NUMERICAL AND ANALYTICAL ANALYSES OF A TORNADO MODEL by PATRICK ALAN SCHMITT, B.S. A THESIS IN MATHEMATICS Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE Approved /^ ^ - i» May, 1999
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NUMERICAL AND ANALYTICAL ANALYSES
OF A TORNADO MODEL
by
PATRICK ALAN SCHMITT, B.S.
A THESIS
IN
MATHEMATICS
Submitted to the Graduate Faculty of Texas Tech University in
Partial Fulfillment of the Requirements for
the Degree of
MASTER OF SCIENCE
Approved
/ ^ ^ - i »
May, 1999
v
' ',.^.j ACKNOWLEDGEMENTS
> I would like to first thank the academy, I mean my parents, whose support has
been the driving force behind all my work. I wish also to sincerely thank Dr. Gilliam
and Dr. Shubov for providing so much of their time to aid me in this endeavor and
for the push to stay on track. I also acknowledge the Faculty and Staff for putting
up with my antics.
I feel I must thank some of my fellow teaching assistants individually. My room
mate Chris, for putting up with me these last few months. April for her ever sunny
and glowing opinions on graduate school. Leah for being herself, and being a good
sport. Clint for keeping the humor low and stupid. Scott for his decisive nature (pick
a degree!). Dan for being about as good a friend as a guy can have. Finally, I would
like to thank all the little people who I have crushed along the way.
11
CONTENTS
ACKNOWLEDGEMENTS ii
ABSTRACT iv
LIST OF FIGURES vi
I INTRODUCTION 1
II NAVIER-STOKES EQUATIONS 2
2.1 General 2
2.2 Cylindrical Coordinates 3
III BURGERS-ROTT MODEL 7
IV STREAMLINES OF THE FLOW 12
4.1 Method 1: An Analytic Approach 13
4.2 Method 2: A Numerical Approach 16
4.3 Pure Burgers-Rott Model 17
4.4 Sources and Sinks 21
V CONCLUDING REMARKS AND FUTURE RESEARCH 31
BIBLIOGRAPHY 32
111
ABSTRACT
During the course of this thesis work, we will be studying and analyzing one of
the first tornado models derived from the Navier-Stokes equations for incompressible
fluid called the Burgers-Rott model. We will completely present the model and its
derivation, but will present this derivation autonomous from the original work. Our
goal is to consider all cases of the model, including those which due to physical
restrictions do not produce a tornado. In doing so, we will go beyond the original
work by using computer simulation to graphically interpret our results.
IV
LIST OF FIGURES
2.1 Cylindrical Coordinates 3
2.2 Translated Curvilinear System 4
3.1 The cylinder Q 8
3.2 The disk CR 10
4.1 Error Analysis of r versus z 16
4.2 Plot of the two methods together 17
4.3 Plot with Too = 2000, a = .02, i^ = 4 18
4.4 Plot with Too = 2000, a = .02, u = l 18
4.5 Phase plot with u = 6-solid, v = 4-dash, v = 1-dot 19
4.6 Plot with Too = 2000, a = .06, z/ = 6 19
4.7 Phase plot with a = .02-solid, a = .06-dash, a = .08 -dot 20
4.8 Plot of Too = 5000, a = .02, ly = 1 20
4.9 Phase plot with Too = 2000-solid, Poo = 5000 -dash, T^ = 7000-dot . . 21
4.10 Plot with Too = 6000, a = .009,6=-.005,1/= 3 23
4.11 Plot with Too = 6000, a = .02,6=- .005, i/ = 3 24
4.12 Plot with Too = 6000, a = .06,6=-.005,1/= 3 24
4.13 Plot with Too = 6000, a = .009,6=-.005, z/ = 1 25
4.14 Plot with Too = 9000, a = .009,6=-.005, z/ = 1 25
4.15 Phase plot with Too = 6000-solid, Too = 9000 -dash. Too = 11000-dot . 26
4.16 Plot with Too = 3000, a = .01,6= .01,1/= 5 26
4.17 Phase plot with ^ 27
4.18 Plot with Too = 3000, a = .01,6= .01,1/= 3 27
4.19 Phaseplot witha = .01,0 = .02 28
4.20 Plot with Poo = 3000, a = .01,6 =.01,1/= 3 28
4.21 Plot with Too = 6000, a = .01,6 =.01,1/= 3 29
4.22 Plot with Poo = 3000, a = . 0 1 , 6 = .005,1/= 3 29
4.23 Phase plot with 6 = .005,6 = .01 30
VI
CHAPTER I
INTRODUCTION
In 1948, J. Burgers presented a model for tornadic motion based upon equations
developed by Navier and Stokes that describe particle motion through an incompress
ible fluid. This work was later extended by Rott in 1958, and took the name of the
Burgers-Rott model for tornados. The Burgers-Rott model was one of the very first
models of this kind developed. At the time of its incarnation, very little was really
known about these violent storms, as is apparent in the assumptions made by the two
mathematicians. The most glaring drawback of this model is the fact that boundary
conditions have not been considered. In effect, making the tornado engulf the entire
planet. Temperature is another physical phenomenon that is not taken into account,
but is known to play a roll in the occurrences of tornados. Even though, there are
limitations to the reality of this model, it is important to understand as much about it
as possible before discussion of more detailed models should take place. In completely
analyzing the Burgers-Rott model, insight may be gained that will aid in the study
of more complicated models.
