Do not turn over until you are told to do so by the Invigilator.

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1 UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination FLUID DYNAMICS - THEORY AND COMPUTATION MTHA5002Y Time allowed: 3 Hours Attempt QUESTIONS 1 and 2, and THREE other questions. Notes are not permitted in this examination. Do not turn over until you are told to do so by the Invigilator. MTHA5002Y Module Contact: Dr Hayder Salman, MTH Copyright of the University of East Anglia Version: 1

2 (i) Write down the continuity equation for an unsteady compressible fluid, with density ρ(x, t) and velocity vector v(x, t) and derive the condition on the velocity for an incompressible fluid. [4 marks] (ii) Consider a pipe that consists of a horizontal segment with a varying cross-sectional area that is aligned along the x -axis and a downward segment of height, h, that is parallel to the z -axis. The two segments are joined together by a smooth bend. Water flowing smoothly within the pipe experiences a gravitational force g = gk where g is the gravitational acceleration and k is the unit normal pointing in the positive z -direction, with z = 0 along the horizonal segment of the pipe. The cross-sectional area at the inlet is given by A 1 while the area at the outlet is A 2. If the pressure at the inlet of the horizontal section is at atmospheric pressure given by p a, and the inlet velocity is V 1, find the pressure at the outlet of the pipe. [8 marks] (iii) Consider a steady three-dimensional flow with a velocity vector field given by u = u(x, y, z)i + v(x, y, z)j + w(x, y, z)k, and with a constant density, ρ, and a pressure field p(x, y, z). If the flow is governed by Euler s equation of motion use the vector identity (u ) u = 1 p gk, ρ ( ) 1 f ( f) = 2 f f (f )f, to show that Bernoulli s equation for a rotational fluid that is given by p ρ u 2 + gz = constant is constant along a streamline. [8 marks] MTHA5002Y Version: 1

3 (i) (a) Obtain an expression for the velocity field, given that the velocity potential is of the form φ(x, y) = 3x 2 3y 2 6xy, (b) Show that the velocity field is incompressible and irrotational. [10 marks] (ii) Determine the streamfunction ψ(x, y) corresponding to the above velocity potential when ψ = 0 at the origin. [6 marks] (iii) Find the equations for the streamlines given by ψ = 0 and hence, sketch the steamfunction by considering lines of constant ψ on the x y planes clearly indicating the direction of the flow. [4 marks] MTHA5002Y PLEASE TURN OVER Version: 1

4 The propagation of waves on a free surface of a fluid contained within a tank is assumed to satisfy the linearized system of equations φ t 2 φ = 0 on h z 0, φ = gη, and z = η on z = 0, t φ = 0 on z = h. z For a fluid bounded by vertical walls located at x = 0 and x = L, a general solution is given by a standing wave of the form where A is a constant. φ(x, z, t) = A cos(kx) cos(ωt)f(z). (1) (i) (a) State which of the above equations are kinematic boundary conditions and which are dynamic boundary conditions. (b) Find the wavenumbers k, such that the velocity potential satisfies the kinematic boundary conditions at the two vertical walls. φ x = 0, [5 marks] (ii) Solve for the velocity potential by starting with the general solution of Laplace s equation given by Equation (1) and solving for the given kinematic boundary conditions. [8 marks] (iii) (a) Determine the discrete form of the dispersion relation for the above system of equations. (b) Hence, assuming a continuous dispersion relation, evaluate the phase speed and group velocity for this system. [7 marks] MTHA5002Y Version: 1

