Tornado on Building

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 to my family members for their

sacrifice, support and encouragement throughout the course of my

doctoral program.

n

TABLE OF CONTENTS

Page

ACKNOWLEDGMENTS ii

LIST OF TABLES v

LIST OF ILLUSTRATIONS vi

1. INTRODUCTION AND RESEARCH PROBLEM 1

1.1 Previous Work 2

1.2 Objectives of the Research 4

2. MATHEMATICAL MODEL 5

2.1 Fluid Dynamics Equations 5

2.2 Tornado Wind Field Models 10

3. 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

4. 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 Forced Vortex Flow 40

4.2.2 Free Vortex Flow 43 5. CONCLUSION AND NEED FOR FUTURE RESEARCH 61

5.1 Conclusion 61

5.2 Directions for Future Research 62

i i i

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 28

3.4 COMPARISON OF FORCES IN THE X-DIRECTION FROM THEORY AND COMPUTER MODEL RESULTS 30

3.5 COMPARISON OF FORCES IN THE Y-DIRECTION FROM THEORY AND COMPUTER MODEL RESULTS 31

4.1 TORNADO FORCE COMPONENTS ON A CIRCULAR BUILDING FROM COMPUTER MODEL IN FORCED VORTEX FLOW 35

4.2 OTHER BUILDING SHAPES CONSIDERED IN APPLYING THE COMPUTER MODEL 36

4.3 TORNADO FORCE COMPONENTS ON A SQUARE BUILDING FROM COMPUTER MODEL IN FORCED VORTEX FLOW 37

4.4 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

4.7 COMPARISON OF FORCES IN Y-DIRECTION BY SIMPLIFIED PROCEDURE AND COMPUTER MODEL FOR CASE 1, FORCED VORTEX FLOW 47

VI





4.12 COMPARISON OF FORCES IN X-DIRECTION BY SIMPLIFIED PROCEDURE AND COMPUTER MODEL FOR CASE 1, FREE VORTEX FLOW 54

4.13 COMPARISON OF FORCES IN Y-DIRECTION BY SIMPLIFIED PROCEDURE AND COMPUTER MODEL FOR CASE 1, FREE VORTEX FLOW 55





v n

CHAPTER 1

INTRODUCTION AND RESEARCH PROBLEM

Each year in the U.S. tornadoes damage many buildings and struc

tures. 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 wind-

resistant 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 post-

storm 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 other alternative is to find

the forces using theoretical methods. The main objective of this

study is to develop a better understanding of tornado forces on build

ings, 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 ) (1.1)

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

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 (1975) attempted to compare inertia forces obtained from

Equation 1.1 and those obtained from a theoretical 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. In reality, tornado-like flow is both viscous and tur-5

bulent. 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 inviscid. In addition, the

structure is assumed to be rigid. 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 physi

cal 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. Next, possible

models of the tornado vortex are discussed. 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 from standard texts (see

Robertson, 1965; Connor and Brebbia, 1976). 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 • X y

p = pressure

Instead of solving Equations 2.1 and 2.2, which are nonlinear, the

velocity and the pressure can be obtained by letting

v x V = W (2.3)

and

V = . ii V = -^ f2 4) X 3y ' > 3x ' ^"^'^^

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 + _3

7 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 1

FIGURE 2 . 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.

§,s. ' " ' ' " ' ' . ' ' ' ' : ' ' ' ' -0 (2.8) 1 3t p d

fZ .w (P2 - Pj) (V2 - V^")

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 . (2.9)

3t X 3x y 3y

which is obtained by taking the curl of Equation 2.2 and using Equa

tions 2.1 and 2.3. Thus it is clear that an arbitrary vorticity cannot

exist in inviscid flow. Any vorticity distribution w^ must satisfy

Equation 2.9. 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 (2.10) 3x dy

Since we are interested in finding the forces on a building, which

happens to be a closed streamline, it is sufficient to know the rela

tive pressure on the building with respect to a particular point.

