APPLICATION OF THE BOUNDARY ELEMENT METHOD FOR TORNADO FORCES ON BUILDINGS by RATHINAM PANNEER SELVAM, B.E., M.E., M.S. in C.E. A DISSERTATION IN CIVIL ENGINEERING Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements of the Degree of DOCTOR OF PHILOSOPHY Approved
Accepted
August, 1985
f
f
^ ' ^
C^^.^
ACKNOWLEDGMENTS
I wish to express my deep appreciation to my committee chairman. Dr. James R. McDonald, for bestowing confidence in me during the course of this challenging
research project.
I also wish to express my
sincere appreciation
to Dr. C.V.G. Vallabhan for teaching me the
numerical technique, and to my remaining committee members, Drs. Kishor C. Mehta, Arun K. Mitra and W. Pennington Vann for their valuable suggestions and comments on this dissertation manuscript.
I thank Dr.
Ernest W. Kiesling for the financial support provided by the Department of Civil Engineering. I thank my aunty and grandmother for instilling in me the value of education, and I am most grateful sacrifice, support
and
to my family members for their
encouragement
doctoral program.
n
throughout
the course of my
TABLE OF CONTENTS Page ACKNOWLEDGMENTS
ii
LIST OF TABLES
v
LIST OF ILLUSTRATIONS 1.
2.
3.
4.
5.
vi
INTRODUCTION AND RESEARCH PROBLEM
1
1.1
Previous Work
2
1.2
Objectives of the Research
4
MATHEMATICAL MODEL
5
2.1
Fluid Dynamics Equations
5
2.2
Tornado Wind Field Models
10
NUMERICAL TECHNIQUES
16
3.1
Formulation of the Boundary Element Equations
17
3.2
Numerical Analysis
20
3.3
Example Problem
24
3.4
Determination of Forces on the Building
26
3.5
Verification of the Model
27
RESEARCH FINDINGS
33
4.1
Tornado Forces on Buildings from Inviscid Flow Theory
33
4.2
Simplified Procedure to Calculate Tornado Forces from Inviscid Flow Theory
40
4.2.1
40
Forced Vortex Flow
4.2.2 Free Vortex Flow CONCLUSION AND NEED FOR FUTURE RESEARCH
43 61
5.1
Conclusion
61
5.2
Directions for Future Research iii
62
LIST OF REFERENCES
64
TV
LIST OF TABLES Table
Page
4.1
LIFT AND INERTIA FORCES IN FORCED VORTEX FLOW
45
4.2
ACCELERATIONS IN FREE VORTEX FLOW
53
LIST OF ILLUSTRATIONS Figure
Page
2.1
DOMAIN AND BOUNDARY CONDITIONS
7
2.2
POINTS ON A STREAMLINE
9
2.3
THREE-DIMENSIONAL TORNADO WIND VELOCITY VECTOR
11
2.4
RANKINE-COMBINED VORTEX MODEL
14
3.1
DISCRETIZATION OF THE BOUNDARY USING CONSTANT BOUNDARY ELEMENT
19
3.2
DISCRETIZATION OF FLUID AND BUILDING BOUNDARY
25
3.3
VELOCITY DISTRIBUTION FOR VERIFICATION OF THE NUMERICAL MODEL COMPARISON OF FORCES IN THE X-DIRECTION FROM THEORY AND COMPUTER MODEL RESULTS COMPARISON OF FORCES IN THE Y-DIRECTION FROM THEORY AND COMPUTER MODEL RESULTS
31
TORNADO FORCE COMPONENTS ON A CIRCULAR BUILDING FROM COMPUTER MODEL IN FORCED VORTEX FLOW
35
OTHER BUILDING SHAPES CONSIDERED IN APPLYING THE COMPUTER MODEL
36
TORNADO FORCE COMPONENTS ON A SQUARE BUILDING FROM COMPUTER MODEL IN FORCED VORTEX FLOW
37
TORNADO FORCE COMPONENTS ON A RECTANGULAR BUILDING FROM COMPUTER MODEL IN FORCED VORTEX FLOW
38
4.5
TORNADO PATH RELATIVE TO BUILDING LOCATION
43
4.6
COMPARISON OF FORCES IN X-DIRECTION BY SIMPLIFIED PROCEDURE AND COMPUTER MODEL FOR CASE 1, FORCED VORTEX FLOW
46
COMPARISON OF FORCES IN Y-DIRECTION BY SIMPLIFIED PROCEDURE AND COMPUTER MODEL FOR CASE 1, FORCED VORTEX FLOW
47
3.4 3.5 4.1 4.2 4.3 4.4
4.7
VI
28 30
4.8
4.9
4.10
4.11
4.12
4.13
4.14
4.15
4.16
4.17
COMPARISON OF FORCES IN X-DIRECTION BY SIMPLIFIED PROCEDURE AND COMPUTER MODEL FOR CASE 2, FORCED VORTEX FLOW
48
COMPARISON OF FORCES IN Y-DIRECTION BY SIMPLIFIED PROCEDURE AND COMPUTER MODEL FOR CASE 2, FORCED VORTEX FLOW
49
COMPARISON OF FORCES IN X-DIRECTION BY SIMPLIFIED PROCEDURE AND COMPUTER MODEL FOR CASE 3, FORCED VORTEX FLOW
50
COMPARISON OF FORCES IN Y-DIRECTION BY SIMPLIFIED PROCEDURE AND COMPUTER MODEL FOR CASE 3, FORCED VORTEX FLOW
51
COMPARISON OF FORCES IN X-DIRECTION BY SIMPLIFIED PROCEDURE AND COMPUTER MODEL FOR CASE 1, FREE VORTEX FLOW
54
COMPARISON OF FORCES IN Y-DIRECTION BY SIMPLIFIED PROCEDURE AND COMPUTER MODEL FOR CASE 1, FREE VORTEX FLOW
55
COMPARISON OF FORCES IN X-DIRECTION BY SIMPLIFIED PROCEDURE AND COMPUTER MODEL FOR CASE 2, FREE VORTEX FLOW
56
COMPARISON OF FORCES IN Y-DIRECTION BY SIMPLIFIED PROCEDURE AND COMPUTER MODEL FOR CASE 2, FREE VORTEX FLOW
57
COMPARISON OF FORCES IN X-DIRECTION BY SIMPLIFIED PROCEDURE AND COMPUTER MODEL FOR CASE 3, FREE VORTEX FLOW
58
COMPARISON OF FORCES IN Y-DIRECTION BY SIMPLIFIED PROCEDURE AND COMPUTER MODEL FOR CASE 3, FREE VORTEX FLOW
59
vn
CHAPTER 1 INTRODUCTION AND RESEARCH PROBLEM Each year in the U.S. tornadoes damage many buildings and structures.
While it is not practical to design most buildings to resist
tornado-induced loads, it is desirable to know the consequences of a tornado strike on a particular building.
For example, the designer of
a high-rise building might ask what would happen if his structure were hit by a tornado.
Would the structure survive or would it collapse?
To answer this very relevant question, it is first necessary to know the loads induced by a tornado on the structure. Even though great progress has been made in research on windresistant buildings and structures, very little is known about tornadic forces on structures.
The types of forces produced by a tornado differ
from those produced by straight wind.
Straight wind typically fluc-
tuates about some mean for a relatively long period of time, whereas tornadic wind changes rapidly in a short duration of time.
