Nuclear Engineering and Design 51 (1979) 295-308 O North-Holland Publishing Company 295 TORNADO MISSILE TRANSPORT ANALYSIS L.A. TWISDALE Research Triangle Institute, Research Triangle Park, NC 27709, USA W.L. DUNN Nuclear Engineering Department, N.C. State University, Raleigh, ArC 27650, USA and T.L. DAVIS Environmental Formulas, Inc., Raleigh, NC 27608, USA Received 5 June 1978 A methodology has been developed to simulate the initial release conditions and subsequent motion of objects transported by tornadoes. A probabilistic three-degree-of-freedom trajectory model which includes drag, lift, and side forces has been developed to simulate rigid body dynamics in turbulent tornado flow fields. Comparisons of this random orientation model to results from ballistic three-degree-of-freedom trajectory analysis are presented and the results suggest that the simpler models are potentially unconservative in predicting missile range and impact velocity. A missile injection methodology has also been developed which treats injection as the composite of all missile interactions in the near-ground domain and relies on a restraint force exceedance criterion to initialize missile release relative to the translating tornado. The aerodynamic forces acting on a potential missile during injection suggest a multi-peaked time history which is significantly influenced by missile offset position from the vortex center. A simulation study of missile injection has been performed to determine a conservative range for the assumed horizontal restraining force. 1. Introduction The consideration of a tornado hazard in the structural design o f a nuclear power plant is achieved through the specification of loading combinations which include the effects of wind, rapid pressure-drop, and wind-propelled missiles. Of these, the impactive missile loading significantly influences the design thickness o f many of the structural walls and barriers at nuclear facilities in the United States. A necessary requirement for the specification of such impactive loads is the capability to predict the transport characteristics of a variety of potential missiles (e.g. [ 1]). Previous investigations of missile transport in tornado flows have generally treated the problem deterministicaUy, employing the simplified three-degree-of-freedom particle trajectory models (e.g. [ 2 - 8 ] or the full six-degree-of-freedom trajectory model . Considering the randomness observed in the displacements of objects transported by tornadoes and the expected turbulence o f tornado winds, an alternative probabilistic approach to tornado missile analysis is suggested herein. The basic methodology distinguishes injection domain transport from free flight missile trajectory and relies on Monte Carlo simulation to generate missile transport characteristics. The methodology can be utilized with a tornado wind model  to generate distributions of impact velocities for assumed missile threats at nuclear power plants [11,12]. Barrier requirements for missile impactive loads can thus be established on the basis of the actual risk to the safety related systems. In this paper the missile trajectory and injection models are presented and simulation results for each of these components in the transport process are summarized. 296 L.A. Twisdale et al. / Tornado missile transport analysis 2. Considerations in tornado missile transport analysis Tornado missile transport is characterized by limited observational data with the most common evidence being the resultant displacement of objects with presumed known initial locations (e.g. [ 13-17 ] ). Although these observations do not provide the inclusive data base necessary for empirical tornado missile assessment, they do suggest several general characteristics of the transport phenomenon. A significant conclusion is that the processes of missile generation and transport produce variable results for similar type objects and initial conditions. This is particularly true for missile transport near the ground surface where flow modification, windfield turbulence, and missile interactions are the greatest. The fact that identical objects having similar initial conditions exhibit significantly different terminal conditions suggests that the variations can be assumed to result from contributing probabilistic mechanisms. This conclusion is qualitatively supported by photogrammetric evidence, which constitutes a second type of observational data. The time history of cloud debris and small tracer objects also suggests erratic trajectory patterns. It is noted, however, there is an apparent total lack of eyewitness photography which documents the trajectory history of potentially damaging missiles of the type suggested in ref. [I]. On the basis of limited tornado missile data and observed transport variability, probabilistic analytical modeling of missile injection and trajectory phenomena is appropriate. The prediction of the trajectories of tornado generated missiles requires knowledge of the tornadic forcing functions, the initial conditions, the governing dynamic and kinematic relations, and the missile aerodynamic coefficients. Because of the complexity of the tornado missile transport process, considerable uncertainty exists in several of these areas and affects the injection and trajectory sophistication suitable for the transport simulation. A brief review of the modeling requirements and considerations for each identified input in the transport analysis follows. 