CFD presenta compresible + incompresible un matrimonio por conveniencia

Este trabajo presenta por un lado una breve síntesis de algunas importantes contribuciones dirigida a la unificación de códigos computacionales para flujos tanto compresible como incompresible y por otro un eficiente precondicionador local para todo el rango de números de Mach y Reynolds implementado sobre un esquema iterativo tipo GMRES con una estrategia que evita el ensamblaje de matrices llamada matriz-free usando como discretización espacial una formulación en elementos finitos. El principal objetivo de esta investigación es lograr un tratamiento unificado de flujo de fluidos tanto compresible como incompresible, viscoso o inviscido apto para simulaciones a gran escala y capaz de ser utilizado sobre plataformas de hardware paralelas.This paper presents a brief review of important contributions towards the unification of compressible and incompressible flow solvers and an efficient local preconditioner for al1 Mach and Reynolds numbers implemented with a matrix-free GMRES iterative scheme and a finite element method. The main goal of this research is the unified treatment of fluid flow at al1 speeds for large scale simulation capable of being implemented over parallel platforms.Peer Reviewe

GMRES physics‐based preconditioner for all Reynolds and Mach numbers: numerical examples

This paper presents several numerical results using a vectorized version of a 3D finite element compressible and nearly incompressible Euler and Navier–Stokes code. The assumptions were set on laminar flows and Newtonian fluids. The goal of this research is to show the capabilities of the present code to treat a wide range of problems appearing in laminar fluid dynamics towards the unification from incompressible to compressible and from inviscid to viscous flow codes. Several authors with different approaches have tried to attain this target in CFD with relative success. At the beginning the methods based on operator splitting and perturbation were preferred, but lately, with the wide usage of time‐marching algorithms, the preconditioning mass matrix (PMM) has become very popular. With this kind of relaxation scheme it is possible to accelerate the rate of convergence to steady state solutions with the modification of the mass matrix under certain restrictions. The selection of the mass matrix is not an easy task, but we have certain freedom to define it in order to improve the condition number of the system. In this paper we have used a physics‐based preconditioner for the GMRES implicit solver developed previously by us and an SUPG formulation for the semidiscretization of the spatial operator. In sections 2 and 3 we present some theoretical aspects related to the physical problem and the mathematical model, showing the inviscid and viscous flow equations to be solved and the variational formulation involved in the finite element analysis. Section 4 deals with the numerical solution of non‐linear systems of equations, with some emphasis on the preconditioned matrix‐free GMRES solver. Section 5 shows how boundary conditions were treated for both Euler and Navier–Stokes problems. Section 6 contains some aspects about vectorization on the Cray C90. The performance reached by this implementation is close to 1 Gflop using multitasking. Section 7 presents several numerical examples for both models covering a wide range of interesting problems, such as inviscid low subsonic, transonic and supersonic regimes and viscous problems with interaction between boundary layers and shock waves in either attached or separated flows

Numerical methods in phase-change problems

This paper summarizes the state of the art of the numerical solution of phase-change problems. After describing the governing equations, a review of the existing methods is presented. The emphasis is put mainly on fixed domain techniques, but a brief description of the main front-tracking methods is included. A special section is devoted to the Newton-Raphson resolution with quadratic convergence of the non-linear system of equations

Improving the convergence rate of the Petrov‐Galerkin techniques for the solution of transonic and supersonic flows

This paper report progress on a technique to accelerate the convergence to steady solutions when the streamline‐upwind/Petrov‐Galerkin (SUPG) technique is used. Both the description of a SUPG formulation and the documentation of the development of a code for the finite element solution of transonic and supersonic flows are reported. The aim of this work is to present a formulation to be able to treat domains of any configuration and to use the appropriate physical boundary conditions, which are the major stumbling blocks of the finite difference schemes, together with an appropriate convergence rate to the steady solution. The implemented code has the following features: the Hughes' SUPG‐type formulation with an oscillation‐free shock‐capturing operator, adaptive refinement, explicit integration with local time‐step and hourglassing control. An automatic scheme for dealing with slip boundary conditions and a boundary‐augmented lumped mass matrix for speeding up convergence. It is shown that the velocities at which the error is absorbed in and ejected from the domain (that is damping and group velocities respectively) are strongly affected by the time step used, and that damping gives an O(N2) algorithm contrasting with the O(N) one given by absorption at the boundaries. Nonetheless, the absorbing effect is very low when very different eigenvalues are present, such as in the transonic case, because the stability condition imposes a too slow group velocity for the smaller eigenvalues. To overcome this drawback we present a new mass matrix that provides us with a scheme having the highest group velocity attainable in all the components. In Section 1 we will describe briefly the theoretical background of the SUPG formulation. In Section 2 it is described how the foregoing formulation was used in the finite element code and which are the appropriate boundary conditions to be used. Finally in Section 3 we will show some results obtained with this cod

