Preconditioning methods for ideal and multiphase fluid flows (original) (raw)
Related papers
42nd AIAA Aerospace Sciences Meeting and Exhibit, 2004
Preconditioning methods can help explicit multistage multigrid flow solvers to achieve fast and accurate convergence for a wide range of Mach numbers including incompressible flows. The implementation of preconditioning methods and the corresponding matrix dissipation terms in existing flow solvers is a challenging task if good convergence rates are to be obtained. This task can be made more computationally efficient through the use of entropy variables and their associated transformation matrices. Even once implemented, the true potential of preconditioning methods cannot be achieved without a properly chosen entropy fix, well-tuned dissipation coefficients, and a modified Runge-Kutta multistage scheme adapted to the discretization stencil and artificial dissipation terms of the particular flow solver. In this paper we expose the crucial aspects of the successful implementation of a squared preconditioner which can be used in a large class of existing flow solvers that use explicit, modified Runge-Kutta methods and multigrid for convergence acceleration. Numerical and analytical optimization techniques are used to obtain optimal parameter values and coefficients for these methods. The results of these optimizations are used to explore the strengths and weaknesses of both the analytical and numerical approaches and to establish whether the results from both methods are well correlated.
Preconditioning methods can help explicit multistage multigrid flow solvers to achieve fast and accurate convergence for a wide range of Mach numbers including incompressible flows. The implementation of preconditioning methods and the corresponding matrix dissipation terms in existing flow solvers is a challenging task if good convergence rates are to be obtained. This task can be made more computationally efficient through the use of entropy variables and their associated transformation matrices. Even once implemented, the true potential of preconditioning methods cannot be achieved without a properly chosen entropy fix, well-tuned dissipation coefficients, and a modified Runge-Kutta multistage scheme adapted to the discretization stencil and artificial dissipation terms of the particular flow solver. In this paper we expose the crucial aspects of the successful implementation of a squared preconditioner which can be used in a large class of existing flow solvers that use explicit, modified Runge-Kutta methods and multigrid for convergence acceleration. Numerical and analytical optimization techniques are used to obtain optimal parameter values and coefficients for these methods. The results of these optimizations are used to explore the strengths and weaknesses of both the analytical and numerical approaches and to establish whether the results from both methods are well correlated.
2002
A preconditioned solution scheme for the computation of compressible flow in turboma- chinery at arbitrary Mach numbers is presented. The preconditioning technique used is applied to a state-of-the-art explicit, time-marching Navier-Stokes code which originally was developed for compressible, high-speed turbomachinery applications. It combines the ideas of low Mach number preconditioning and artificial compressibility method into a unified approach where principally fluids with arbitrary equations of state can be simulated. As shown by the test cases presented, it allows the code to simulate flows eciently and accurately independent of the Mach number. A description of the Navier-Stokes equations for rotating coordinate systems, along with the solution scheme and the details of the preconditioning method is given. Since turbomachinery computations are often performed on truncated domains, the solution scheme should be used in conjunction with non-reflecting boundary conditions. A ch...
Assessment of preconditioning methods for multidimensional aerodynamics
Computers & Fluids, 1997
We consider the steady state equations for a compressible fluid. For low speed flow the system is stiff since the ratio of the convective speed to the speed of sound is very small. To overcome this difficulty we alter the time dependency of the equations while retaining the same steady state operator. In order to achieve high numerical resolution we also alter the artificial dissipation (or Roe matrix) of the numerical scheme. The definition of preconditioners and artificial dissipation terms can be formulated conveniently by using other sets of dependent variables rather than the conservation variables. The effects of different preconditioners, artificial dissipation and grid density on accuracy and convergence to the steady state of the numerical solutions are presented in detail. The numerical results obtained for inviscid and viscous two-and three-dimensional flows over external aerodynamic bodies indicate that efficient multigrid computations of flows with very low Mach numbers are now possible. 0 1997 Elsevier Science Ltd.
Preconditioning algorithms for the computation of multi-phase mixture flows
39th Aerospace Sciences Meeting and Exhibit, 2001
Preconditioned time-marching algorithms are developed for a class of isothermal compressible multi-phase mixture flows, relevant to the modeling of sheet-and 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.
