A dimensionless velocity profile, u = u/U = Co + Ce-Czy, where, y = y/8, is...

  

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A dimensionless velocity profile, u = u/U = Co + Ce-Czy, where, y = y/8, is proposed to approximate the laminar boundary layer solution for flow around a corner. The outer flow velocity can be expressed as, U = Cx, where, C, is constant. Boundary layer has the thickness of, 8, where, Co, C, and C, are the constants to be determined in order to match the boundary conditions in the boundary layer, including the no slip condition, match the outer flow velocity and zero shear stress at the edge of the boundary layer. Apart from that, an additional boundary condition is proposed. Ju y2 == dx y=0 Given the fluid kinematic viscosity, v = 1.46 x 10-5 m/s and the constant, C = 0.09 s-1. a. Validate the proposed additional boundary condition. [3 marks] b. Find the velocity profile, u", by evaluating the constant, Co, C, and C2. [8 marks] C. Determine the displacement thickness and momentum thickness in term of boundary layer thickness. [6 marks] d. Show that the general momentum integral equation can be rewritten as, d8m (d8 + d8 \dx/ (28m + 8v) = 2Cx where, 8m, and, 8m, are the displacement and momentum thickness respectively. e. Calculate the boundary layer thickness. [3 marks] [5 marks] A dimensionless velocity profile, u = u/U = Co + Ce-Czy, where, y = y/8, is proposed to approximate the laminar boundary layer solution for flow around a corner. The outer flow velocity can be expressed as, U = Cx, where, C, is constant. Boundary layer has the thickness of, 8, where, Co, C, and C, are the constants to be determined in order to match the boundary conditions in the boundary layer, including the no slip condition, match the outer flow velocity and zero shear stress at the edge of the boundary layer. Apart from that, an additional boundary condition is proposed. Ju y2 == dx y=0 Given the fluid kinematic viscosity, v = 1.46 x 10-5 m/s and the constant, C = 0.09 s-1. a. Validate the proposed additional boundary condition. [3 marks] b. Find the velocity profile, u", by evaluating the constant, Co, C, and C2. [8 marks] C. Determine the displacement thickness and momentum thickness in term of boundary layer thickness. [6 marks] d. Show that the general momentum integral equation can be rewritten as, d8m (d8 + d8 \dx/ (28m + 8v) = 2Cx where, 8m, and, 8m, are the displacement and momentum thickness respectively. e. Calculate the boundary layer thickness. [3 marks] [5 marks]