Our goal in writing this thesis, is to provide some graphical representation to the
streamlines of the Burgers-Rott model, especially the cases not solvable analytically.
We plan to use these graphs to visually explain the effects of individual parameters
on the system as a whole.
CHAPTER II
NAVIER-STOKES EQUATIONS
2.1 General
In order to study any type of tornado model, we must first begin with a discussion
of some basic fluid dynamics. We will be mainly interested in viscous fluid. When we
deal with these flows, we also need to take into account parameters such as pressure
{P{^^y,z,t)), viscosity (dynamical, /z ), density {p = constant for incompressible
flow), velocity of the flow (v), and external force (F) [2]. Using these parameters,
Euler argued that the velocity field of an "ideal" fluid would satisfy.
^ + ( v - v ) 7 ; = - v p - f /
where
p = —J = -• p p
Both Navier and Stokes later added a viscosity term to the flow to obtain the equations
dv
where
^ -i-(v-V)v-iyAv = -Vp + f at
ly = — (kinematic viscosity).
Together with the continuity equation, the Navier-Stokes system becomes
Tr + (v-V)v = -Vp + f + i/Av at ^ '
V 'V = 0.
If we assume the external body force is potential (i.e., / = — Vf/), the system
becomes
dv — + {v • V)v = uAv - V (p -\- U) ^ (2.1) V-v = 0
In this system, {v • V)v represents convection, and describes energy transport as
a result of particle motion, uAv represents diffusion in the system and describes
dissipation of energy [2]. System (2.1) will form the basis for the rest of the discussion
as we turn to a specific tornado model.
2.2 Cylindrical Coordinates
In this section, we turn to a study of models. In this work, we are interested
in studying solutions of (2.1) which exhibit rotation about a vertical axis. If we
think about the path a particle would take once caught in the tornado, we expect
to see a circular motion about a vertical axis. We naturally then want to be able
to describe this type of motion. For this reason it is useful to express the system
(2.1) in cylindrical coordinates, which better describe the circular motion about the
z-axis by combining the usual polar coordinates for x and y, and the usual z-axis.
The coordinates (x, y, z) translate to cylindrical coordinates using the substitutions
X — r cos 9, y = r sin 9, and z = z
r - lines, z,9 = constant
9 - lines z, r = constant
z - lines r.,9 — constant
z -Une
r - Une 0 -Une (circle)
Figure 2.1: Cylindrical Coordinates
At every point of R^ space we have three coordinate unit vectors, e'r, e*e, and e*
—* -*
Cr = cos i -I- sin ^ j —• —*
60 = — sin ^ i + cos 9 j
6r K.
We also have dcr ^ dee
= ee, = —e*r, and all other derivatives are equal to zero. The 09 ' d9
coordinate vectors e'r, e , and e* form an orthonormal system as seen in (2.2).
^0 AB,
zk = zez(R)
*• y
Figure 2.2: Translated Curvilinear System
With this and a lengthy calculation, the Navier-Stokes system (2.1) written in
terms In cylindrical coordinates, the Navier-Stokes system becomes
dVr ^ ^ ^ VQ -\- V • VVr =
r
V0Vr
dt dve dt
+ V ' VVQ +
-\- V • VVz =
dp^ f. Vr 2 dve dr \ ^ r^ 7-2 QQ
I dp ( . 2 dvr V0
r r d9 I dp
r dz -\-i/Av,
dt ^ d ldv0 dv, r dr r dr oz
(2.2)
(2.3)
(2.4)
(2.5)
Which is
dvr ~dt
d d d VI + Vr^ + Ve^^ + V,^- ]Vr- -^
dr
dp
dv0
dt
dr
d_
dr
dp
+ z/
dz dVr
dr
dr + z/
d9
ld_
r dr d_ d9
1 d
+
V0 -
1 d^Vr
VrV0
+ d\ dz^
Vr
dVr
r' 12,
r dr \ dr
1 d'^V0 d'^V0
'^^W'^'d^ V0_
dvz dt
d d d + VrTT + VOT^T: + Vzl^ Vz "
VI
dr dp dz
+ 1/
d9
1 d
dz dv.
r dr \ dr + 1 d^v, + d^v. V,
r2 d9'^ dz^
The continuity equation is of the form
I d . , dvz (rvr) + r dr dz
0.