The short,
intense, rapidly changing action of tornadic wind produces both drag and inertia forces (Wen, 1975).
The predominant forces produced by
straight winds are drag forces, the inertia forces being essentially negligible. Most knowledge of tornado effects on buildings comes from poststorm damage investigations (McDonald, 1970; Mehta, et al., 1971; Minor, et al., 1972).
However, experience with tornadoes striking high-rise
buildings is limited.
Other methods for predicting wind forces on
buildings are wind tunnel and theoretical or numerical modeling.
Since
both wind speed and wind direction change rapidly with time in a tornado, it is difficult, if not impossible, to simulate tornado winds in a wind tunnel.
Some modeling of tornado vortices has been accom-
plished in the laboratory (Chang, 1971; Davies-Jones, 1976), but the state of the art has not progressed to anywhere near that of wind tunnels for modeling straight winds. the forces using theoretical
The other alternative is to find
methods.
The main objective of this
study is to develop a better understanding of tornado forces on buildings, especially inertia forces, through numerical techniques such as the the boundary element method. 1.1.
Previous Work
Wen (1975) attempted to calculate drag and inertia forces produced by tornadoes on buildings by applying Morrison's equation.
This semi-
empirical equation had been used previously by Sarpkaya, et al. (1963, 1981) to calculate wave forces on structures, for uniformly accelerated flow.
Wen assumed that the equation was applicable to rotational flow
as found in a tornado vortex.
Morrison's equation takes the form
Q(t) = ^PCpDV(t)^ + J p c y a ( t ) where Q(t) = tornado forces per unit height of a building at time t V(t) = velocity at time t a(t) = acceleration at time t Cpj = aerodynamic drag coefficient C = inertia coefficient m p = mass density of fluid
(1.1)
D = instantaneous projected width of the building normal to the direction of the velocity The first term in Equation 1.1 is the drag force and the second is the inertia force.
Values of the drag and inertia coefficients for a
circular cylinder immersed in a two-dimensional uniformly accelerated flow were found by Sarpkaya (1963) to be 1.2 and 1.3, respectively.
Wen
recognized that in the case of tornadic winds with unsteady flow and varying acceleration, the values of C-. and C would not necessarily be u m "' the same.
In the absence of experimental data, he assumed values of 1.1
and 1.0 for C^ and C^, respectively. Hunt Equation
(1975) attempted
to compare
inertia forces obtained
1.1 and those obtained from a theoretical
from
solution for a
simple, time dependent inviscid flow problem, for which a closed form solution exists.
However, in making the two calculations, he inadver-
tently used different flow conditions in the two cases.
Hence, his
comparison of inertia forces is not based on equivalent flow conditions. Furthermore, the problem considered is a hypothetical one that has no direct application to a tornado-like flow. In view of the above discussion, there is a need for additional study of the problem of determining the forces on buildings produced by tornado-like flow.
To determine values of the drag and inertia coef-
ficients experimentally under tornado-like flow conditions is difficult, if not impossible.
The only alternative appears to be a theoretical
approach using the principles of fluid dynamics and available numerical techniques. bulent.
In reality, tornado-like flow is both viscous and tur5 The Reynolds number for wind flow is of the order of 10 . To
solve the problem numerically with such a high Reynolds number, even without considering turbulence, is a difficult task.
To include the
effects of turbulence would, perhaps, take several years.
In view of
the difficulty of the proposed problem, as a starting point, it seems reasonable
to assume
the fluid
to be
structure is assumed to be rigid.
inviscid.
In addition, the
A mathematical model is developed
with which the tornado forces on a building can be determined for conditions of inviscid flow.
The boundary element method is used to
solve the applicable equations.
Specific objectives of the research are
described in the next section. 1.2.
Objectives of the Research
The basic objective of this research is to estimate tornado forces on buildings
by applying
basic fluid dynamic
principles.
Specific
objectives are: 1.
To mathematically
model
the tornado
and
tornado-structure
interaction through fluid dynamic principles 2.
To solve the mathematical equations by the boundary element method to permit determination of forces produced by tornadoes on a building
3.
To develop a simplified method for calculating tornado forces and comparing the results with numerical model.
CHAPTER 2 MATHEMATICAL MODEL To solve any physical problem by numerical techniques, the physical phenomena must be represented in terms" of mathematical equations. Wind flow is both viscous and turbulent, but to solve the applicable mathematical equations is difficult, if not impossible, at this point in time.
One approach is to assume the fluid is inviscid.
In the case
of inviscid flow, the drag forces are zero and the resulting forces are inertia and lift forces.
First the governing differential equations
for a two-dimensional, inviscid flow are discussed. models of the tornado vortex are discussed.
Next, possible
A tornado model
that
satisfies the two-dimensional inviscid flow condition is selected and the resulting forces are found through numerical techniques. 2.1.
Fluid Dynamics Equations
For two-dimensional, incompressible, inviscid flow the governing fluid
dynamics
equations
can be written
Robertson, 1965; Connor and Brebbia, 1976).
from
standard
texts
(see
The equations in vectorial
notation are: Continuity Equation:
v • V = 0
(2.1)
Momentum Equation:
||- + (V-v)V = - - vp
(2.2)
where V = vector differential operator V = velocity vector, v i + v j for two-dimensional flow problems •^
p = pressure
X
y
Instead of solving Equations 2.1 and 2.2, which are nonlinear, the velocity and the pressure can be obtained by letting vx V = W
(2.3)
and VX = . ii V = -^ 3y ' > 3x '
f2 4)
^"^'^^
where W is the vorticity vector and ^^, is a stream function. Equation 2.4 automatically satisfies the equation of continuity (Eq. 2.1). Substituting the elements of Equation 2.4 into Equation 2.3, Poisson's Equation is obtained: 2 V Tjj = w over a region, n,
(2.5)
where 2
2
V 2 = 9 7 + _3 3x^ 3y'
and w
is the vorticity about the z axis. The boundary conditions required to solve Equation 2.5 are (1)
^ - ^ on r,
(2) |i = v^ on r, whe re
r, and
r^ represent complementary positions on the total
boundary r, -rr-is the partial derivative with respect to the outward normal and v is the velocity at the surface, as shown in Figure 2.1. By solving Equation 2.5 with proper boundary conditions, the velocity is found at any time t.
Even though the solved velocity is
3il/
sTT = ^ ° " ^2
'I' = ^ on r
FIGURE 2 . 1 .
1
DOMAIN AND BOUNDARY CONDITIONS
8 time dependent, the procedure for solving Equation 2.5 is independent of time.
Knowing the velocity, the pressure on a streamline is cal-
culated by writing the equation of motion (Eq. 2.2) along a streamline.
3t ^ 9? ^ T ^
-^3?
(2.7)
Integrating from point 1 to point 2 (Fig. 2.2), a modified form of Bernoulli's equation is obtained, which is applicable for unsteady flow along a streamline. fZ .w §,s.
(P2 - Pj) (V2 - V^") ' " ' ' " ' ' . ' ' ' ' : ' ' ' ' -0
1 3t
(2.8)
d
p
Subscripts 1 and 2 refer to two arbitrary points on a particular streamline.
For Equation 2.8 to be valid in inviscid flow, the vor-
ticity w must satisfy the relationship 3w
3w, 3w, z +V -1+ V ^ = 0 .
3t
X 3x
(2.9)
y 3y
which is obtained by taking the curl of Equation 2.2 and using Equations 2.1 and 2.3. Thus it is clear that an arbitrary vorticity cannot exist in inviscid flow. Equation 2.9.
