worksheet1 - 4. The Euler equation is u t + ( u ) u =-1 p,...

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Fluid Dynamics 3 2011/12 Sheet 1 Homework to be handed in Friday 21st October: 1,3,4,9. 1. Let a closed loop of particles C ( t ) be defined by x = a (cos s + αt sin s, sin s, 0) , 0 s < 2 π, where each value of s corresponds to a different fluid particle, and a,α > 0. (i) Sketch how C ( t ) changes as function of time, starting from t = 0. Which points remain stationary ? (ii) Find the Lagrangian velocity field as function of the particle marker s . (ii) Show that the Eulerian velocity field is u = ( αy, 0 , 0) . 2. Find the general equation and sketch (a) the streamlines (at t = 0 , 1 2 , 1); and (b) the particle paths (for particles released at the point (1 , 1 , 0) at t = 0 and t = 1), for the following time dependent, two-dimensional flow fields, given in a Cartesian coordinate system u = ( u,v, 0) (i) u = a cos( πt ), v = a sin( πt ) ( a constant) (ii) u = x - V t , v = y ( V constant) (iii) u = ktx , v = - kty ( k constant) (iv) u = xt , v = - y 3. A two-dimensional fluid velocity field u = ( u,v, 0) has components given by u = u 0 sin ω ( t - y/v 0 ) , v = v 0 . Find and sketch (i) the pathlines for a particle released into the flow from ( x,y ) = (0 , 0) at t = t 0 . Consider different values of t 0 . (ii) the streamlines of the flow. Again consider different instants in time.
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Unformatted text preview: 4. The Euler equation is u t + ( u ) u =-1 p, where u = ( u,v,w ) is the velocity and p the pressure. Write out the second component of the Euler equation in components. 5. Simplify (i) ijk ijm ; (ii) ijk ijk . 6. Demonstrate that for ( x ), a scalar function of position = 0 7. Prove the vector identity, for vector eld u ( x ) and v ( x ), ( u v ) = u ( v )-v ( u ) + ( v ) u-( u ) v 8. Show that for a vector eld u ( x ) u ( | u | 2 / 2) = u ( u ) u 9. For each of the following functions, ( x ), nd and 2 4 using sux notation and the summation convention. (i) = f ( r ); f is an arbitrary scalar function (ii) = a x /r 3 , where a is a constant vector and r 2 = x x ....
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worksheet1 - 4. The Euler equation is u t + ( u ) u =-1 p,...

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