A primitive variable, strongly implicit calculation procedure for two and three-dimensional unsteady viscous flows: applications to compressible and incompressible flows including flows with free surfaces

A coupled solution procedure is described for solving the time-dependent Navier-Stokes equations in body-fitted nonorthogonal curvilinear coordinates for both compressible and incompressible flows in two and three dimensions;For the two-dimensional compressible form of equations, this approach employs the strong conservation law form of the governing equations but uses primitive variables (u, v, p, T) rather than the more traditional conservative variables ([rho], [rho]u, [rho][upsilon], E[subscript]t) as unknowns. A coupled modified strongly implicit procedure (CMSIP) is used to efficiently solve the Newton-linearized algebraic equations. It appears that this procedure is effective for Mach numbers ranging from the incompressible limit (M[subscript][infinity]~ 0.01) to supersonic. Generally, smoothing was not needed to control spatial oscillations in pressure for subsonic flows despite the use of central differences. Dual time stepping was found to further accelerate convergence for steady flows. Sample calculations, including steady and unsteady low Mach number internal and external flows and a steady shock-boundary layer interaction flow, illustrate the capability of the present solution algorithm;For three-dimensional incompressible liquid flows with the presence of free surfaces, this approach, coupled with the artificial compressibility method, is used to solve the Newton-linearized algebraic equations for the primitive variables (u, [upsilon], w, p). A true unsteady three-dimensional flow simulation has been obtained for liquid sloshing flow in a partially filled spherical container undergoing a general motion characteristic of that experienced by a spin-stabilized satellite. It appears that this unified approach can handle compressible and incompressible flows very effectively