P.K. Khosla

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.

Global PNS solutions for subsonic strong interaction flow over a cone-cylinder-boattail configuration

Abstract The solution of the semi-elliptic or so-called parabolized Navier-Stokes equations is considered for large Reynolds numbers and subsonic flows with strong pressure interaction. Flow past a cone-cylinder-boattail configuration is investigated as a prototype of strong viscous-inviscid interaction. A global boundary-layer relaxation procedure is utilized for the formulation of the discrete boundary-value problem. The resulting marching procedure does not require a sub-layer type of approximation. Furthermore, the method does not restrict the step size in the marching direction and is free from any departure effects. Solutions with large recirculation regions are calculated.

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.

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.

Global pressure relaxation procedure for compressible turbulent strong interaction flows

Abstract Numerical turbulent solutions of the compressible RNS equations are obtained by means of the global pressure relaxation procedure. The ellipticity of the RNS equations in this computational approach is conveniently but appropriately represented by the streamwise pressure gradient term in a multi-pass boundary-layer-like marching technique. The numerical formulation includes the full effect of the elliptic pressure interaction and, therefore, allows computation of strong viscous—inviscid interacting flows. The flow regimes considered contain many of the characteristic features of complex fluid flows, e.g. recirculation, embedded shock waves and mixing shear layers. Simple algebraic turbulent closure models are employed. Results are presented for axisymmetric geometries. Comparison is made with available experimental and numerical data.

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.

Pressure flux-split viscous solutions for swirl diffusers

Abstract A method for predicting flow in diffusers with inlet swirl has been developed. Solutions of the pressure flux-split reduced Navier-Stokes equations are obtained for flows in axisymmetric diffusers. Viscous-inviscid interactions and flow field with toroidal recirculation regions are efficiently captured. The computational model is verified by comparison with experimental data and other computations. The extreme sensitivity to grid, turbulent closure model and inlet profiles is discussed.

A review of strongly coupled algorithms for viscous flow problems

Abstract Numerical solution of the fluid dynamic equations requires appropriate discretization, quasi-linearization and choice of algorithm. For the latter, approximate factorization procedures put forth by Beam and Warming have played a significant role in the successful completion of this task. The present authors have demonstrated that another element, strong coupling of the discrete equations with the discrete form of the boundary conditions, can play an equally important role. In the present study, the significance of this coupling on the effectiveness of several approximate factorization techniques for the solution of the Navier–Stokes equations is reviewed. It it shown that, even with strong coupling, no single iterative solution algorithm is optimal for the entire spectrum of the flow problems considered.

Inviscid steady/unsteady flow calculations

Abstract The solution of the Euler equations using a flux splitting procedure is considered for low subsonic to high supersonic flows. Steady and unsteady, internal and external flow fields, are computed. For transient flow, a direct sparse matrix solver is applied to compute the flow field at each instant of time. Oscillation free normal and oblique shocks are captured. Unstart and restart of a simplified two-dimensional inlet is investigated.