Mid_Sem_Exam_A_2016

2
Q2
Velocity Potential - Stream function
f (x, y ) = xA y + By A+1 , A ≥ 0, B ≥ 0
(20 points)
A function possibly describing a two-dimensional flow is given as
→ − V = vr e ˆ ˆ ˆ r + vθ e θ + vz e z

Polar
→ − Divergence (dot product) of a Vector, V → − ∂u ∂v ∂w ∇· V = + + ∂x ∂y ∂z → − ∇× V = Cartesian
2
∂ 2ψ ∂x∂y
2
,
where ∇2 is the Laplacian operator. The above equation can be non-dimensionalized using reference length L, reference velocity U and reference pressure ρU 2 . Determine the functional form of reference stream-function ψ0 .
CIVL3612/9612 - Mid-Semester Exam
4
Q4
Potential flow over a half-body
(20 points)
The bottom of a river has a 4 m high bump that approximates a Rankine half-body, as in the figure. The pressure at point B on the bottom is 130 kPa, and the river velocity is 2.5 m/s. Use inviscid theory equations provided, estimate the pressure at point A on the bump, which is 2 m high in elevation. For water at 20◦ C, ρ = 998 kg/m3 .
OR
Determine velocity profile An infinitely long, solid, vertical cylinder of radius R is located in an infinite mass of viscous incompressible fluid. Consider the flow in which the cylinder is rotating about a fixed axis with constant angular velocity, ω . Neglect body forces and assume that the flow is axisymmetric.
CIVL 3612/9612 - Fluid Mechanics, Semester 1 2016 Mid-Semester Exam - TUESDAY Last name First name SID University email Instructions 1. The exam will last 1 hour and 50 minutes, inclusive of 5 minutes reading time. 2. Don’t forget to write your name on this page as well as your student number. 3. This is a closed book exam. You are NOT allowed to bring additional material of any kind. 4. Relevant equations to solve the problems are at the back of this booklet. 5. A standard scientific calculator is permitted. No graphical or programmable calculators are allowed. Similarly, phones, tablets or computers are strictly not allowed. 6. Read the questions carefully and start with the easiest problem. 7. Clearly and neatly show each step of your solution in the space provided. If your final answer is incorrect, you may receive partial points if your method is deemed to be correct. 8. If you have the correct answer without showing each step of your solution, you will receive a mark of zero. 9. Do NOT remove any pages from this booklet. If you need extra pages to write answer, ask a tutor for blank pages. 10. Use FRONT AND BACK of this answer booklet. 11. Good luck. For examiners only: @.au
(a) Simplify the Navier-Stokes equations stating all assumptions clearly (b) State the proper boundary conditions. (c) Derive an expression for the velocity distribution. (d) Determine the only non-zero component of shear stress in this flow. Where is it maximum?
x 2 − a0 γ+1 t
(15 points)
A one-dimensional unsteady flow is given by the following velocity u=
and density field ρ = ρ0 γ−1 x 1 2 + γ + 1 t a0 γ + 1
2 γ −1
5
Q5
Navier-Stokes equations
(25 points)
Determine pressure distribution A two-dimensional flow field for is given as u = Kx, v = −Ky, w = 0, K = constant.
(a) Neglect gravity and show that the velocity field is an exact solution to the incompressible Navier-Stokes equations. (b) Compute the pressure field p(x, y ) and relate it to the absolute velocity V 2 = u2 + v 2 . Interpret the result. (c) Briefly state how your approach to solve this problem will change if gravity is not neglected and z is ‘up’ (gx = 0, gy = 0, gz = −g )? Limit your answer to 2-3 sentences.
Inviscid theory for half-body Stream function = U r sin θ + m θ 2π Half-width = πb b= m 2πU Equation of surface: r = b(π − θ) sin θ
CIVL3612/9612 - Mid-Semester Exam
Determine (1) A and B for f (x, y ) to be a valid velocity potential, i.e. φ = f (x, y ). (2) A and B for f (x, y ) to be a valid stream function, i.e. ψ = f (x, y ). (3) If A = 0 and φ = f (x, y ), sketch at least two distinct streamlines.
CIVL3612/9612 - Mid-Semester Exam
3
Q3
Solve ONLY ONE of the following
(20 points)
Non-dimensional governing equation The Pressure-Poisson Equation (PPE) relating pressure (p) and stream function (ψ ) in a two-dimensional incompressible flow is given as ∂ 2ψ ∂ 2ψ ∇ p = 2ρ − ∂x2 ∂y 2
OR
Froude Number Similarity An open channel with a rectangular cross-section has a 8 m width and 2 m depth with water flowing at 5 m3 /s. A model is designed to discharge 1/1024 that of the prototype. (a) Find width and depth of the model using Froude number similarity. (b) Is Reynolds number similarity achieved for tests in (a) if the working fluid is the same for model and prototype? Why?