Finite-amplitude equilibrium solutions for plane Poiseuille-Couette flow

Two-dimensional nonlinear equilibrium solutions for the plane Poiseuille-Couette flow are computed by directly solving the full Navier-Stokes equations as a nonlinear eigenvalue problem. The equations are solved using the two-point fourth-order compact scheme and the Newton-Raphson iteration technique. The linear eigenvalue computations show that the combined Poiseuille-Couette flow is stable at all Reynolds numbers when the Couette velocity component sigma(2) exceeds 0.34552. Starting with the neutral solution for the plane Poiseuille flow, the nonlinear neutral surfaces for the combined Poiseuille-Couette flow were mapped out by gradually increasing the velocity component sigma(2). It is found that, for small sigma(2), the neutral surfaces stay in the same family as that for the plane Poiseuille flow, and the nonlinear critical Reynolds number gradually increases with increasing sigma(2). When the Couette velocity component is increased further, the neutral curve deviates from that for the Poiseuille flow with ar. appearance of a new loop at low wave numbers and at very low energy. By gradually increasing the sigma(2) values at a constant Reynolds number, the nonlinear critical Reynolds numbers were determined as a function of sigma(2). The results show that the nonlinear neutral curve is similar in shape to a linear case. The critical Reynolds number increases slowly up to sigma(2) similar to 0.2 and remains constant until sigma(2) similar to 0.58. Beyond sigma(2) > 0.59, the critical Reynolds number increases sharply. From the computed results it is concluded that two-dimensional nonlinear equilibrium solutions do not exist beyond a critical sigma(2) value of about 0.59.