Sankaran Venkateswaran

Computation of Flows with Arbitrary Equations of State

The extension of time-marching computations to fluids with arbitrary equations of state is demonstrated by means of stability analyses, simplified problems, and practical applications. Most of the examples use the properties of supercritical hydrogen for which the density varies by more than an order of magnitude for small changes in pressure and temperature, but representative computations for incompressible fluids and perfect gases are also given to demonstrate the generality of the procedure. Because representative flow velocities in typical supercritical fluids applications are much lower than the speed of sound, convergence enhancement through eigenvalue control is often necessary. This is accomplished through a generalization of earlier preconditioning methods that enables efficient computation of arbitrary equation of state fluids, perfect gases, and incompressible fluids by a single procedure. The present approach thus provides a single method that is uniformly applicable to all equations of state

Effect of grid aspect ratio on convergence

The effects of high-aspect-ratio grids on algorithm convergence are considered by means of vector stability theory and computational experiments. The results indicate that approximately factored implicit schemes experience convergence deterioration because of nonoptimum local time-stepping procedures and increased need for viscous preconditioning. Based on this insight, an enhanced algorithm is devised using improved selection of the local time step, appropriate definition of the viscous preconditioning matrix, and proper implementation of the boundary conditions. The new algorithm provides uniformly efficient convergence at all aspect ratios for both Euler and Navier-Stokes computations for a variety of test problems.

Influence of Stagnation Regions on Preconditioned Solutions at Low Speeds

We have examined the effects of different preconditioning definitions on robustness in the vicinity of stagnation regions. Specifically, we have utilized a series of NACA0012 test cases and two different computer codes. Interestingly, the two codes—an in-house research code called GEMS and the wellknown NASA Ames research code ARC2D—are observed to show different degrees of sensitivity to the stagnation region. Based on the studies, we propose a modified definition referred to as local maximum preconditioning, which is shown to maintain robust performance over a range of parametric variations.

Convergence Assessment of General Fluid Equations on Unstructured Hybrid Grids

A general finite volume formulation for arbitrary fluids on generalized grids is presented and solved by implicit methods. Primary focus is on a line GaussSeidel method modified for unstructured grids, and a GMRES method. The effects of grid aspect ratio and control volume shape on convergence rates are addressed parametrically by means of simple problems. An effort is made to define an appropriate Courant number that allows various problems and various flow regimes to converge at near-optimum speeds for a single CFL value. Results show that both the LGS and GMRES methods give excellent aspect-ratio-independent convergence for rectangles, and good convergence for triangles. Results for hexahedrons are likewise good while for tetrahedrons or prisms much wider variations in convergence rates remain. Gauss-Seidel and LU methods are very effective for isotropic grids but slow dramatically as grid aspect ratios are increased. Representative applications of the modified LGS method to flows past airfoils on hybrid grids show excellent convergence for high and low Reynolds numbers and for subsonic and transonic Mach numbers.

Preconditioning Algorithms for the Computation of Multi-Phase Mixture Flows

Preconditioned time-marching algorithms are developed for a class of isothermal compressible multi-phase mixture flows, relevant to the modeling of sheetand super-cavitating flows in hydrodynamic applications. Using the volume fraction and mass fraction forms of the multi-phase governing equations, three closely related but distinct preconditioning forms are derived. The resulting algorithm is incorporated within an existing multi-phase code and several representative solutions are obtained to demonstrate the capabilities of the method. Comparisons with measurement data suggest that the compressible formulation provides an improved description of the cavitation dynamics compared with previous incompressible computations.

COMPARISON OF COMPUTATIONAL CODES FOR MODELING HYDROGEN-OXYGEN INJECTORS

Computational solutions from three different groups are compared to ascertain current capabilities for using CFD techniques in the design of injector elements for gas-gas rocket engines. Results are compared for a single element injector configuration for which detailed local flowfield measurements are available. The results show that all computations are similar in quality, and that all provide reasonable predictions of the resulting flowfield. Specific issues of concern in the computations are summarized. Overall, capabilities for single element modeling are deemed to be acceptable as long as the solutions remain axisymmetric. Practical extension to three-dimensions appears to be feasible in the near future, while the prediction of configuration s that use liquid propellents requires additional physical model development.

COMPUTATION OF COMPRESSIBLE MULTIPHASE FLOWS

A computational model capable of capturing fully compressible multiphase flow including energy conservation is presented. The model is a finite volume form based on the Reynolds Averaged Navier-Stokes Equations and is capable of considering fully general equations of state. The preconditioning matrix is presented and ensures a well conditioned eigensystem, essential for efficient and accurate multiphase computations. Solutions are given representing both the utility of the preconditioning method as well as the ability of the model to capture highly compressible multiphase flow fields. When considering compressible flows where thermal effects are significant and a general equation of state is necessary, energy conservation in a multiphase flow field is shown to be a requirement.

Multiphase Computations for Underwater Propulsive Flows

An algorithm for modeling compressible phenomena in multi-phase, reacting flows is developed. Time-marching preconditioned methodology is used as the algorithmic framework because of its inherent capability of handling multiple flow regimes, such as the incompressible bulk liquid flow, low Mach number compressible vapor flow and transonic/supersonic twophase mixture flow with predominant thermal effects. A preconditioning system is developed based on the onedimensional inviscid subsystem and shown to yield a well-conditioned eigensystem. Computational results representative of a hypothetical high-speed supercavitating-vehicle propulsion plume are presented to verify the capabilities of the formulation.

Computational fluid dynamic analysis of liquid rocket combustion instability

The paper presents a computational analysis of liquid rocket combustion instability. Consideration is given to both a fully nonlinear unsteady calculation as well as a new CFD-based linearized stability analysis. An analytical solution for the linear stability problem in a constant area combustion chamber with uniform mean flow is developed to verify the numerical analyses.

Multi-dimensional analysis of combustion instabilities in liquid rocket motors

A three-dimensional analysis of combustion instabilities in liquid rocket engines is presented based on a mixed finite difference/spectral solution methodology for the gas phase and a discrete droplet tracking formulation for the liquid phase. Vaporization is treated by a simplified model based on an infinite thermal conductivitiy assumption for spherical liquid droplets of fuel in a convective environment undergoing transient heating. A simple two parameter phenomenological combustion response model is employed for validation of the results in the small amplitude regime. The computational procedure is demonstrated to capture the phenomena of wave propagation within the combustion chamber accurately. Results demonstrate excellent amplitude and phase agreement with analytical solutions for properly selected grid resolutions under both stable and unstable operating conditions. Computations utilizing the simplified droplet model demonstrate stable response to arbitrary pulsing. This is possibly due to the assumption of uniform droplet temperature which removes the thermal inertia time-lag response of the vaporization process. The mixed-character scheme is sufficiently efficient to allow solutions on workstations at a modest increase in computational time over that required for two-dimensional solutions.