No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
HLLC+: Low-Mach Shock-Stable HLLC-Type Riemann Solver for All-Speed Flows
... In the field of aeronautics, there are many situations where both incompressible flow regimes with very low Mach numbers and compressible flow regimes with shocks coexist at the same time. Thus, many all Mach number AUSM-family, 44,45 HLLCtype, 46,47 Roe-type, 12,48 and HLLEM-type 49 schemes have been proposed in recent years to handle the mixed high and low speed flow problems. In these all Mach number schemes, fixes for resolving low Mach numbers flow features and suppression of numerical shock instabilities are implemented. ...
... where z n is the face normal Mach number function defined in Eq. (47), and fp1 is the multi-dimensional pressure function. The pressure function f p1 includes all the neighboring interfaces for shock detection and is estimated as ...
A low diffusion version of the Harten–Lax–Leer convective pressure split (HLL-CPS) scheme for resolving the shear layers and the flow features at low Mach numbers is presented here. The low diffusion HLL-CPS scheme is obtained by reconstructing the velocities at the cell interface with the face normal Mach number and a pressure function. Asymptotic analysis of the modified scheme shows a correct scaling of the pressure at low Mach numbers and a significant reduction in numerical dissipation. The robustness of the HLL-CPS scheme for strong shock is improved by reducing the contribution of the contact wave in the vicinity of the shock. The improvement in robustness for strong shock is demonstrated analytically through linear perturbation and matrix stability analyses. A set of numerical test cases are solved to demonstrate the efficacy of the proposed scheme over a wide range of Mach numbers.
... Most of the analyses and solutions for numerical shock instabilities at high Mach numbers and inaccuracies at low Mach numbers have been developed for the Roe, 46 AUSM, 47,48 HLLC, 49,50 and HLLEM 51 schemes. The HLL-CPS scheme 10 has been shown to preserve positivity, satisfy entropy conditions, and accurately capture contact discontinuities. ...
This paper compares the Harten–Lax–Leer–Einfeldt Modified (HLLEM) and Harten–Lax–Leer Convective Pressure Split (HLL-CPS) schemes for Euler equations and proposes improvements for all-Mach number flows. Enhancements to the HLLEM scheme involve adding anti-diffusion terms in the face normal direction and modifying anti-diffusion coefficients for linearly degenerate waves near shocks. The HLL-CPS scheme is improved by adjusting anti-diffusion coefficients for the face normal direction and linearly degenerate waves. Matrix stability, linear perturbation, and asymptotic analyses demonstrate the robustness of the proposed schemes and their ability to capture low Mach flow features. Numerical tests confirm that the schemes are free from shock instabilities at high speeds and accurately resolve low Mach number flow features.
... To avoid grid aligned shock instabilities at the curved Mach-disks, numerical fluxes between FV elements and mixed DG/FV interfaces are computed with a local Lax-Friedrichs solver. The Lax-Friedrichs solver was chosen over grid-aligned shock stabilization techniques proposed by Fleischmann et al. [13] and Chen et al. [8], since these methods require an additional tuning parameter. The setup is computed for ∈ (0, 500 s] on procs = 16384 processors of the high performance computing (HPC) cluster HAWK, requiring a total of 911 thousand CPU hours. ...
In this paper, we present an hp-adaptive hybrid Discontinuous Galerkin/Finite Volume method for simulating compressible, turbulent multi-component flows. Building on a previously established hp-adaptive strategy for hyperbolic gas-and droplet-dynamics problems, this study extends the hybrid DG/FV approach to viscous flows with multiple species and incorporates non-conforming interfaces, enabling enhanced flexibility in grid generation. A central contribution of this work lies in the computation of both convective and dissipative fluxes across non-conforming element interfaces of mixed discretizations. To achieve accurate shock localization and scale-resolving representation of turbulent structures, the operator dynamically switches between an h-refined FV sub-cell scheme and a p-adaptive DG method, based on an a priori modal solution analysis. The method is implemented in the high-order open-source framework FLEXI and validated against benchmark problems, including the supersonic Taylor-Green vortex and a triplepoint shock interaction, demonstrating its robustness and accuracy for under-resolved shock-turbulence interactions and compressible multi-species scenarios. Finally, the method's capabilities are showcased through an implicit large eddy simulation of an under-expanded hydrogen jet mixing with air, highlighting its potential for tackling challenging compressible multi-species flows in engineering.
... Being able to solve the Riemann problem in both cases required us to choose an all-speed Riemann solver. We chose HLLC+ from Chen et al. (2020), that is based on the original approximate Riemann solver of Harten-Lax-van Leer with contact restoration and includes multiple wave fixes for both low and high Mach numbers. We also tested the regular HLLC Riemann solver, as well as the HLLE for isothermal system of equations. ...