(2.6)
(2.7)
(2.8)
(2.9)
(2.10)
(2.11)
Here Vr, V0, and v,, are the components of the vector v with respect to the basis e ,
60, 6,, i .e . ,
v{r, z) = Vr(r)6r + V0(r)60 -I- Vz(z)ez.
We also need the vorticity form of these equations. Vorticity is defined as w = V x -y,
and the equation for vorticity is
dw 'dt
-\- (v • V) w = (w • V) V.
The vorticity equations in cylindrical coordinates take the form
dWr d d d
dt + r a ^ + a + ^a '"^ d d
= I Wr-^-\-Wz— ]Vr-\-U Id f dw
dwf d d d r dr \ dr
VeWr
r I 1 d Wr d Wr Wr r^r- 1 + -r^rr^T- + r2 d9^ dz"^
dt + r a ^ + ^ ^ ^ + ^ ^ ^ r ^ + r
d d\ weVr = Wr-^ + Wz— V0 H h 1/
or oz J r
1 5 / dw0\ 1 d'^W0 d'^W0 r dr \ dr I r^ d9'^ dz"^
W0
dwz dt
d_ dr
d
d_ d
o' d
+ |^.Tr- + - . ^ + - . ^ l ^ .
= I Wr-^-\-Wz— ]Vz-\-iy 1 5 / dwz\ 1 d'^Wz d'^Wz r dr \ dr I r^ dz^ dz"^
We end this preliminary discussion by giving some relationships between the compo
nents of V and w that will be used later.
1 dv^ dvf, Wr =
W0 =
r d9 dz dVr dvz dz dr ' 1
w^. = -d . . dvr
d-r^'"'^ - -de
6
CHAPTER III
BURGERS-ROTT MODEL
In this chapter, we turn to the main object of interest in this thesis, the Burgers-
Rott tornado model. This model was first studied by Burgers (1948) and was later dv
extended by Rott (1959). We seek an axially symmetric (i.e., ^ = 0) stationary (i.e.
dv
'di = 0) solution to Navier-Stokes equations. Which implies that v(r, 9, z, t) = v{r, z).
Burgers had the idea to seek solutions to Navier-Stokes equations in the form
v{r, z) = Vr{r)er + V0{r)60 + Vz{z)ez (3.1)
Theorem: Solutions to (3.1) must be of the form
Vr{r)
veir)
Vz{z)
—ar +
r CXD
27rr2
2a{z — ZQ)
b
r
1 — exp ar 2u
c + -r
(3.2)
(3.3)
(3.4)
where a > 0, Too > 0, 6 and c are constants. The Burgers-Rott solution corresponds
to the case 6 = 0, c = 0. We present a proof given in [6].
proof: We first note the under the assumptions in (3.1) the Navier-Stokes system
becomes
Vr dVr dr
Vc dP
dr + 1/
dV0 VrV0 Vr— \-
dr = —z/
V
r dVz
^ dz dv,.
1 d
r dr
dVr
r
dP
r dr \ dr
d'^v.
dVr\ dr J
V0' fpZ
Vr
IY*£i
^y-
\ d , ,
r dr dz
dz ' ' dz^ '
0, (continuity equation).
We will also need the vorticity form of these equations where
dw
'dt -\- {v-V)w = (w •S/)v-\- i/Aw.
Writing this equation out we find that only one term provides new information:
dw, V ^ dr ^ dz
dVz V d ( dwz H r
rdr\ dr
where
w, Wz{r)
M'r) = --^{rv0{r))
Wr = W0 = 0.
The continuity equation (3.5) can be written as
1 d dv. r dr dz
= constant = 2a
which implies two things
Vz{z) = 2a{z- zo),
r
There are several possibilities for the Vr term dependent on 6.
6 < 0 =^ a sink of fluid on the z-axis
6 = 0 ^ a tornado
6 > 0 =^ a source of fluid on the z-axis
(3.5)
(3.6)
(3.7)
Recall, that the Burgers-Rott model corresponds to the case 6 = 0. We will focus on
this case and leave the other two cases of 6 for later discussion.
Consider Q, a cylinder with dO, its surface shown in the following figure.
Q
>• z.
Z2
Figure 3.1: The cylinder Q,
8
Recall, we are dealing with an incompressible fluid. This implies that the flux of
V through the surface is zero, if there are no sources or sinks inside Q.. The flux of v