A Mach-uniform preconditioner for incompressible and subsonic flows
International Journal for Numerical Methods in Fluids, 2014
In this study, a novel Mach-uniform preconditioning method is developed for the solution of Euler equations at low subsonic and incompressible flow conditions. In contrast to the methods developed earlier in which the conservation of mass equation is preconditioned, in the present method, the conservation of energy equation is preconditioned, which enforces the divergence free constraint on the velocity field even at the limiting case of incompressible, zero Mach number flows. Despite most preconditioners, the proposed Mach-uniform preconditioning method does not have a singularity point at zero Mach number. The preconditioned system of equations preserves the strong conservation form of Euler equations for compressible flows and recovers the artificial compressibility equations in the case of zero Mach number. A two-dimensional Euler solver is developed for validation and performance evaluation of the present formulation for a wide range of Mach number flows. The validation cases studied show the convergence acceleration, stability, and accuracy of the present Mach-uniform preconditioner in comparison to the non-preconditioned compressible flow solutions. The convergence acceleration obtained with the present formulation is similar to those of the well-known preconditioned system of equations for low subsonic flows and to those of the artificial compressibility method for incompressible flows.
Ocean Engineering, 2016
This paper presents the application of a dual-time stepping scheme to the computation of unsteady cavitating and ventilated flows. The flows are modeled for multidimensional problems based on fully-compressible multicomponent, multi-phase mixture Navier-Stokes equations. In order to handle the sharp discontinuity flows that previously developed preconditioning methods fail to do, the dual-time stepping scheme introduces a modified preconditioning parameter. This preconditioning parameter is defined using a "pressure-based" form so that accurate, efficient, and robust computations can be performed without dependence on the Mach number. The system of equations is solved on multi-block structured curvilinear grids with a high-resolution upwind scheme. Both the convergence performance and validation of the computational results are examined for various test cases including inviscid gaseous mixture flows in a tube, two-phase shock tube problem, free-surface flow in a nozzle, single-phase water flow, cavitating flows, transonic water flow, and ventilated flows over underwater vehicles. The results obtained with the modified form are in good agreement with the exact solutions and experimental data. In terms of accuracy, efficiency, and robustness, the modified form is strongly recommended for use in mixture flow computations when sharp discontinuities are present.
2008
The conservative-form, pressure-based PCICE numerical method (Martineau and Berry, 2004) (Berry, 2006), recently developed for computing transient fluid flows of all speeds from very low to very high (with strong shocks), is simplified and generalized. Though the method automatically treats a continuous transition of compressibility, three distinct, limiting compressibility regimes are formally defined for purposes of discussion and comparison with traditional methods-the strictly incompressible limit, the nearly incompressible limit, and the fully compressible limit. The PCICE method's behavior is examined in each limiting regime. In the strictly incompressible limit the PCICE algorithm reduces to the traditional MACtype method with velocity divergence driving the pressure Poisson equation. In the nearly incompressible limit the PCICE algorithm is found to reduce to a generalization of traditional incompressible methods, i.e. to one in which not only the velocity divergence effect, but also the density gradient effect is included as a driving function in the pressure Poisson equation. This nearly incompressible regime has received little attention, and it appears that in the past, strictly incompressible methods may have been conveniently applied to flows in this regime at the expense of ignoring a potentially important coupling mechanism. This could be significant in many important flows; for example, in natural convection flows resulting from high heat flux. In the fully compressible limit or regime, the algorithm is found to reduce to an expression equivalent to density-based methods for high-speed flow.
PRECONDITIONING APPLIED TO EULERIAN–EULERIAN GAS-SOLID FLOW CALCULATIONS WITH VARIABLE DENSITY
2006
Local preconditioning for Eulerian-Eulerian gas-solid flow calculations with variable gas density is investigated. The performance of simultaneous solution algorithms for low-Mach calculations is strongly related to preconditioning. The gas-solid drag source terms are at the origin of a solid volume fraction and frequency dependency of the mixture speed of sound. Whereas the solid volume fraction dependency of the mixture speed of sound is straightforward to account for in the gas-solid preconditioner, this is not the case with the frequency dependency, the frequency not being an extra calculation variable. The preconditioners used so far in the gas-solid flow literature do not account for or eliminate the frequency dependency of the mixture speed of sound and transfer the problem to the numerical speed of sound. Not accounting for the frequency dependency of the mixture speed of sound in the gas-solid preconditioner results, however, in drastic convergence slow down, even when a fully implicit treatment of the drag source terms is taken. Possible approaches to account for the frequency dependency of the mixture speed of sound in the gas-solid preconditioner are investigated. It is shown that the gas-solid preconditioner does not need to remove the frequency dependency of the mixture speed of sound, but should account for it and be scaled according to the mixture speed of sound at the highest frequency calculated, that is the filter frequency mixture speed of sound, which logically depends on the local mesh resolution. Accounting for the filter frequency mixture speed of sound in the gas-solid preconditioner as good as eliminates the reduction of the convergence speed by the gas-solid drag source terms. The addition of a drag history force to the gas-solid preconditioner to properly rescale frequencies lower than the filter frequency hardly alters the convergence behavior. The convergence speed is determined by propagation at the highest, i.e. filter frequency.