Any vorticity distribution w^ must satisfy
If it does not, the flow is viscous and the differential
equations presented are not valid. A method to find the pressure on any point 2 on a streamline, knowing the pressure at a particular point 1 is presented in Equation 2.8.
If it is necessary to find the pressure on any arbitrary point in
FIGURE 2.2. POINTS ON A STREAMLINE
10 the domain, knowing the pressure at any point on the boundary, one needs to numerically integrate Equation 2.2 from the point of known pressure to the point of interest, i.e., P2 - Pi = 1
!fd>^^l^dy 3x dy
(2.10)
Since we are interested in finding the forces on a building, which happens to be a closed streamline, it is sufficient to know the relative pressure on the building with respect to a particular point. Hence, Equation 2.8 is used for the analysis. 2.2. Once
the
physical
Tornado Wind Field Models phenomena
are
represented
by
mathematical
equations, the next step is to describe the tornado wind field before the building
interferes with
the flow.
Before any discussion
of
different tornado models, the basic properties of tornado wind fields are described. Tornadoes are high-velocity, narrow-path windstorms, consisting of rotating columns of air with a low pressure core.
The vortex extends
down to the ground surface, or near the surface, and travels over the ground at rates of up to 70 mph.
The air within the tornado vortex
moves with a translational speed v. and has three additional velocity components in the tangential, radial
and vertical
directions
(Fig.
2.3). The radial distribution of tangential velocity
VQ
is assumed to
be symmetric about a vertical line through the tornado center.
The
tangential velocity is zero at the center and increases linearly from
11
Velocity Components: v^ = Translational
v = Tangential V
r
V.
=
Radial
=
Vertical
FIGURE 2.3.
THREE-DIMENSIONAL TORNADO WIND VELOCITY VECTOR
12 the center until it reaches a maximum value v excess of 200 mph.
which may be in e ,max -^
The distance from the center to the location of
^e,max ^'^ called the core radius r^.
Outside the core radius, the
tangential velocity decreases rapidly as the distance from the center increases and approaches ambient pressure at the vortex boundary. The radial and vertical wind velocity components in a tornado are modeled in various ways depending on whose model is being considered. Because these components are not considered in subsequent analyses, they are not discussed further. There are many different models proposed to represent tornado wind fields. gories:
They are grouped for discussion purposes into three cate(1)
meteorological
(3) laboratory
models.
models,
Meteorological
(2)
engineering
models
attempt
models, and to
satisfy
thermodynamic and hydrodynamic balances associated with tornado dynamics.
The available models are discussed extensively in the recent
reports by Lewellen, et al. (1980) and Redman, et al. (1983).
The
objective of an engineering model is to represent the tornado wind field in a simplistic manner that bounds the magnitude of the various wind components.
Extensive lists of available engineering models were
given by Seniwongse (1977) and Lewellen (1976).
Laboratory models are
attempts to create small scale vortices which may or may not be representative
of an
actual
tornado.
Davies-Jones
(1976) presented a
critique of the various laboratory models that have been published prior to that date. In the research described in this dissertation, as a starting point, we are interested in a two-dimensional inviscid flow.
For this
13 purpose, a simple
Rankine-Combined
vortex model
is adequate.
The
tangential velocity VQ varies as shown in Fig. 2.4; the radial and vertical velocities are assumed to be zero.
The variable a in Fig. 2.4
is a constant which influences the tangential velocity distribution. In addition, it is assumed that the tornado vortex moves with a uniform translational velocity v..
The region where v^ = ar is called a forced
vortex region and the stream function equation in this region is v^i|» = 2a
(2.11)
ar.2 The region where VQ = — ^ is called a freee vortex region and the governing equation in this region is v2,(, = 0
(2.12)
For a valid solution for forces in inviscid flow, the vorticity w satisfy Equation 2.9.
Since w
is either zero or a constant over the
entire region, when the tornado model Equation 2.9. v., w
must
is not moving, w
satisfies
When the tornado is moving with a translational velocity
satisfies Equation 2.9 in the separate free and forced vortex
regions but not at the interface between them (that is at the core radius r ) , where a discontinuity in w
occurs.
As a result, there is
a valid solution for velocities with the Rankine-Combined vortex model, but these velocities do not yield valid pressures and forces.
Solu-
tions with this vortex model moving through the domain showed erratic pressure values at successive time steps.
To avoid this problem and to
obtain a valid solution, the region for numerical solution is assumed to be in either the forced vortex or the free vortex field.
The
14
. Symmetrical about (j_
Free Vortex Region
Forced Vertex Region
Free Vortex Region
a is a constant ''c Inner ' Core
FIGURE 2.4.
RANKINE-COMBINED VORTEX MODEL
Tornado moves at translational velocity v. .
15 assumption that the solution region is in the forced vortex field is reasonable if the tornado core diameter is large compared to the size of the building.
The free vortex field could engulf a building if the
tornado passed the building at a distance greater than r . The numerical techniques for solving the applicable fluid dynamic equations in conjunction with the tornado wind field model are described in the next chapter.
CHAPTER 3 NUMERICAL TECHNIQUES The most commonly used numerical techniques for solving physical problems having irregular domains are finite difference, finite element and boundary element methods.
Of these, the first one approximates the
governing equations of the problem using local
expansions for the
variables, generally a truncated Taylor series.
It is difficult to
solve any problem with an irregular domain using this method. other two methods
deal with equivalent
integral
equations.
The These
integral equations allow one to represent even irregular regions if small enough elements are used.
Because of advancements in computer
technology, these methods are attractive to engineers and scientists for the solution of many physical problems. The three methods also can be classified as domain and boundary methods.
The finite difference and finite element methods are domain
methods, wherein the unknowns are in the domain.
In the case of the
boundary element method the unknowns are on the boundary.
Of the three
methods, the boundary element method is preferred for the tornadostructure interaction problem for the following reasons: 1.
The boundary integral equation itself is a statement of the exact solution to the problem posed and errors arise only because of the inability to carry out the required integration in a closed form.
2.
The velocities at the surface of the building are the unknowns and can be solved directly by the boundary element method. 16
17 3.
The boundary element method is efficient, if the areas of domain are large compared to the length of the boundary, as in this case.
4.
Data preparation is simpler because in this problem input data are only required on the boundary.
The two-dimensional
problem reduces to a one-dimensional line integral problem. 3.1.
Formulation of the Boundary Element Equations
To find the velocity distribution at any time t, Poisson's equation must be solved with proper boundary conditions (Eqs.
2.5 and
2.6).
integral
However, as an alternative approach, the boundary
equations may be formulated using a weighted residual
technique as
proposed by Brebbia et al. (1978 and 1984) or Banerjee, et al. (1981). Both sides of Equation 2.5 are multiplied by a weighting function (t) , which
is sufficiently
continuous to be differentiable as often as
required. Integrating over the whole domain yields
(v ifj) (t)ds^ =
w^^^dn
(3.1)
Integrating the Laplacian in the left expression twice by parts or using Green's second identity yields
2
3X
^'^ an
, 3*
^ an'
Q.
J n ^*^"
(3.2)
In order to eliminate the first domain integral in Equation 3.2, the weighting function is selected such that it yields V^(t> + 5. = 0
(3.3)
18 where 6 is a Dirac delta function, which has the following properties: 6.. = 0 for every point in the domain except point i 5. = » at point i and T{;
6. d Q = ij*.
