A high‐order discontinuous Galerkin method for all‐speed flows

In this article, we present a discontinuous Galerkin (DG) method designed to improve the accuracy and efficiency of steady solutions of the compressible fully coupled Reynolds‐averaged Navier–Stokes and k ? ω turbulence model equations for solving all‐speed flows. The system of equations is iterated to steady state by means of an implicit scheme. The DG solution is extended to the incompressible limit by implementing a low Mach number preconditioning technique. A full preconditioning approach is adopted, which modifies both the unsteady terms of the governing equations and the dissipative term of the numerical flux function by means of a new preconditioner, on the basis of a modified version of Turkel's preconditioning matrix. At sonic speed the preconditioner reduces to the identity matrix thus recovering the non‐preconditioned DG discretization. An artificial viscosity term is added to the DG discretized equations to stabilize the solution in the presence of shocks when piecewise approximations of order of accuracy higher than 1 are used. Moreover, several rescaling techniques are implemented in order to overcome ill‐conditioning problems that, in addition to the low Mach number stiffness, can limit the performance of the flow solver. These approaches, through a proper manipulation of the governing equations, reduce unbalances between residuals as a result of the dependence on the size of elements in the computational mesh and because of the inherent differences between turbulent and mean‐flow variables, influencing both the evolution of the Courant Friedrichs Lewy (CFL) number and the inexact solution of the linear systems. The performance of the method is demonstrated by solving three turbulent aerodynamic test cases: the flat plate, the L1T2 high‐lift configuration and the RAE2822 airfoil (Case 9). The computations are performed at different Mach numbers using various degrees of polynomial approximations to analyze the influence of the proposed numerical strategies on the accuracy, efficiency and robustness of a high‐order DG solver at different flow regimes. Copyright © 2014 John Wiley & Sons, Ltd.     

A numerical method for 3D barotropic flows in turbomachinery

A numerical method for the simulation of 3D inviscid barotropic flows in rotating frames is presented. A barotropic state law incorporating a homogeneous-flow cavitation model is considered. The discretisation is based on a finite-volume formulation applicable to unstructured grids. A shock-capturing Roe-type upwind scheme is proposed for barotropic flows. The accuracy of the proposed method at low Mach numbers is ensured by ad-hoc preconditioning, preserving time consistency. An implicit time advancing only relying on the algebraic properties of the Roe flux function, and thus applicable to a variety of problems, is presented. The proposed numerical ingredients, already validated in a 1D context and applied to 3D non-rotating computations, are then applied to the 3D water flow around a typical turbopump inducer.     

Accuracy of high‐order density‐based compressible methods in low Mach vortical flows

At low Mach numbers, Godunov‐type approaches, based on the method of lines, suffer from an accuracy problem. This paper shows the importance of using the low Mach number correction in Godunov‐type methods for simulations involving low Mach numbers by utilising a new, well‐posed, two‐dimensional, two‐mode Kelvin–Helmholtz test case. Four independent codes have been used, enabling the examination of several numerical schemes. The second‐order and fifth‐order accurate Godunov‐type methods show that the vortex‐pairing process can be captured on a low resolution with the low Mach number correction applied down to 0.002. The results are compared without the low Mach number correction and also three other methods, a Lagrange‐remap method, a fifth‐order accurate in space and time finite difference type method based on the wave propagation algorithm, and fifth‐order spatial and third‐order temporal accurate finite volume Monotone Upwind Scheme for Conservation Laws (MUSCL) approach based on the Godunov method and Simple Low Dissipation Advection Upstream Splitting Method (SLAU) numerical flux with low Mach capture property. The ability of the compressible flow solver of the commercial software, ANSYS FLUENT , in solving low Mach flows is also demonstrated for the two time‐stepping methods provided in the compressible flow solver, implicit and explicit. Results demonstrate clearly that a low Mach correction is required for all algorithms except the Lagrange‐remap approach, where dissipation is independent of Mach number. © 2013 Crown copyright. International Journal for Numerical Methods in Fluids. © 2013 John Wiley & Sons, Ltd.     