Hence, Equation 2.8 is used for the analysis.

2.2. Tornado Wind Field Models

Once the physical 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

<L OF TORNADO CORE

Velocity Components:

v = Translational

v =

V = r

V. =

Tangential

Radial

Vertical

FIGURE 2.3. THREE-DIMENSIONAL TORNADO WIND VELOCITY VECTOR

12

the center until it reaches a maximum value v which may be in e ,max -^

excess of 200 mph. The distance from the center to the location of

ê,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. They are grouped for discussion purposes into three cate

gories: (1) meteorological models, (2) engineering models, and

(3) laboratory models. Meteorological models attempt to satisfy

thermodynamic and hydrodynamic balances associated with tornado dy

namics. 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 repre

sentative 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

vi|» = 2a (2.11)

2 ar. 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 must

satisfy Equation 2.9. Since w is either zero or a constant over the

entire region, when the tornado model is not moving, w satisfies

Equation 2.9. When the tornado is moving with a translational velocity

v., w 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

Inner

''c

' Core

a is a constant

Tornado moves at translational velocity v. .

FIGURE 2.4. RANKINE-COMBINED VORTEX MODEL

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 de

scribed 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. The

other two methods deal with equivalent integral equations. 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 tornado-

structure 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 integra

tion in a closed form.

2. The velocities at the surface of the building are the un

knowns 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 equa

tion must be solved with proper boundary conditions (Eqs. 2.5 and

2.6). However, as an alternative approach, the boundary integral

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

Q.

3X , 3* ^'^ an ^ an' 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

Q.

T{; 6. d Q = ij*. (3.4)

The solution of Equation 3.3 is called the fundamental solution. For

the two-dimensional case the fundamental solution for the Laplace

Equation is

* = ^ '" <F^ (3.5)

where r is the distance from the point of application of the delta

function to the point under consideration in the integration (Fig.

3.1). Substituting Equation 3.3 into Equation 3.2, the following

expression for a singularity at "i," is

^^ + ^ M. + 3n Q.

w (|)dfi = ^ 3n (3.6)

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 n w (})dn = 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^ÂKT

20

On a smooth boundary, C . = 1/2. For interior points, C. = 1, and for

points outside n, C . = 0. On nonsmooth boundaries, C. is proportional

to the solid angle. 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

where

(j> = V b

b = ^ [.n (i) . 1]

Then using Green's second identity, i.e.,

(3.8)

V bdf = Q.

3n (3.9)

Equation 3.7 becomes

C.^. + ^ iidr + ^ 3n

w 2b dr = z 3n 3n ^ (3.10)

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 and N^ belong to r^. The

values of ^ and v (v = ^) are constant on each element, and equal s s 3n'

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 ' -• N C.ij^. + S T 1 i=l }

ij, |i dr + i: r " j=l -^j ^ J

w f^ dr = I r. ^ '" J=l

^ | ^ * d r (3.11)

For constant elements the boundary is always "smooth." Hence the

coefficient C. is identically equal to 1/2. The length of element j is

r.. Equation 3.11 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. The '1' and

V 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 Jr. 3n

N ij; + z

^ j=l >.3n

N /- 1 3i|'-

j = H . J J

The integrals l^dr relate the i^^ node to element j over which the in-3n

tegration is carried out. Hence these integrals can be noted as H..

Similarly, the integrals (|)dr can be called G.. and the summation

of w. 3b i°- dr can be called B.. Hence Equation 3.12 can be written as: 3n T

ij . N . N

^ + z H ^ + 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^) (3.15)

/=1 "iJ 'j ' 'i ^ j=l ' ^ ^ 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

dependent problem. Note that N values of ^ and N„ values of v are

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 corres

ponding to the node under consideration): 91 dr = ^ z (1^) w

*dr = J- ^ M^\ (3.18) r. m=l J

H. . IJ

G. . IJ

and

1^ dr = w^ z [--i z (| ) w ] - 3n z . T 2 1 3n m m-* r. j-i m=l

and 3^1 ^ and jî j=i

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 prob

lems.

