Homework set6. Problems 8.31,39,43,64,70

**8.23 Find the resultant velocity vector at point A due to the combination of uniform stream, vortex and line source.

Add vectorially, to get v = 11.3 m/s at @ = 44.2 degrees.

 

8.26 Find the resultant vector velocity at A by the uniform stream, line source, line sink, and line vortex.

Solution: The source and sink are each (5)^/12 = 2.24 m from point A, so the sink velocity is 10/2.24 = 4.47 m/s and the source velocity is 12/2.24 = 5.37 m/s. The vortex velocity is 9/2 = 4.5 m/s. The net horizontal component is 6.44 m/s. The net vertical component is -6.85 m/s (down). Then the resultant induced velocity at A is v = (6.85^2 + 6.44^2)^1/2 = 9.4 m/s at \theta = tan^-1 (-6.85/6.44) = -46.8 deg.

8.31 A Rankine half body is formed by uniform flow of 7 m/s and a source at 0,0 , with dividing streamline at 0,3 m. Find a. the source strength m in m^2/s; b. the distance a of the stagnation point; c. the distance h of the dividing streamline at point 4,0; d. the total velocity at A = 4,h .

The equation for the body is r_body = m(\pi - r)/Usin@ . Thus, for @ = \pi / 2 and y=r_body = 3 m we find m = 13.4 m^2 / s. Also a = 1.91 m. At x=4 m, r_body = m(\pi - r)/Usin@ = 13.4(\pi - @)/7 sin@ = 4/cos@ . Solve for @= 47.8 degrees.

r_A = 4.0/cos47.8 = 5.95 m; h = r sin@ = 4.41 m. The velocity at A is computed from 8.18:

V_A = U(1 + a^2/r^2 + 2a/r cos @_A )^{1/2} = 8.7 m/s

 

8.39 Sketch the streamlines of a uniform stream U_infinit pas a line source sink pair aligned vertically with the source at +a and the sink at -a on the y axis. Does a closed body shape appear?

No closed body shape appears.

8.43 Consider water at 20 C flowing past a 1 m diameter cylinder. What doublet strength in m^2/s is required to simulate this flow? If the stream pressure is 200 kPa use inviscid theory to estimate the surface pressure at a. 180 deg; b. 135 deg; and c. 90 deg.

The doublet strength \lambda = U_infinity a^2 = 6 0.5^2 = 1.5 m^3/s. Surface pressures from Bernoulli's equation, with V_surface = 2 U_infinity sin@ a. at 180 deg p_surf = 218000 ; b. 182000 ; c. 146000 Pa.

 

8.64 Determine qualitatively from boundary layer theory whether separation will occur.

In 8.61 for \psi = A r^n sin n@ the velocity component along the x axis with @=0 and r=x v_r = u = const x^{n-1}. Thus, for a. n=3, so U = const. x^2 ; b. n=2, U = const. x^1 ; c. n= 3/2, U = const. x^1/2 ; all give favorable pressure gradients, so there should be no separation.

 

8.70 Show that the complex potential f(z) = U[z + a/4 coth \pi z / a ] represents flow past an oval shape placed midway between two parallel walls y = a/2. What is the practical application?

The stream function is \psi = U[ y - ( a/4 sin 2 \pi y / a ) / ( cosh 2 \pi x / a - cot 2 \pi y / a )]

The streamlines are trapped between the wall and an oval body shape, with width a/2 and height 0.51 a. An application is the estimate of wall blockage effects when a body in a wind tunnel is trapped between walls.