, z) = a
1
~p Ur(P, z) + r Uv(P, z).
(14)
re(p) =Pm [ 1 -- exp [-- 1.25643Colp m)2 ] , 0 < p
pm~p
re(p)= 0 ,
P > Po,
Po = -b/a,
a < 0,
po "*°°
a ~ O,
,
G6~, z) = I - exp {-~
The fo~m, obtained by integration,is
(s)
u=(o, z)= Uo3' ~ ( p ) , nLo, ~ z) + m(p)
,(15)
where
(9)
H(p, z) = 6a~--~)e-a~lS(P) { l + 6a~)
z < ~6o),
c,~, z) = 1 - exp {--a[~(p) + f]l~(p) ), z> ~(p),
(1o) and the parameters a and b are determined for a specific tornado to match path width (W'), as discussed in section 4. The radial velocity is given by
Although no additional random variables have been utilized in the specification of Uz, variations in Pmo, 3', ~'o, and'S provide for significant variations in the vertical velocity profdes.
t
u,(p, z) = Uo'r [m(p)/p] e'~, z), where
4. Calculation pcocedurc
~%o, z) = (z + f)/~(p)exp[-a(z + ~/6(p)], z<8(p),
Using the windfield model, a ~ c i f t c tornado is defined by the parameters ~ , o , U~, W',Pmo, S', 7", and 6~. The dependent varisbles 7' and Uo are then calculated, the former by eq. (13) and the ktter according
~Tp, z)=O
z > ~(p), (12)
140
W.L. Dunn and L.A. Twisdale / Windfield model for tornado missile transport
to
along an axis defined by
U 0 = Uro,o/moG(Pmo, Zo)(1 + 72) I/2 ,
(l 7)
where m o is the constant mo = m(Pm0)/Pmo = 1 - e x p ( - 1.25643) = 0.71533. (18) Determination of the parameters a and b in eqs. (8) and (9) which provide model compatibility with the tornado intensity and width variables follows from velocity vector geometry considerations. Tornado path width (W*) is defined as the distance between constant horizontal velocities (Uh = [U2to + U2 ] 1/2) corresponding to gale intensity winds (vb = 73 mph) at elevation z o such that Uh ~< Ub outside the tornado path. This wind boundary definition follows from the FPP tornado classification system [29] which utilizes damage caused by near-ground windspeeds to estimate W for each tornado. From fig. 2, it is noted that the effect of translational velocity is a loss of windfield symmetry such that the distance from the stationary vortex center (point O) to the left boundary (Rt) is always less than the distance to the right boundary (Rr). It can be shown that the new position of zero horizontal velocity (O') is always shifted to the second quadrant for cyclonic flow and
R~ - - - w ~ , . . ~
~= tan-lfUr(Pr,Zo)/Uo(Or, Zo)r ~ tan--l~/.
(19)
For given distance p from the stationary vortex center, we note that on the left side the maximum Uh always occurs at point d whereas the minimum value of Uh always occurs at point b. For p > Pro, the line defined by the points c and O provides an accurate and simple reference axis to determine the left boundary Uh = Ub. Thus, R t is related to vb by {[v0(R~, z o ) - vt] 2 + u~(&, Zo)) '/2 = vb = J07 fps. (20) On the right-hand side, the maximum Uh always occurs at point e and the minimum always occurs at point a for given P and constant % The net horizontal windspeed at point e is larger than that at point f since the rotational and translational velocity vectors are parallel there. Thus, as shown in fig. 2, we treat the boundary as occurring at a radial distance Or from the tornado origin, where Rr = Or cos ~,
(21)
and the right-hand boundary condition is Ur0 (,Or, Zo) + Ut = Ob •
(22)
For 7 = 0 we note that R r = Pr ; also, depending on the variation of 7 with p, it is possible for the net horizontal velocity at some point between e and f to exceed that at R = Rr. However, an involved treatment is required to determine the exact point, and the approximation is good to place the right boundary at R = Rr. We then need only the identities
Rr -------~.
lw
W* = R t +Rr
(23)
and O~
LEFT BOUNDARY
f
I CENTER LINE
Uh=vb
RIGHT BOUNDARY
[ Uh=v b
Fig. 2. Tornado windspeed specification at path width boundary.
m(Pmo)/Pmo = m o = a + b/Pmo ,
(24)
with eq. (18) through eq. (22) to find the appropriate a and b parameters. This is accomplished by an iteratire procedure beginning with an assumed minimum value o f R t = Pmo and incrementing outward until the set of equations is satisfied. This procedure "fits" the model to the given tornado path width, forcing the windfield to assume the proper magnitude at the boundaries. For some low-intensity, high-translational velocity tornadoes, the net horizintal velocity may
W.L. Dunn and L.A. Twlsdale / g~ndfleld model for tormdo mm~ie transport ue
not reach the boundary velocity at the left edge of the core. In these cases, we take simply R~ = 0 and t h m R , = W'.