(3.4)
Q.
The solution of Equation 3.3 is called the fundamental solution. the two-dimensional
case the fundamental
solution
for the
For
Laplace
Equation is * = ^
'"
(3.5)
where r is the distance from the point of application of the delta function to the point under consideration in the integration 3.1).
Substituting
Equation
3.3
into Equation
3.2, the
(Fig.
following
expression for a singularity at "i," is
^^ +
^
M. + 3n
w (|)dfi =
^ 3n
(3.6)
Q.
Equation 3.6 is valid for any point in the domain, but in order to formulate the solution as a boundary problem it is necessary to apply it at the boundary.
The general boundary integral equation for both
domain and boundary can be written as (Brebbia, 1978; Brebbia, et al ., 1984; Banerjee and Butterfield, 1981):
C.ii;. +
^ 3n
w (})dn =
n
li*dr an
(3.7)
where C. is a constant depending upon the position of the singularity.
19
Element
FIGURE 3.1.
USING CONSTANT BOUI^^AKT
20 On a smooth boundary, C^. = 1/2. points outside n, C^. = 0. to the solid angle.
For interior points, C. = 1, and for
On nonsmooth boundaries, C. is proportional
The boundary integrals are calculated over the
line enclosing the two-dimensional domain and the domain integrals are calculated over the area.
To evaluate the domain integral, the domain
must be divided into integration cells or elements, which is tedious. In the present problem (Eq. 3.7),
since w
is a constant, the
domain integral can be converted to a boundary integral (Brebbia, et al., 1984; Fairweather, et al., 1979; Danson and Kurch, 1983) by substituting (3.8)
(j> = V b
where b = ^
[.n (i) . 1]
Then using Green's second identity, i.e., V bdf^ =
(3.9)
3n
Q.
Equation 3.7 becomes
C.^. +
^ iidr +
w
2b dr =
z 3n
3n ^
(3.10)
^ 3n
Since all the integrations are boundary integrals, the discretization of the domain is simple and computationally efficient. 3.2.
Numerical Analysi_s
Because it is difficult to find an analytical solution to Equation 3.10 for a particular geometry and boundary conditions, a suitable
21 reduction of the equation to an algebraic form is required that can be solved by numerical methods.
The integral equation (Eq. 3.10) can be
discretized into a series of elements called boundary elements as shown in Figure 3.1.
The points where the unknown values are considered are
called "nodes" and are taken to be in the middle of each element for so-called "constant" elements.
While other elements are available,
constant elements are considered to be sufficient for the problem considered herein.
It is possible to utilize higher order elements
wherein the unknowns may vary linearly or quadratically from one node to another (Brebbia et al., 1984). With
the
constant
element
the
boundary is discretized into N
elements, of which N. of them belong to r values of ^ and v
(v s
= ^) are s 3n'
and N^ belong to r^.
The
constant on each element, and equal
to their values at the mid node of the element.
At each element the
value of one of the two variables, i.e., ^ or v^, is known.
Discretiz-
ing Equation 3.10 yields N
N ij, | i dr + i: C.ij^. + S T 1 i=l } r ^" j=l ^j ^ For constant
.
elements
' -r. J
the boundary
-•
^ '"
Equation
3.11
^|^*dr
(3.11)
J=l
is always
coefficient C. is identically equal to 1/2. r..
N
w f^ dr = I
"smooth." Hence the
The length of element j is
represents, in discrete form, the
relationship
between node i at which the fundamental solution is applied and all j elements (including the one in which i=j) on the boundary. V
The '1' and
values inside the integral in Equation 3.11 are constant within each
22 element and w^ is a constant or zero as assumed.
Consequently, all of
these can be taken outside of the integrals. This gives N Z
+
j=l The integrals
N Jr. 3n
N /-
ij; +
z
^
j=l
>.3n
j=H.
1 3i|'J
J
l^dr relate the i^^ node to element j over which the in3n
tegration is carried out.
Hence these integrals can be noted as H..
Similarly, the integrals
(|)dr
of w.
3b dr can be called B.. i°-
3n
T
ij^.
N
.
^ +
z
H
can
be
called G.. and the summation
Hence Equation 3.12 can be written as: N
^
+ B
= Z G.. (|i).
(3.13)
The integrals in this case can be evaluated analytically, as the fundamental solution and element geometry are very simple.
In general,
it is necessary or more convenient to integrate them numerically. Rewriting Equation 3.13 for the i
node and defining
H.. when i f j H.. = ^ . "J , ^-^ ' H. . + ^ when i = j
(3.14)
then Equation 3.14 becomes N
N
(1^)
/=1 "iJ 'j ' 'i ^ j=l '^^ ^
(3.15)
j
Applying the above equations at all N nodes, a set of equations is obtained that can be expressed in matrix form as H U + B = GQ
(3.16)
23 dr\)
where U and Q are vectors of ^ and -^ at all the nodes. Here the H and an G matrices and the vector B depend only on the geometry of the problem. They need not be calculated at each time step, even if it is a time Note that N values of ^ and N„ values of v are
dependent problem.
known on r (N^ + N2 = N) and hence one has a set of only N unknowns in Equation 3.16, which can now be reordered in accordance with the unknown under consideration.
Reordering Equation 3.16 with all the
unknowns on the left hand side and a vector on the right hand side that is obtained by multiplying the matrix elements by the known values of ip and V one obtains an equation of the form AX = Y
(3.17)
where X is the vector of unknown Vs and v 's. Because the A matrix is the same at each time step, it is decomposed only once using Cholesky's decomposition method.
Equation 3.17 is solved for the unknowns at each
time step by forward and backward substitution. The integrals H.., G.. and B. can be calculated using simple Gaussian quadrature rules for all elements (except the one corresponding to the node under consideration): 91 dr = ^ z (1^) w H. . IJ
*dr = J-
G. . IJ
r.
^ M^\ m=l
J
and j=i
and 3^1
z 1(|^) -r. 13n^ dr = w^z j-i .z T [--i 2 m=l 3n m wm-*] ^ and j^i
(3.18)
24 where ^^ is the element length and w^^ is the weight associated with the numerical integration point m. Usually, four integration points are sufficient to provide the required accuracy for two-dimensional problems. The integrals corresponding to the singular elements, H.., G.. and 3b
TTT dr can be computed
analytically.
Here the H.. and
— l^dr
terms, for instance, are identically zero due to orthogonality between the normal and the surface of the element, i.e..
|bdr =
1^ in dr = 0
3n
'j^i
and
l^dr =
"ii
3n
3r 3n
(3.19)
|i |I dr = 0 3r 3n J=i
J=i
3r
because 3]^= 0 ^o'^ i^J The G.. term can be derived as (Brebbia et al, 1984):
n Gii=7
I T I C^" 'i 1 + 1]
(3.20)
where Ji. is the element length.
3.3.
Example Problem
As an example, consider an 80 ft diameter circular building as shown in Figure 3.2.
The stream function ^is specified on the surface
of the building, which is the boundary r^. Since there is no flow perpendicular to the wall of the building, the wall becomes a streamline.
Any arbitrary value can be specified for ij;, because this is the
only region where essential boundary conditions are required.
In this
25
= v^ on T^ Boundary
^ H
1
I
1
1
I
H
V ij; = 2a or 0 in fi
9
Building Shape ij; = 0 in r^ Boundary o o
H
i
1
••
H
1-
400'
FIGURE 3.2.