2.1. Tornado f l o w characteristics Tornado windfield characteristics are discussed in the companion paper [ 10], which utilizes a probabilistic approach to model the inherent variability in tornado dynamics. In addition to the tornado flow specification, there are several additional considerations regarding the tornado forcing functions in missile transport analysis. The effect of suspended particles on the fluid-missile interaction is not known; however, a 5% increase in ambient air density is assumed in a conservative attempt to simulate particle entrainment in the near ground missile transport region. Secondly, flow field turbulence (e.g. [18,19]) appears to be a significant factor in tornado missile transport analysis. As a missile accelerates and approaches the local wind speed, the unsteady velocity fluctuations due to turbulence could approach or even exceed the steady relative velocity component. This effect tends to randomize the resultant aerodynamic forces and moments acting on the body. Another consideration is the result of apparent flow acceleration as the missile moves through the turbulent field. Inertial forces due to tornado flow field acceleration have been studied by Wen  and Sarpkaya [21 ] and are apparently significant. Since the required flow inertial coefficients for a variety of potential missiles (bluff bodies) in a tornado vortex do not exist, the inertial forces have been neglected in tornado missile investigations. However, a net effect is postulated to be a spreading of the missile trajectories. 2.2. Missile release coriditions The initial conditions and release mechanisms are an important component of missile transport since the impact velocities and ranges are significantly dependent upon these initial values. Because of the complexities of the near ground transport analysis, most investigators (e.g. [3,4,7,22])have followed the approach of Bates and Swanson  by identifying an injection phase and a trajectory phase. During the injection phase the missile is assumed to be acted upon by vertical forces which lift the missile to an injection height position. Others have either positioned the missile initially at some height and initiated the trajectory analysis (e.g. ) or utilized a mechanistic treatment involving missile lift-off from a smooth frictionless ground plane (e.g. ). An injection methodology which recognizes the potential for variable missile restraints, the effect of ground interactions which randomize the missile orientation and near-ground motion, and the amplification of the drag and lift forces near the ground plane has not been previously utilized. L.A. Twisdale et al. / Tornado missile transport analysis 297 2.3. Missile tra/ectory models 3. Random missile orientation m o d e l The simplest form of dynamic equations of motion are those that are used to describe the motion of a particle, and these are in the form of integrated force equations. Such three-degree-of-freedom models ignore the angular motion about the mass center and generally assume that only the drag force (fD) is acting in the direction of the wind-missile relative velocity. For nonspherical shapes, the neglection of the lift (fL) and side (fs) forces introduces a bias in the trajectory analysis which could be particularly important in non-rectilinear flow fields. The excursions due to these lateral aerodynamic forces would be expected to significantly affect the trajectory characteristics in a three-dimensional tornado flow field. The six-degree-of-freedom trajectory models of rigid bodies consider drag, lift, and side forces in which both force and angular momentum equations, in addition to other kinematic equations, are required to solve the initial value problem. In view of tornado windfield turbulence considerations, the degree of randomness inherent in the missile injection phase, the aerodynamic data and computational requirements of a 6D model, and the transport 2.4. Aerodynamic considerations Z 8 / II z S) Yt / "ut xt For the ballistic particle model, missile aerodynamic characterization can be accomplished with the quantity known as the flight parameter (CDA/W) or its reciproMCL / 1 / I cal, the ballistic coefficient (W/CDA). If the missile is assumed to tumble during flight, the flight parameter is modified accordingly. Several investigators have derived ,, tumbling drag coefficients (Cot), which are intended to account for random tumbling (RTM) by averaging over ~ V missile orientation. This quantity represents a mathematical expectation which depends both on the form of the drag coefficient as a function of missile attitude and on the averaging process (e.g. the assumed probability distribution on missile attitude). The aerodynamic characteristics required for sixdegree-of-freedom (6D) trajectory analysis include drag, lift, and side force coefficients as a function of missile attitude relative to the wind. In addition, pitching moment and damping coefficients as a function of missile