Computational Fluid Dynamics
John F. Wendt (Ed.)
Computational Fluid Dynamics
An Introduction
With Contributions by
John D. Anderson Jr., Joris Degroote, Gerard Degrez, Erik Dick,
Roger Grundmann and Jan Vierendeels
Third Edition
123
Editor
Prof. Dr. John
Problem 3.1
[Difficulty: 2]
3.1 Compressed nitrogen (63.5 kg) is stored in a spherical tank
of diameter D = 0.75 m at a temperature of 25C. What is the
pressure inside the tank? If the maximum allowable stress in
the tank is 210 MPa find the minimum theor
Problem 4.12
[Difficulty: 3]
Given:
Data on velocity field and control volume geometry
Find:
Surface integrals
z
4m
3m
Solution:
5m
First we define the area and velocity vectors
r
dA = dydzi + dxdzj
r
r
V = axi + byj + ck or V = 2 xi + 2 yj + k
We will ne
Problem 2.1
Given:
Viscous liquid sheared between parallel disks.
Upper disk rotates, lower fixed.
Velocity field is:
r
r z
V = e
h
Find:
a.
Dimensions of velocity field.
b.
Satisfy physical boundary conditions.
r
r
To find dimensions, compare to V = V (
Problem 5.2
Given:
Velocity fields
To find:
Which are 3D incompressible
Solution:
Governing equation:
2u 2v 2w 0
x
y
z
t
Assumptions:
Incompressible flow
(Continuity equation)
is constant
This is the creation against which we will check all of the flow
Engineering Mathematics (I)
Homework#7
1.
Due: Thu., Dec. 24, 2015
Please find the Laplace transform of the following functions
(i)
f (t ) 2t 3U t 1
(ii) f (t ) e 3t sin 5t
(iii) f (t ) t 2 sin 5t
(iv) The square wave given below.
(v) The triangular wave
Sec. 3.3]
simultaneous linear differential equations
65
the equations of the given system (2) are all linear, sums of solu
tions will also be solutions.
Hence we can combine the three particular
solutions into the general solution
Since
Xi +
x2
x3 =
Ai +
ADVANCED ENGINEERING
64
MATHEMATICS
Thus
[Sec.
3.3
Mi
2
and thus the first of the three particular solutions is
xi
Vi
For nij
= 1, we have
=
i
similarly, from Eq. (3),
9.42
A2

6B2
 3C2
= 0
+ 2Bi + 3C2 =
4At
If
A
 3^!
+
0
2C2 = 0
we solve for B2 and C2
simultaneous linear differential equations
Sec. 3.3]
63
Now let us attempt to find solutions of this system of the form
y = Be",
x = Aeml,
z = Cemt
Substituting these values into the equations in (2) and dividing out
the common factor emt leads to the set
ADVANCED ENGINEERING
62
MATHEMATICS
[Sec. 3.3
Thus four relations must exist among the original eight arbitrary con
stants.
Solving these equations, we find (among equivalent possibilities
E
=
A,
F
=
G =
2(C + D),
H
=
2(C
 D)
It is tedious but perfectl
Sec. 3.2]
simultaneous linear differential equations
A particular integral is obviously
Y
= 3; hence the general solution is
y = Ae~' + Be*' + C cos
(6)
61
21
+ D sin
21

3
To find x, we can substitute for y in either of the original differential
equati
CHAPTER
3
SIMULTANEOUS LINEAR DIFFERENTIAL EQUATIONS
In many applied problems there are not one, but
Introduction.
several dependent variables, each a function of a single independent
The formulation of such problems in mathemati
variable, usually time.
c
LINEAR DIFFERENTIAL EQUATIONS
Sec. 2.6]
57
be neglected,
determine the natural frequency of the oscillations that occur when
the system is slightly disturbed.
(Hint: Use the energy method to obtain the
Fio.
Fig.
2.7.
2.8.
of very small motions, determine
ADVANCED ENGINEERING
58
Determine
each one.
Am.
the possible deflection
If P
=
BI,
n =
in a deflected position given by x
curves and the loads required
1, 2, 3
MATHEMATICS
A sin
[Sec.
2.6
to produce
the column will be in equilibrium
y)
*
A sin
For all
oth
Substituting
equation
[SEC. 2.6
MATHEMATICS
ADVANCED ENGINEERING
56
these moments into the basic equation (12), we obtain the differential
Ely"
=
cos 8)y
(F

