Steady flow in a sudden expansion at high Reynolds numbers

The sudden expansion of a laminar flow in a two-dimensional channel is examined theoretically in the limit of large Reynolds number R. Previous investigators found, from experiment and from numerical solutions of the equations of motion, that a region of closed streamlines is formed whose streamwise length is linearly related to R for R = O(102). It is desired to determine if the steady solutions to the Navier–Stokes equations continue to exhibit this relationship indefinitely for increasing R. Since solutions are sought for which the longitudinal length scale is O(R) and that in the tranvserse direction is O(1), the equations of motion reduce to the boundary-layer equations as R→∞. These equations are solved numerically using a finite difference technique for selected values of λ, the ratio of the upstream channel half-width to the step height. Steady solutions are found for all values of λ when the inlet velocity profile is parabolic. However, a uniform inlet velocity profile yields steady solutions with an O(R) wake length only for λ⩽λc = 1.54. Analogous results apply in the axisymmetric case for which λc is found to be equal 3.67.

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