attitude and attitude rates are needed for the rotational equations. For most missile shapes found in the vicinity of nuclear power plants, this full aerodynamic definition is not available. Recent experimental work [9,26] Fig. 1. Coordinate systems and missile speeifieation. (a) Plant provides 6D data for a cylinder, rectangle, and vehicle and tornado frames; Co) MCL specification; (e) wind frame and shape. missile orientation. _ _ _ Z y 298 L.A. Twisdale et al. / Tornado missile transport analysis bias implicit with the ballistic 3D approach, a modified 3D random missile orientation model has been developed. In this model the actual rigid body orientation of the missile is considered and the aerodynamic specification includes drag, lift, and side forces. The missile centerline orientation is specified by two randomly determined angles (4, ~) measured from a (u, v, w) coordinate system as defined in fig. l(b). The relative velocity vector defines the v direction while t~ = (b × k)/Ib × kl and ~ = (fi X b). Once the missile orientation is established for a time step, wind axis unit vectors are determined by forming the vector cross product of the missile centerline position nnit vector (MCL) with the relative velocity vector (6) to establish the pitch axis (/5). The missile diameter unit vector (M/)) is rotated through a randomly selected angle (6) from the pitch axis. The relative velocity unit vector (6) is then combined with the pitch axis (/5) in a vector cross product to establish the lift unit vector (L). This approach defines the wind axis system (b,/3, and £) for each time step which provides the respective directions for the three aerodynamic force components: drag, side, and lift. This approach provides an aerodynamic force that is properly oriented for the missile attitude. The magnitudes of the three translational forces are taken as proportional to the three static aerodynamic coefficients (CD, Cs, and CL), which may each be functions of total wind angle of attack (a) and roll angle (5). These angles (o~,6) are both shown in fig. l(c). The missile angles, and hence vectors MCL and MD, are updated at selected intervals according to = cos-I (1 -- 2~1), ¢ = rr(2~2 - 1), 0<¢<~, ] fo -mg Lj :E where ~1, ~2, and ~3 are random numbers selected from a uniform distribution on the unit interval. The time between missile orientation updates is termed the update period, and its reciprocal, update frequency. The angles t~ and/3 are used as input to the aerodynamic coefficient determination. Once the three coefficients are determined, they are combined with the dynamic pressure, reference area (A), and the three appropriate wind axis unit vectors to form the total aerodynamic force for a single time step. dt p dVo + V,V dt p (2) dVz 1I LJ dt The random orientation model includes drag, lift, and side forces which are assumed proportional to the square of the magnitude of the wind-missile relative velocity (v) (1) 0<8~2, dVr =m -n~<n, ot = cos-1 (sin ff cos ¢), 8 = 2zr~3 , For missile transport simulation at a particular plant site, an appropriate location is chosen to situate a righthand cartesian frame (Fp) such that the axes are preferably parallel or perpendicular to major safety related plant structures. Fig. 1 shows this reference frame Fp along with other reference frames that will be used in the development of the model. As indicated in fig. 1(a), the tornado is assumed to touch down at point S(Ps, 0s, Zs), the centerline of the path of the tornado will intersect the Xp-axis at an approach angle (r) measured positive counterclockwise from the positive Xp-aXis. For the kinematic equations, an Ft cartesian frame is established with its origin 0t(,O0t , 00t , z0t ) attached to the center of the tornado windfield and moving along with it at a uniform translational speed of UT. Associated with each of these reference frames are two corresponding cylindrical frames Fpc and Etc. The mass center of the missile is tracked relative to the plant cylindrical frame, Fpc, according to the dynamic equations of motion, which take the form fL - fs PaAv 2 2 IcDl CL • (3) Cs A standard transformation is used to obtain fr, fo, and fz in the Fp system. These equations form a set of six coupled, nonlinear, ordinary differential equations which define an initial value problem for a set of prescribed initial conditions. Shampine's method  is used to integrate these equations. L.A. Twisdale et aL / Tornado missile transport analysis 3.1. Missile aerodynamics Since complete aerodynamic characteristics do not exist for the variety of possible missile shapes, a modified cross-flow theory has been applied to develop the aerodynamic coefficients for the random orientation model. This approach has been successfully used to develop the wind axis aerodynamic forces as a function of angle of attack for slender cylinders knowing only the drag force coefficients for the body in normal flow to the major body axes . The basic theory assumes the superposition of two flows perpendicular to the missile axis (axial and cross flow) in which the magnitude of the mutually orthogonal flows is determined vectorially knowing freestream velocity and angle of attack. The aerodynamic forces acting on the missile