Improvements to compressible Euler methods for low‐Mach number flows

In the present study improvements to numerical algorithms for the solution of the compressible Euler equations at low Mach numbers are investigated. To solve flow problems for a wide range of Mach numbers, from the incompressible limit to supersonic speeds, preconditioning techniques are frequently employed. On the other hand, one can achieve the same aim by using a suitably modified acoustic damping method. The solution algorithm presently under consideration is based on Roe's approximate Riemann solver [Roe PL. Approximate Riemann solvers, parameter vectors and difference schemes. Journal of Computational Physics 1981; 43 : 357–372] for non‐structured meshes. The numerical flux functions are modified by using Turkel's preconditioning technique proposed by Viozat [Implicit upwind schemes for low Mach number compressible flows. INRIA, Rapport de Recherche No. 3084, January 1997] for compressible Euler equations and by using a modified acoustic damping of the stabilization term proposed in the present study. These methods allow the compressible Euler equations at low‐Mach number flows to be solved, and they are consistent in time. The efficiency and accuracy of the proposed modifications have been assessed by comparison with experimental data and other numerical results in the literature. Copyright © 2000 John Wiley & Sons, Ltd.     

An implicit high-order preconditioned flux reconstruction method for low-Mach-number flow simulation with dynamic meshes

A fully implicit high-order preconditioned flux reconstruction/correction procedure via reconstruction (FR/CPR) method is developed to solve the compressible Navier-Stokes equations at low Mach numbers. A dual-time stepping approach with the second-order backward differentiation formula (BDF2) is employed to ensure temporal accuracy for unsteady flow simulation. When dynamic meshes are used to handle moving/deforming domains, the geometric conservation law is implicitly enforced to eliminate errors due to the resolution discrepancy between BDF2 and the spatial FR/CPR discretization. The large linear system resulted from the spatial and temporal discretizations is tackled with the restarted generalized minimal residual solver in the PETSc (portable, extensible toolkit for scientific computation) library. Through several benchmark steady and unsteady numerical tests, the preconditioned FR/CPR methods have demonstrated good convergence and accuracy for simulating flows at low Mach numbers. The new flow solver is then used to study the effects of Mach number on unsteady force generation over a plunging airfoil when operating in low-Mach-number flows. It is observed that weak compressibility has a significant impact on thrust generation but has a negligible effect on lift generation of an oscillating airfoil.     

UNIFIED COMPUTATION OF FLOW WITH COMPRESSIBLE AND INCOMPRESSIBLE FLUID BASED ON ROE''''S SCHEME

A unified numerical scheme for the solutions of the compressible and incompressible Navier-Stokes equations is investigated based on a time-derivative preconditioning algorithm. The primitive variables are pressure, velocities and temperature. The time integration scheme is used in conjunction with a finite volume discretization. The preconditioning is coupled with a high order implicit upwind scheme based on the definition of a Roe's type matrix. Computational capabilities are demonstrated through computations of high Mach number, middle Mach number, very low Mach number, and incompressible flow. It has also been demonstrated that the discontinuous surface in flow field can be captured for the implementation Roe's scheme.     

A preconditioned dual time-stepping procedure coupled with matrix-free LU-SGS scheme for unsteady low speed viscous flows with moving objects

In this paper an effective method is developed to solve unsteady low speed viscous flow problems with moving objects by using the governing equations of compressible fluids. The method is based on a dual time-stepping scheme, combined with low Mach number preconditioning and an implicit matrix-free Lower-Upper Symmetric Gauss-Seidel iteration on unstructured dynamic meshes. Because preconditioning modifies the governing equations, that induces the change of system's eigenvalues and eigenvectors, characteristic boundary conditions are also modified to suit the preconditioned characteristic system. Several test cases are simulated, including an in-line oscillating cylinder in a fluid at rest, flow over a flapping NACA0014 airfoil and low speed flow past a flapping-wing micro-air vehicle. Compared with experimental results whenever possible, the computed results indicate that this algorithm shows satisfactory improvement of solution efficiency and accuracy for low speed flow problems.     