DISCRETIZATION OF FLUID AND BUILDING BOUNDARY
26 case ^l, is taken as zero on r^.
This boundary condition alone will not
yield a valid solution. So, an outer fluid boundary Fg, where the velocity distribution is not disturbed by the building, has to be considered. Theoretically the outer fluid boundary r^ has to be an infinite distance from the center of the building.
For practical purposes, a distance of about five
times the radius of the building from the building's center is considered satisfactory. The velocity v on the surface r« is calculated from the relationship v^ = -v^n^ + v n, where n, and n, are the direc^ s x Z y l 1 c tion cosines of the outward normal with respect to the x and y axes, respectively. For the current work, the r elements.
boundary is divided into 64 equal
From a convergence study conducted on the number of ele-
ments, it was found that 96 elements gave essentially the same result with large number of boundary elements.
When giving the input, the
outer nodal numbering has to be given in counterclockwise direction and the inner nodal numbering in clockwise direction (Brebbia, 1978). 3.4 Determination of Forces on the Building After solving for the unknown velocities, v^, from Equation 3.17 at each time step, the pressure on the building is computed from Equation 2.8. Since we are interested in finding the resultant forces created on the building, it is sufficient to know the relative pressure on the building with respect to a particular point. So the pressure is assumed to be zero at an arbitrary point on the building to start with. To evaluate the term J, i^ ds the partial derivative |^ is first calcu-
27 lated using central finite difference and then Equation 2.8 cally
integrated from
point 1 to point 2.
is numeri-
A time step of 0.1 sec is
considered sufficient for computing ||. using central finite difference. From the known pressure on the surface, the forces in the x and y directions per unit height of the building are calculated by resolving the forces into the x and y components at each time step. 3.5.
Verification of the Model
Any numerical model developed should be verified by comparing the numerical solution to a closed form solution or a known solution by some other method, for a standard problem.
Only then can the model be
applied to an arbitrary problem with the expectation of obtaining a correct solution.
To verify the proposed model, the following unsteady
flow problem is considered (Hunt, 1975).
It is assumed that the fluid
is flowing with a velocity v. over a circular cylinder of radius a as shown in Figure 3.3.
In addition it is assumed that the fluid has a
rigid body rotation from the center of the circular cylinder (forced vortex).
The origin of time is taken as zero when v^ acts along the
positive X-axis. For inviscid flow, the standard solution for the velocity around a circular cylinder for a particular radius r, angle e and time t is as follows: 2 V = v j l - K) cos (e - 3) r t ^^ ^2 v. = V. (1 - K)
sin (e - a) + ar
28
Building Shape
6 = tan" (-ta)
FIGURE 3.3.
VELOCITY DISTRIBUTION FOR VERIFICATION OF THE NUMERICAL MODEL
29 where a is a constant and 6 = tan"^(-ta).
Knowing the velocity, the
pressures on the surface of the cylinder are calculated using Equation 2.8.
From the known pressures, the forces in the x and y direc-
tions per unit height of the building are calculated by integrating the resolved pressures in the x and y directions. The forces in the x and y directions can be derived as follows: -2iTpv.aa 1/1 + at^
1 + a t
Taking a = 1.333, v^ = 70 ft/sec, P = 0.002376 lb sec^/ft^ and a = 40 ft, values of Q and Q are evaluated using Equation 3.22. Values of ^ y QX and Qy are plotted and labeled "theory" in Figure 3.4 and 3.5, respectively. For the same flow and for the same parameters, the geometry is discretized as shown in Figure 3.2 in order to solve the problem using the numerical model. The velocities v and v at the outer boundary r X y c are calculated as follows: V = v.cos B- y a A
U
V = V.sin 3 + X a y t Knowing the velocities in the x and y directions, the velocity v^ on the surface r^ is calculated as follows: V3 = -v^n^ + v^n^
(3.24)
where n^ and n« are the direction cosines of the outward normal with respect to the x and y axes, respectively. Using this as the boundary condition on r
and keeping ^ = 0 on r^, the forces in the x and y
directions are determined by computer using the numerical model at
30
O
UJ
Q: OO •-I I —
o _J I =3
z a "O
c
I-l o
CO UJ Q : CJ UJ Q: I—
o o 00 I
o => u- o. u- o o o z o o z oo <: Qc: >-
Q2E O O
CO LU C3
(U/sdL>|) U0L:i09ULa-X 9M^ UL saojoj
O LU ^ I—
31
O
00
O =D LU in cc UJ
I-l a :
o1
_i
>- UJ
o 3: z: Ho: ^ UJ UJ o
HH h-
in
T3
=3 00 o.
O U
o C£. o O
c
1 O)
E 1—
UJ SL
O
U- O Z^
u. <: o >-
z o: o o oo UJ HH 3= OC 1—
0
•
ro Ul Q:
C3
(:^^/sdL>|) uoL:i03JLa-A 9M^ UL saoaoj
32 specific time intervals. These forces are also plotted in Figure 3.4 and 3.5 and are labeled "computer."
Comparison of the values of Q and ^x
Q^ by "theory" and "computer" indicates good agreement in this case. Thus, the method is validated.
CHAPTER 4 RESEARCH FINDINGS In Chapter 3 a numerical model for solving the two-dimensional inviscid flow problem for a tornado is developed and verified.
Using
the model, the tornado forces on a building with a circular cross section are calculated for both forced vortex and free vortex flow. The model can be used to calculate forces on any arbitrary shape.
To
show the capability of the model to calculate forces on any cross sectional shape, forces are calculated on a square and a rectangular cross section for forced vortex flow. By properly interpreting the various force components obtained from the numerical model, a simplified procedure is derived to calculate
tornado
Morrison's
forces
Equation
using
a semi-empirical
for tornado-like
flow.
equation The
results
similar to from
the
simplified procedure compare favorably with those obtained from the numerical model. 4.1.
Tornado Forces on Buildings from Invisci'd Flow Theory
Suppose a tornado vortex is translating along the x axis from left to right. V
If a is a constant that defines the forced vortex flow and
is the translational
velocity of the tornado, then the velocity
components in the x and y directions are given by V
= V. - ya X t V = (x - V.)a y ^
(4.1)
The origin of time t is taken when the tornado center coincides with the center of the building.
Knowing the velocities in the x and y 33
34 directions, the velocity v^ on the surface of the building r^ (Fig. 3.2) is calculated using Equation 3.24. knowing that
Knowing v^ on surface r^ and
* = 0 on the r^ boundary, the resultant forces on the
building in the x and y directions can be calculated for each time step using the previously described numerical procedure. As an example, the tornado forces on a circular building having a diameter of 80 ft are calculated for the case where a = 1.333 and v^ = 70 ft/sec. The x and y components of the resultant forces are plotted as a function of time in Figure 4.1. To show the capability of the computer model to calculate forces on any cross sectional shape, forces are calculated on a square and a rectangular cross section having the same area of cross section as that of a circular cylinder having a diameter of 80 ft as shown in Figure 4.2.
The flow is assumed to be the same forced vortex flow as that for
the circular cylinder in this section.
The x and y components of the
resultant forces are plotted as a function of time for the square and rectangular cross section in Figures 4.3 and 4.4, respectively. Because Wen (1975) proposed to use Morrison's Equation for calculating tornado loads on structures, the results obtained from the use of Morrison's Equation are compared with results from the computer model.
