A comparison of two methods for solving 3-D unsteady compressible viscous flows

Numerical solutions of three-dimensional unsteady compressible viscous flows for several cases have been obtained using two time-marching procedures: a noniterative alternating direction implicit (ADI) scheme and a generalized minimal residual (GMRES) method using an iterative version of the ADI scheme as a preconditioner. Results obtained are compared with each other and also with experimental data. The GMRES scheme can reduce CPU time by as much as 60% compared to the noniterative ADI scheme

The numerical simulation of the compressible flow passed a NACA 0012 airfoil has been undertaken using the Naviers-Stokes and total energy equations. The resolution of the governing equations is based on a Roe’s approximate Riemann solver. The integration of these equations is performed by means of an implicit upwind Total Variation Diminishing (TVD) scheme, developed by Harten and Yee. The resolution of the set of the governing equations is based on an Alternating Direction Implicit (ADI) formulation. A description of the numerical integration of the equations is first given. The resolution of the governing equations is then described. Two configurations has been investigated. First, this numerical code was checked on the calculation of the compressible flow in the vicinity of a NACA 0012 airfoil without incidence. Secondly, the interaction between a vortex and a NACA 0012 airfoil has been investigated. In these two calculations, the Mach and Reynolds numbers are respectively equal to 0.6 and 22000., the CFL number is 20. The numerical results are reviewed. A quite good agreement with numerical results obtain by other authors on similar configuration has been obtained.

A three-dimensional unsteady compressible viscous flow solver has been developed, and applied to a low speed centrifugal compressor configuration. Good agreement with experiments have been obtained for a compressor tested at NASA Lewis Research Center. Comparisons with measured surface pressures, and performance data are given. For this compressor, stall was found to occur in the diffuser region. It is demonstrated that this stall may be eliminated, and stable operation may be restored by the use of bleed valves located on the diffuser walls. INTRODUCTION In many army applications (tanks and rotorcraft) small gas turbine engines are often used. The mass flow rate in these engines is small (below 10 Ibs/sec). The compression is provided by one or two centrifugal compressor stages, sometimes in combination with an axial compressor stage. Centrifugal compressors are well suited for handling such low mass flow rates. They are rugged and compact. However, the geometric complexity of the blades and the twisting blade passages give rise to a host of aerodynamic problems such as unsteadiness, establishment of cross flows, flow separation at the trailing edge of the compressor blade, separation over diffuser guide vanes, and unsteady meandering motion of a 'tip vortex' that originates in the vicinity of the gap between the rotor and the casing. In many instances, this unsteadiness leads to rotating stall and surge. Considerable interest exists in the engine community in understanding and eventually controlling these phenomena, particularly compressor stall and surge. Computational fluid dynamics methods provide an efficient way of studying the complex flow phenomena within centrifugal compressors. Thanks to the availability of post-processing and scientific visualization software, the computed flow fields can be extensively analyzed. It is also convenient to test within a numerical model various control strategies such as bleeding, energizing of incoming flow using jets, excitation /wiggling of guide vanes, etc. Passive mechanisms such as slots, slats and flaps can also be easily studied within the scope of a numerical method. These simulations are thus a useful and economical first step to detailed flow control experiments. Finally, CFD simulations provide a wealth of numerical and flow physics information which are useful in developing reduced order models. For these reasons, a 3-D CFD code capable of analyzing centrifugal compressor configurations has been developed by the present investigators. Rather than solving the flow equations in a rotating coordinate