Viscous Boundary Layers in Rotating Fluids Driven by Periodic Flows

Abstract

The boundary layers formed in a rotating fluid by oscillating flow over an infinite half-plate are analyzed in detail. The undisturbed flow is composed of a steady velocity parallel to the plate plus a small oscillating component, both of which are uniform in space. The governing, first-order, boundary-layer equations are obtained through an asymptotic expansion in the parameter E/Ro2 (Ekman number divided by the Rossby number squared) that is appropriate when no geometric scaling length is present. The problem is analyzed as the sum of a nonlinear, steady solution and a linearized, unsteady solution. The steady solution produces a boundary layer whose thickness oscillates (with wavenumber close to the Coriolis frequency f divided by the free-stream velocity to ?0) along distance x (dimensionless) from the leading edge and asymptotically approaches the classic Ekman layer for large x. If scaling is lost through f=0 (no rotation), the self-similar Blasius solution is obtained. The x-oscillating, steady boundary layer plus the unsteady oscillating free-stream at frequency ?? drive the unsteady problem. The results show that the solution is predominantly composed of two inertial-wave-vector components, one circularly polarized to the left (CPL) and the other circularly polarized to the right (CPR). The CPL component is asymptotically scaled by the factor ??+f, while the factor |???f| similarly scales the CPR component. Near resonance (??≈f) these widely different scaling factors cause two distinct boundary layers to be formed. At resonance (??=f), the ?critical latitude effect,? the CPR component acts self-similar (in an asymptotic sense) through loss of scaling and contributes a purely diffusive (no Coriolis turning) boundary layer that grows in x without bound in height, much like other self-similar boundary layers (the Blasius solution and Stokes first problem). Therefore this problem does not contain the phenomena of ?erupting boundary layers? and ?infinite vertical velocities? that have been associated with resonance in the literature. Of course, the remnant, CPR-component boundary layer is infinitely thick at infinite x, and the leading edge of the plate remains a singular point where the boundary-layer equations are not valid. The displacement thickness is discussed along with its frequency dependence to show the driving term for the second-order outer problem. Solutions for large and small ?? are briefly discussed in an appendix since they are, by comparison, rather straightforward.