The fluid is incompressible, inviscid and has constant density p. Gravitational acceleration is denoted by g throughout.

1. (a) Define the terms streamline, particle path and streakline.

(b) A fluid moves two-dimensionally so that its velocity u is given by

u = e^2t i + e^3t j,

where i and j are the unit vectors for the Cartesian coordinates (x, y) and t is time.

Obtain equations in terms of x and y alone for the following:

(i) the streamline through (1,1) at time t = 0,

(ii) the particle path for a particle released from (1,1) at time t = 0,

(iii) the streakline, at time t = 0, through (1,1) formed by particles released from (1,1) at times t W 0.

(c) Sketch carefully the three loci describing the fluid motion identified in (b) on the same diagram in the (x, y) plane. Where two or more curves coincide, mark the tangents to the curves. Indicate clearly the start and end points, or extent, of each curve.

(d) Show that in steady flow the quantity

H = p + 2 p|u|2 + pG

is constant along streamlines.

You may use the momentum equation

Du

p 瓦=-Vp + P^F,

where p is the pressure. Here F is a conservative external force per unit mass with potential G defined through

F 二一VG.

You may also use the result

(u.V)u = ¹ V|u|² + (Vx u) x u.

2. An impermeable, solid cylinder of radius a lies in a two-dimensional irrotational flow field with the centre of the cylinder lying at the origin of the Cartesian coordinate system Oxy. The Cartesian components (u,v) of the velocity far from the cylinder are given by

u T Ux, v T -Uy as r = x² + y² T x.

The circulation about the cylinder is k. The streamfunction,由 for the motion defined through

dp dp

^u = k, ^v = ^—k,

dy dx

satisfies Laplace's equation.

(a) What boundary conditions does p satisfy (i) at large distances, (ii) on the cylinder?

(b) Solve for p in terms of polar coordinates (r, 0).

(c) Obtain expressions for the polar components of velocity, u_r and u们 on the cylinder.

(d) Show that the stagnation points on the cylinder occur for polar angles 0 satisfying

sin(20) = K/4na²U.

(e) By considering intersections of the graph of the function fi(0) = sin(20) with the horizontal line 走(0) = K/4na² U, when plotted on the same axes as a function of 0, discuss the location of the stagnation points for the cases (i) k = 0, (ii) 0 < k < 4na²U, and (iii) k = 4na²U.

(f) What happens to the location of the stagnation points when k > 4na²U ? By writing down the complex velocity potential obtain an equation whose roots give the location of the stagnation points.

You may use without proof the relations

1 dp dp

^ur = — y, ^uo = 际—.

r d0 dr

3. The circle theorem states that the complex velocity potential for potential flow outside the solid cylinder |z| = a, of radius a > 0, can, in certain circumstances, be written in the form

^w(^z) = / (^z) +1 ^(a2/z).

(a) Give sufficient conditions for the validity of the theorem, defining f and f and proving that there is no normal flow through the circle |z| = a.

(b) A line source of strength 2nm lies at z = zo outside the cylinder and has complex potential given by mlog(z — zo) (where a and m are real constants, and zo is a complex constant with |zo| > a and imaginary part, Rz°, positive). The circulation about the cylinder | z| = a is zero.

(i) Use the circle theorem to obtain the complex potential for this system in the form

a²  

w(z) = mlog(z — zo) + mlog( zo).

(ii) Rearrange this expression and so describe the image system inside the cylinder in terms of simple singular flows.

(iii) A second line source of the same strength is introduced at z =布. Write down the complex potential, in terms of simple singular flows, for this new flow consisting of two vortices external to the cylinder.

(iv) Consider the curve C shown below. C coincides with the real-z axis for |z| > a and, in |z| W a, C coincides with the semi-circle |z| = a, Rz 2 0. In terms of simple singular flows, what is the image in C of a line source of strength 2nm lying at z = zo, above C, as shown on the figure?

(v) A line source of strength 2nm lies at z = 2ia above a solid flat surface with a solid semi-circular bump, as given by curve C below. What is the velocity on the real-z axis, i.e. Rz = 0,况z 2 a?

(vi) The line source at zo is replaced by a line vortex of strength k. What is the image in C of the line vortex?

 

4. A stream of local depth h flows along a channel of constant width. The channel floor rises slowly from a constant height zero to reach a small constant height k. Suppose that the fluid depths upstream and downstream of the rise are hi and h respectively.

(a) Explain carefully how to use Bernoulli's equation to show that

$(hi)=収 h2)+ k,

where ①(h) = h + Q2/(2gh²) and Q = uh is the volume flux per unit width. Distinguish carefully between the fluid depth and the surface height where necessary.

(b) Sketch the graph of $(h) for h > 0, finding the position, h = h_m, of the minimum of Define the Froude number, F, at any point in the flow and note a physical interpretation of F. Find the value of F when h = h_m. Show that the Froude number when the depth is h can be written F = (h_m/h)³/². Find the ranges of h for subcritical and supercritical flow.

(c) Show that in terms of the depth ratio A = h/h_m the function $ can be written

以 h) = h + hm/(2h²) = hm[A + 1/(2A²)],

and the Froude number F = A^-3/².

(d) Further downstream the channel floor returns to zero height. The ultimate downstream depth is h3 = 1 h_m-

(i) Using part (c), or otherwise, derive the equation

2A³ - 5A² + 1 = 0,

satisfied by the depth ratio A = h/h_m.

(ii) Using the factorization, 2A³ — 5A² + 1 = (2A — 1)(A² — 2A — 1), find, in terms of h_m, the two possible values of the upstream depth, hi.

(iii) Using part (c), or otherwise, find the downstream Froude number and the two possible values of the upstream Froude number. Briefly describe the two possible flows.