The integrals corresponding to the singular elements, H.., G.. and

3b l^dr TTT dr can be computed analytically. Here the H.. and —

terms, for instance, are identically zero due to orthogonality between

the normal and the surface of the element, i.e..

and

|bdr = 3n

'jî

1^ in dr = 0 3r 3n

"ii l^dr = 3n

|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

G i i = 7 I T I C^" 'i + 1] 1

where Ji. is the element length.

(3.19)

(3.20)

3.3. Example Problem

As an example, consider an 80 ft diameter circular building as

shown in Figure 3.2. The stream function îs 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 stream

line. 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

o o

Building Shape

ij; = 0 in r^ Boundary

9

H i 1 ••

400'

H 1-

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 con

sidered 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 boundary is divided into 64 equal

elements. 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 is numeri

cally integrated from point 1 to point 2. 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 = vjl - 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 Equa

tion 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

Q and Q are plotted and labeled "theory" in Figure 3.4 and 3.5, X y

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- ya 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 c o o u- o. u- o o o z o o z oo <: Qc: >-

Q- O 2E LU O ^ O I—

CO

LU

C3

(U/sdL>|) U0L:i09ULa-X 9M^ UL saojoj

31

O 00

in T3 c O U - UJ

o UJ o 3: z: H-

o: ^ UJ HH h-

=3 00 o. UJ SL o o C£. O O U- O

Z^ u. <: o

>-z o: o o oo UJ HH 3= OC 1— <c ck. :^ 2: 0 0 01 0 u-

•

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 cal

culate tornado forces using a semi-empirical equation similar to

Morrison's Equation for tornado-like flow. The results 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. If a is a constant that defines the forced vortex flow and

V 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 (4.1)

V = (x - V.)a y ^

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 v^ on surface r^ and

knowing that * = 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 calcu

lating 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 VS(t)

35

o o •r— 't—

u o S- &-

• r - ' ^ o o

I I X > -

n-f

iZI

p

C2i

/

1 ^

... /

na

,0

0 4-

ca

P

.Ei

d

(C o - rt « -* o -• I •in I

:P-

I

w

(/) "O c= o u

CO

• E

w

I

C3

o - J I—I

=3 3 CO o

_ l cc u-<: _J X => U l o »— on QC •-< O o >

U l z o o on o 00 u_

0 - J 0 1 U l z o o o <-) :s U l oc C_} LU

O =J Ll . Q .

o o

z z: o on

a: ID

:^j./sdL>| UL aojoj

36

Direction of Tornado Travel

a) Square Cross Section

00

Direction of Tornado Travel

58'

b) Rectangular Cross Section

FIGURE 4.2. OTHER BUILDING SHAPES CONSIDERED IN APPLYING THE COMPUTER MODEL

37

c o • ^ +J (J (U L.

•r-Q

1 X

c o • r -4-> U

•r-O

1 > -

1

1 1 1

1 1

D H-

0

^

m

. /

p-

/

ef

/ P

d

P d

.P^'

.? '

N I ft I

N

in •a c o u O)

oo OJ

E

W

I

CO

I-H

o - J 2 H-l O => _ J CO u_ LU X QC LU •a: I— =3 Q: c r o oo > •a: Q

LU Z CJ o on

o 00 Li_

O - J O . LU 2 1 Q O O O 2 1

UJ on O LU Q i I— O =3

u. a. 2:

o o o o

a: o o Q:

CO

LU

on cu

:;^/sdL>| UL aouoj

38

in •a c o 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 : I— z on

o =3 Q- O.

o o

o o on on o u-L L .

o z Q •-• <C Q Z _J 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. The other variables are as defined in Chapter 1 (Eq.

