Exercise 14
20000530-1
a) (3) Derive an expression for the enstrohpy 2 = i i in terms of the tensor vi /xj . Here i is the
vorticity.
b) (3) Derive an expression for the dissipation function = ij vi /xj . Here ij is the viscous stress
tensor. Assume incom

Fluid Mechanics, SG2214, HT2010
October 4, 2010
Exercise 12: Drag on Circular Cylinder and Wall Jet
Example 1: Drag on a Circular Cylinder
Use the integral form of the NavierStokes equations, Consider a x volume in a uid with velocity u. The
continuity eq

Fluid Mechanics, SG2214, HT2010
September 30, 2010
Exercise 11: Similarity and Wake Flow
Example 1: Converging Channel
Consider the high Reynolds-number ow in a converging channel. Compute the boundary layer over the
surface at y = 0.
Assume the free stre

Fluid Mechanics, SG2214, HT2010
September 28, 2010
Exercise 10: Potential Flow
Example 1: Half body over a wall
A line source of strength Q is located at (0, a) above a at plate that coincides with the x-axis. A uniform
stream with velocity U ows along th

Fluid Mechanics, SG2214, HT2010
September 24, 2010
Exercise 8: Vorticity, Bernoulli and Stream Function
Example 1: Solid-Body Rotation
Consider the ow in an uniformly rotating bucket with velocity
u = (r, 0, 0)
a) Use Bernoulli equation to determine the f

Fluid Mechanics, SG2214, HT2010
September 22, 2010
Exercise 8: Natural convection in channel
Example 1: Natural convection in channel
Consider the ow in a vertical long channel with heated and cooled walls. The x-axis is vertical and parallel
to the chann

Fluid Mechanics, SG2214, HT2010
September 20, 2010
Exercise 7: Exact solution for energy equation and axisymmetric
Flow
Example 1: Exact solution for energy equation
Consider plane Poiseuille ow in a straight channel with walls at y = h. The temperature a

Fluid Mechanics, SG2214, HT2010
September 17, 2010
Exercise 6: Exact Solutions to the Navier-Stokes Equations II
Example 1: Stokes Second Problem
Consider the oscillating Rayleigh-Stokes ow (or Stokes second problem) as in gure 1.
1
0
1
0y
1
0
1
0
1
0
1
0

Fluid Mechanics, SG2214, HT2010
September 14, 2010
Exercise 5: Exact Solutions to the Navier-Stokes Equations I
Example 1: Plane Couette Flow
Consider the ow of a viscous Newtonian uid between two parallel plates located at y = 0 and y = h. The
upper plan

Fluid Mechanics, SG2214, HT2010
September 10, 2010
Exercise 4: Adimensional form and Rankine vortex
Example 1: adimensional form of governing equations
Calculating the two-dimensional ow around a cylinder (radius a, located at x = y = 0) in a uniform stre

Fluid Mechanics, SG2214, HT2010
September 9, 2010
Exercise 3: Conservation Equations and Stress Tensor
Example 1: conservative form of continuity equation
a) Consider a xed, closed surface S in a uid. Show that conservation of mass implies
+
(ui ) = 0 .
t

Fluid Mechanics, SG2214, HT2010
October 6, 2010
Exercise 13
1
Compute the far-eld two-dimensional turbulent wake U (x, y ) behind a cylinder.
Let
U1 (x, y ) = U0 U (x, y ),
Us (x) = U1 (x, y = 0)
Assume
Us U0
U1 U0 ,
u, v Us ,
x L,
yl
Continuity
U
V
+
=0

Homework 1 Fluid Mechanics 5C1214
Due 23/9, 2005
Be careful to explain and motivate every non-trivial step of the solutions to these problems.
1. Use tensor notation to show the following relations
a)
(F G) = (
b)
(
F) =
c) u = r
F) G F (
(
G)
F ) F

SCI, Mekanik, KTH
1
Exam, SG2214 Fluid Mechanics
11 January 2011, at 09:00-13:00
Examiner: Anders Dahlkild (ad@mech.kth.se)
Copies of Cylindrical and spherical coordinates, which will be supplied if necessary, can be
used for the exam as well as a book of

SCI, Mekanik, KTH
1
Exam, SG2214 Fluid Mechanics
23 October 2010, at 09:00-13:00
Examiner: Anders Dahlkild (ad@mech.kth.se)
Copies of Cylindrical and spherical coordinates, which will be supplied if necessary, can be
used for the exam as well as a book of

Exam in Fluid Mechanics 5C1214
Final exam in course 5C1214 12/1 2006
Examiner: Prof. Dan Henningson
The point value of each question is given in parenthesis and you need more than 20 points to
pass the course including the points obtained from the homewor

S oFunop ( ' T-o F A .4 t N
o
?
O = - 3 n v rZu
l
Sfrzz cfw_ r D c \ . 2 3 2 d 0 e
tt.(o) 4
t
?bz
- -\
,1 = u (1)
,rtn -
=\
3
,l
u ( h ) 2u o
= A LJ 1 - C r
t
u
= \ , ^ =3 S
+Crb rLz-
V2
r,t( " ) - - C , - -O :
u(t^) =
I
7v
+.> ( rtg) : 0 ":
h
r'.'
tcfw_

Fluid Mechanics, SG2214, HT2010
September 6, 2010
Exercise 2: Euler/Lagrange Coordinates and Relative Motion
Streamline (Eulers representation)
Instantaneously (i.e. at xed time t), a uid particles displacement dx is tangential to its velocity u:
u
dx
=
.

Fluid Mechanics, SG2214, HT2010
September 2, 2010
Exercise 1: Tensors and Invariants
Tensor/Index Notation
Scalar (0th order tensor), usually we consider scalar elds function of space and time
p = p(x, y, z, t)
Vector (1st order tensor), dened by directio

Due 22/9, 2006
Problem set 1
Be careful to explain and motivate every non-trivial step of the solutions to these problems.
1. A wind at constant velocity U0 is blowing towards south (x1 -direction say). Due to heating, material
uid elements increasse thei

c) (3) Solve the problem and show that the solution approaches the potential vortex for small
values of
and an appropriate similarity variable.
u=
u
/2r
b) (6) Transform the problem to an ordinary dierential equation by dening
a) (3) Write down the gover

Exam in Fluid Mechanics 5C1214
Final exam in course 5C1214 18/10 2004 09-13 in L 31, M 37, M 38
Examiner: Prof. Dan Henningson
The point value of each question is given in parenthesis and you need more than 20 points to
pass the course including the point

Exam in Fluid Mechanics 5C1214
Final exam in course 5C1214 13/01 2004 09-13 in Q24
Examiner: Prof. Dan Henningson
The point value of each question is given in parenthesis and you need more than 20 points to
pass the course including the points obtained fr