Inviscid Flow Final Exam
2017 Spring Hyungmin Park
1
1. (10 points) Explain the Helmholtz instability and Rayleigh instability.
2. (15 points) For the potential flow around a sphere (radius R), obtain the pressure variation along the sphere surface in terms of θ (which is the angle from the front stagnation point). Use the idea that the flow around a sphere can be represented by the superposition of uniform flow and point doublet.
3. (30 points) Potential flow and conformal mapping
(a) (5 points) Show that the Joukowski transformation maps a circle with radius a in z-plane onto a flat plate with length 4a in z-plane.
(b) (15 points) Consider the problem of uniform flow (U) oriented at an angle of 10° to a flat- plate airfoil. Determine the location on the airfoil of all stagnation points both (i) with enforcement of the Kutta condition at the trailing edge and (ii) with the condition of no circulation around the plate.
(c) (10 points) Determine the net force acting on the airfoil in the former case.
4. (20 points) There is a point source of strength Q [m3/s] at the origin, and a uniform line sink of strength k = Q/a extending from z = 0 to z = a. The two are combined with a uniform stream U parallel to the z-axis. Show that the combination represents the flow past a closed surface of revolution of airship shape, whose total length is the difference of the roots of:
z2 . a2
z a±1
!
"
# $
%&= Q
4πUa2
a2
z z
= + z
Inviscid Flow Final Exam
2017 Spring Hyungmin Park
2
5. (25 points) Let’s consider a small-amplitude plane wave is traveling along the liquid surface with velocity c with the liquid depth of h. The governing equation for velocity potential (f) and associated linearized boundary conditions are as follows.
(a) (5 points) Explain the physical meaning of each boundary conditions.
(b) (5 points) Let’s assume the profile of the surface wave as . Then, the solution of given Laplace equation can be assumed to be the form given below.
Derive the dispersion relation and find the values of non-dimensionalized propagation speed (c2/gh) for shallow and deep liquids conditions, respectively.
(Note: you can ignore the effect of surface tension.)
(c) (10 points) Describe the dispersion of arbitrary-shaped waves using the dispersion relation obtained in (b).
(d) (5 points) What happens if one additionally considers the effects of surface tension?
2 2
2
2 2
2 2
0
1 ( , )
( , ) ( ,0, ), ( ,0, ) ( ,0, ) 0 @
( , , ) 0 @
y h
x y
P x t
x t x t x t g x t y
t y t t y
x h t y h
y y
f f
f
h f f f h
r
f f
=-
¶ ¶
Ñ = + =
¶ ¶
¶ =¶ ¶ + ¶ + ¶ = =
¶ ¶ ¶ ¶ ¶
ö
¶ ¶
= - = =-
¶ ÷ø ¶
1 2
2 2 2
( , , ) cos ( ) sinh y cosh y
x y t p x ct C p C p
f l l l
æ ö
= - çè + ÷ø
( , )x t sin2p(x ct)
h e
= l -
Inviscid Flow Final Exam
2017 Spring Hyungmin Park
3
6. (30 points) Consider a Schwartz-Christoffel transformation of a channel flow with a vertical plate extending part way (in z-plane) across to the upper half of the ς-plane. Let’s assume that the points are to be mapped as indicated (e.g., A to a) in the figure below. The velocity far upstream in the channel is U, and the channel has a uniform height H, except at the plate, which is of height h.
(a) (10 points) Obtain the differential equation, dz/dz, which defines the mapping. You don’t need to integrate the equation to get the explicit form of the mapping function.
(b) (10 points) Determine the complex potential in the ς-plane, F(ς).
(c) (10 points) Determine the constant K in the differential equation in (a) and also a
relationship between the unknown values of d and f. Use the proper boundary conditions.
You may have to assign a negative value to the square root of a positive quantity to obtain the relation between d and f.
APPENDIX
In three-dimensional potential flow (in spherical coordinates), stream function for a uniform flow: 1
2U∞2r2sin2θ for a source located at the origin: φ=− Q
4πr,ψ=− Q
4π(1+cosθ) for a line sink (length L): ψ= q
4π(L+r−η) From Blasius integral law: X−iY=iρ
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