FLUID DYNAMICS, THEORY AND COMPUTATION MTHA5002Y

Transcription

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. You will not be penalised if you attempt additional 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 Mark Blyth, MTH Copyright of the University of East Anglia Version: 1

2 (i) Consider a flow with velocity field u(x,y) = yi+q(x)j, for some function q(x), where i and j are the unit vectors in the x and y directions respectively. (a) Is this flow incompressible? Justify your answer. (b) Give the condition for a flow to be irrotational. (c) Determine the form of q(x) such that this flow is irrotational. Your result should contain an arbitrary constant. (d) Determine the arbitrary constant in q if the flow has a stagnation point at (x,y) = (0,0). (e) Assuming the form of q determined in part (i)(d), calculate Du/Dt. [13 marks] (ii) A river flows over a cliff edge to create a large waterfall. Immediately before the cliff edge, the river is flowing horizontally at speed U. The total vertical distance descended by the waterfall is H. (a) Write down the steady form of Bernoulli s equation in a gravitational field with acceleration g. (b) Assuming steady flow in the waterfall, and assuming that the pressure in the water is everywhere equal to atmospheric pressure, p a, show that V 2 U 2 = 2gH, where V is the speed of the waterfall on impact at the bottom. [7 marks] MTHA5002Y Version: 1

3 (i) Consider the flow with velocity field u(x,y) = ui + vj, where i and j are the unit vectors in the x and y directions respectively, and where u = 2y b 2, v = 2x a 2, for constants a > 0 and b > 0. (a) Identify any stagnation points for this flow. (b) Compute the vorticity ω = u for this flow. (c) Given that u = ψ y and v = ψ x, determine the streamfunction ψ for this flow. (d) Sketch the streamline ψ = 1, including an arrow to show the direction of flow. (e) Solve the particle path problem by determining x(t) and y(t) which satisfy dx dt = u, dy dt = v, with initial condition x(0) = a, y(0) = 0. (Hint: Differentiate the first of these equations with respect to time.) [17 marks] (ii) Let C be the streamline corresponding to ψ = 1 for the flow in part (i). State, with reason, if the circulation Γ = u dx C is zero or non-zero. [3 marks] MTHA5002Y PLEASE TURN OVER Version: 1

4 An applied pressure excites waves of small amplitude on the surface of a liquid of infinite depth so that the free surface is located at y = η(x,t), where y measures vertical distance up from mean level y = 0. The velocity potential in the liquid, φ(x,y,t), satisfies Laplace s equation, φ xx + φ yy = 0 over < y < 0, subject to the linearised surface conditions φ t and φ y = η t = gη P sin(βx)sin(σt) on y = 0, ( ) on y = 0, where P, β and σ are all positive constants, and g is the acceleration due to gravity. Also we have that φ 0 as y. ( ) (i) (a) Assume that η = Asin(βx)sin(σt). State the physical meaning of the constants A, β and σ. (b) Assume also that φ = sin(βx) cos(σt)f(y). Starting from Laplace s equation, show that f β 2 f = 0. Obtain the general solution for f(y) involving two arbitrary constants. Eliminate one of these arbitrary constants using the given condition as y. (c) Use conditions ( ) and ( ) to find the other arbitrary constant and hence determine A in terms of P, g, σ, and β. (d) What happens to the wave amplitude as P 0? [17 marks] (ii) What happens mathematically if P 0 and σ 2 = βg? Interpret this case physically. [3 marks] MTHA5002Y Version: 1

5 (i) Describe the purpose of each block of the following Matlab program, including the method being implemented: (a) (x.ˆ2+1)/((x+2). sqrt(1 2 x)); a=0; b=.5; N=100; h=(b a)/n; (b) It=0; for k=1:(n 1) x=a+h k; It=It+feval (f,x); end It=h (f(a)+f (b))/2+h It ; (ii) Given the interpolation points x 0 = 0, x 1 = 1 4, x 2 = 1 : [5 marks] (a) Give the formula for the Lagrange interpolating polynomial of degree two for a general function f(x), where the Lagrange interpolating functions are given by L k (x) = 2 j=0 j k x x j x k x j. (b) Let f(x) = (1+x) 1. Usetheformulain(a)tofindaninterpolatingpolynomial of degree two for f(x). THIS QUESTION CONTINUES ON THE NEXT PAGE [7 marks] MTHA5002Y PLEASE TURN OVER Version: 1