system, the present calculations are carried out in an inertial system, with grid velocity terms used to simulate a rotor. Inflow and outflow boundary conditions appropriate for internal flows in general and centrifugal compressors in particular are used. This paper is organized as follows. The flow solution algorithm is first described. Next, results in the form of velocity fields, surface pressure data and performance map are presented for a NASA low-speed centrifugal compressor. The performance of this compressor for off-design low mass flow ratio conditions is subsequently studied. Methods for improving the compressor performance during such conditions are discussed. NUMERICAL FORMULATION The 3-D unsteady compressible Reynolds averaged Navier-Stokes equations are solved numerically using a time marching scheme. This involves solving the governing equations at each time step by marching in time from an initial flow condition with appropriate boundary conditions. The governing equations are cast in a curvilinear body-fitted coordinate system (£,T|,£) and solved using an Alternating Direction Implicit (ADI) scheme 1. This implicit scheme has been validated for a number of fixed wing and rotary wing configurations both in steady and unsteady flight conditions^'*. The partial derivatives in the governing equations are calculated using finite difference approximations. The Graduate Research Assistant, Student Member AIAA. * Regents' professor, Associate Fellow AIAA. Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc. inviscid flux terms are calculated using a third order upwind scheme and the viscous stress terms are calculated using standard central difference representations. The details of the upwind formulation are given by Roe. The time derivative is approximated using a two point first order backward difference. The finite difference representation of the governing Navier-Stokes equations at time level 'n' is components at the inflow boundary were set to zero, assuming that there was no swirl at the inlet. It is possible within the framework of the present boundary conditions to prescribe swirl, non-uniform total pressure and total temperature at the inlet. At every time step, the speed of sound a and the normal component of velocity un were found at the inflow boundary from the relations: AT 2a 2a — u = n — u n = I 8 f R + S S + S , T (1) Interior where Aq* is the change in the flow property vector q during adjacent time levels and At is the time step. The operators 8£, 5r| and 8^ are the standard central difference operators. The viscous terms are evaluated explicitly and the inviscid flux vectors F, G and H are calculated implicitly at time level n+1. A three factor approximate factorization is used to factorize the implicit coefficient matrix operator into block tri-diagonal matrices. The factorized system of equations is solved using the Thomas algorithm. Turbulence effects are modeled using the SpalartAllmaras model. INITIAL AND BOUNDARY CONDITIONS The centrifugal compressor is shown in figure 1. The boundary grids are also shown for a single blade passage. The volume grid is made of three segments: inlet region, the rotor, and the diffuser. The following boundary conditions were applied: Periodicity: In the present study, the flow through only a single blade passage has been studied, assuming blade-toblade periodicity. At the periodic boundaries, the flow properties are computed by averaging the properties on either side of the periodic boundary. Solid walls: At the solid walls, the no slip boundary condition was used. The velocity on the walls of the inlet and the diffuser were set to zero, while the velocity for grid points on the compressor blades and the shaft were set equal to Q x r. The pressure, density and temperature values at the solid surfaces were extrapolated from the interior using relationships such as 9p/3n=dT/9n=3p/3n=0, where n represents the direction normal to the solid surface. Inflow Boundary: At this boundary, the stagnation temperature T0 and the total pressure p0 were assumed known. In the simulation of a compressor drawing the air from the atmosphere, these quantities may be set based on ambient conditions. In a multistage compressor analysis, these quantities will be known at the downstream boundary of the previous stage. The tangential (vand w-) (2)