z
ool
//
\ \
• =
Some typical results are given in order to demonstrate some of the characteristics of the windfield model and its utilization in tornado missile simulations. Radial, tangential, vertical and rotational velocities are plotted in figs. 3 - 6 for the case Ur0,o = 406.27 fps, Ut = 7.33 fps, and ~o = 500 ft., with the remaining perameter values noted on the figures. The radial velocities given in fig. 3 demonstrate the variation in magnitude obtained for the stated conditions due to variation in the parameter 7. The effects of ground roughness and the extent of ground interaction are accounted for by allowing variation of the parameters 7 and 6 o. It is noted that the vertical velocity model is also directly proportional to 7, so that significant variation of both radial and vertical ue
~oo =
f
~oo
/l
":
5. Results
300
141
_ oo
5
200 I00
e (z)
o
I 100
160 rt ,
=
p
ioo
2o0
3oo
~oo
5oo
60o
~)
o
ioo
~oo
3oo
~oo
5oo
~t'ps )u~
F ~ 4. Romtion~ wlodty mad bou~=~tryt,ye~thlckneu.
magnitudes is possible, simulating gross ground interaction effects. The boundary layer thickneu is also affected by ground condRions. Smaller localboundary layer thicknesses genzrafiy remit in smaller verticalvelocities, but the effect is not large. Figs. 3 and fig. 4 indicate that radial, tangential (and hence rotational) velocities are nonzero at z = 0 and that rotational velocity is nearly constant with height above about 100 ft. The radial distances to maximum tangential velocity in fig. 3 differ for different heights as a consequence of the use of S = 0.I76 in the model. The two cases of infinite(figs.3 and 4) and finite (fig. 5) radius to zero windfield are demonstrated. Note that the vertical velocity in fig. 3 remains positive, though small, outside the core, whereas in fig. 5 it becomes negative for large p. This
lOO
o
i0o
2OO I
300 !
-ioo
~00 !
5OO 4
60o 0
7OO I
~f ' P S ~ A
z=O y=0.6
~ t , Pmv =
Ur Uz (~s) 200 loo
~
0o ft
md~l ~ .
ers
160 ft
S
~ 0.176
W
= 1668 ft
•
= .o9267
b
~ 99.6256
Y
~ c.~5
I1\
i
\
°0
o
I
Fig. 3. Tae~nfial, ~
Fsa'Ieters Omo = 30 tt
-2oo
(t~,)
Velocity
"p (et)
I
~100
s.
.: 165 t't
"
'
(~1
% = 3hO f t
and vertical velodt~sas functiom of Fig. 5. Veloc/ty profiles fat a ~
tornado of flMte extent.
142
W.L. Dunn and L.A. Twisdale / Windfield model for tornado missile transport z
Omo = 160 f% S = 0.176, V = O.L5 W
400
1668 ft
a
.09267
b
99.6256
3 ~ 30C
]
i00
Ur
(fps)
-200
-I00
200
300
400
500
(fgs)
1;0
i00
200
300
(fps )
Uz
Fig. 6. Tangential, radial and vertical velocities as functions of height for 0 = 160 ft. is determined by the values of a and b which result from the boundary fitting process (the parameter "a" is positive in one case and negative in the other), since d m / d p = a for large p. The boundary layer thickness is also plotted in fig. 4.