are parallel to each flow component direction and are proportional to the directional dynamic pressure. For other shapes, flow field similarity in the cross flow regime as the angle of attack changes is the major requirement for the cross flow theory to be applicable. Thus, it is reasonable to consider extension of the theory to sharp edged beams (e.g. rectangular or ' T ' cross sections) which force boundary layer separation at a fixed point and therefore produce similar potential cross flow fields for all angles of attack. In principle, this concept allows the generation of lift, drag, and side forces for certain sharp edged planar symmetric sections, if the drag coefficients are known for flow impacting normal to the three major faces of each shape. Normal flow coefficients can be found in the literature (e.g. [29,30,31 ]) for a variety of shapes. Near-ground correction factors for these coefficients have been approximated from Hoerner  to simulate observed drag increases over the free stream value. Pretransition Reynolds number range coefficients are conservatively utilized, a tip loss correction for finite missile dimensions is approximated, and missile face porosity is considered. Comparisons  of the developed aerodynamics with recent experimental data  for a plank missile indicates the validity of the extension of cross-flow theory. Although the random orientation trajectory model does not directly employ RTM coefficients, it is compared with ballistic 3D RTM results for cylindrical missiles. Because of the differences in the RTM coefficients presented in the tornado missile literature, a properly developed form has been derived. The expression for the expected value of the drag coefficient, Cd, of a tumbling 299 missile is 7r 21r 21r C'd'f f f o o o Cd(Ct,/3, 8 ) / ( t~'/3'8)dSd/3da' (4) where t~, ~ and 8 are orientation angles as specified in fig. 1, and f(ol, fl, 8) is the joint probabiliyt density function describing orientation likelihood. For a cylinder of diameter d and length L, cross-flow theory indicates that CD(Ot, fl, 8) = CD(Ot) = CDc sinaot +-~L CDalC°Sat~l .(5) Assuming uniformly random spatial orientation, f(ot,/3, 6) = (I/81r 2) sin a and the expression for the RTM coefficient is derived as 1 (3ff + ffd CDa) ' CD = 4 \ 7 CDc 4L (6) where the subscripts a and c refer to axial and crossflow directions, respectively. This expression yields a significiantly higher expected value than the previously published results of Bates and Swanson  and Redmann et al. . However, Bates and Swanson used a drag coefficient the terms of which varied as sin2ct and cos2c~ instead of the cubes of these quantities, and their averaging process is suspect. Redmann et al. employed a formulation similar to that given by eq. (4) but apparently evaluated one of the integrals improperly. It is noted that this general formulation agrees with that given by Sentman and Niece . 3.2. Model comparison and results The random orientation trajectory model has been developed such that it can operate in any of three modes: 3D ballistic, random orientation with drag force only (random drag), or random orientation, with drag, lift, and side forces (full random). This approach has facilitated model verification and hypothesis testing; in particular, verification of the 3D ballistic mode has been achieved by comparison of published results [8,9]. There are no comparable random orientation models to test against directly, so a series of tests was devised to indicate its validity and applicability. 3.2.1. Random drag model An expected feature of the random drag model is 300 L.A. Twisdale et al. / Tornado missile transport analysis Table 1 Comparison of ballistic 3D and random drag models Case no. Number Mode of trials (n) Initial orientation a 1 2 3 1 t 50 ballistic 3D random drag random drag max CD max CD random 4 50 random drag 5 50 6 50 7 1 Update freq. (Hz) Impact velocity d Ground impact point b ~ 33 Range D (ft) VH (fps) VZ (fps) 0 1 76.8 76.8 51.3 30.5 30.5 19.9 random 2 50.5 19.5 random drag random 10 49.8 19.1 random drag random 100 49.0 18.8 ballistic 3D RTM CD - 48.7 18.7 82.6 82.6 55.1 (26.4) c 54.1 (22.4) 53.4 (8.7) 52.5 (2.9) 52.2 115.6 115.6 8O.7 (31.9) 78.0 (22.4) 80.2 (8.3) 79.7 (3.4) 79.7 -31.4 -31.4 -32.7 (1.1) -33.0 (0.9) -32.8 (0.4) -32.7 (0.2) -32.7 a Missile initial position; x = 0,y = 0, z = 20 ft. b For multiple trial runs the impact point is the centroid of the impact pattern. c Numbers in parentheses are standard deviations. d VH and VZ are horizontal and vertical velocity components, respectively. that it should duplicate the corresponding ballistic case if the update frequency is zero since a fixed-attitude trajectory is obtained. By varying the initial orientation, the range of ballistic trajectories (from minimum to maximum drag coefficient) should be predicted. Another expected feature of the random drag model is that in the extreme limit of high update frequency the trajectory should approach the ballistic random tumbling mode trajectory. To assess these hypotheses, the random drag tests summarized in table 1 were conducted for the standard utility pole missile . The results verify the expected features of this model. Comparison of cases 1 and 2 indicates that the zero update frequency random drag model is equivalent to a ballistic (fixed-attitude) model. Comparison of cases 3 - 6 with case 7 indicates that the random drag model tends to converge to the ballistic random tumbling mode case as update frequency is increased. For high update frequency the variance is small, whereas for sufficiently low update frequency, a maximum spread among the trajectories is obtained, corresponding to the range of ballistic 3D fixed-attitude trajectories for random initial orientations. 