Two‐phase flow modelling in gas‐stirred liquid vessels with SUPG‐stabilized equal‐order interpolations

The modelling of liquid flow in gas‐stirred vessels is described. A simple two‐phase model accounts for the buoyancy effect of bubbles. Friction between liquid and gas is modelled with the hypothesis of independent bubbles. The resulting PDE system is discretized with an original version of the SUPG‐FEM technique which stabilizes both the convection term and equal‐order interpolations for velocity and pressure, which are known to be unstable for incompressible flows. The resulting steady state discrete system is solved via pseudotemporal explicit iteration with a local time step and a preconditioning to homogenize the temporal scales for liquid and gas

Adaptive refinement criterion for elliptic problems discretized by FEM

In a recent paper we presented a data structure to be used with multigrid techniques on non‐homogeneously refined FEM meshes. This paper focuses on the adaptive refinement techniques used there. The error estimate is obtained from standard Taylor series. For each element we compute its efficiency in terms of the size, the norm of the second derivatives of the unknown and the parameter p, where Lp is the chosen norm. The way the norm influences the optimal mesh is studied. The number of elements to be refined at each step is such to produce a fast convergence to the optimal mesh, followed by successive homogeneous refinements. We hope that the analysis of these two subjects could be of value for people working with other (perhaps very dissimilar) adaptive refinement techniques (error estimate and data structure, for instance)

Making curved interfaces straight in phase‐change problems

This paper presents a method for straightening curved interfaces arising in phase‐change problems. The method works on isoparametric finite elements, performing a second transformation which maps the master element onto a new one in which the interface looks like a straight line. This allows using the current Gaussian integration technique for squares to evaluate the integrals over each phase. The method provides a better estimation of the contribution of latent heat effects to the residual vector, compared to those obtained by using the assumption that the interface is straight. Appropriate guidelines for solving the non‐linear system of equations arising in this kind of problem are also given. Several numerical examples are presented to show the performance of the method

A general algorithm for compressible and incompressible flow. Stability analysis and explicit time integration

Addresses two difficulties which arise when using a compressible code with equal order interpolation (non‐staggered grids in the finite‐difference nomenclature) to capture a steady‐state solution in the incompressible limit, i.e. at low Mach numbers. Explains that, first, numerical instabilities in the form of spurious oscillations in pressure pollute the solution and, second, the convergence to the steady state becomes extremely slow owing to bad conditioning of the different speeds of propagation. By using a stabilized method, allows the use of equal‐order interpolations in a consistent (weighted‐residual) formulation which stabilizes both the convection and the continuity terms at the same time. On the other hand, by using specially devised preconditioning, assures a rate of convergence independent of Mach number

Computing ship wave resistance from wave amplitude with a non‐local absorbing boundary condition

A method for computing ship wave resistance from a momentum flux balance is presented. It is based on computing the momentum flux carried by the gravity waves that exit the computational domain through the outlet plane. It can be shown that this method ensures a non‐negative wave‐resistance, in contrast with straightforward integration of the normal pressure forces. However, this calculation should be performed on a transverse plane located far behind the ship. Traditional Dawson‐like methods add a numerical viscosity that dampens the wave pattern so that some amount of momentum flux is lost, and resulting in an error in the momentum balance. The flow field is computed, then, with a centred scheme with absorbing boundary conditions

Steady state incompressible flows using explicit schemes with an optimal local preconditioning

Solving large systems of equations from CFD problems by the explicit pseudo-temporal scheme requires a very low amount of memory and is highly parallelizable, but the CPU time largely depends on the conditioning of the system. For advective systems it is shown that the rate of convergence depends on a condition number defined as the ratio of the maximum and the minimum group velocities of the continuum system. If the objective is to reach the steady state, the temporal term can be modified in order to reduce this condition number. Another possibility consists in the addition of a local preconditioning mass matrix. In this paper an optimal preconditioning for incompressible flow is presented, also applicable to compressible ones with locally incompressible zones, like stagnation points, in contrast with the artificial compressibility method. The preconditioned system has a rate of convergence independent from Mach number. Moreover, the discrete solution is highly improved, eliminating spurious oscillations frequently encountered in incompressible flows