6 - 6 - (iii) A fluid particle moves in a flow so that its path (x(t),y(t)) satisfies dx dt = u(x,y,t) = x, dy dt = v(x,y,t) = y, (x(0),y(0)) = (0.1,1). (a) Determine the streamfunction, ψ(x, y), for the above streamline equations and write the condition that the streamfunction satisfies along streamlines. (b) The Trapezium rule for dy dt = f(y) is given by y k+1 = y k + h 2 (f(y k)+f(y k+1 )), where h is the step size. Apply this rule to dy dt an expression of the form y k+1 = g(h)y k, where g(h) is a function depending on h. = y and rearrange to obtain (c) Use the initial condition to compute y k for k = 1, 2 with h = 0.5. Using the fact that ψ(x,y) is constant along streamlines, compute the corresponding x k values for the streamline passing through (x,y) = (0.1,1) and ψ(x,y) = 1.1. [8 marks] MTHA5002Y Version: 1

7 (i) The complex potential for an irrotational, incompressible flow is given by w(z) = φ(z)+iψ(z), where z = x+iy, φ(z) is the velocity potential and ψ(z) is the streamfunction. Write down the relation between the velocity field of the flow and the complex potential. [2 marks] (ii) Consider two vortices each of circulation Γ with vortex 1 located at (a,0) and vortex 2 located at ( a,0). (a) Write down the complex potential for the flow and use this to evaluate the velocity field of the flow. (b) Determine the stagnation points of the flow. (c) Show that, at very large distances from the origin, the flow is like that of a single point vortex with circulation equal to the sum of the two vortices. (d) By expressing each of z a and z + a in complex polar form or otherwise, obtain an expression for the stream-function. Hence, sketch the streamlines of the flow. [12 marks] (iii) (a) Calculate the induced velocity on vortex 1 and on vortex 2. (b) The two vortices can be brought to rest by adding a third vortex located at the origin. Show that with a suitable choice of the circulation of the third vortex, the induced velocities of the three vortex system will be zero. [6 marks] MTHA5002Y PLEASE TURN OVER Version: 1

8 (i) Amappingfromthecomplex z planetothecomplex ζ planeisgivenby ζ = z+z 1 where z = x+iy. (a) Using the complex polar representation for z, show that this mapping transforms the circle z = 1 to a line on the real axis. (b) Show that the same transformation maps a circle z = a, where a > 1 is a constant, to an ellipse in the ζ plane of the form ( ) 2 ( ) 2 ζr ζi + = 1, c d such that ζ = ζ R +iζ I, where ζ R and ζ I are real. Hence, determine the constants c and d. [6 marks] (ii) An aerofoil can be obtained by mapping a circle centred off the origin at z = δ and with radius 1+δ where δ > 0. The circle has the equation z +δ = 1+δ. (a) Bywritingtheequationforthecircleinpolarcomplexformwith arg(z+δ) = θ, show that the real part of the image of the circle in the complex ζ plane is given by ζ R = 2cosθ +δ( 1+cos2θ)+O(δ 2 ), for small δ and obtain a similar expression for ζ I. (b) Hence, determine the coordinates for the leading and trailing edges of the aerofoil up to and including O(δ). Also, find the value of θ at which the maximum thickness of the aerofoil occurs up to the same order. [8 marks] (iii) A flow past a cylinder in the z complex plane is given by ] w(z) = U [(z +δ)+ (1+δ)2 + iγ ln(z +δ), (z +δ) 2π for real constants U and Γ. Use Blasius formula to determine the lift force on the corresponding aerofoil in the ζ-plane. [6 marks] END OF PAPER MTHA5002Y Version: 1

