Oscillatory channel flow with large wall injection

Abstract

In the presence of small–amplitude pressure oscillations, the linearized Navier–Stokes equations are solved to obtain an accurate description of the time–dependent field in a channel having a rectangular cross–section and two equally permeable walls. The mean solution is based on Taylor's classic profile, while the temporal solution is synthesized from irrotational and rotational fields. Using standard perturbation tools, the rotational component of the solution is derived from the linearized vorticity transport equation. In the absence of an exact solution to rely on, asymptotic formulations are compared with numerical simulations. In essence, the analytical formulation reveals rich vortical structures and discloses the main link between pressure oscillations and rotational wave formation. In the process, the explicit roles of variable injection, viscosity and oscillation frequency are examined. Using an alternative methodology, both WKB and multiple–scale techniques are applied to the linearized momentum equation. The momentum equation is of the boundary–value type and contains two small perturbation parameters. The primary and secondary parameters are, respectively, the reciprocals of the kinetic Reynolds and Strouhal numbers. The multiple–scale procedure employs two fictitious scales in space: a base and an undetermined scale. The latter is left unspecified during the derivation process until flow parameters are obtained in general form. Physical arguments are later used to define the arbitrary scale, which could not have been conjectured a priori. The emerging multiple–scale solution offers several advantages. Its leading–order term is simpler and more accurate than other formulations. Most of all, it clearly displays the relationship between the physical parameters that control the final motion. It thus provides the necessary means to quantify important flow features. These include the corresponding vortical wave amplitude, rotational depth of penetration, near–wall velocity overshoot and surfaces of constant phase. In particular, it discloses a viscous parameter that has a strong influence on the depth of penetration, and furnishes a closed–form expression for the maximum penetration depth in any oscillation mode. These findings enable us to quantify the location of the shear layer and corresponding penetration depth. By way of theoretical verification, comparisons between asymptotic formulations and numeric predictions are reassuring. The most striking result is, perhaps, the satisfactory agreement found between asymptotic predictions and data obtained, totally independently, from numerical simulations of the nonlinear Navier–Stokes equations. In closing, a standard error analysis is used to confirm that the absolute error associated with the analytic formulations exhibits the correct asymptotic behaviour.

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