Chebyshev spectral method for incompressible viscous flow with boundary layer control via suction or blowing

Abstract

The MISES quasi 3-D design/analysis code implements a two-equation integral method with empirical closure relations to solve the boundary layer flow problem with or without suction, but lacks the option of flow control via blowing. The integral method is parameterized with the shape parameter H _ 6*/0 which cannot be applied to the blowing problem since 0 - 0 downstream of the injection slot causing H -, co - a computational disaster. In this thesis, two alternate approaches are proposed to solve the blowing problem. First, a two-equation integral method parameterized with the profile parameters of a multi-deck representation of a turbulent jet based on Coles' law of the wake was formulated. The appearance of spurious singularities in the Jacobian matrices associated with the system of equations and the vector of unknowns prevented this method from being implemented. Second, a Chebyshev spectral method using the wall function technique was applied to the defect form of the incompressible viscous momentum equation. A turbulent jet profile was computed with N = 40 modes, a number low enough to allow the method's implementation into the MISES framework.

(cont.) For the spectral approach, a stand-alone code was developed to solve laminar and turbulent flow over a flat plate with the following configurations: solid wall, porous wall with vertical suction/blowing, and fluid injection from an inclined slot. For the turbulent case, the Reynolds stress was replaced with a composite model for the eddy viscosity based on Spalding's law of the wall for the inner layer and Clauser's outer layer formulation. In the laminar regime, N - 10 modes are required for an accurate solution whereas the two-layer structure of a turbulent boundary layer increases this number to N 100 modes. The incorporation of a wall function, consistent with the inner layer eddy viscosity model, in the approximation of the streamwise velocity, reduced the required number of modes by an order of magnitude - a major computational advantage. The more general Spalart-Allmaras turbulence model was implemented in the spectral formulation to investigate the effect of using a wall function based on Spalding's law of the wall.

(cont.) For the flat plate case (solid wall), a small inconsistency between the wall function and the eddy viscosity model produced an erroneous shear stress near the wall. Nevertheless, the velocity profile was in close agreement with an accurate representation constructed from Spalding's law of the wall and Coles' law of the wake.