9 MTHA5002Y Fluid Dynamics Exam Feedback 2017/18 Question 1 The first compulsory question. The first part of this question as answered very well. A few students did not recall the definition of the convective derivative, but many who did applied it correctly and obtained the right result. In part (ii) almost all students remembered the required form of Bernoulli s equation, and the vast majority correctly applied it to derive the given formula connecting the two speeds V and U. Question 2 The second compulsory question. Most students recalled that a stagnation point is one where the fluid velocity vanishes, and correctly computed the vorticity. While very many students determined the streamfunction correctly, the sketch of the required streamline(an ellipse with semi-major axis a and semi-minor axis b) was less successful. A large number of students could not solve the particle path problem, which was almost identical to one seen during the module - a very common mistake here was to overlook the fact that x and y are functions of t and to attempt to integrate the two ODEs directly. In part (ii) some students correctly applied Stokes theorem to note that Γ is non-zero. In fact it was not required to remember Stokes theorem. This was noticed by a few students who correctly reasoned that since ψ = 1 is a streamline, and since the only stagnation point has already been identified at the origin, the given line integral must be non-zero. Question 3 This question on small amplitude waves was attempted very well by a large number of students. Part (ii) is rather tricky and could be answered either by reference to the results from part (i) or as a standalone. Where students did go wrong with this question was either in solving the given ODE for f(y) - this is year 1 work, and the solution was discussed in lectures - or else in applying the boundary condition at minus infinity correctly. Aside from a few algebraic slips, a large number of students successfully determined A and correctly reasoned that the wave amplitude goes to zero as the pressure forcing goes to zero. 1

10 Question 4 Few students answered question 4. For part (i), most students correctly identified the code from part (a), but did not recognize the code in part (b) to be the Trapezoidal rule. For part (ii), few students were able to give the general Lagrange interpolation formula. Students did well on part (iii), which was an example from the lecture notes. Question 5 (i) This required recalling the definition. Straightforward bookwork but there were students who dropped marks on this. (ii) (a) was generally well attempted although a number of students immediately assumed the two vortices represent a vortex and an image even though there is no mention of a wall in the question. It seemed that they were simply replicating an answer from what they considered to be a similar question that was covered in the module. (b) was straightforward with only one stationary point although assuming a vortex an image would have led to the incorrect result. (c) Most people managed to recover the correct answer for this part. (d) Many students struggled here. Most of the attempts were based on the polar form but rather than expressing z a and z + a in polar form, i.e. z a = r 1 exp(iθ 1 ) and z a = r 2 exp(iθ 2 ) people assumed the polar form for z and were not able to proceed very far. Consequently, almost everyone struggled to sketch the streamlines correctly. This last part was the most poorly attempted of the entire question. (iii) This was generally answered well and for those who knew how to answer (a) went on to correctly answer (b) also. One of the main errors that students madewasnottoobtainexpressionforthe velocityatthepositionsofthe vortices but rather to leave their answers as functions of z. Question 6 This question was very poorly attempted. (i)(a) This was bookwork which most students managed to get. For (b) some students struggled with this although on the whole many students found it straightforward. (ii) Here, so many students struggled to recover the correct equation for the circle z+δ = 1+δ which is in fact z = δ+(1+δ)e iθ since this is a circle with origin at ( δ,0) and radius (1 + δ). Once this is known the remainder of the calculation involved substituting into the expression for ζ given in the question and using a Binomial expansion to approximate the z 1 term and thereafter separating real and imaginary parts. For (b), it would have been possible to determine at least the coordinates along the real axis of the leading and trailing 2

11 edges from the expression for ζ R that is already given in the question but hardly anyone attempted to do that. (iii) Only a few students recalled the key results needed for this part of the question, namely Blasius formula how to use the residue theorem to evaluate the integral. Even fewer students then made an attempt to evaluate the lift force on the body. 3