A weakly magnetized neutron star (NS) undergoing disk accretion should release about a half of its power in
a compact region known as the accretion boundary layer. Latitudinal spread of the accreted matter and efficient
radiative cooling justify the approach to this flow as a two-dimensional spreading layer (SL) on the surface of
the star. Numerical simulations of SLs are challenging because of the curved geometry and supersonic nature
of the problem. We develop a new two-dimensional hydrodynamics code that uses the multislope second-order
MUSCL scheme in combination with an HLLC+ Riemann solver on an arbitrary irregular mesh on a spherical
surface. The code is suitable and accurate for Mach numbers at least up to 5-10. Adding sinks and sources
to the conserved variables, we simulate constant-rate accretion onto a spherical NS. During the early stages of
accretion, heating in the equatorial region triggers convective instability that causes rapid mixing in latitudinal
direction. One of the outcomes of the instability is the development of a two-armed ‘tennis ball’ pattern rotating
as a rigid body. From the point of view of a high-inclination observer, its contribution to the light curve is seen
as a high-quality-factor quasi-periodic oscillation mode with a frequency considerably smaller than the rotation
frequency of the matter in the SL. Other variability modes seen in the simulated light curves are probably
associated with low-azimuthal-number Rossby waves.
... Since shock instabilities severely limit the application of shock-capturing methods in supersonic or hypersonic flow simulations, numerous works have tried to eliminate the instability problem. [26][27][28][29][30][31][32][33][34][35][36][37][38] Readers are referred to these papers for detailed information. ...
Modern shock-capturing schemes often suffer from numerical shock instabilities when simulating strong shocks, limiting their application in supersonic or hypersonic flow simulations. In the current study, we devote our efforts to investigating the shock instability problem for second-order schemes, which has not gotten enough attention in previous research but is crucial to address. To this end, we develop the matrix stability analysis method for the finite-volume Monotone Upstream-centered Schemes for Conservation Laws (MUSCL) approach, taking into account the influence of reconstruction. With the help of this newly developed method, the shock instability of second-order schemes is investigated quantitatively and efficiently. The results demonstrate that when second-order schemes are employed, whether shock instabilities will occur is closely related to the property of Riemann solvers, just like the first-order case. However, enhancing spatial accuracy still impacts the shock instability problem, and the impact can be categorized into two types depending on the dissipation of Riemann solvers. Furthermore, the research emphasizes the impact of the numerical shock structure, highlighting both its role as the source of instability and the influence of its state on the occurrence of instability. One of the most significant contributions of this study is the confirmation of the multidimensional coupled nature of shock instability. Both one-dimensional and multidimensional instabilities are proven to influence the instability problem, and they have different properties. Moreover, this paper reveals that increasing the aspect ratio and distortion angle of the computational grid can help mitigate shock instabilities. The current work provides an effective tool for quantitatively investigating the shock instabilities for second-order schemes, revealing the inherent mechanism and thereby contributing to the elimination of shock instability.
... All problems use the newly proposed linearized primitive variables from Section 3.4.2. As for numerical fluxes, the standard Harten, Lax, and van Leer (HLL) approximate Riemann solver (Harten et al. 1983;Toro 2009) and the HLLC+ (where "C" stands for contact discontinuity) approximate Riemann solver from Chen et al. (2020) are both utilized. In each case, wave speeds are estimated following Batten et al. (1997). ...
This paper presents a fully multidimensional kernel-based reconstruction scheme for finite volume methods applied to systems of hyperbolic conservation laws, with a particular emphasis on the compressible Euler equations. Nonoscillatory reconstruction is achieved through an adaptive-order weighted essentially nonoscillatory (WENO) method cast into a form suited to multidimensional reconstruction. A kernel-based approach inspired by radial basis functions and Gaussian process modeling, which we call kernel-based finite volume method with WENO, is presented here. This approach allows the creation of a scheme of arbitrary order of accuracy with simply defined multidimensional stencils and substencils. Furthermore, the fully multidimensional nature of the reconstruction allows for a more straightforward extension to higher spatial dimensions and removes the need for complicated boundary conditions on intermediate quantities in modified dimension-by-dimension methods. In addition, a new simple yet effective set of reconstruction variables is introduced, which could be useful in existing schemes with little modification. The proposed scheme is applied to a suite of stringent and informative benchmark problems to demonstrate its efficacy and utility. A highly parallel multi-GPU implementation using Kokkos and the message-passing interface is also provided.
... It can be seen from equations (36,49,50) while the density perturbation in the proposed HLLEM-FP1D scheme has settled to a value less than the initial perturbation. The pressure perturbation is unable to feed the density perturbations in the proposed scheme, unlike the HLLEM scheme. ...