Typical vertical variations of the three wind components are shown in fig. 6. It is noted that U0 is not monotonically increasing with height for constant p, as a consequence of the fact that this specific tornado is non-cyclindrical (S = 0.176); m fact, Uo does monotonically increase with height for constant P / P m , All three components are constant above the boundary layer. For a narrow, high intensity tornado, as shown in fig. 6, the vertical velocity is quite high inside the small core since all the radial inflow becomes a concentrated vertical updraft distributed over a small area. Velocity characteristics of the six standard missiles postulated by the NRC [30] have been assessed using this windfield model and tile probabilistic missile transport model given by Twisdale, Dunn, and Davis [24]. Twenty tornadoes were simulated using the random variable probability density functions described herein. The tornado intensity, translational velocity, and path width variables were characterized
Table 1 Missile velocity parameter estimators Missile
NRC region
No. of observations Vm
Vi
Impact o, Vi (if/s) o
#
o
~
o
Postulated design velocity [30] (ft/s)
Max. o, V m (ft/s)
Equiv. impact o, V~ (ft/s)
Wood beam
I II III
1661 1642 1640
199 210 254
73.40 83.75 75.91
33.91 37.55 33.02
96.07 104.32 100.84
58.74 59.33 36.47
53.34 65.52 53.99
41.23 54.13 28.32
272.6 229.9 190.5
6" Pipe
I II III
1423 1173 1014
160 106 75
91.93 83.90 74.14
30.84 30.41 20.07
97.87 68.38 67.58
38.47 29.60 18.89
49.75 33.90 28.31
28.95 21.95 14.76
170.8 137.9 32.8
1" Steel rod
I II III
1425 1257 1173
190 118 101
95.85 76.02 70.17
34.35 29.34 20.00
98.83 78.82 64.88
47.75 30.95 33.32
54.07 36.81 25.92
37.42 26.55 18.00
167.5 131.4 26.3
Utility pole
I II III
1441 1315 1181
159 121 93
95.32 80.37 72.75
37.55 30.07 18.71
110.46 76.67 62.16
51.87 28.12 29.98
51.74 37.72 35.28
38.73 25.26 20.64
180.6 157.6 85.4
12" Pipe
I
II III
1217 894 673
108 58 27
86.86 71.00 67.14
31.26 19.36 24.27
92.57 57.86 66.67
34.93 20.17 25.40
45.95 22.95 39.53
23.24 15.87 12.57
154.3 91.9 23.0
I II III
1274 1033 799
71 32 27
54.25 46.36 38.07
19.95 15.46 14.83
56.66 54.11 40.27
33.17 18.76 11.31
* * *
Vehicle
o = velocity.
* * *
193.8 170.8 134.6
If.L. Dunn and L.A. Twisdale / P/indfleld model for tornado missile transport
from available data records [25]. A total of 10 000 missile histories (500 per tornado) were simulated for the maximum F'-scale intensity range [25] in each of the three NRC tornado regions [25]. A hypothetical layout of six safety-related structures (targets) was postulated with the missiles uniformly distributed within a rectangular zone centered on the plant. The missile initial elevations were uniformly distributed between five and fifty feet above the ground, except the vehicle missile which was injected between five and ten feet. The maximum velocity attained by each missile during its trajectory was determined; for those missiles striking one of the targets, impact velocity and effective normal-collinear impact veocity were calculated. The means and standard deviations of these quantities are given in table 1 ; also listed is the reference velocity postulated by the NRC [30] for a normal-collinear impact of each missile type in each region. These results suggest that the currently acceptable design velocities are very conservative when compared to an actual plant case study using the windfield model described herein. 6. Discussion A probabilistic tornado windfield model has been synthesized which includes the variable flow characteristics that appear to be important in missile transport analysis. The inherent random behavior of tornadoes and the fundamental modelling uncertainty resulting from incomplete knowledge of tornado dynamics have been treated through the characterization of several windfield parameters as random variables. The approach assumes that a prior/reasoning cannot be utilized exclusively to specify conservative flow characteristics for a variety of missile shapes and initial conditions. By introducing random variables to account for the major sources of uncertainty and significant flow variations, and synthesizing functions consistent with theoretical considerations and observational information, a model was developed which potentially envelopes the expected variety of tornado flow characteristics. The resulting model exhibits the following significant characteristics: (a) The asymptotic behavior of windfield components with radial distance can be adjusted to match the right and left boundaries to the tornado intensity, path width, and translational velocity variables,
143
as specified from their respective probability density functions. Positions for tornado path width wind boundary velocity comparisons have been identified for a translating vortex and a calculational procedure developed to insure tornado wind field compatibility with path width constraints. @) The shape of the envelope of the core region can by cylindrical or conical, with the radius to maximum tangential windspeed, Pro, a random variable dependent on tornado path width. (c) The vertical updraft within the core satisfies continuity with the radial inflow and depends on the magnitude of the radial velocity, the core size, and the boundary layer thickness. This variability allows the possibility of large updrafts, particularly for narrow tornadoes. Tornado missile ILmulations indi. care that such updrafts for high-intensity tornadoes may be capable of lifting certain missiles to altitudes of several hundred feet. (d) The model applies to the observed range of intensity, width, and translational speeds and the dependent tornado parameters are evaluated consistent with these specified tornado characteristics. Also, the vortex windfield components behave reasonably at limiting values o f p and z. At p = 0, radial and tangential velocities vanish but a vertical updraft exists; for large p, all velocities approach zero or some small finite value; above the boundary layer the windfield is a Rankine-like vortex; and at z = 0, radial and tangential velocities are conservatively taken as non-zero (in effect considering the phne z = 0 as the top of a sublayer) whereas vertical velocity is zero. Utilizing this model, wind velocity components have been presented, as functions of radial and vertical distances, for selected values of the input param. eters. In addition, missile velocity statistics were generated by sampling from a variety of high-inten. sity tornadoes and simulating missile trajectories. Average impact velocities are generally lower than those postulated by the NRC (see table 1) and the mean effective normal collinear impact velocities are significantly below NRC postulated values. In addition, the model has been used in an extensive simulation study of missile transport and damage probability assessment, as described by Twisdale, Dunn and Chu [25], for which existing deterministic models were not well suited.
