Exam in Fluid mechanics 5C1214

Final exam in course 5C1214 23/10 2003 14-18 in Q14, Q15, Q21Examiner: Prof. Dan HenningsonThe 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 dt is

dri(dt) = dx0i +

∂ui

∂x0j

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, Rδi1, is deformed into

r(1)i = R(δi1 +

∂ui

∂x01

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 x0

i = 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 ω.

(10) Find the velocity field of the water driven by the rotating rod and the gravitationalacceleration g.

z

g

r

θ

a b

Figure 1: Water and a concentric rotating rod inside a cylinder.

3. Bernoulli’s equation

a) (3) Derive the following formula:

uj

∂ui

∂xj

=1

2

∂

∂xi

(ujuj) + εijkωjuk.

b) (3) Use this in the momentum equation to derive Bernoulli’s equation for unsteadypotential flow.

c) (3) Use the result in (a) to derive Bernoulli’s 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

U∂u1

∂x= ν

∂2u1

∂y2.

b) (2) Show that the mass flux Q1, in the direction of the wake is constant in x

Q1 =

∫

∞

−∞

ρu1dy. (∗)

c) (7) A similarity solution can be found by scaling the wake velocity with Us(x) = u1(x, 0)so that f(η) = u1(x, y)/Us(x), where η = y/δ(x) is the similarity variable. Show, using (∗),that Us = k/δ(x), where k is a given constant. Find the similarity solution for u1.

Y

X

UU+u1

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

= κ∂2T

∂xj∂xj

.

(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.

uθ =ωa2

b2− a2

(

b2− r2

r

)

uz =1

4

g

ν

(

r2− a2

− (b2− a2)

ln(r/a)

ln(b/a)

)

3. See Kundu & Cohen pp. 110-114.

4. Laminar wake flow.

u1 =Q1

2ρ

√

U

πνxe−

Uy2

4νx

5. See Kundu & Cohen pp. 511-512.

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