SCI, Mekanik, KTH 1

Exam, SG2214 Fluid Mechanics1 November 2013, at 14:00-18:00Examiner: Anders Dahlkild (ad@mech.kth.se)

Copies of Cylindrical and spherical coordinates, which will be supplied if necessary, can beused for the exam as well as a book of basic math formulas, a calculator and an English dictio-nary. The point value of each question is given in paranthesis and you need more than 20 pointsto pass the exam including the points obtained from the homework problems.

1. (10 p.) Consider the complex velocity potential

F (z) =U0

2a

(z2 + (

a2

z)2).

a) Show that the coordinate axes and the circle |z| = a are streamlines.b) Obtain the velocity components in polar coordinates.c) Let’s assume that this 2D flow field describes the flow towards a half circular cylinder

resting on a plane wall at y = 0, see figure 1. Make a rough sketch of the streamlines. Underwhat conditions of the physical parameters is the given velocity field for |z| ≥ a a reasonableflow model of a real fluid? Are there any particular areas of the real flow field where you wouldexpect large differences compared to the given velocity field?

d) Now, think of the half cylinder as hollow, like a thin shell of radius a, and with a smallhole in the shell at the highest stagnation point. This small pressure hole evens out the constantpressure of the stagnant fluid inside the shell to the same value as the stagnation pressure ofthe flow outside. Calculate the vertical component of the net force on the whole shell due to thepressures distribution on both sides of the shell obtained from the present flow model.

Figure 1

2. (10 p.) Consider the boundary layer in a viscous, incompressible fluid around the stagna-tion point of a slowly curved surface where the velocity at the edge of the boundary layer isapproximated by

Ue(x) = U0(x

L− 2

3(x

L)3), for|x| < L,

where the velocity U0 and the length L are given constants.a) Roughly, in which region along the surface |x| ≤ L should one expect separation of the

boundary layer? Explain why?b) The velocity in the boundary layer can be written in the form

u(x, y) = U0

(x

Lf ′1(η)− 2

3(x

L)3f ′

3(η) +O(

(x

L)5))

,

where f1 and f3 are given functions of the boundary layer coordinate η = y√νL/U0

. Neglecting

the terms of order 5 or higher in x/L < 1, determine the y-component of the velocity in theboundary layer if f1 and f3 and their derivatives are regarded as known. Also, what boundaryconditions for f1, f

′1, f3 and f ′

3 should hold at η = 0 and for f ′1 and f ′

3 as η →∞?c) It is known from a more detailed analysis that f ′′

1 (0) = 1 and f ′′3 (0) = 2.5. Determine the

point of separation of the boundary layer.

2 SCI, Mekanik, KTH

3. (10 p.) In a Rankine vortex the fluid in the core rotates like a solid body, whereas outsidethe core the flow is that of an irrotational line vortex. Consider a tornado that can be idealizedin 2D as a Rankine vortex with a core of radius R = 15 m. The pressure at this radius is 1.5kPa below that of the atmospheric pressure at large distance from the tornado.

a) Calculate the circulation around any circuit enclosing the whole core of the tornado.b) Consider such a tornado which center translates at a constant velocity of 20 m/s due to the

generally stormy weather conditions. If the tornado is moving straight towards your position,find the time required for the pressure to drop from 0.06 kPa to 1.5 kPa below the atmosphericpressure. You can neglect effects of compressibility and assume the density to be constant atρ = 1.2 kg/m2. The atmospheric pressure at large distances from the tornado is now that of thewind blowing at the constant speed of 20 m/s.

c) You have a manometer mounted outside your house to register the approach of the tornado.Calculate the time derivative of the measured pressure as function of the distance to the tornado’scentre as it approaches. You may assume the core of the tornado has not yet reached your house.

(The rapid drop of pressure in combination with the resulting excess pressure inside the wallsof buildings may cause a house to virtually explode.)