1.1). The acceleration terms that have been previously used by Wen in

Morrison's Equation are:

3V 3V 3v

'x ' - - n * \ ^ * ' y ^

(4.3) 3v 3V 3V

a = — ^ + V — s _ + w y. y 3t X 3x y 3y

Substituting v^ and v^ from Equation 4.1 into Equation 4.3, the ac

celerations 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 = °

^x' PV« \*" • y^" (4.4)

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 agree

ment 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. The drag forces in

inviscid flow are zero. The lift forces are produced by the rotation

of the fluid. 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 de

scribed 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 = -

'y'--

3 v , 3V 3V ' + V + V

3 t X 3x y 3 y _

'^\ ^\ 5v ' J' + V ^ + V -TT^

3t X 3x y ^y _

(4.8)

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

a = 3t

3V

y 3t

3V^ 9V 3V - A + V — ^ + V ^ 3t X 3x y 3y (4.9)

3v 9v^

3t x 3x

3V + V

y 9y J

42

After simplification. Equation 4.9 becomes

I = - V — - + V -\

X [x 3x V 3yJ

[ 3V„ 3V 1 V - ^ + V — ^ X 3x y 3y

(4.10)

In order to verify the above relationships, inertia force com

ponents 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). For rotational

flow the value of C is not available from theory. At this stage, it

is assumed that the value of C^ is 2.0 for rotational flow. At the end

of this chapter, after comparing the results of the 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. To

calculate the accelerations 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 veloci

ties in the x and y directions at any time t for forced vortex flow are V = v.cose - y'a X t

V = v.sine + x'a y t

(4.11)

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. Instead,

Equation 4.3 is used to calculate the forces. 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:

r2= (x')2+ (y")^ ( -1 )

45

cc o

cc o

oo LU C..3 CC O

3 o

X

s . o

• o

o & -o

u s . o

•r-4-> s-

4 J CM

C5

CsJ <o

E O Q. t=

CVJ I

CVJ

II

E o Q.

CVJ

II

CSJ CVJ

ro fd CVJ

CVJ

CVJ CVJ

^ x >>

E C J

Q. t=

CM

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CVJ X

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CVJ 1

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CVJ

>> ar

m

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(U o s . o

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CVJ

II

CVJ X

ar

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^ CVJ

1

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CVJ

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CVJ ta o Q. ^

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td a Q. t=

CVJ

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1—1

>> ar

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o u +->

> CVJ

m s CJ. t=

CVJ

II

t—1

X

ar

o LD

<* £

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4->

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(TJ 3 Q. t=

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1—1

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46

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z o 1—I

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Q O LU 1

X

^ •—1

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a LU >

LU C£ CJ CC a u. u. o •^ a 00 1—1

Q : <: Q-^ O o

• to "^ LU CC r3 o

=3 Q a LU LU <-» o cr: o o Q: U .

Q •-• LU 1—1 UJ LL. 00 I - l < :

_ i o Q. 21 Q^ I - l O 00 u .

ft

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:;^/SdL>| *U0L^39- lLa-X UL SDUOJ

47

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w

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in • o

c o a OJ oo

1 OJ E

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oo

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Q X Z LU < : 1—

cc a

LU => LU on <_> on

=3 C3 O LU

O LU (.} L l .

u. t_) cx: o o cc u.

o a. •z. a I - l a LU 00 1—1

cc < o. ST

o o

r«» * : ! •

LU CC Z3 to

1—1 LU Ll_ I / ) •-• «c _J <_) O. s: cc I- l o 00 u.

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48

CVJ

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2> - O T3 •r- O)

" ^ O ' I - J-(u to M- a» S- > . - 3

o c Q. a. U . I - l E E

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z 1—1

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a LU Z h -<ion

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u. o QE: o p a: u. O Q .

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«:»•

LU on Z3 O

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49

to "O c o u • LU 03 O

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o I-l on I— UJ o t— LU = ) 3 on a.a •—I s —J Q O U.

