Semidiscrete Galerkin modelling of compressible viscous flow past a circular cone at incidence
Abstract
A numerical study of the laminar and compressible boundary layer, about a circular cone in a supersonic free stream, is presented. It is thought that if accurate and efficient numerical schemes can be produced to solve the boundary layer equations, they can be joined to numerical codes that solve the inviscid outer flow. The combination of these numerical codes is competitive with the accurate, but computationally expensive, NavierStokes schemes. The primary goal is to develop a finite element method for the calculation of 3D compressible laminar boundary layer about a yawed cone. The proposed method can, in principle, be extended to apply to the 3D boundary layer of pointed bodies of arbitrary cross section. The 3D boundary layer equations governing supersonic free stream flow about a cone are examined. The 3D partial differential equations are reduced to 2D integral equations by applying the Howarth, Mangler, Crocco transformations, a linear relation between viscosity, and a Blasiustype of similarity variable. This is equivalent to a Dorodnitsyntype formulation. The reduced equations are independent of density and curvature effects, and resemble the weak form of the 2D incompressible boundary layer equations in Cartesian coordinates. In addition the coordinate normal to the wall has been stretched, which reduces the gradients across the layer and provides high resolution near the surface. Utilizing the parabolic nature of the boundary layer equations, a finite element method is applied to the Dorodnitsyn formulation. The formulation is presented in a PetrovGalerkin finite element form and discretized across the layer using linear interpolation functions. The finite element discretization yields a system of ordinary differential equations in the circumferential direction. The circumferential derivatives are solved by an implicit and noniterative finite difference marching scheme. Solutions are presented for a 15 deg half angle cone at angles of attack of 5 and 10 deg. The numerical solutions assume a laminar boundary layer with free stream Mach number of 7. Results include circumferential distribution of skin friction and surface heat transfer, and cross flow velocity distributions across the layer.
 Publication:

Ph.D. Thesis
 Pub Date:
 October 1989
 Bibcode:
 1989PhDT........35M
 Keywords:

 Circular Cones;
 Compressible Flow;
 Flow Distribution;
 Galerkin Method;
 Incidence;
 Viscous Flow;
 Boundary Layer Equations;
 Compressible Boundary Layer;
 Differential Equations;
 Finite Element Method;
 Incompressible Boundary Layer;
 Integral Equations;
 Laminar Boundary Layer;
 NavierStokes Equation;
 Supersonic Flow;
 Three Dimensional Boundary Layer;
 Two Dimensional Boundary Layer;
 Viscosity;
 Fluid Mechanics and Heat Transfer