s
ME304 1 /5
SCHOOL OF ENGINEERING
MODULAR HONOURS DEGREE COURSE
LEVEL 3
SEMESTER 1
2005/2006
FLUID DYNAMICS
Examiner: Prof S Sazhin
Attempt FOUR questions only
Time allowed: 2 hours
Total number of questions = 6
All questions carry equal marks
The figures in brackets indicate the relative weightings of
parts of a question
Special Requirements:
Prandtl Meyer Expansion Tables
Tables for Plane Oblique Shocks
Formulae Sheets
ME304 2 / 5
1) a) Explain the meaning of the following terms:
i) streamline
(1)
ii) stream function
(1)
iii) velocity potential
(1)
iv) potential flow
(1)
v) irrotational flow
(1)
vi) doublet
(1)
The flow of air around the tip of a rocket can be approximated by a potential flow
with the following stream function
Ψ  v o rsinθ 
qθ
2π
where v o  200 m/s; q  80 m 2 /s
b) Find the location of the stagnation point.
(9)
c) Determine the value of flow velocity at the points:
i)
ii)
r  10 cm; θ  120 
r  5 cm; θ  180 


(4)
(2)
d) Draw schematically the streamlines and equipotential lines for this flow and
explain their main properties.
(4)
ME304 3 / 5
2)
a) Air with a free stream velocity of 30 m/s flows past a smooth thin rectangular plate
which is 2 m wide and 30 m long in the flow direction. Taking the kinematic viscosity
of air equal to 1.5 x 10-5 m2/s and its density equal to 1.2 kg/m3:
i) Establish whether the flow is turbulent or laminar.
(2)
ii) Determine the average shear stress on the plate.
(5)
iii) Determine the boundary layer thickness 10 m from the leading edge.
(3)
iv) Explain the difference between displacement thickness and momentum
v)
thickness
(2)
Derive the expression for the displacement thickness
(3)
b) The boundary layer described in part (a) can be modelled using Prandtl mixing length
theory.
i) Describe the assumptions of this theory.
(4)
ii) Using these assumptions, derive an expression for the velocity of the flow as
a function of the distance from the surface. Start your analysis with the
relation τ
turbulent
the (x, y) plane.
 - ρνx vy  constant. Assume that the flow is confined to
(6)
ME304 4 / 5
3) All engineering CFD codes are focused on the solution of the following general equation:


ρφ  div  ρ u φ   div Γgrad φ  S
t


a) Simplify this equation assuming that all parameters change in one direction only, do
not explicitly depend on time and assume that the source term is equal to zero.
(8)
b) Discretise this simplified equation using the central-differencing finite-volume
approach.
(12)
c) Describe the range of applicability of the central differencing.
(5)
4) Consider a plane wall 30 cm thick, one side of which is kept at T1 = 300 K while
another is kept at T2 = 600 K. Find the distribution of temperature inside this wall
using the finite-volume technique, for 3 computational cells. The distribution of
temperature inside the wall is controlled by the equation
d 2T
 0.
dx 2
(25)
ME304 5 / 5
5) Figure Q5 shows a 2-D aerofoil positioned at 3 incidence in a flow of dry air ( =
1.4) at a Mach number of 1.4 in an ambient static pressure of 0.9 bar.
Calculate the coefficient of lift using shock-expansion theory.
(25)
0.6 c
3
0.4 c
M1= 1.4
p = 0.9 bar
5

Figure Q5
6) A normal shock wave moves into still air with a velocity of 1600 m/s. The air is at
13C and 90 kPa. Calculate the stagnation pressure and temperature behind the wave
assuming that γ  1.4 and R  287
J
.
kgK
Use the following relation between Ma1 and Ma2:
Ma 2 
Ma 12  2/ γ  1
2 γ γ  1Ma 12  1
(25)