High order numerical simulation of the thermal load on a lobed strut injector for scramjet applications

In this paper, the thermal load on an actively cooled lobed strut injector for scramjet (supersonic combustion ramjet) applications is investigated numerically. This requires coupled simulations of the strut internal and external flow fields together with the heat conduction in the solid injector body. In order to achieve a fast mixing, the lobed strut is positioned at the channel axis to inject hydrogen into the core of a Mach 3 air stream. There it is exposed to the extremely high temperatures of the high speed flow. While the external air and hydrogen flows are supersonic, the strut internal hydrogen flow is mainly subsonic, in some regions at very low Mach numbers. To enable a simulation of the internal flow field which ranges from very low to very high Mach numbers (approximately Mach 2.25 at the nozzle exit), a preconditioning technique is employed. The compressible finite‐volume scheme uses a spatially fourth order multi‐dimensional limiting process discretization, which is used here for a first time to simulate a geometrically and fluid mechanically highly complex problem. It will be demonstrated that besides its high accuracy the multi‐dimensional limiting process scheme is numerically stable even in case of demanding practical applications. The coupled simulation of the lobed strut injector delivers unique insight into the flow phenomena inside and outside the strut, the heat fluxes, the temperature distribution in the solid material, the required hydrogen mass flux with respect to cooling requirements and details concerning the conditions at the exit of the injector. Copyright © 2016 John Wiley & Sons, Ltd.     

UNIFIED COMPUTATION OF FLOW WITH COMPRESSIBLE AND INCOMPRESSIBLE FLUID BASED ON ROE＇S SCHEME

A unified numerical scheme for the solutions of the compressible and incompressible Navier-Stokes equations is investigated based on a time-derivative preconditioning algorithm. The primitive variables are pressure, velocities and temperature. The time integration scheme is used in conjunction with a finite volume discretization. The preconditioning is coupled with a high order implicit upwind scheme based on the definition of a Roe＇s type matrix. Computational capabilities are demonstrated through computations of high Mach number, middle Mach number, very low Mach number, and incompressible flow. It has also been demonstrated that the discontinuous surface in flow field can be captured for the implementation Roe＇s scheme.     

Continuous and discrete adjoint approaches for aerodynamic shape optimization with low Mach number preconditioning

Discrete and continuous adjoint approaches for use in aerodynamic shape optimization problems at all flow speeds are developed and assessed. They are based on the Navier–Stokes equations with low Mach number preconditioning. By alleviating the large disparity between acoustic waves and fluid speeds, the preconditioned flow and adjoint equations are numerically solved with affordable CPU cost, even at the so‐called incompressible flow conditions. Either by employing the adjoint to the preconditioned flow equations or by preconditioning the adjoint to the ‘standard’ flow equations (under certain conditions the two formulations become equivalent, as proved in this paper), efficient optimization methods with reasonable cost per optimization cycle, even at very low Mach numbers, are derived. During the mathematical development, a couple of assumptions are made which are proved to be harmless to the accuracy in the computed gradients and the effectiveness of the optimization method. The proposed approaches are validated in inviscid and viscous flows in external aerodynamics and turbomachinery flows at various Mach numbers. Copyright © 2007 John Wiley & Sons, Ltd.     

Van Leer格式在全速域范围的推广

通过对格式耗散项的修正将Van Leer格式推广至全速域流场求解范围.对格式耗散项的分析表明,在低马赫数流动情况下格式耗散项中不应包含声速项,以此为依据对Van Leer迎风分裂格式提出了耗散项的修正方法.结合对控制方程时间导数项的预处理,修正后的格式能够成功地模拟低速流动问题,同时在其他马赫数范围内也不损失格式的收敛...     

