ch05 - Problem 5.3 Problem 5.4 Problem 5.6 Problem 5.9 The...

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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 uxy , () Ax x 2 y 2 + = For incompressible flow du dx dv dy + 0 = Hence vxy , y x , d d d = du dx Ay 2 x 2 x 2 y 2 + 2 = so , y 2 y 2 x 2 y 2 + 2 d = , 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 uxy , () U 3 2 y δ x 1 2 y δ x 3 = and δ x () c x = so , U 3 2 y cx 1 2 y 3 = For incompressible flow du dx dv dy + 0 = Hence vxy , y x , d d d =
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du dx 3 4 U y 3 c 3 x 5 2 y cx 3 2 = so vxy , () y 3 4 U y 3 c 3 x 5 2 y c x 3 2 d = , 3 8 U y 2 3 2 y 4 2c 3 x 5 2 = , 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 5mm = 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 0.2 0.4 0.6 0.8 1.0 0.0000 0.0005 0.0010 0.0015 0.0020 v / U y / δ The solution is v U 3 8 δ x y δ
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This note was uploaded on 04/07/2008 for the course MECHENG 530.327 taught by Professor Su during the Fall '08 term at Johns Hopkins.

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ch05 - Problem 5.3 Problem 5.4 Problem 5.6 Problem 5.9 The...

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