Steady flow past sudden expansions at large Reynolds number. Part I: Boundary layer solutions

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Abstract
In the steady laminar flow past a sudden channel expansion at large Reynolds number R, the longitudinal length scale of the separated eddy typically increases linearly with R whereas the transverse length scale is O(1). If these trends were to continue indefinitely, the equations of motion in the bulk flow would reduce to the boundary layer equations as R→∞. Previous investigators obtained steady finite difference solutions to the latter for all values of the expansion ratio when the inlet velocity profile was parabolic; however, when the inflow was uniform, numerical instabilities were encountered below a certain value of the expansion ratio. In the present work these instabilities are avoided through the use of a global Newton method, and uniform inflows to several sudden expansion geometries are considered. In each case steady solutions are found only above a critical value of the expansion ratio where the pressure gradient becomes singular near the reattachment point. These results suggest that for uniform inflows and smaller values of the expansion ratio, the eddy length will no longer increase linearly with R when the latter is sufficiently large.