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Twisdale1979-TornadoMissileTransport.pdf
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 [9]. 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 [10] 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 [20] 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 [23] 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.
[24]) or utilized a mechanistic treatment involving missile lift-off from a smooth frictionless ground plane
(e.g. [5]). 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 [27] 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 [28]. 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 [28]
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 [12] of the developed
aerodynamics with recent experimental data [32] 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 [23] and Redmann et al. [9]. 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 [25].
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 [1]. 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.
[9] 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 [12] 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.
References
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Natural Phenomena, Section 3.5.1.4, Standard Review
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Fly UP