3. 2. 2. Full random model Comparison of the full random model to the random drag model permits a determination of the effects of the addition of lift and side forces on the trajectory characteristics. It is frequently argued, with respect to the use of random tumbling mode coefficients, that lift and side forces will tend to have negligible net effect because the directions in which these forces act will vary as the missile randomly tumbles. However, the results in table 2 indicate that not only is the variance larger in the full random case, but the trajectories tend to be longer. Hence, the 3D random orientation model, with drag, lift, and side forces included, is considered to be the more appropriate model for tornado missile transport analysis. It is noted that the full random model exhibits the same type of behavior with update frequency as the random drag model, but has larger variances and different impact points. The recently developed 6D model by Redman et al.  permits comparison of rotational rigid body flight with the full random 3D model in a laminar tomadic field. Comparison of a series of cases with different update frequencies with 50 6D trajectories for the utility pole missile is presented in table 3. It is noted that trajectory horizontal path length tends to increase with decreasing update frequency and that the random model results tend to change little for update frequencies below 1 Hz. In fig. 2 the impact positions of the 50 trials at the 1 Hz update frequency are illustrated with those corresponding to the 50 6D trajectories. Among the 6D trajectories there are a few unusually 301 L.A. Twisdale et al. / Tornado missile transport analysis Table 2 Comparison of random drag and full random models a Case no. Number of trials (n) Mode 1 50 2 50 random drag (2 Hz) full random (2 Hz) Impact velocity Ground impact point x (ft) y (ft) 50.5 19.5 60.7 26.2 (fps) 78.0 (22.4) 83.8 (28.6) (fps) -33.0 (0.9) -35.5 (7.0) a Notes a - d of table 1 apply. and resulted in an average horizontal range o f 723.3 ft. long ones, two covering horizontal distances o f 805.4 Based on the preceding results and considerations, and 612.0 ft. These missiles remained airborne on the an update frequency o f 1 Hz is suggested for use in the order o f 4 . 8 - 5 . 8 s and result from favorable initial transport model. As evident from fig. 2, this provides orientations and slow tumbling during flight. Randomadequate lateral scatter, and considering flow turbulence izing effects (such as tornado turbulence, flow modiand multivortex phenomena which tend to randomize fications, and missile interactions) would tend to ,missile rotation, the 3D random orientation model at 1 enhance missile tumbling and thus shorten these traHz is considered an appropriate probabilistic simulation jectories. It is interesting to note that if the single longest of rigid b o d y flight in a tornado. trajectory is ignored, the mean range of the 6D trajectories (D in table 3) is 93.5 ft. It is also noted that the full random model is capable o f generating very long 4. Missile injection methodology trajectories, if very low update frequencies are employed. For instance, ten missiles were flown from an initially The missile trajectory methodology discussed prefavorable orientation at an update frequency of 0.2 Hz viously is appropriate for missiles in free flight subjected Table 3 Comparison of full random and 6D models a Case no. Number of trials Mode Update frequency (Hz) Ground impact point x y (ft) (ft) Range D fit) Impact velocity i7,H VZ (fps) (fps) 1 50 full random 5 48.1 18.9 2 50 full random 2 64.5 28.4 3 50 full random 1.25 71.1 27.9 4 50 full random 1.0 81.1 31.9 5 50 full random 0.667 81.2 38.2 6 50 6D - 98.4 32.5 52.8 (28.1) 72.4 (58.5) 80.5 (65.0) 91.8 (72.3) 93.1 (92.1) 107.7 (147.5) 76.0 (18.8) 87.9 (34.2) 85.9 (34.3) 91.9 (32.8) 88.4 (46.5) 102.7 (45.2) a Notes of table 1 apply. -33.5 (4.9) -36.2 (6.8) -36.0 (7.6) -37.0 (8.8) -36.3 (7.4) -36.8 (8.6) 302 L.A. Twisdaleetal./Tornado missHe ~ansportana~sis Legend O Full Random Model (1 Hz) JPL 6-D Model Tornado Origin at Injection: (0,'528',0) Tornado translating in x-direction Missile Position at Injection: (0,0,20') 3o0 A z~ o 200 £ O 6~ /,, lOO 0 O 0 o o o ~ ^ ~ o8~ ~ a-' a~ ~ 0 A o~ ~ dPO &O ~o o o ,., Eo o ~o~ o 0 I I 2oo 3oo I 4o I 6oo [ 70o IL x Track Distance (ft) Fig. 2. Impact point distribution for utility pole missile. to gravitational and aerodynamic accelerations. In general, the initial acceleration of a stationary object by tornadic winds requires that certain restraining forces be overcome before motion is possible. In addition to gravity forces, these restraints can consist of structural, frictional, or interlocking