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Exam in Fluid mechanics 5C1214
Final exam in course 5C1214 23/10 2003 14-18 in Q14, Q15,
Q21Examiner: Prof. Dan Henningson
The point value of each question is given in parenthesis and you
need more than 20 points topass the course including the points
obtained from the homework problems.Copies of appendix B from Kundu
& Cohern can be used for the exam as well as a book ofbasic
math formulas and a calculator.
1. Relative motion.
Consider the relative motion of two fluid particles initially
separated by the distance dx0i .(Here x0i and t are the Lagrangian
coordinates).
a) (5) Show that the separation at time dtis
dri(dt) =dx0i +
uix0j
dx0jdt.
b) (2) Use this relation to write down the expression for the
deformation of the sides ofa small cube aligned with the coordinate
directions. Thus, show that the components ofthe side aligned with
the x-direction, Ri1, is deformed into
r(1)i =R(i1+
uix01
dt),
with analogous expressions for the other two sides.
c) (4) Show that the deformation rate of the volume of the cube
is given by the divergenceof the velocity field. Note that x0i =xi
at t= 0.
2. Flow between a rod and a cylinder.
Consider a long hollow cylinder with radius b, a concentric rod
with radius a inside thecylinder, and water occupying the space
between the rod and the cylinder, see figure 1.The rod is rotating
with the constant angular frequency .
(10) Find the velocity field of the water driven by the rotating
rod and the gravitationalacceleration g.
z
g
r
a b
Figure 1: Water and a concentric rotating rod inside a
cylinder.
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3. Bernoullis equation
a) (3) Derive the following formula:
ujuixj
=1
2
xi(ujuj) +ijkjuk.
b) (3) Use this in the momentum equation to derive Bernoullis
equation for unsteadypotential flow.
c) (3) Use the result in (a) to derive Bernoullis equation for
steady inviscid flow. Discussthe validity of the derived
relation.
4. Laminar wake flow.
Consider the laminar wake flow downstream of a two-dimensional
streamlined body athigh Reynolds number, see figure 2. Assume that
the wake is so weak that we can writeu= U+u1 where u1 is negative
and much smaller than the free-stream velocity U.
a) (3) Motivate the usage of the equation
Uu1x
=2u1y2
.
b) (2) Show that the mass flux Q1, in the direction of the wake
is constant in x
Q1=
u1dy. ()
c) (7) A similarity solution can be found by scaling the wake
velocity with Us(x) =u1(x, 0)so thatf() =u1(x, y)/Us(x), where
=y/(x) is the similarity variable. Show, using (),
that Us=k/(x), where k is a given constant. Find the similarity
solution for u1.
Y
X
UU+u
1
Figure 2: Wake downstream of a flat plate.
5. Mean heat equation.
The heat equation for the instantaneous turbulent flow is
T
t + uj
T
xj=
2T
xjxj.
(8) Derive the heat equation governing the mean flow.
Good Luck!
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Answers to exam in Fluid mechanics 5C1214, 2003-10-23
1. See Kundu & Cohen p. 57.
2. Flow between a rod and a cylinder.
u = a2
b2 a2
b2 r2
r
uz =1
4
g
r2 a2 (b2 a2)
ln(r/a)
ln(b/a)
3. See Kundu & Cohen pp. 110-114.
4. Laminar wake flow.
u1=
Q12 U
xeUy2
4
x
5. See Kundu & Cohen pp. 511-512.
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