To compare the results from the two methods, the same flow
conditions (Eqn. 4.1) are used.
The forces in the x and y directions
can be written as follows:
Q,(t) =ipcA(t) (4.2)
Qy(t) = r V S ( t )
35
w
:P/
...
C3
na o
o
o
-J
u o S•r-
&'^
I X
I >-
o
I—I
,0
•r— 't—
0
=3 CO
4-
3 o _l
cc u<:
o
_J X => U l
o »— on QC •-< O
ca
n-f
o
P
>
Ul
z o o on o
.Ei
00 u_
d (/)
"O
c= o u
iZI
CO
• E
p
0 01
-J Ul
z o o o <-) :s U l oc C_} LU
O =J
C2i
Ll.
Q.
o
o
z z: o on /
a: ID
w
1^
(C
o
-^
rt
«
-*
o
-•
•in
I
:^j./sdL>| UL aojoj
I
I
I
36
Direction of Tornado Travel
a) Square Cross Section
Direction of Tornado Travel
00
58' b) Rectangular Cross Section FIGURE 4.2.
OTHER BUILDING SHAPES CONSIDERED IN APPLYING THE COMPUTER MODEL
37
N
pCO
/
c •o ^
c o •r-
+J (J
4-> U
L. •r-
•r-
Q
O
1 X
ef
1
o -J 2 H-l O => _ J
1 1 1 1 1
(U
I-H
CO u_ LU X QC LU
/
1
•a: I— =3 Q :
P
>-
cro oo > •a: Q
d
D H-
Z
o on o
P
0 0 Li_
d
in
•a c o u O)
0
oo OJ
^
E
O -J O . LU 21 Q O O O 21
UJ on O LU Q i I— O =3
u. o o a: o
m
CO
./ .P^'
LU
.?^' W N
LU CJ
I
:;^/sdL>| UL aouoj
ft I
I
on cu
a. 2: o o o
Q:
38
on on o _i =D Q CD LU Z O
«a: Q: I— o O Ll_ LU
Q:
Z
I-H
•a: O
Q O
00 z : in
•a c o o
I— z on
o =3 Q- O.
o o o o on on o uLL.
o z
Q •-•
on *-t o ^ I— OQ
LU
en Z3 C5
Co
;D
ift
"^
ffJ
W
"H
^J./SdL>| UL 90J0J
39 where a^(t) and 9^(1) are accelerations in the x and y directions, respectively. 1.1).
The other variables are as defined in Chapter 1 (Eq.
The acceleration terms that have been previously used by Wen in
Morrison's Equation are: 3V
'x
3V
3v
'--n*\^*'y^ (4.3) 3v
a
3V
3V
=— ^ + V —s_ + w
y
3t
X 3x
y.
y 3y
Substituting v^ and v^ from Equation 4.1 into Equation 4.3, the accelerations a^ and a at the center of the cylinder and the inertia force can be written as follows:
\
lx=y=0 = \ t " ••
^ lx=y=0 = ° (4.4)
^x' PV« \*" • y^" Comparing the results from Morrison's Equation given in Equation 4.4 with results from the computer model as shown in Figure 4.1, it is clear that the two results do not agree.
When t < 0, Q is negative
according to Equation 4.4, whereas Q is positive according to the computer
model.
The force in the y direction is zero from Equation
4.4, whereas the force component in the y direction calculated from the numerical model is not zero.
From this we can conclude that the use of
acceleration terms given in Equation 4.3 do not give correct values for tornado flow assumed.
40 1^1:—Simplified Procedure to Calculate Tornado Forces rrom inviscid Flow Theory An approach for applying Morrison's Equation, which is in agreement with results from the numerical model -for a tornado-like flow is proposed herein. Tornado forces are obtained for both forced and free vortex flow from the results of the numerical model and by applying basic fluid dynamics principles. 4.2.1
Forced Vortex Flow
The resultant forces produced by a forced vortex in inviscid flow can be classified as inertia and lift forces. inviscid flow are zero. of the fluid.
The drag forces in
The lift forces are produced by the rotation
The lift forces may be derived using the Kutta-Joukowski
theorem (Robertson, 1965; Eskinazi, 1962).
The theorem states that the
total force per unit length on a cylinder placed in a uniform stream velocity v. is equal to the product of the density of the fluid P, the circulation around the cylinder and the stream velocity v^ of the fluid.
The direction of the force is normal to v^ and the axis of the
cylinder.
The three vectors, free-stream velocity, vorticity, and lift
force form a right-hand triad in the coordinate system.
The circula-
tion, c is defined as the integral of the surface velocity around a closed loop.
C^fv^ds
(4.5)
When the forced vortex moves from left to right along the x axis, the lift force acts in the negative y direction. Q
= -pcv^ = -2iipaa v^
(^-6)
41 The inertia force in tornado-like flow may be calculated using the second term of Morrison's Equation (Eqn 1.1). ^ " ^m P^^*"^^ 0^ C'^oss Section) x (acceleration)
(4.7)
where C^ is the inertia coefficient and p is the density of air.
The
modification required for this case involves the calculation of the acceleration
components.
Under conditions of rotational
flow, the
acceleration of each fluid particle is toward the center of rotation. This acceleration is the same, even if the flow is unsteady, as described
in Equation
4.1.
This acceleration
toward
the center of
rotation can be calculated using Equation 4.3, the result of which is Equation 4.4.
Because of the circular motion of the fluid particles,
centrifugal forces are exerted on the building by the fluid particles. These forces point in a direction away from the center of rotation, and hence must have an opposite sign to that of Equation 4.3.
*x = -
3v, 3V '^ + V 3t X 3x
'^\
+ V
+
3t
V
(4.8)
5v ' ^
X
3y_
y
^\ J'
'y'--
3V
+
3x
-TT^
V
y
^y _
Furthermore, when the entire tornado is moving, the change in velocity with respect to time for a point must be added.
The final equations
for the acceleration components thus become
3V^
a
= 3t 3V
y
3t
9V
3V
- A + V —^ + V ^ 3t X 3x y 3y 3v 3t
9v^ x 3x
3V + V
y 9y J
(4.9)
42 After simplification. Equation 4.9 becomes
I
= -
X
—-
V
V 3yJ
[x 3x
[
V
-\
+ V
3V„
(4.10)
3V 1
- ^ + V —^ X 3x y 3y
In order to verify the above relationships, inertia force components in the x and y directions were calculated using Equations 4.7 and 4.10.
The results are compared for several cases. The value of C m
is 2.0 for constant accelerated flow over a circular cylinder, in the case of inviscid flow theory.
This has been established through a
closed form solution (Sarpkaya and Isaacson, 1981). flow the value of C
is not available from theory.
For rotational
At this stage, it
is assumed that the value of C^ is 2.0 for rotational flow. of this chapter, after comparing the results of the
At the end
simplified pro-
cedure with computer results, it can be concluded that the C value is •^ m 2.0 for rotational flow.
Using C
= 2.0 in Equation 4.7 and adding the
lift forces yields values of the x and y components of the forces. calculate the accelerations
To
in the x and y directions, a general
expression for velocities in the x and y directions are required for a tornado moving
a perpendicular distance D from the center of the
building and at an angle e from the x axis (see Fig. 4.5). The velociV = v.cose at- y'a ties in the x and y directions any time t for forced vortex flow are X t (4.11) V = v.sine + x'a y t
43
Tornado Path
^x
FIGURE 4.5. TORNADO PATH RELATIVE TO BUILDING LOCATION
44 where x' = X + D sine - v.t cose (4.12)
y' = y - D cose - v^t sine
In the above equations it is assumed that the time t is zero when the tornado center coincides with point A in Fig. 4.5.
