Question

. An experimentalist has measured the u-velocity component of a steady, two- imensional flow field. It is approximated by u 3x2y x +10 It is also known that the v-velocity is zero along the line y-0. a) Find an expression for the v-velocity in the entire field b) Find an expression for the streamfunction, v, for this flow c) Determine the location of any stagnation points in the flow (stagnation means -0) d) Calculate the acceleration field (a and ay) for this flow field.

Answer 1

(a): Apply continuity equation for 2D flow as follows: Ou avo * oy 818 818 818 = (3x²y+x+10) =-(6xy+1) =-bry - 1 â=(-6x-1)ây integrate both side v=-68"-y+c v=-3xy2 – y +c ...... (1) At y=0, v=0 From equation (1) 0=-3.rx 0-0+C C=0 Substitute 0 for c in equation (1). v=-3xy? – y +0 v=-3xy? - y ...... (2) Thus, the expression for the v-velocity is -3xy2 - y).

(b): Write the expression for the stream function for incompressible, irrotational, and steady 2D flow as follows: Oy ov OY = 3.x²y +x+10 @y=(3r’y+x+10)@y integrate on both side y = 3.rº." + xy +10y+f (x) y=1.5.rº y2 + xy +10y+f(x) ...... (3) Write the expression for the stream function for incompressible, irrotational, and steady 2D flow as follows: v=-oay ax =-(-3xy? – y) OV = 3xy? + y y=(3xy2 + y)ax integrate on both side y =372 5* + xy + f(y) y=1.5rºy2 + xy + f(y) Comparing the equation (3) and (4). w=1.5x+y2 + xy +10y Thus, the expression for the stream function is 1.5.x y2 + xy +10y.

(c): At stagnation point, v=0 -3.xy? - y=0 y (3xy +1)=0 y=0 At stagnation point, u=0 3r- x0+x+10=0 x+10=0 r=-10 Thus, the location of the stagnation point is (d): Write the expression for the x-component of the acceleration as follows: ou ди a =U +1 ax Oy az =(3r’y+x+10) 8 (3r’y+x+10)+(+3xy? - y) (3r’y+x+10) &z=(3r’y+x+10)x(6xy +1)+(-(3xy? + y))*(3r”) a =(3r’y+x+10)x(6xy +1)-(3xy? + y)<(3.rº) ą, =9x?y2 +6r+y +60.xy+x+10 Write the expression for the y-component of the acceleration as follows: ovov a, =U — +1 a= (3x’y+x+10) (-3xy? – y) +(-3xy? –Y) (-3xy? –Y)| a, = (3x²y +x+10)*(+3y2)+(3xy2 + y)<(6xy +1) a, = 9x?y? +6x42 – 30y2 + y