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Tentamen Str omningsmekanik a.k. 5C1203
Final exam in course 5C1203 27/5 2002 8-13 in E31Examiner: 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
pages 348-355 in Acheson can be used for the exam as well as a book
of basic mathformulas and a calculator.
1. Incompressible ow of a viscous uid between two innite,
parallel surfaces is drivenby a constant pressure gradient p/x = F
. The surfaces are located at y = 0 and
y = h and the upper surface is moving with the constant velocity
U in the x-direction.a) (7) Calculate the velocity eld.b) (3)
Calculate the upper surface velocity U for which there is no
friction force onthe upper plate.
2. A hill with the height h has the shape of a half circular
cylinder as shown in Figure 1.Far from the hill the wind U is
blowing parallel to the ground in the x-direction andthe
atmospheric pressure at the ground is p0 .a) (5) Assume potential
ow and show that the stream function in cylindrical coor-dinates is
of the form
= f (r )sin ,
where f (r ) is an arbitrary function. Calculate the velocity
eld above the hill.b) (3) Derive an equation for the curve with
constant vertical wind velocity V .c) (2) Assume that the density
and the gravitational acceleration g is constant.Calculate the
atmospheric pressure at the top of the hill.
y
x
Figure 1: Streamlines above a hill with h = 100 m and U = 5 m/s
. A paraglider pilot with a sink of 1 m/swill nd lift in the area
within the dotted line, while soaring along the hill.
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3. Turbulent ow.a) (4) Derive the Reynolds average equation
valid for turbulent ow.b) (4) Assume that the Reynoldsstress can be
modelled as a turbulent viscosity andintroduce this into the
Reynolds equation and simplify.
4. a) (3) Show that
u ju ix j
= 12
x i
(u j u j ) + ijk j uk
b) (4) Derive the vorticity equation starting with the
Navier-Stokes equation forincompressible ow.c) (3) Show that the
following relation holds for incompressible ow and discuss
itsimplication for inviscid ow
ijx j
= ijkkx j
5. A uid is owing in a 2D jet out from a slit in a wall and into
a large space lled withthe same uid that is practically motionless.
Let x be the coordinate in the directionof the jet and y parallel
to the wall.a) (2) Assume that the ow is governed by the boundary
layer equations and motivatethe usage of the equation,
u ux
+ vuy
= 2 u
y 2 ()
b) (4) Show by integrating ( ) that the momentum ux, M , in the
direction of the jet is constant in x. Hint: Integrate by parts and
use incompressibility.
M =
u2 dy ()
c) (6) Assume that the stream function is ( x, ) = um (x)g(x)f
() where the simi-larity variable is = y/g (x). Show, using (),
that the maximum velocity of the jet
at x is um (x) = C/ g(x), where C is a given constant. Show that
the thickness of the jet grows proportional to x2 / 3 , and that f
() satises,f + kf f + kf
2= 0 ,
where k is a constant.
Good Luck!
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Solutions to exam in Str omningsmekanik a.k. 5C1203,
2002-05-27
1. Poiseuille-Couette ow.a) 2 uy 2
= F
u(0) = 0u(h) = U
u = F h 2
2 yh
yh
2
+ U yh
b)uy
(h) = 0 U = F h 2
2
2. Potential ow.a)
ur = U 1 h2
r 2cos
u = U 1 + h2
r 2sin
b)
r = h 2U V sin cos c)
p = p0 32
U 2 gh
3. Turbulent ow.
U it
+ U jU ix j
= 1
P x i
+ 2 U i x j
(u iu j )
4. Vorticity.
it + u j
ix j = j
u ix j +
1R
2
i
5. See Acheson problem 8.4.