A high-resolution numerical scheme based on the MUSCL–Hancock approach is developed to solve unsteady compressible two-phase dilute viscous flow. Numerical considerations for the development of the scheme are provided. Several solvers for the Godunov fluxes are tested and the results lead to the choice of an exact Riemann solver adapted for both gaseous and dispersed phases. The accuracy of the scheme is proven step by step through specific test cases. These simulations are for one-phase viscous flows over a flat plate in subsonic and supersonic regimes, unsteady flows in a low-pressure shock tube, two-phase dilute viscous flows over a flat plate and, finally, two-phase unsteady viscous flows in a shock tube. The results are compared with well-established analytical and numerical solutions and very good agreement is achieved. Copyright © 1999 John Wiley & Sons, Ltd.

A control-volume -based finite element method with a new formulation for the convecting integration-point velocity for the implicit solution of unsteady compressible and incompressible flows is presented in this article. The method creates a strong coupling between pressure and velocity in incompressible flow on a colocated grid. A pressure-based approach is used to generalize the method for both compressible and incompressible flows. Several quasi-one-dimensional test problems provide validation of the method for Mach numbers ranging from very incompressible flow to supersonic flow with no penalty in accuracy or rate of convergence. Strong shock waves are captured without introducing artificial viscosity.

The generalized minimal residual (GMRES) method is a common choice for solving the large nonsymmetric linear systems that arise when numerically computing solutions of incompressible nonlinear elasticity problems using the finite element method. Analytic results on the performance of GMRES are available on linear problems such as linear elasticity or Stokes’ flow (where the matrices in the corresponding linear systems are symmetric), or on the nonlinear problem of the Navier–Stokes flow (where the matrix is block-symmetric/block-skew-symmetric); however, there has been very little investigation into the GMRES performance in incompressible nonlinear elasticity problems, where the nonlinearity of the incompressibility constraint means the matrix is not block-symmetric/block-skew-symmetric. In this short paper, we identify one feature of the problem formulation, which has a huge impact on unpreconditioned GMRES convergence. We explain that it is important to ensure that the matrices are perturbations of a block-skew-symmetric matrix rather than a perturbation of a block-symmetric matrix. This relates to the choice of sign before the incompressibility constraint integral in the weak formulation (with both choices being mathematically equivalent). The incorrect choice is shown to have a hugely detrimental effect on the total computation time.

A discontinuity-capturing scheme of finite element method (FEM) is proposed. The unstructured-grid technique combined with a new type of adaptive mesh approach is developed for both compressible and incompressible unsteady flows, which exhibits the capability of capturing the shock waves and/or thin shear layers accurately in an unsteady viscous flow at high Reynolds number. In particular, a new testing variable, i.e., the disturbed kinetic energyE, is suggested and used in the adaptive mesh computation, which is universally applicable to the capturing of both shock waves and shear layers in the inviscid flow and viscous flow at high Reynolds number. Based on several calculated examples, this approach has been proved to be effective and efficient for the calculations of compressible and incompressible flows.

Two techniques are proposed in the preconditioning for the Krylov sub-space method called the Generalized Minimal RESidual (GMRES) method to accelerate inner iterations based on the method of characteristics (MOC). The GMRES method is an iterative method to solve a linear algebraic system byminimizing the norm of the residual vector. The proposed preconditioning technique is based on the first flight collision probability which is efficiently made by the multi-dimensional MOC code. To simplify the preconditioner, slight couplings among regions are ignored by considering the mean free path. And another proposed technique makes simplified preconditioner by the scaling matrix which can homogenize and de-homogenize the fuel region and the cladding region. The scaling technique reduces the size of the matrix and also reduces the calculation time of inverse matrix. Numerical results show that the preconditioner simplified by the mean free path efficiently reduces the number of iterations for the GMRES algorithm. And the scaling technique keeps the efficiency of preconditioner even in the multi-dimensional geometry. The total calculation time is found to be reduced when these techniques are employed.

We present a numerical investigation on the mixed precision Generalized Minimal Residual (GMRES) method, preconditioned by deflation. GMRES is a widely used iterative solver for sparse, non-symmetric linear systems. The mixed precision version of restarted GMRES, or mixed precision GMRES(m), is known to achieve double precision accuracy while delivering substantial speedups by using single precision (32-bit) arithmetic for most computations and double precision (64-bit) for key corrective steps only. In this study, the mixed precision GMRES(m) is enhanced with a deflation-based preconditioner to accelerate its convergence. All preconditioning operations are carried out in single precision. We demonstrate the effectiveness of the mixed precision GMRES(m) preconditioned by deflation in terms of both convergence (iterations required) and computational performance. This is validated through numerical experiments on a multi-core environment using a variety of matrices. The results show that the preconditioner remains highly effective in reducing iterations even under reduced precision. Additionally, we compare the speedups achieved by mixed precision over double precision in the preconditioned GMRES(m) with those in the non-preconditioned version. For the majority of tested matrices, the speedups in the preconditioned and non-preconditioned cases are comparable.

The Generalized Minimum Residual (GMRES) method is one of the most widely-used iterative methods for solving nonsymmetric linear systems of equations. In recent years, techniques to avoid communication in GMRES have gained attention because in comparison to floating-point operations, communication is becoming increasingly expensive on modern computers. Since graphics processing units (GPUs) are now becoming crucial component in computing, we investigate the effectiveness of these techniques on multicore CPUs with multiple GPUs. While we present the detailed performance studies of a matrix powers kernel on multiple GPUs, we particularly focus on orthogonalization strategies that have a great impact on both the numerical stability and performance of GMRES, especially as the matrix becomes sparser or ill-conditioned. We present the experimental results on two eight-core Intel Sandy Bridge CPUs with three NDIVIA Fermi GPUs and demonstrate that significant speedups can be obtained by avoiding communication, either on a GPU or between the GPUs. As part of our study, we investigate several optimization techniques for the GPU kernels that can also be used in other iterative solvers besides GMRES. Hence, our studies not only emphasize the importance of avoiding communication on GPUs, but they also provide insight about the effects of these optimization techniques on the performance of the sparse solvers, and may have greater impact beyond GMRES.

