Initial value study of effect of distributed roughness on boundary layer transition

The technique of the initial value problem is used to study the effect of wall roughness on boundary layer transition. Using a Laplace transform in x and a Fourier transform in time as is appropriate for consideration of spatial growth, the equation that results is a nonhomogeneous Orr-Sommerfeld equation for the doubly transformed normal velocity variable. The roughness at the wall is introduced through a nonhomogeneous boundary condition at y = 0. After solving the nonhomogeneous Orr-Sommerfeld equation with the appropriate boundary conditions, the final solution in terms of the primitive variables is obtained through Laplace and Fourier inversion of the transformed solution. The final solution is the summation of the contributions from the poles and branch cuts in the s-complex plane. The final solution shows clearly that within the present frame of linear analysis, wall roughness generates standing waves only, and therefore does not directly excite the growing Tollmier-Schlichting wave.