mechanisms which tend to resist motion. They are important in characterizing the initial release conditions of the object relative to the moving tornado. The physical environment of the injection region suggests that: (1) flow field turbulence and flow interference modifications are considerably increased over free flight flow; (2) the presence of potential sources of missile interaction, such as ground surface perturbations, other missiles, and small structures, contributes to the randomization of injection domain transport; and (3) the restraining mechanisms, including both initial restraint conditions as well as subsequent "trapping" or wedging restraints, exhibit considerable variability in type, location, and magnitude. Each of these general" hypotheses regarding the transport environment in the injection domain contributes to the statistical variability and the complexity of modeling missile injection events. Mechanistic deterministic modeling is thus not considered appropriate for missile injection and a probabilistic approach is adopted in the following which considers variable missile restraint forces and random initial missile orientations. The mechanistic treatment allows for the inclusion of both horizontal and vertical restraints and considers tornado translatory effects. 4.1. Injection domain variables and missile release criterion As illustrated in fig. 3, the injection domain is the region in which the missile is generated. Since most potential missile sources are at or near the ground surface, the injection domain extends laterally over the ground surface and roughly parallel to the ground/structure profile. As suggested in fig. 3, there are a number of possible event sequences in which potential missiles L.A. Twisdale et al. / Tornado missile transport analysis 303 Tr&jectory Domain (c) "~ mj /J-- -- (d) ¢"~" / ~ ~ ~-- ~I (a)// /,'' ~" fo ii I / _.-[.-'~__~f-->e- --.(f) (e) ,-4-- i InjectionDomain :~0 mi Fig. 3. Hypothetical injection domain events. could remain in the injection domain or possibly escape to the trajectory domain: missile m i could interact with missile m/and terminate within the injection domain (event a); its aerodynamic lifting forces could not sustain flight (event b); or it could become blocked by another object (event f). Less likely, missile m i could experience favorable wind gusts and lifting forces (event c), be subjected to a favorable missile collision (event d), or experience a ramp injection (event e) and thereby escape the injection domain. These hypothetical event sequences imply that missile transport within the injection domain involves a multitude of non-aerodynamic probabilistic event sequences. The methodology developed to simulate missile transport in the injection domain relies on probabilistic characterization of variables which specify the initial release conditions. Missile injection location in the horizontal plane is assumed to be randomly distributed within the specified missile origin zone in the plant vicinity  and a distribution for missile elevation is also assumed. Missile orientation relative to the tornado windfield is possibly the most critical factor affecting successful missile escape from the injection domain. On the basis of the previously discussed effects which tend to randomize missile orientation during the injection sequence, random initial missile orientation is assumed. For a particular missile position and orientation, motion can be achieved when at least one of the restraining forces acting on the object is exceeded. The characterization of these restraints is specified by the random variable f o , as depicted in fig. 3. By postulating a probability distribution offo, conditions which depict the original missile availability modes, subsequent wed- ging forces, missile weight, and friction forces can be considered in this approach. The effects of potential missile interactions and multiple missile contact forces suggest that fo is at least partially dependent upon the events of other potential missiles. This raises questions of missile dependence in the injection methodology and suggests that conservative models must be utilized in this part of the analysis because of the unfeasibility of explicitly modeling multiply correlated missile injection. The approach adopted here is to specify restraining force over a range which tends to optimize missile transport. Minimum restraint specification is not necessarily conservative since the missile may tend to fall before the maximum tornadic forces have arrived; whereas maximum restraint specification will result in many objects which do not displace at all. The question of optimum fo specification has been evaluated in a simulation study and is reported in section 4.3. A missile injection criterion of first calculated exceedance of the restraining force is selected. That is, given the missile location and orientation, the missile is released to the trajectory model with zero initial velocity at a tornado position where the calculated aerodynamic forces exceed the restraining forcefo *, where the asterisk denotes a particular sampled value from the appropriate density functionf(fo). This criterion ensures that the tornado is sufficiently close to the missile such that horizontal or vertical motion is imminent. 