5 (i) Describe the function of each block of the following piece of Matlab code, and the overall purpose of the code: N=16; xleft = 0; xright = 2*pi; xlen = xright - xleft; varm=0.5; dxuni=xlen/n; xnonuni(0)=xleft; xnonuni(n)=xright; for i=1:n-1 xu=xleft+dxuni*i; xnonuni(i)=xu+varm*sin(xu); dx(i)=xnonuni(i)-xnonuni(i-1); end dx(n)=xb(n)-xb(n-1); dx(0)=dx(n); dx(n+1)=dx(1); dxmin=1.0; x(0)=xleft-dx(0)*0.5; for i=1:n x(i)=x(i-1)+0.5*(dx(i-1)+dx(i)); dxmin=min(dx(i),dxmin); end [5 marks] (ii) A fluid particle moves in a flow so that its path x(t) satisfies dx dt = cos(2πt), x(0) = 0. (1) (a) Determine the exact solution, x(t), and evaluate the exact solution at t n n = 0, 1, 2, 3 and for a time-step size h = You can assume that t 0 = 0. (b) Evaluate an approximation, x n, n = 0, 1, 2, 3, along with the error, for e n = x(t n ) x n for n = 1, 2, 3, 4, using Euler s method where the time-step size is taken to be h = 0.25 and x 0 = x(t 0 ) = 0. [8 marks] MTHA5002Y PLEASE TURN OVER Version: 1

6 - 6 - (iii) The trapezium rule for approximating the solution to the system given in Equation (1) is x n+1 = x n + h 2 (u(t n+1) + u(t n )) (a) Taking a time-step size of h = 0.25, use the trapezium rule to construct an approximate discrete path x n, for n = 0, 1, 2 3, and calculate the error. (b) State the difference between the Euler approximation and the trapezium rule in terms of type of scheme, function evaluations and error. [7 marks] MTHA5002Y Version: 1

7 (i) Letting z = x + iy, define the complex potential, w(z), for an incompressible, irrotational two-dimensional flow in terms of the velocity potential φ(x, y) and the. stream function ψ(x, y). [2 marks] (ii) It is given that w = K log z, 2π where K is a real constant. By writing z in complex exponential form, show that the streamlines are straight lines passing through the origin. [3 marks] (iii) A source of strength K is located at the point (0, a), with a solid wall at y = 0, < x <. Using the method of images, show that the complex potential is given by w(z) = K 2π log(z2 + a 2 ). and calculate the fluid velocity on y = 0. [7 marks] (iv) Show that the transformation z = Z 3 maps the region 0 <arg(z) < π/3 to the half plane Im (z) > 0, and hence calculate the complex potential for a source strength K located at Z = be iπ/6 between rigid walls arg(z) = 0, arg(z) = π/3. [5 marks] (v) Show that at a point P, far from the source at Z = be iπ/6 and the origin, the fluid velocity takes the form u 3K πr ˆR, where OP = R ˆR. Explain this result in terms of volume flux. [3 marks] MTHA5002Y PLEASE TURN OVER Version: 1

8 (i) For a two-dimensional steady, irrotational flow, the complex potential is given by w(z) where z = x + iy. Write down an expression for the cartesian components of velocity, u and v, in terms of dw/dz, and explain what is meant by a stagnation point. [4 marks] (ii) Show that w = Ue iα z, where U, α are real constants, represents uniform flow, magnitude U, at angle α to the positive x -axis. (iii) By writing z = ae iθ, show that if ( ) w = Ue iα z + a2 e 2iα z iγ log z, ( ) 2π [3 marks] where U, α, a, Γ are real constants, then z = a is a streamline, and prove that if Γ = 4πUa, then there is only one stagnation point. [6 marks] (iv) The force on a body in a flow is F = F x x + F y ŷ, where x and ŷ are unit vectors in the x and y directions respectively. Write down Blasius formula giving the force on the body in terms of the complex potential of the flow. A circle z = a is placed in a two-dimensional flow with complex potential given by ( ). Ignoring body forces, use Blasius s formula to show that F.U = 0, where U = U cos αˆx + U sin αŷ. [7 marks] END OF PAPER MTHA5002Y Version: 1

Candidates must show on each answer book the type of calculator used. Only calculators permitted under UEA Regulations may be used.

UNIVERSITY OF EAST ANGLIA School of Mathematics May/June UG Examination 2011 2012 FLUID DYNAMICS MTH-3D41 Time allowed: 3 hours Attempt FIVE questions. Candidates must show on each answer book the type