UNIVERSITY OF EAST ANGLIA School of Mathematics Spring Semester Examination 2004 FLUID DYNAMICS Time allowed: 3 hours Attempt Question 1 and FOUR other questions. Candidates must show on each answer book the type of calculator used. Log Tables, Statistical Tables and Graph Paper are available on request. Do not turn over until you are told to do so by the Invigilator. Copyright of the University of East Anglia
2 You may use the following vector identity, 2 u = ( u) ( u). In cylindrical polar coordinates, when u = u r + v θ +w ẑ, u = 1 r r θ ẑ r r θ z u rv w, 2 = 2 r + 1 2 r r + 1 2 r 2 θ 2, u = 1 r ψ θ, v = ψ, for streamfunction ψ. r In Cartesian coordinates, when u = ui + vj + wk, the Navier-Stokes equations are: ρ(u t + uu x + vu y + wu z ) = p x + µ(u xx + u yy + u zz ), ρ(v t + uv x + vv y + wv z ) = p y + µ(v xx + v yy + v zz ), ρ(w t + uw x + vw y + ww z ) = p z + µ(w xx + w yy + w zz ), u x + v y + w z = 0.
3 1. A plane jet of fluid of unit density penetrates an ambient gas, which is at atmospheric pressure p a. The undisturbed jet is confined between the lines y = a and y = a, where a > 0, and moves in the x direction at speed U at pressure P(x, y). Fix axes moving with the jet so that the fluid appears to be stationary. Let λ and ε be parameters, with ε 1. Perturb the jet such that its upper surface and lower surfaces are now at y = f(x, t) = ±a + εa 1 (t) cos(λx). Inside the jet, let εu(x, y, t), εv(x, y, t) be the perturbation velocities in the x and y directions respectively, and εp(x, y, t) the perturbation pressure, so that the new jet pressure is p(x, y, t) = P + εp. The velocity perturbation satisfies v t = p y, and the pressure perturbation p(x, y, t) satisfies Laplace s equation 2 p x + 2 p 2 y = 0. 2 At the upper and lower surfaces, the pressure discontinuities are p y=a p a = κ, p a p y= a = κ, respectively, with the curvature given by κ = f xx (1 + f 2 x) 3/2. (i) Neglecting terms of size ε 2 and higher, show that κ = ελ 2 a 1 cos(λx). Hence show that p(x, ±a, t) = ±λ 2 a 1 cos(λx). (ii) Write u = û(y, t) cos(λx), v = ˆv(y, t) cos(λx) and p = ˆp(y, t) cos(λx). Neglecting terms of size ε 2 and higher, use the kinematic conditions at y = ±a and the results from (i) to derive the conditions ( 1 ) 2ˆp λ 2 t ± ˆp 2 y = 0 at y = ±a. (iii) Write ˆp = e st g(y). Confirm by direct substitution that g(y) = A cosh λy + B sinh λy. By satisfying the conditions derived in (ii), show that s 2 = λ 3 tanh(λa) or s 2 = λ 3 coth(λa). TURN OVER
4 2. Define the rate of strain tensor e ij and the vorticity tensor ξ ij. Explain their meaning in terms of the kinematics of the flow. By considering a small cube of fluid, explain why, if e 11 + e 22 + e 33 = 0, the local rate of change of volume is zero and the fluid is incompressible. For a fluid of density ρ which is flowing with velocity u, the continuity equation is given by Dρ + ρ u = 0. Dt State the Reynolds transport theorem. By considering a moving fluid parcel of volume V (t) and surface area S(t), derive the Navier-Stokes equations for an incompressible Newtonian fluid in the absence of a body force.
5 3. Viscous fluid flows along an open rectangular channel which is bounded by two stationary parallel semi-infinite walls at y = 0, y = b and a bottom wall at z = 0. The fluid occupies the region 0 y b and 0 z <. The bottom wall moves at speed U in the positive x direction. The flow is assumed to be steady and unidirectional in the x direction with velocity component u. The fluid is at rest a long way from z = 0. Use the continuity equation to show that u = u(y, z). Write down the boundary conditions to be satisfied by u on z = 0, at y = 0 and y = b, and as z. By considering the momentum equations, show that the horizontal pressure gradient p/ x must be constant. Assuming this constant to be zero, deduce that the fluid velocity u satisfies the two-dimensional Laplace s equation, 2 u y + 2 u 2 z = 0. 2 By seeking a separable solution which satisfies the boundary conditions, show that in the fluid domain u = 4U π (2k + 1) 1 sin{(2k + 1)πy/b}e (2k+1)πz/b. k=0 Show that the volume flux Q along the channel is given by Q = 8Ub2 π 3 k=0 1 (2k + 1) 3. TURN OVER
6 4. Show that the steady two-dimensional equations of incompressible Stokes flow, given by 0 = p + µ 2 u, u = 0, are satisfied provided that where ψ(x, y) is the streamfunction. 4 ψ = 0, (1) A very viscous fluid in Stokes flow occupies the infinite region 0 < r <, which is bounded by two solid walls at θ = 0, θ = α, where r and θ are plane polar coordinates. The wall θ = 0 moves at constant speed U in the negative x direction. The other wall θ = α is fixed. Write down appropriate boundary conditions for the fluid velocity on θ = 0 and θ = α. Assuming that the flow is modelled by equation (1), seek a solution in the form ψ = Urf(θ), and show that f(θ) satisfies the equation f + 2f + f = 0. Solve this equation and apply the boundary conditions to show that f(θ) = θ sin2 α cosθ (α 2 kθ) sin θ α 2 sin 2, α where k = 1 (2α sin 2α). 2
7 5. Incompressible fluid of kinematic viscosity ν flows at high Reynolds number over a flat rigid plate which occupies 0 < x <, y = 0. Justify the reduction of the steady Navier-Stokes equations to the boundary layer form u u x + v u y = U(x) U (x) + ν 2 u y 2, u x + v y = 0, where (u, v) are the velocity components in the (x, y) directions, and U(x) is the external slip velocity at the top of the boundary layer. Explain why the pressure does not vary across the boundary layer. Suppose that the external slip velocity U(x) = U, which is constant. Introducing a streamfunction, ψ, defined by ψ = (νu x) 1/2 f(η), where η = show that the momentum equation reduces to the form f + 1 2 ff = 0. ( U νx ) 1/2 y, State the boundary conditions to be satisfied by f(η) at the solid plate and as η. If the boundary layer momentum thickness, θ, is defined as show that θ = 0 u ( 1 u ) dy, U U ( νx ) 1/2 θ = 2 f (0). U TURN OVER
8 6. Define the vorticity of a fluid motion and explain what is meant by a Stokes layer at a solid wall in an oscillating flow. An incompressible Newtonian fluid occupies the region y > 0 above a stationary flat wall at y = 0, < x <. A flow is driven in the fluid by a pressure gradient p/ x = G cosωt, where G and ω are constants. The fluid velocity is bounded as y. If the flow is unidirectional, with u = u(y, t) and v = 0, show that u satisfies the equation u t = (G/ρ) cosωt + u ν 2 y, 2 where ν, ρ are the kinematic viscosity and density of the fluid respectively. Supply the boundary condition at the wall y = 0. Let σ = (ω/2ν) 1/2. Find the solution for u(y, t) and demonstrate that it is π/2 out of phase with the pressure gradient. Compute the vorticity in the fluid and state whether it is in or out of phase with the pressure gradient. Deduce the existence of a Stokes layer at the wall when σ is large. Give an estimate for the thickness of this layer. END OF PAPER