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Introduction-Fluid-Mechanics-Solution-Chapter-05

Introduction-Fluid-Mechanics-Solution-Chapter-05 - Problem...

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Problem 5.9 The x component of velocity in a steady incompressible flow field in the xy plane is u = Ax /( x 2 + y 2 ), where A = 10 m 2 /s, and x and y are measured in meters. Find the simplest y component of velocity for this flow field. Given: x component of velocity of incompressible flow Find: y component of velocity Solution u x y , ( ) A x x 2 y 2 + = For incompressible flow du dx dv dy + 0 = Hence v x y , ( ) y x u x y , ( ) d d d = du dx A y 2 x 2 ( ) x 2 y 2 + ( ) 2 = so v x y , ( ) y A x 2 y 2 ( ) x 2 y 2 + ( ) 2 d = v x y , ( ) A y x 2 y 2 + =
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Problem 5.13 A useful approximation for the x layer is a cubic variation from u = 0 at the surface ( y = 0) to the freestream velocity, U , at the edge of the boundary layer ( y = δ ). The equation for the profile is u / U = 3/2( y / δ ) - 1/2( y / δ ) 3 , where δ = cx 1/2 and c is a constant. Derive the simplest expression for v / U , the y component of velocity ratio. Plot u / U and v / U versus y / δ , and find the location of the maximum value of the ratio v / U . Evaluate the ratio where δ = 5 mm and x = 0.5 m. Given: Data on boundary layer Find: y component of velocity ratio; location of maximum value; plot velocity profiles; evaluate at particular point Solution u x y , ( ) U 3 2 y δ x ( ) 1 2 y δ x ( ) 3 = and δ x ( ) c x = so u x y , ( ) U 3 2 y c x 1 2 y c x 3 = For incompressible flow du dx dv dy + 0 = Hence v x y , ( ) y x u x y , ( ) d d d =
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du dx 3 4 U y 3 c 3 x 5 2 y c x 3 2 = so v x y , ( ) y 3 4 U y 3 c 3 x 5 2 y c x 3 2 d = v x y , ( ) 3 8 U y 2 c x 3 2 y 4 2 c 3 x 5 2 = v x y , ( ) 3 8 U δ x y δ 2 1 2 y δ 4 = The maximum occurs at y δ = as seen in the corresponding Excel workbook v max 3 8 U δ x 1 1 2 1 = At δ 5 mm = and x 0.5 m = , the maximum vertical velocity is v max U 0.00188 =
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Problem 5.13 (In Excel) A useful approximation for the x component of velocity in an incompressible laminar boundary layer is a cubic variation from u = 0 at the surface ( y = 0) to the freestream velocity, U , at the edge of the boundary layer ( y = d ). The equation for the profile is u / U = 3/2( y / d ) - 1/2( y / d ) 3 , where d = cx 1/2 and c is a constant. Derive the simplest expression for v / U , the y component of velocity ratio. Plot u / U and v / U versus y / d , and find the location of the maximum value of the ratio v / U . Evaluate the ratio where d = 5 mm and x = 0.5 m. Given: Data on boundary layer Find: y component of velocity ratio; location of maximum value; plot velocity profiles; evaluate at particular point Solution To find when v / U is maximum, use Solver v / U y / δ 0.00188 1.0 v / U y / δ 0.000000 0.0 0.000037 0.1 0.000147 0.2 0.000322 0.3 0.000552 0.4 0.00082 0.5 0.00111 0.6 0.00139 0.7 0.00163 0.8 0.00181 0.9 0.00188 1.0 Vertical Velocity Distribution In Boundary layer 0.0
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