144
W.L. Dunn and L.A. Twisdale
Windfield model for tornado missile transpor~
Acknowledgements The assistance of Dr. J. Hsu in reviewing the existing tornado models and suggesting Kuo's formulation as a starting point for this analysis is appreciated. This work was sponsored by the Electric Power Research Institute as Research Project 616-1, with Dr. B.B. Chu serving as project manager.
References [1] E. Kessler, Proc. Symp. on Tornadoes, Texas Tech. Univ. June (1976) p. 431. [2] W.S. LeweUen, Proc. Symp. on Tornadoes, Texas Tech. Univ., June 1976 p. 107. [3] R.P. Davies-Jones, Proc. Symp. on Tornadoes, Texas Tech. Univ., June, 1976 p. 151. [4] W.H. Hoecker, Jr., Monthly Weather Rev. 88 (1960) 167. [5 ] W.H. Hoecker, Jr., Monthly Weather Rev. 89(12) (1961) 533. [6] F.C. Bates and A.E. Swanson, Res. Rep. Black & Veach, (1967). [7] D.F. Paddleford, Westinghouse Elec. Corp., WCAP-7897, April 1969. [8] R.C. Iotti, Velocities of Tornado Generated Missiles, Ebasco Sere., ETR-1003, February 1975. [9] A.J.H. Lee, ASCE Speciality Conf. on Structural Design of Nuclear Plant Facilities, Chicago, IL, December 17-18, 1973. [10] B.R. Morton, Geophysical Vortices, Progress in Aeronautical Sciences, Vol. 7, (Pergamon Press, Oxford, 1966). [ 11 ] D.R. Beeth and S.H. Hobbs, Topical Rept. B&R-001, Brown & Root, Houston, TX (May 1975). [12] P. Dergarabedian and F. FendeU, J. Atmospheric Sei. XXI, (1973) 26. [13] Emil Simiu and M. Cortes, Tornado Borne Missile Speeds, Nat. Bur. Stds., NBSIR 76-10-50, April 1976.
[ 14] A.K. Bhattacharyya, R.C. Boritz, and P.K. Niyogi, 2nd ASCE Specialty Conf. on Structural Design of Nuclear Power Plant Facilities, New Orleans, LA, December 8 - 1 0 1975 p. 562. [151 Y.K. Wen, J. Struct. Div., Proc. ASCE, Vol. 101, (1975) 169. [16] H.L. Kuo, Axisymmetric Flows in the Boundary Layer of a Maintained Vortex, J. Atmospheric Sci., 23, (1971) 20. [ 17 ] G.H. Redmann et al., EPRI 308, Te chn. Rep. 1 (February 1976). [18J G.F. Carrier, J. Fluid Mech., 49 (1971) 133. [19] G.N. Sun, E.G. Burdestle, and R.O. Barnett, Nucl. Eng. Des. 44 (1977) 407. [20] J.A. Shanahan, Proc. Syrup. on Tornadoes, Texas Tech. Univ., June 1976 p. 251. [21] R.P. Davies-Jones and G.T. Vickers, NOAA Techn. Mem. ERL NSSL-57, November 1971. [221 S.W. Chi, Tellus, XXVI, (1974) 444. [23] H. Schlichting, Boundary Layer Theory (McGraw-Hill, New York, 1965). [241 LA. Twisdale, W.L. Dunn, and T.L. Davis, Nucl. Eng. Des. 51 (1978) 295. [25 ] L.A. Twisdale, W.L. Dunn, and J. Chu et al., EPRI NP-768 and NP-769, Elec. Power Res. Insti., Palo Alto, CA, May 1978. [26] Nuclear Regulatory Commission, Design Basis Tornado for Nuclear Power Plants, Regulatory Guide 1.76, April 1974. [27] J.R. Eagleman, V.U. Muinhead, and N. Willems, Thunderstorms, Tornadoes, and Building Damage (D.C. Heath, 1975). [28] J.F. Costello, Proc. on Tornadoes, Lubbock, TX June, 1976, p. 349. [29] T.T. Fujita and A.D. Pearson, 8th Conf. on Severe Local Storms, October 1973. [30] Nuclear Regulatory Commission, Section 3.5.1.4, Standard Rev. Plan, June 1975.