4.2. Envelope o f injection and injection zone The potential for initial motion of an object in the injection domain is governed by the injection variables L.A. Twisdale et al. / Tornado missile transport analysis 304 Possible f0H (Ib) Tornado Legend 3000 Approaches t n ~t ~ • x; . on Envelope ' .': --.-- Offset = -r - - Offset = 0 m - - __ O f f s e t = +r m 2000 of Yt lOO0 i000 0 , Xp(ft) 2000 Fig. 5. F o r c e d i s t r i b u t i o n f o r b o a r d - t y p e m i s s i l e . Fig. 4. Envelope of injection and injection zone. and the relative tornado position and strength. In fig. 4 a tornado with direction defined by the angle r is approaching an object located at point m. The object displaces when the release criterion is met; this occurs when the center of tornado has moved from an assumed touchdown point S to the injection point I. If the direction of this tornado is maintained but its touchdown point is allowed to vary laterally, a set of such injection points is formed. This set will form a closed curve (or curves) called the envelope of injection. The region enclosed by the envelope is defined as the injection zone; for given fo*, the tornado wind force can inject the missile only if the tornado origin (0t) passes through this zone. The zone can be singly or doubly connected depending on the missile orientation and characteristics, tornado description, and the value o f f o. A doubly connected injection zone is shown in fig. 4. With the tornado center at the injection point I and the missile at the polar coordinates (,oi, 0i, Zo) relative to Ft, the injection criterion implies that f ~j~', (7) where f is the aerodynamic force. In this investigation this release criterion is simplified by specifying that the missile restraints consist of two force component, f0v and foil, which represent the restraints in the vertical and horizontal directions, respectively. It is then assumed that release occurs when either [/'2 +f21 1/2 ~ f O H (8) or fz >fov, (9) are satisfied where fr, fo, and fz are the Fpc components of the aerodynamic force. The decoupling of the restraining forces is conservative in the sense that it does not require that both restraints be exceeded. In addition, it simplifies the fo specification consistent with the concept of optimizing fo for maximum missile transport. Solutions for the injection envelope and missile injection profiles have been obtained for a beam-shaped missile subjected to a 242 mile/h tornado. The missile, resting on its long edge, was placed at different offsets relative to the tornado traveling along the x-direction. The horizontal aerodynamic force distribution for three offset cases are plotted in fig. 5. The force-time histories all show double spikes, but they are not of equal value. This is due to the rotational nature of the wind, the translational movement, and the inward radial flow towards the tornado center. From such force distributions, a family of horizontal injection envelopes has been generated and are shown in fig. 6 for horizontal constraints varying from 200 to 3000 lb. The contours indicate a multipeaked injection force surface. 4.3. Specification of restraining force A simulation study was performed to determine the optimum range of missile restraining force for use in the injection methodology. Using the missiles identified in 305 L.A. Twisdale et al. / Tornado missile transport analysis 0.2 k 300. __.( 8OO ~00 (ft) 40~ Possible Approaches at rm/3 Increment T Offset (ft) Fig. 6. I~ection envelope forboard missile. table 4, transport ranges and impact velocities were evaluated for a tornado with a Umax = 282 mile/h and a mean translation velocity of 40 mile/h. In this study the missiles were injected uniformly over the height from 0 to 20 ft within the 73 mile/h tornado wind boun- Table 4 Missile descriptions for restraint study Number Missile Lib d/b Reference Weight area (lb) (ft2) 1 2 3 4 5 30 20 96 24 14 1 4 50 1.2 1 30 80 32 20 15 utility pole beam or box plywood sheet wide flange steel pipe 950 1600 100 1160 750 daries. As summarized in table 5, for the total attempted 6665 missile injections, the number of unsuccessful attempts increased as the horizontal restraining force increased. For the heavier missiles (wide flange beam and pipe) the number of unsuccessful trials began to dominate the results as the horizontal restraint approaches 5W, where W is the missile weight. For example, out of a total of 2531 trials for the pipe, only 50 successes were obtained. The study indicates that, given restraint exceedance, the conditional range and impact velocity of the trajectory increases with increasing restraint. In all cases, the conditional range (/)) and the impact velocity (l~i) increase asfoH is varied from W to 5W. Use of fOH = 0.5W was also investigated for the two cylindrical missiles to simulate rolling friction and the results follow the same pattern as previously indicated. For the purpose of specifying optimal horizontal restraint forces, the mean unconditional ranges presented in table 5 L.A. Twisdale et al. / Tornado missile transport analysis 306 Table 5 Restraining force simulation results Percent successful injection