4. (10 p.) Consider the 2D flow under a sloped bearing pad which slides horizontally with aconstant velocity U on a thin layer of oil resting on a horizontal surface. The gap between thebearing pad and the surface is given by

h(x) = h0(1 + αx/L)

where the slope α << 1 and the lengths h0 << L are a given constants.In a coordinate system following the bearing pad, see figure 2, one can show that the approx-

imate momentum equations in the gap as α << 1 are given by

0 = −∂p∂x

+ µ∂2u

∂y2, 0 = −∂p

∂y,

with body forces neglected. The vertical component of the velocity, v << u, can be neglected inthe following. Experimentally one has found that the pressure can be described approximatelyby the formula

p(x) = patm + α3µLUh2

0

x

L(1− x

L),

where patm is the exterior atmospheric pressure.

a) Derive an expression for the velocity u(x, y) in the coordinate system following the bearingpad using the approximations above.

b) Calculate the stress vector acting in a position on the surface of the bearing pad. You mayneglect terms that are quadratic in the slope, i.e. α2 << α. The unit normal to the surface canbe expressed en = n/|n| where n = ∇(h(x)− y).

Figure 2

SCI, Mekanik, KTH 3

5. (10 p.) A viscous liquid film of constant thickness h flows due to gravity, g, down a planeinclined wall at an angle α with respect to the horizontal plane. For fully developed flow thevelocity component along the plate is given by

u(y) = U0y

h(2− y

h),

where y is the distance from the plate measured in the normal direction. The plane wall is inthermal contact with a heat reservoir at constant temperature Twall and at the free surface youcan assume adiabatic conditions (i.e. there is no heat exchange).

a) Determine the constant U0 if the viscosity of the liquid, µ, and the density, ρ, are consideredas constants.

b) Make an energy analysis of the system at the fully developed steady state and perform thefollowing tasks:

• Consider a 2D control volume of length L along the plate and with thickness h. Evaluateeach of the integrals of the total energy equation and try to determine the heat flux,if any, through the plane wall of the control volume. If your result is non-zero, try toexplain physically where this energy (the heat flux) comes from. The energy equationcan be written

d

dt

∫Vρ(e+

12uiui)dV+

∮Sρ(e+

12uiui)uknkdS =

∫VρgiuidV+

∮Sui(−pδik+τik)nkdS−

∮SqknkdS.

• Also, calculate the total dissipation in the control volume, i.e. calculate∫V ΦdV , where

for an incompressible fluid Φ = 2µeijeij and eij = 12( ∂ui∂xj

+ ∂uj

∂xi).

x

y

z

r

θ

z

• P (r, θ, z)

Figure A.11: Cylindrical polar coordinates

A.3 Curvilinear coordinates

Cylindrical Polar Coordinates

The cylindrical polar coordinates are (r, θ, z), where θ is the azimuthal angle, see figure A.11. The velocitycan be written as

u = urer + uθeθ + uzez, (A.2)

where the unit vectors are related to Cartesian coordinates as

er = ex cos θ + ey sin θ,eθ = −ex sin θ + ey cos θ,

ez = ez.(A.3)

Non-zero derivatives of unit vectors∂er

∂θ= eθ,

∂eθ

∂θ= −er. (A.4)

Gradient of a scalar p

∇p =∂p

∂rer +

1r

∂p

∂θeθ +

∂p

∂zez. (A.5)

Laplacian of a scalar p

∇2p =1r

∂

∂r

(r∂p

∂r

)+

1r2

∂2p

∂θ2+

∂2p

∂z2. (A.6)

Divergence of a vector u

∇ · u =1r

∂(rur)∂r

+1r

∂uθ

∂θ+

∂uz

∂z. (A.7)

Advective derivative of a scalar p

(u · ∇)p = ur∂p

∂r+

uθ

r

∂p

∂θ+ uz

∂p

∂z. (A.8)

Curl of a vector u

∇× u =(1r

∂uz

∂θ− ∂uθ

∂z

)er +

(∂ur

∂z− ∂uz

∂r

)eθ +

1r

(∂(ruθ)

∂r− ∂ur

∂θ

)ez. (A.9)

Incompressible Navier-Stokes equations with no body force

∂ur

∂t+ (u · ∇)ur −

u2θ

r= −1

ρ

∂p

∂r+ ν

(∇2ur −

ur

r2− 2

r2

∂uθ

∂θ

), (A.10)

∂uθ

∂t+ (u · ∇)uθ +

ur uθ

r= − 1

ρ r

∂p

∂θ+ ν

(∇2uθ +

2r2

∂ur

∂θ− uθ

r2

), (A.11)

∂uz

∂t+ (u · ∇)uz = −

1ρ

∂p

∂z+ ν∇2uz. (A.12)