Stanley G. Rubin

Analysis of global pressure relaxation for flows with strong interaction and separation

Abstract A global or pressure relaxation formulation for the reduced form of the Navier-Stokes equations, frequently referred to as semi-elliptic or partially parabolized or just “parabolized” Navier-Stokes (PNS), ‡ is presented. Difference procedures and relaxation solutions for the ( u , v , p ) system are described. The continuity equation is satisfied exactly at each grid point and a poisson pressure equation is not required explicitly. Several model problems, e.g. finite flat plate, trough, boattail and airfoil, are considered. Strong pressure interaction is evident in each case and axial flow separation occurs for several of the problems. The questions of accuracy, stability, convergence rate, and implied difference forms of the pressure and vorticity equations are addressed. Solutions for laminar and turbulent modelling are presented.

Navier-Stokes calculations with a coupled strongly implicit method—I: Finite-difference solutions☆

Abstract Stones unconditionally stable, strongly implicit numerical method is extended to the 2 x 2 coupled vorticity-stream function form of the Navier-Stokes equations. The solution algorithm allows for complete coupling of the boundary conditions. Solution for arbitrary large time steps, and for cell Reynolds numbers much greater than two have been obtained. The method converges quite rapidly without adding artificial viscosity or the necessity for under-relaxation. This technique is used here to solve for a variety of internal and external flow problems. Moderate to large Reynolds numbers are considered for both separated and unseparated flows. The procedure is extended to higher-order splines in Part 2 of this study.

A conjugate gradient iterative method

Abstract A strongly implicit pre-conditioned form of the conjugate gradient method is considered. The resulting iterative technique is applicable for sparse systems of difference equations arising from boundary value problems. The method is used to solve two- and three-dimensional potential flows. In addition, it is extended to a 2 x 2 coupled system to solve the Navier-Stokes equations in stream function-vorticity form.

Consistent strongly implicit iterative procedures for two-dimensional unsteady and three-dimensional space-marching flow calculations

Abstract First and second-order accurate time consistent versions of the coupled strongly implicit procedure (CSIP) have been developed and investigated for diffusion, potential flow and reduced Navier-Stokes (RNS) equations. Typical examples for the flow over an airfoil and a flat plate at incidence are presented. The method is also applicable to space marching for 3-D flows. Primitive variable forms of the (RNS) and the boundary region equations are considered for low speed flow near the trailing edge of a finite-span plate and for supersonic flow over a cone at incidence, respectively. The composite velocity formulation is considered for flow over a cylinder of rectangular cross-section.

Incompressible Navier-Stokes solutions with a new primitive variable solver

Abstract A new primitive variable RNS/NS formulation for incompressible viscous flow is presented. This formulation, which has been applied for both two- and three-dimensional flows, is a semi-implicit, time-marching method using standard primitive variables. For this scheme the discrete continuity equation is satisfied directly at each time step, without artificial compressibility or pressure Poisson concepts . All the linear terms (continuity, pressure gradients, diffusion) are treated implicitly and all the non-linear convection terms are treated explicitly. Since all acoustic behavior is implicit, a convection-only CFL stability condition results even for incompressible flow . This is true without a P t term added to the continuity equation. The formulation is robust and provides accurate solutions that compare well with experimental data.

Transonic flow solutions using a composite velocity procedure for potential, Euler and RNS equations

Abstract Solutions for transonic viscous and inviscid flows using a composite velocity procedure are presented. The velocity components of the compressible flow equations are written in terms of a multiplicative composite consisting of a viscous or rotational velocity and an inviscid, irrotational, potential-like function. This provides for an efficient solution procedure that is locally representative of both asymptoptic inviscid and boundary layer theories. A modified quasi-conservative form of the axial momentum equation that is required to obtain rotational solutions in the inviscid region is presented and a combined quasi-conservation/non-conservation form is applied for evaluation of the reduced Navier-Stokes (RNS), Euler and potential equations. A variety of results are presented and the effects of the approximation on entropy production, shock capturing and viscous interaction are discussed.

A Flux-Split Solution Procedure for Unsteady Inlet Flow Calculations

The solution of reduced Navier Stokes (RNS) equations is considered using a flux-split procedure. Unsteady flow in a two dimensional engine inlet is computed. The problems of unstart and restart are investigated. A sparse matrix direct solver combined with domain decomposition strategy is used to compute the unsteady flow field at each instant of time. Strong shock-boundary layer interaction, time varying shocks and time varying recirculation regions are efficiently captured.

Application of the reduced Navier–Stokes methodology to flow stability of Falkner–Skan class flows

Abstract This investigation ascertains the ability of the reduced Navier–Stokes (RNS) methodology to model linear flow stability. This is accomplished through development and investigation of two reduced forms of the Orr–Sommerfeld equation and of a second order RNS direct numerical simulation (DNS). The stability of five Falkner–Skan flows ( β =1.0, 0.2, 0.0, −0.1, and −0.1988) is investigated for these modified forms of the Orr–Sommerfeld equation (OSE). Neutral stability curves are numerically generated and compared for three forms of the OSE, viz. full Navier–Stokes equations, two-dimensional thin-layer Navier–Stokes equations which exclude only axial diffusion, and two-dimensional reduced Navier–Stokes equations which exclude all axial diffusion, as well as all diffusion in the normal momentum equation. Effects of a deferred corrector to include these terms are also investigated. Results of the computations demonstrate that the reduced forms of the OSE are consistent with the full OSE. With confirmation that the reduced Navier–Stokes equations contain the information required to properly model flow stability, development of a new class of asymptotic theories, stability methods, and approaches to direct numerical simulations, based on the RNS methodology, becomes feasible. Results from full DNS calculations using the RNS equations demonstrate the proper characteristics for disturbance growth and decay of the velocity disturbances. Velocity disturbance profiles are also of the required shape and magnitude.

Streamline based grid adaption for Euler and Navier-Stokes: Direct and inverse design applications

A numerical method for solving inviscid steady-state two-dimensional flows is presented and extended to viscous flows. A system of quasi-one-dimensional governing equations applicable to both direct as well as inverse design problems for computing incompressible, subsonic and supersonic flows is derived. A pressure based flux split finite difference approximation is used along the streamlines. The method does not require any complicated grid generation technique. Applications are presented for both direct and inverse problems. The results are compared with exact solutions whenever available.

Segmented Domain Decomposition Multigrid Solutions for Two and Three-Dimensional Viscous Flows

Several viscous incompressible two and three-dimensional flows with strong inviscid interaction and/or axial flow reversal are considered with a segmented domain decomposition multigrid (SDDMG) procedure. Specific examples include the laminar flow recirculation in a trough geometry and in a three-dimensional step channel. For the latter case, there are multiple and three-dimensional recirculation zones. A pressure-based form of flux-vector splitting is applied to the Navier-Stokes equations, which are represented by an implicit, lowest-order reduced Navier-Stokes (RNS) system and a purely diffusive, higher-order, deferred-corrector. A trapezoidal or box-like form of discretization insures that all mass conservation properties are satisfied at interfacial and outflow boundaries, even for this primitive-variable non-staggered grid formulation. The segmented domain strategy is adapted herein for three-dimensional flows and is extended to allow for disjoint subdomains that do not share a common boundary.