Jonathan Weiss

Preconditioning Applied to Variable and Constant Density Flows

A time-derivative preconditioning of the Navier-Stokes equations, suitable for both variable and constant density fluids, is developed. The ideas of low-Mach-number preconditioning and artificial compressibility are combined into a unified approach designed to enhance convergence rates of density-based, time-marching schemes for solving flows of incompressible and variable density fluids at all speeds. The preconditioning is coupled with a dual time-stepping scheme implemented within an explicit, multistage algorithm for solving time-accurate flows. The resultant time integration scheme is used in conjunction with a finite volume discretization designed for unstructured, solution-adaptive mesh topologies. This method is shown to provide accurate steady-state solutions for transonic and low-speed flow of variable density fluids. The time-accurate solution of unsteady, incompressible flow is also demonstrated.

Implicit Solution of Preconditioned Navier- Stokes Equations Using Algebraic Multigrid

We describe an algorithm for the solution of the Navier-Stokes equations on unstructured meshes that employs a coupled algebraic multigrid method to accelerate a point-implicit symmetric Gauss-Seidel relaxation scheme. The equations are preconditioned to permit solution of both compressible and incompressible flows. A cell-based, finite volume discretization is used in conjunction with flux-difference splitting and a linear reconstruction of variables. We present results for flowfields representing a range of Mach numbers and Reynolds numbers. The scheme remains stable up to infinite Courant number and exhibits CPU usage that scales linearly with cell count

IMPLICIT SOLUTION OF THE NAVIER-STOKES EQUATIONS ON UNSTRUCTURED MESHES

In this paper we describe an algorithm for the solution of the Navier-Stokes equations on unstructured meshes. The algorithm employs a coupled algebraic multigrid (AMG) method to accelerate a two-sweep point implicit Gauss-Siedel relaxation scheme. The equations are preconditioned to permit solution of both compressible and incompressible flows. A cellbased, finite-volume discretization is used in conjunction with flux difference splitting and a linear reconstruction of variables. We present results for flow fields representing a range of Mach numbers and Reynolds numbers. The scheme remains stable for infinite CFL number and exhibits CPU usage that scales linearly with cell count.

Propulsion-related flowfields using the preconditioned Navier-Stokes equations

A previous time-derivative preconditioning procedure for solving the Navier-Stokes is extended to the chemical species equations. The scheme is implemented using both the implicit ADI and the explicit Runge-Kutta algorithms. A new definition for time-step is proposed to enable grid-independent convergence. Several examples of both reacting and non-reacting propulsion-related flowfields are considered. In all cases, convergence that is superior to conventional methods is demonstrated. Accuracy is verified using the example of a backward facing step. These results demonstrate that preconditioning can enhance the capability of density-based methods over a wide range of Mach and Reynolds numbers.

Continuity Convergence Acceleration of a Density-Based Coupled Algorithm

Density-based algorithms that employ low-Mach number preconditioning techniques to eliminate numerical stiffness and improve accuracy of low speed flow regimes may sometimes experience slow convergence and increased diffi culty to establish the final mass flow. This is especially true for complex flow situations, where the presence of a widely disparate range of flow speeds and highly non-linear physics can impair the effi ciency of the iterative algorithm. The cause of such decreases in convergence rate of the numerical solution can be traced back to the modified pressure wave speeds arising in the preconditioned equation system. In this work we propose a novel accelerator for density-based, time-marching algorithms to alleviate the numerical problems mentioned above. The technique employs intermittent updates to pressure based on the solution of an elliptic pressure-correction equation derived from continuity, with associated velocity corrections formulated consistently from the linearization of the coupled equations. Introducing additional pressure and velocity corrections in this manner significantly improves both the local and global mass balance, as well as the overall convergence behavior of the algorithm. Numerical examples are presented to demonstrate the performance of a coupled, density-based solver combined with a continuity based convergence acceleration for a few representative complex flow applications. Nomenclature p static fluid pressure, N m2 v fluid velocity vector, ms T static fluid temperature, K ρ fluid density, Kg m3 E fluid total energy, E = CvT + v.v 2 , J Kg H fluid total enthalpy, H = E + p/ρ, J Kg τ pseudo-time, s ∆τ pseudo-time interval, s − →x coordinate vector, m −→ dx coordinate vector displacement, m − → A surface area vector, m V volume, m.

Prediction of engine and near-field plume reacting flows in low-thrust chemical rockets

A computational model is employed to study the reacting flow within the engine and near-field plumes of several small gaseous hydrogen-oxygen thrusters. The model solves the full Navier-Stokes equations coupled with species diffusion equations for a hydrogen-oxygen reaction kinetics system and includes a two-equation q-omega model for turbulence. Predictions of global performance parameters and localized flowfield variables are compared with experimental data in order to assess the accuracy with which these flowfields are modeled and to identify aspects of the model which require improvement. Predicted axial and radial velocities 3 mm downstream of the exit plane show reasonable agreement with the measurements. The predicted peak in axial velocity in the hydrogen film coolant along the nozzle wall shows the best agreement; however, predictions within the core region are roughly 15 percent below measured values, indicating an underprediction of the extent to which the hydrogen diffuses and mixes with the core flow. There is evidence that this is due to three-dimensional mixing processes which are not included in the axisymmetric model.