Low‐cost implicit schemes for all‐speed flows on unstructured meshes

Matrix‐free implicit treatments are now commonly used for computing compressible flow problems: a reduced cost per iteration and low‐memory requirements are their most attractive features. This paper explains how it is possible to preserve these features for all‐speed flows, in spite of the use of a low‐Mach preconditioning matrix. The proposed approach exploits a particular property of a widely used low‐Mach preconditioner proposed by Turkel. Its efficiency is demonstrated on some steady and unsteady applications. Copyright © 2008 John Wiley & Sons, Ltd.     

A time‐accurate pseudo‐compressibility approach based on an unstructured hybrid finite volume technique applied to unsteady turbulent premixed flame propagation

A preconditioning approach based on the artificial compressibility formulation is extended to solve the governing equations for unsteady turbulent reactive flows with heat release, at low Mach numbers, on an unstructured hybrid grid context. Premixed reactants are considered and a flamelet approach for combustion modelling is adopted using a continuous quenched mean reaction rate. An overlapped cell‐vertex finite volume method is adopted as a discretisation scheme. Artificial dissipation terms for hybrid grids are explicitly added to ensure a stable, discretised set of equations. A second‐order, explicit, hybrid Runge–Kutta scheme is applied for the time marching in pseudo‐time. A time derivative of the dependent variable is added to recover the time accuracy of the preconditioned set of equations. This derivative is discretised by an implicit, second‐order scheme. The resulting scheme is applied to the calculation of an infinite planar (one‐dimensional) turbulent premixed flame propagating freely in reactants whose turbulence is supposed to be frozen, homogeneous and isotropic. The accuracy of the results obtained with the proposed method proves to be excellent when compared to the data available in the literature. Copyright © 2004 John Wiley & Sons, Ltd.     

A high‐order discontinuous Galerkin solver for low Mach number flows

In this work, we present a high‐order discontinuous Galerkin method (DGM) for simulating variable density flows at low Mach numbers. The corresponding low Mach number equations are an approximation of the compressible Navier–Stokes equations in the limit of zero Mach number. To the best of the authors'y knowledge, it is the first time that the DGM is applied to the low Mach number equations. The mixed‐order formulation is applied for spatial discretization. For steady cases, we apply the semi‐implicit method for pressure‐linked equation (SIMPLE) algorithm to solve the non‐linear system in a segregated manner. For unsteady cases, the solver is implicit in time using backward differentiation formulae, and the SIMPLE algorithm is applied to solve the non‐linear system in each time step. Numerical results for the following three test cases are shown: Couette flow with a vertical temperature gradient, natural convection in a square cavity, and unsteady natural convection in a tall cavity. Considering a fixed number of degrees of freedom, the results demonstrate the benefits of using higher approximation orders. Copyright © 2015 John Wiley & Sons, Ltd.     

Comparison of low Mach number models for natural convection problems

 We investigate in this paper two numerical methods for solving low Mach number compressible flows and their application to single-phase natural convection flow problems. The first method is based on an asymptotic model of the Navier–Stokes equations valid for small Mach numbers, whereas the second is based on the full compressible Navier–Stokes equations with particular care given to the discretization at low Mach numbers. These models are more general than the Boussinesq incompressible flow model, in the sense that they are valid even for cases in which the fluid is subjected to large temperature differences, that is when the compressibility of the fluid manifests itself through low Mach number effects. Numerical solutions are computed for a series of test problems with fixed Rayleigh number and increasing temperature differences, as well as for varying Rayleigh number for a given temperature difference. Numerical difficulties associated with low Mach number effects are discussed, as well as the accuracy of the approximations. Received on 17 January 2000     

A Mach‐uniform preconditioner for incompressible and subsonic flows

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. Copyright © 2013 John Wiley & Sons, Ltd.     