(F sin
$)x
or
 B sinh cfw_y^P
*)
and substituting,
.
cosh
V(F
cos
9)
Ly
"\
V^cos
.1
.
(tan
9/A
Sec.
linear differential equations
2.6]
and the moment at x + Ax is
M(x +
Ax) =
(ii
 x + Ax)Fi +
+
(xn
55
 x + Ax)Fn
Subtracting these, we find
M(x +
Ax)
 Mix)
=
AM
Ax(Fi
=
+
+Fn)
=
VAx
Hence, dividing by Ax and letting Ax > 0, we have
V
dcfw_EI
ADVANCED ENGINEERING
54
But
z2
[Sec. 2.6
of inertia of the crosssection
dA is simply the moment
about the neutral axis.
MATHEMATICS
area
Hence the internal moment of the fiber forces
EI
(11)
is
This internal moment must hold in equilibrium the external m
LINEAR DIFFERENTIAL EQUATIONS
Sec. 2.6]
63
have by Hooke's law, Eq. (9),
n
e
ZE
F
On the other hand, if the crosssection
actual force in the fiber is
AF
(10)
area of the fiber
is
AA, the
Ez AA
R
AA
= a
UV
We can now form a picture of the distribution o
52
ADVANCED ENGINEERING
MATHEMATICS
as the elastic curve, or deflection curve, of the beam.
[Sec. 2.6
The line in which
the neutral surface is cut by any plane cross section of the beam is called
It is in the determination of the deflec
tion curve that di
LINEAR DIFFERENTIAL EQUATIONS
Sec. 2.6]
51
or, substituting from Eqs. (5), (6), (7), and (8),
5 W*
+
(~
Differentiating this with respect
Wi
2ff
.
vy +
(Wl+g
^
+
2aW*)
+ 2key) ~ cfw_Wl + W,)V ~ C
to time gives us
) VS+
Uyy +
2k ey
 (W,
+ W,)y
= 0
or, div
ADVANCED ENGINEERING
50
MATHEMATICS
[Sec. 2.6
With these facts in mind we can calculate the total instantaneous energy of the system
without difficulty.
The potential energy consists of two parts: (a) the potential energy of the weights
Wi and Wt due to t
LINEAR DIFFERENTIAL EQUATIONS
Sec. 2.6]
the given conditions into Eqs. (1), (2), (3), and (4), we find
Substituting
0

a
+
0 =
0 =
0
Solving these simultaneously

b
b

+
6
+2
+2
c
+ 2d
2c
b +3c 6d
for o,
12,
a =
and
47
and d gives
6, c,
6 = 12,
c =
LINEAR DIFFERENTIAL EQUATIONS
Sec. 2.6]
ence level.
49
Its potential energy is therefore
(<*+a + jf)!
(4)
Substituting from (1), (2), (3), and
(4) into the relation
 constant
K.E. + P.E.
we have
;
f (*)'
+
sf @y + a 
^
()'Ik.
+
 iA^ 
c + . + >ic
46
ADVANCED ENGINEERING
MATHEMATICS
[SEC. 2.5
but also these terms multiplied by x, and is therefore
ax
ci cos 2x + cj sin 2x + Cjx cos 1x +
To find a particular integral, we try Y
into the differential equation gives
cos x
(.4
+ B sin x) + 8(
A
i + 9S si
ADVANCED ENGINEERING
44
6. Using
MATHEMATICS
[SEC. 2.5
the method of variation of parameters, find a particular integral of the
y" + w*y = sin kx. Discuss the limiting case when k*u, and show
equation
Y
that it leads to
7. Show
=
1 eB
uX.
that the genera
linear differential equations
Sec. 2.5]
The characteristic
equation in this case is
m
+
5m
+ 9m +

5
45
0
By inspection*
m =
is seen
to be a root.
1
Hence
must be one term in the complementary function.
When the factor corresponding
to this root is divi