For various values
of 9 and D the expressions for lift and inertia forces in the x and y directions are designated as Cases 1, 2 and 3 in Table 4.1. From these values, the forces in the x and y directions are calculated using Equation 4.5 and adding lift forces.
These forces are compared with
computer results in Figures 4.6 through 4.11.
From these figures one
can conclude that the results from the simplified procedure agree very well with computer results for a forced vortex flow. 4.2.2
Free Vortex Flow
For free vortex flow, it is found that Equation 4.10 is not valid to compute accelerations, since the fluid is irrotational. Equation 4.3 is used to calculate the forces.
Instead,
For a general free
vortex flow moving as in Figure 4.2, the velocity distribution in the x and y directions at any time t can be derived as follows: ar^y' V = V. cose - — « — X t r^ (4.13) ar^ x' V = V. sine + — « y t r^ where x' and y' are defined by Equation 4.12 and r is given by: r 2 = (x')2+ (y")^
(^-1^)
45
4J CM
C5
u s. o
cc o
3
CVJ
CsJ
•r4->
s-
o
O Q. t=
Q.
o
LU
&-
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CVJ
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46
N
Simplified Computer
Case 1 Forced Vortex Inviscid Flow
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OC *-• « c c i —I CJ O . C3.
/
4
^^ S
o •—' o CJ oo U -
fl
f
JV
LU
/f
a:
/ #
C!i
#
G
,:^
lO
C*
150
U5
"^
•rt
:;^/sdL>i «uoL:t39JLa-x UL gojoj
51
r W
> - LU CQ O O O »—I \O LU
on LU I— =3
3
on o.
a
HH s : - I o O ll. I CJ >X a LU
to
c o CJ OJ
(/) I
E
sc-z. yI—I ef oc O CO LU > LU CC
CJ Z3 O Q : O LU O LU CJ U. CJ o^
o LJ.
o
cc Lu
o a. Z O CO I—I
Q CO LU »—• LU u. to
a : I-l «C eC —I CJ Q. O. S S QC
O •—I O CJ 00 U .
CC =3 CD
7 ^i/sdi>| 'uon^a-iia-A "t 33-'°J
52 The expressions for acceleration components in free vortex flow for all three cases can be derived as in forced vortex flow using Equation 4.3. Because the procedure is somewhat lengthy, though straightforward, the general expressions for accelerations are listed in Table 4.2 without derivation.
For free vortex flow the circulation is zero around the
cylinder; hence, the lift force is zero.
The forces obtained by the
simplified procedure are compared with those obtained by the computer model in Figures 4.12 to 4.17, assuming r = 100 for each case. The values from the simplified procedure differ somewhat from the results of the computer model for forced vortex flow.
This difference
may be because of error caused by time discretization in the computer results.
The difference appears large compared to those for a forced
vortex flow, but is due mainly to a difference in scale of the ordinates. Thus we can conclude that the results from the simplified procedure agrees very well with computer results for a circular cylinder. Calculating forces using the simplified procedure looks simple because the value of C is the same for any direction of flow due to axisymm metry of the circular cylinder. For any other cross section the values of C vary with the direction of flow; hence, further research is m needed to establish an appropriate value of C^^.
The computer model
will be useful to calculate the C^ coefficient for any cross section and for any direction of flow. compute
Also, the computer model is useful to
resultant forces on any arbitrary cross section due to forced
or free vortex flow in any direction.
53
TABLE 4.2 ACCELERATIONS IN FREE VORTEX FLOW
Keeping k = ar
3v^
3v 3t
kv^sine
2ky'v^(x'cose + y'sine)
kv^cose ' "~^
2kx'v^(x'cose + y'sine) ^
"^
^^x _ 2ky'x' !!x_ _ k ^ 2ky'2 "?x "^ » 3y " " •;:7 r4
^
_ k
3x ~ 7 ^
2kx'2 "Pi
^ '
_
3y
^
and
3v^ dy 3v aX= - 3t r ^ + vX ^3x+ v . y 3y^ 3Vy I
y
= — ^
3t
av + V
——
X 3x
3v + V
2kx'y'
—-
y 3y
54
T s o 1—
X
i f
•!->
J- - a
O -1f—1
> o
to Qi (U • ! in Qi > (0 i. c CJ U_ I - l
> - LU CO Q
O
• ^
I
z z o
t-
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LU I— ^ O. O O CJ _ J Ll.
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56
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o z z: o HH »— CJ LU OC Q I
X
oc LU »— =3 Q. O O O _l
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to •o c o o OJ
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O LU LU U . CJ LU O OC U . OC U O Q. Z O CVJ O LU t/0 I—• LU
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.—I
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=) C3
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vl
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UL 9 3 J 0 J
57
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o o
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OC UJ I— =3
OC a .
o oo >-
o
u-
Z Z X HH < U J CO UJ CJ OC O U.
h-
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<3
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*
58
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CO
_i LU
o o
z o ^
"^ t^
1—1
oc
1— LU
O 1— LU =3 O.
oc HH
3 ^ ~^
Q1
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t*
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o o u.
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X <3: LU
to •o
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c o u
OJ 00 1 0)
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LU o: o>
o =3 cc Q
O
LU LU CJ LU O OC U . CC u . O O. LL.
A
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ITJ
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irt -*
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o n
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im
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H.I LU oo 1—1 u. in cc I - l < _ l C3 < CL. a .
4.16.
<0 irt
1—
59
«
N -J >- LU QO O
^
z o z: o 1—1 on
•
»- LU
t^
CJ t—
LU ID OC O. I-H
z: Q1 CJ O -JO3 >-
f*
z
u. o Z X
cC LU »— 00 LU QC LU QC O CJ ID > QC LU LU O a U. CJ LU t—<
to T3 C
«
6
o a
.
w d
on cc Lu U- o O. o
A
z O
Q
CO
LU 00 1—1 LU t-^ U. 00 —> < : en _l CJ
^
id
E 4.17.
«
•^
o c ID
6
iO
t^^/sqL *uoL:;o9wiLa-A
UL 9 0 J 0 J
o o
z: OC HH o 00 LJ_
60 In reality, fluid flow is viscous and turbulent.
The value of C m
for any cross section will be reduced from the value for inviscid flow. Sarpkaya (1963) found that the C^^ value reduced from 2 to 1.3 for a straight, constant accelerated flow over a circular cylinder. rotational flows found in tornadoes, the C viscous flow.
For
values are not known for
It is believed that C^^ value will be close to one or
slightly greater than one for most cross sections. The effect of viscosity and turbulence on lift forces is not known.
Further research is needed in this area.
In view of the above
discussions, it can be concluded that the magnitude of the forces due to inertia and lift in tornadic flow are of the same order of magnitude as drag forces.
Hence, the inclusion of these forces in the design of
buildings may be important.
CHAPTER 5 CONCLUSION AND NEED FOR FUTURE RESEARCH 5.1 A numerical
model
with
Conclusion
sufficient
simplifying
assumptions
to
represent interaction between a tornado and a rigid structure has been developed.
A numerical solution for the real situation requires the
consideration of both viscosity and turbulence.
However, to include
the effects of viscosity alone is very complex; treatment of turbulence is even more difficult.