Efficient Large-eddy Simulations of Low Mach Number Flows Using Preconditioning and Multigrid

In the present paper, an implicit time accurate approach combined with multigrid, preconditioning and residual smoothing is used for the large-eddy simulation (LES) of low Mach number flow. In general, due to the restriction imposed on the time step by the physics of the flow, the advantage of an implicit method over an explicit one for LES is not obvious. It is shown that for the test cases considered in this paper, the present approach allows an efficiency gain of a factor 4–7 compared to the use of a purely explicit approach. The efficiency varies according to the test case, grid clustering, physical time step and requested residual drop. Numerical difficulties are catalogued and mitigatory procedures are introduced. Several problems with available experimental and DNS data are employed to verify the efficiency of the method.     

Implicit preconditioned high‐order compact scheme for the simulation of the three‐dimensional incompressible Navier–Stokes equations with pseudo‐compressibility method

This paper combines the pseudo‐compressibility procedure, the preconditioning technique for accelerating the time marching for stiff hyperbolic equations, and high‐order accurate central compact scheme to establish the code for efficiently and accurately solving incompressible flows numerically based on the finite difference discretization. The spatial scheme consists of the sixth‐order compact scheme and 10th‐order numerical filter operator for guaranteeing computational stability. The preconditioned pseudo‐compressible Navier–Stokes equations are marched temporally using the implicit lower–upper symmetric Gauss–Seidel time integration method, and the time accuracy is improved by the dual‐time step method for the unsteady problems. The efficiency and reliability of the present procedure are demonstrated by applications to Taylor decaying vortices phenomena, double periodic shear layer rolling‐up problem, laminar flow over a flat plate, low Reynolds number unsteady flow around a circular cylinder at Re = 200, high Reynolds number turbulence flow past the S809 airfoil, and the three‐dimensional flows through two 90°curved ducts of square and circular cross sections, respectively. It is found that the numerical results of the present algorithm are in good agreement with theoretical solutions or experimental data. Copyright © 2011 John Wiley & Sons, Ltd.     

Improving Euler Computations at Low Mach Numbers

Abstract

This paper consists of two parts, both dealing with conditioning techniques for low-Mach-number Euler-flow computations, in which a multigrid technique is applied

In the first part, for subsonic flows and upwind-discretized linearized 1-D Euler equations, the smoothing behavior of multigrid-accelerated point Gauss-Seidel relaxation is investigated. Error decay by convection over domain boundaries is also discussed. A fix to poor convergence rates at low Mach numbers is sought by replacing the point relaxation applied to unconditioned Euler equations by locally implicit “time” stepping applied to preconditioned Euler equations. The locally implicit iteration step is optimized for good damping of high-frequency errors. Numerical inaccuracy at low Mach numbers is also addressed. In the present case it is not necessary to solve this accuracy problem

In the second part, insight is given into the conditions of derivative matrices to be inverted in point-relaxation methods for 1-D and 2-D, upwind-discretized Euler equations. Speed regimes are found where ill-conditioning of these matrices occurs, and 1-D flow equations appear to be less well-conditioned than 2-D flow equations. Fixes to the ill-conditioning follow more or less directly, when thinking of adding regularizing matrices to the ill-conditioned derivative matrices. A smoothing analysis is made of point Gauss-Seidel relaxation applied to discrete Euler equations conditioned by such an additive matrix. The method is successfully applied to a very low-subsonic,     

Development of the generalized MacCormack scheme and its extension to low Mach number flows

The low Mach number performance of the MacCormack scheme is examined. The inherent dissipation in the scheme is found to suffer from the degradation in accuracy observed with traditional, density‐based methods for compressible flows. Two specific modifications are proposed, leading to the formation of the generalized MacCormack scheme within a dual‐time framework (called GMC‐PC). The first modification involves reformulating the flux by splitting it into particle convection and acoustic parts, with the former terms treated using the traditional MacCormack discretization and the latter terms augmented by the addition of a pressure‐based artificial dissipation. The second modification involves a reformulation of the traditional nonlinear fix introduced by MacCormack in 1971, which is found to be necessary to suppress pressure oscillations at low Mach numbers. The new scheme is demonstrated to have superior performance, independent of Mach number, compared with standard MacCormack implementations using several canonical test problems. Copyright © 2017 John Wiley & Sons, Ltd.