I (.3 > - X

O LU Z Z I— I-l <£ cx: o t o LU > LU Q^ O I D Q cx: o LU O LU t_) Li. O O^

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LU on t 3

:t^/SdL>| *U0L:i09ULa-A UL 9 0 J 0 J

50

CO

-o • 1 —

•o o OJ to O ' l -&- > o c U - i - i

T3 0)

• r -H -• r -

— O . E

•^ CO

&. (U

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to "O c o u OJ oo

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4 f

fl

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•>• U J 03 O

o Z Z o I-l oc I— LU CJ I— LU = 3 3 CC Q-a •-• Z - J Q O L i .

I CJ X X

a LU Z Z l — I- l <C a^ o t/> LU > LU CC t J =3 Q O l O LU O LU O U . CJ Q l

O O LL. Q : Ll. O Q-

f t Z Q CO O LU CO I—I LU I—I U . CO OC *-• «c c i —I CJ O . C3.

^ ^ S o •—' o CJ oo U-

/f

#

/

#

LU

a:

C!i

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:; /sdL>i «uoL:t39JLa-x UL gojoj

51

r W

to c o CJ OJ (/)

I

E

>- LU CQ O

O

O »—I on \- LU O I— LU =3 3 on o. a HH s : - I o O l l .

I CJ > - X

a LU sc-z. y-I—I ef oc

O CO LU > LU CC CJ Z3 O Q: O LU O LU CJ U. CJ o^

o o LJ. cc Lu

o a. Z Q CO O LU CO »—• LU I—I u . t o a : I-l «C eC —I CJ Q. O . S S QC O •—I O CJ 00 U .

CC = 3 CD

7

î/sdi>| 'uonâ-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 or-

dinates.

Thus we can conclude that the results from the simplified pro

cedure 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 axisym-m

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. Also, the computer model is useful to

compute 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^ kv^sine 2ky'v^(x'cose + y'sine)

3v kv^cose 2kx'v^(x'cose + y'sine)

3t ' "~^ " ^

^^x _ 2ky'x' !!x_ _ k ^ 2ky'2 "?x "^ » 3y " " •;:7 r4

^ _ k 2kx'2 ^ _ 2kx'y' 3x ~ 7^ "Pi ' 3y ^

and

3v^ dy 3v a = - r ^ + v ^ + v . ^ X 3t X 3x y 3y

3Vy av 3v I = — ^ + V — — + V — -

y 3t X 3x y 3y

54

f—1

Qi in (0

CJ

s o X 1—

<D U . • ! - >

J- -a O - 1 -

> o to (U • ! -Qi > i. c

U_ I - l

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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. For

rotational flows found in tornadoes, the C values are not known for

viscous flow. 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 Conclusion

A numerical model with 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. The fluid is

assumed to be inviscid. 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. By properly interpreting

the various force 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 irst step

in the process of solving for the forces on buildings during tornadoes

through f luid dynamic principles. The boundary element method has been

used. The assumption of inviscid flow results in only obtaining the

inertia and l i f t forces acting on a structure immersed in a tornado

like 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 struc

tures 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 struc

tures. The technology and computer hardware are available to im

mediately pursue these problems.

LIST OF REFERENCES

1. Banerjee, P.K. and Butterfield R IQAI D ^

2. Brebbia, C.A., 1978, The Boundary Elempnt Vothnw ^. r • Pentech Press, L o n d o n j - W T i t i i T ^ i l T N i S ^ ^

3. Brebbia, C A . , Telles, J.C.F. and Wrobpl i r loo/i D . Element Techniques. Spînger-Verlag, New YoVk^;: ' ^^^^^^^^

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 BuiIdingsând Structures art I I , No. 6, Tokyo, Japan, pp. 231-240. uccureb, rart

^' ÎT%\''A'C< ^"K. ^'êbbia, C.A., 1976, Finite Element Techniques for Fluid Flow, Newnes-Butterworth, Lond5H: ^"^^

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. 31, 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. 147-143.

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.

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 2455-

16. 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.

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Tornado on Building