In addition, a tornado wind flow model con-
sidering viscosity and turbulence is itself not well established.
In
view of these difficulties, as a first step, the following assumptions are made in developing the numerical solution presented. assumed to be inviscid.
The fluid is
The tornado is modeled by a Rankine-Combined
Vortex and the fluid flow is assumed to be two-dimensional.
With the
assumption of inviscid flow, the drag forces on the building become zero.
Only the inertia and lift forces are present.
The use of the
Rankine-Combined Vortex model implies free and forced vortex flow in different regions, with a discontinuity in the flow field between them as reported in Chapter 2.
Mathematical difficulties in determining
forces which are presented by this discontinuity in the tornado model are avoided by assuming that the building is located either entirely in the free or entirely in the forced vortex region. Applicable mathematical equations are solved using the boundary element method.
Details of the numerical procedure are described in
61
62 Chapter 3.
Tornado forces on rigid buildings of any arbitrary cross
section may be computed using the numerical procedure. Results using the numerical method have been obtained for various tornado path directions and building shapes. the various
force
By properly interpreting
components obtained from the numerical model, a
simplified procedure is derived to calculate the same forces using equations similar to Morrison's Equation for tornado-like flow.
Al-
though the numerical model is relatively easy to apply, use of the simplified procedure eliminates
the need for computer calculations.
The results from the simplified procedure compare favorably with those obtained from the numerical model as presented in Chapter 4. Thus, the principal contribution of this research is a f i r s t step in the process of solving for the forces on buildings during tornadoes through f l u i d dynamic principles. used.
The boundary element method has been
The assumption of inviscid flow results in only obtaining the
inertia and l i f t forces acting on a structure immersed in a tornadolike vortex.
A secondary contribution is the development of a simpli-
fied procedure for obtaining tornado l i f t
and inertia forces in a
building in inviscid flow conditions. 5.2
Directions for Future Research
A complete numerical solution of the tornado-structure interaction problem for the real situation could possibly take five to ten years. The use of
supercomputers also may be necessary to carry out the
voluminous calculations.
The boundary element method shows consider-
able promise as an approach to solving the problem.
Possible future
63 subjects
that must
be
investigated
in the course of solving the
tornado-structure interaction problem include: 1.
A mathematical model of the tornado wind field which includes the effects of viscosity and turbulence.
2.
A numerical
model
of tornado-structure
interaction which
includes the effects of viscosity and turbulence. 3.
A relationship between C^ and C^ from the numerical model which can be used in a form of Morrison's Equation.
4.
An empirical method for determining design loads on structures subjected subjected to tornadoes which can be obtained from the results of the numerical model.
Future attention to the above listed topics could lead to an extensive research program that no doubt would open up many other areas of research for the effects of wind forces on buildings and other structures.
The technology and computer hardware are available to im-
mediately pursue these problems.
LIST OF REFERENCES 1.
Banerjee,
P.K.
and
Butterfield
R
2.
Brebbia, C.A., 1978, The Boundary Elempnt Vothnw ^. Pentech Press, L o n d o n j - W T i t i i T ^ i l T N i S ^ ^
r
3.
Brebbia, C A . , Telles, J.C.F. and Wrobpl i r Element Techniques. Sp^inger-Verlag, New Y o V k ^ ; :
D . ' ^^^^^^^^
4.
Chang, .CC 1971, "Tornado Wind Effects on Buildings and Strur tures with Laboratory Simulation," Proceedings. Third^Tntp.n.t-"'" a Conference on wind Effects on BuiIdings^and Structures art I I , No. 6, Tokyo, Japan, pp. 231-240. uccureb, rart
^'
^IT%\''A'C<
6.
Danson, D.J. and Kurch, G, 1983, "Using BEASY to Solve Torsion Problems in Boundary Elements," 5th International Seminar, Hiroshima, Japan, Springer-Verlag, New York, NY.
7.
Davies-Jones, R.P., 1976, "Laboratory Simulation of Tornadoes," Proceedin9S, Symposium on Tornadoes: Assessment of Knowledge and Implications for Man, Texas Tech University, Lubbock, TX, pp. 151-
8.
Eskinazi, S, 1962, Principles of Fluid Mechanics, Allyn and Bacon, Inc., Boston.
9.
Fairweather, G., Rizzo, F . J . , Shippy, D.J. and Wu, Y.S., 1979, "On the Numerical Solution of Two-Dimensional Potential Problems by Improved Boundary Integral Equation Method," Journal of Computational Physics, Vol. 3 1 , pp. 96-112.
10.
Hunt, J.C.R., "Discussion on Dynamic Tornadic Wind Loads on Tall Buildings," Journal of the Structural Division, ASCE, No. STll, pp. 2446-2449.
11.
Lewellen, W.S., 1976, "Theoretical Models of the Tornado Vortex," Proceedings, Symposium on Tornadoes: Assessment of Knowledge and Implications for Man, Texas Tech University, Lubbock, TX, pp. 147143.
12.
Lewellen, W.S. and Sheng, Y.P., 1980, "Modeling Tornado Dynamics," Final Report (January 1976-May 1980), submitted to U.S. Nuclear Regulatory Commission, Washington, DC.
IQAI
D
^
loo/i
•
^"K. ^'^ebbia, C.A., 1976, Finite Element Techniques for Fluid Flow, Newnes-Butterworth, Lond5H: ^"^^
64
65 13. McDonald, J.R., 1970, "Structural Response of ;, Tu,on^. c^ Building to the Lubbock Tornado," S t o ^ Research Repor^^^^^^^^^ Texas Tech University, Lubbock, TX. '^eporL :)KKUI, 14. Mehta, K.C., McDonald. J.R., Minor, J.E. and Sanger A l IQ7I "Response of of Structural Systems to the Lubbock Storm " Itlk Research Report SRR03, Texas Tech University, Lubbock. TX^^ 15. Minor, J.E., Mehta, K.C. and McDonald, J.R., 1972, "Failure nf Structures Due to Extreme Winds," Journal of the Structural Division, ASCE, Vol. 98, No. STll, Proc. Paper 9324, pp 245516. Redmann, G.H., et al., 1983, "Windfield and Trajectory Models for Tornado-Propelled Objects," report submitted to Electric Power Research Institute, Palo Alto, CA. 17.
Robertson, J.M., 1965, Hydrodynamics in Theory and Application Prentice-Hall, Inc., EngIewood Cliffs, NJ. '
18.
Sarpkaya, T., 1963, "Lift, Drag and Added-Mass Coefficients in a Time-Dependent Flow," Journal of Applied Mechanics, ASME, Vol. 30, No. 1, pp. 13-15.
19. Sarpkaya, T. and Garrison, C.J., 1963, "Vortex Formulation and Resistance in Unsteady Flow," Journal of Applied Mechanics, ASME, pp. 16-24. 20.
Sarpkaya, T. and Isaacson, M., 1981, Mechanics of Wave Forces on Offshore Structures, Van Nostrand Reinhold Co., New York, NY.
21.
Seniwongse, M.N., 1977, "Inelastic Response of Multistory Buildings to Tornadoes," Ph.D. Dissertation, Texas Tech University, Lubbock, TX.
22.
Wen, Y.K., 1975, "Dynamic Tornadic Wind Loads on Tall Buildings," Journal of the Structural Division, ASCE, No. STl, Proc. Paper 11045, pp. 169-185.