A pressure‐based semi‐implicit procedure has been developed for the computation of compressible flows on a system with collocated finite volume formulations. The method includes the implementation of a high‐order accurate weighted essentially non‐oscillatory scheme within the pressure‐based algorithm which employs an approximate Riemann solver for the computation of the cell face inviscid fluxes. The developed scheme is applied to the computation of unsteady compressible flow through a shock tube which experiences various Mach numbers, and the computations are compared with an analytical solution. The computations indicate that the developed procedure leads to sharp discontinuity representation without creation of spurious oscillations. Then, the developed numerical method is utilized to investigate unsteady two‐dimensional flow for a lax configuration and the results are compared with the results of the similar scheme in the density‐based algorithm. At the end, the investigation of steady compressible flows in bump and wedge channel is considered. The simulations show a considerable improvement over the existing pressure‐based method. The investigations show that the developed method is suitable for studying compressible steady and unsteady flows.

Numerical simulation of 3D fluid–structure interaction flow using an immersed object method with overlapping grids

We present a qualitative model for the convergence behaviour of the Generalised Minimal Residual (GMRES) method for solving nonsingular systems of linear equationsAx =b in finite and infinite dimensional spaces. One application of our methods is the solution of discretised infinite dimensional problems, such as integral equations, where the constants in the asymptotic bounds are independent of the mesh size.Our model provides simple, general bounds that explain the convergence of GMRES as follows: If the eigenvalues ofA consist of a single cluster plus outliers then the convergence factor is bounded by the cluster radius, while the asymptotic error constant reflects the non-normality ofA and the distance of the outliers from the cluster. If the eigenvalues ofA consist of several close clusters, then GMRES treats the clusters as a single big cluster, and the convergence factor is the radius of this big cluster. We exhibit matrices for which these bounds are tight.Our bounds also lead to a simpler proof of existing r-superlinear convergence results in Hilbert space.

In this article we consider the iterative solution of the linear system of equations arising from the discretisation of the poly-energetic linear Boltzmann transport equation using a high-order/hp–version discontinuous Galerkin finite element approximation in space, angle, and energy. In particular, we develop preconditioned Richardson iterations which may be understood as generalisations of source iteration in the mono-energetic setting, and derive computable a posteriori bounds for the solver error incurred due to inexact linear algebra, measured in a relevant problem-specific norm. We prove that the convergence of the resulting schemes and a posteriori solver error estimates are independent of the mesh size h and polynomial degree p. We also discuss how the poly-energetic Richardson iteration may be employed as a preconditioner for the generalised minimal residual (GMRES) method. Furthermore, we show that standard implementations of GMRES based on minimising the Euclidean norm of the residual vector can be utilized to yield computable a posteriori solver error estimates at each iteration, through judicious selections of left- and right-preconditioners for the original linear system. The effectiveness of poly-energetic source iteration and preconditioned GMRES, as well as their respective a posteriori solver error estimates, is demonstrated through numerical examples arising in the modelling of photon transport. While the convergence of poly-energetic source iteration is independent of h and p, we observe that the number of iterations required to attain convergence when employing GMRES only depends mildly on h and p. Moreover, this latter approach is highly effective in the low energy regime.

GMRES(m) method, the restarted version of the GMRES (generalized minimal residual) method, is one of the major iterative methods for numerically solving large and sparse nonsymmetric problems of the form Ax=b . However, the information of some eigenvectors that compose the approximation disappears and then the good approximate solution cannot be obtained, because of this restart. Recently, in order to improve such a weak point, some algorithms which named MORGAN, DEFLATION and DEFLATED-GMRES algorithm, have been proposed. Those algorithms add the information of eigenvectors that can be obtained in the previous restart frequency. In this paper, we study those algorithms and compare their performances. From the numerical experiments on the distributed memory machine Fujitsu AP3000, we show that DEFLATED-GMRES( m, k ) method performs the good reduction of residual norms in these algorithms.