Conditional Unconditional /) D (ft) fit) Impact Number of trials velocity F'i Initial Unsuccessful successes restraint (fps) exceedances Utility pole 0.5W W 2W 5W 2.72 6.77 19.05 146.22 2.72 6.77 8.50 19.73 18.15 23.16 39.71 105.35 50 50 50 100 0 0 62 641 50 50 112 741 100 100 45 13 Box W 2W 5W 112.70 176.55 273.80 77.19 88.28 99.93 41.03 44.40 106.96 50 100 100 23 100 174 73 200 274 68 50 36 Sheet W 2W 5W 23.65 45.65 67.69 23.65 44.75 59.38 29.44 39.33 65.19 50 50 100 0 1 14 50 51 114 100 98 88 Wide flange W 2W 5W 8.45 29.36 128.10 3.28 7.73 4.16 24.32 46.15 84.96 50 50 50 79 140 1489 129 190 1539 39 26 3 0.5W W 2W 5W 2.68 5.14 24.09 113.40 1.49 2.01 3.51 2.24 18.05 22.51 42.17 92.86 50 50 50 50 40 78 293 2481 90 128 343 2531 56 39 15 2 Totals 1050 5615 6665 are useful. These values are based upon the total number o f attempted injections and thus provide a measure of optimality. For the heavier missiles, the unconditional mean transport range (/)) peaks within the range o f f o n = [W, 5W]. For the lighter missiles, the restraints are exceeded even at fOH = 5 I4/ and thus the total range increases with increasing foil. However, since horizontal restraints in excess o f 5W are not expected for the majority o f missile availability modes and the unconditional ranges o f the heavier missiles (with better damage capability) peak f o r f o n < 5W, the range [I41, 5W] is suggested as the bounds for optimal fOH specification. This range provides for wide limits in the percentage o f successful injections, as noted in the last column o f table 5. In this methodology, the vertical restraining force is conservatively specified as the missile weight. The use o f f o v > W would not appreciably affect the results for the case o f f o n ~ 2W since horizontal exceedance generally occurs first. For example, in the case of foil = W, only 0, 4, 5, 2, and 0 histories out o f the total number of trials for the utility pole, box, sheet, wide flange, and pipe, respectively, experienced vertical restraint exceedance first. For the cases o f high horizontal restraints (e.g. fort = 5W) the number o f unsuccessful trials significantly affects the unconditional ranges and the use of a higher vertical restraint would reduce the values further. Missile type Pipe Horizontal restraint fOH Transport range Total 5. Conclusions A probabilistic three.degree-of-freedom trajectory model which includes drag, lift, and side forces can efficiently simulate tornado transport for a class of assumed missile shapes. By varying the time interval between missile updates and the aerodynamic forces considered, the model can simulate 3D ballistic fixed attitude, 3D L.A. Twisdale et al. / Tornado missile transport analysis ballistic random orientation, 6D ballistic transport, and 6D random orientation transport. For high update frequency, the variance of the trajectory characteristics is small and converges to the respective random tumbling mode results; for low update frequency, a maximum spread among the trajectories is obtained. On the basis of a series of trajectory comparisons, an update frequency of 1 Hz for the random orientation model is shown to provide the necessary transport variance expected of rigid bodies in three-dimensional tornadic flows. The results also suggest that 3D ballistic transport underestimates the velocity and range characteristics of tornado-generated missiles. A random-tumbling mode drag coefficient for cylindrical missile shapes has been derived from probabilistic considerations. The result is higher than several such coefficients reported in the tornado missile literature and thus may affect the conclusions of previous investigators. However, ballistic RTM transport is shown to be unconservative when compared to the expected values and variances of the missile velocities and ranges of the random orientation model. A modified cross-flow theory has been applied to develop the aerodynamic coefficients which are used in the standard formulae for drag, lift, and side forces. The aerodynamic coefficients are increased for missile locations at or below 5ft to account for near-ground amplification. In the transport analysis, the standard air density has been increased by 5% to account for effects such as entrained dust particles and suspended moisture in the tornado. A missile injection methodology has been developed which treats injection as a composite of all interactions in the near-ground injection domain. In a simulation study of over 6000 attempted injections, the interval of restraining force which maximize missile transport was estimated as one to five times the missile weight. A force exceedance criterion is utilized to initialize the missile release relative to the moving tornado. The aerodynamic forces are thus evaluated at each time step in the injection process; the results suggest the potential for multiply-peaked time histories. Acknowledgement The work reported herein represents part of a tornado missile investigation performed at Carolina Power & 307 Light Company and sponsored by the Electric Power Research Institute (EPRI). The authors gratefully acknowledge this support and the suggestions of Dr. B.B. Chu of EPRI. 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