... Xie et al. [39] have proposed an HLLC scheme for all Mach numbers by introducing a pressure diffusion term to suppress shock instability and implementing Dellacherie's fix [35] for low Mach numbers. Chen et al [40] have proposed an all-speed HLLC-type scheme that introduces an anti-dissipative pressure fix to overcome the accuracy problem in low Mach number limits and a shear viscosity term to overcome shock instability. ...
... The first solvers based on these findings were still in the realm of preconditioning, like the solver by Guillard and Murrone [13], but the ideas were transferred over time to the Riemann solvers themselves as in [3,8,20,27,28,30,33,34] to name just a few of them. A comparison of some of these solvers can be found in [26]. ...
Many low-Mach or all-Mach number codes are based on space discretizations which in combination with the first order explicit Euler method as time integration would lead to an unstable scheme. In this paper, we investigate how the choice of a suitable explicit time integration method can stabilize these schemes. We restrict ourselves to some old prototypical examples in order to find directions for further research in this field.
... To this end, a velocity-based sensor is introduced in this paper. This sensor can be computed in a localized manner as opposed to the non-local shock sensors used in most of the carbuncle cures reported in past studies [6,[8][9][10]12,[15][16][17][18] . The paper is organized as follows. ...
A hybrid numerical flux scheme is proposed by adapting the carbuncle-free modified Harten-Lax-van Leer contact (HLLCM) scheme to smoothly revert to the Harten-Lax-van Leer contact (HLLC) scheme in regions of shear. This hybrid scheme, referred to as the HLLCT scheme, employs a novel, velocity-based shear sensor. In contrast to the non-local pressure-based shock sensors often used in carbuncle cures, the proposed shear sensor can be computed in a localized manner meaning that the HLLCT scheme can be easily introduced into existing codes without having to implement additional data structures. Through numerical experiments, it is shown that the HLLCT scheme is able to resolve shear layers accurately without succumbing to the shock instability.
... [12,14,24]. For low Mach flows, preconditioning techniques are usually applied to release the small time step condition and cure large numerical viscosities in the shock capturing schemes [11,15,41,54,57] h t + ∇ · (hu) = 0, (hu) t + ∇ · (hu ⊗ u) + 1 ε 2 h∇H = 0. We start with the following single-scale expansions of the solutions h and u, in terms of ε, ...
In this paper, high order semi-implicit well-balanced and asymptotic preserving finite difference WENO schemes are proposed for the shallow water equations with a non-flat bottom topography. We consider the Froude number ranging from O(1) to 0, which in the zero Froude limit becomes the "lake equations" for balanced flow without gravity waves. We apply a well-balanced finite difference WENO reconstruction, coupled with a stiffly accurate implicit-explicit (IMEX) Runge-Kutta time discretization. The resulting semi-implicit scheme can be shown to be well-balanced, asymptotic preserving (AP) and asymptotically accurate (AA) at the same time. Both one- and two-dimensional numerical results are provided to demonstrate the high order accuracy, AP property and good performance of the proposed methods in capturing small perturbations of steady state solutions.
High-order Godunov methods for gas dynamics have become a standard tool for simulating different classes of astrophysical flows. Their accuracy is mostly determined by the spatial interpolant used to reconstruct the pair of Riemann states at cell interfaces and by the Riemann solver that computes the interface fluxes. In most Godunov-type methods, these two steps can be treated independently, so that many different schemes can in principle be built from the same numerical framework. Because astrophysical simulations often test out the limits of what is feasible with the computational resources available, it is essential to find the scheme that produces the numerical solution with the desired accuracy at the lowest computational cost. However, establishing the best combination of numerical options in a Godunov-type method to be used for simulating a complex hydrodynamic problem is a nontrivial task. In fact, formally more accurate schemes do not always outperform simpler and more diffusive methods, especially if sharp gradients are present in the flow. For this work, we used our fully compressible Seven-League Hydro SLH ) code to test the accuracy of six reconstruction methods and three approximate Riemann solvers on two- and three-dimensional (2D and 3D) problems involving subsonic flows only. We considered Mach numbers in the range from 10^ to 10^ , which are characteristic of many stellar and geophysical flows. In particular, we considered a well-posed, 2D, Kelvin--Helmholtz instability problem and a 3D turbulent convection zone that excites internal gravity waves in an overlying stable layer. Although the different combinations of numerical methods converge to the same solution with increasing grid resolution for most of the quantities analyzed here, we find that (i) there is a spread of almost four orders of magnitude in computational cost per fixed accuracy between the methods tested in this study, with the most performant method being a combination of a low-dissipation Riemann solver and a sextic reconstruction scheme; (ii) the low-dissipation solver always outperforms conventional Riemann solvers on a fixed grid when the reconstruction scheme is kept the same; (iii) in simulations of turbulent flows, increasing the order of spatial reconstruction reduces the characteristic dissipation length scale achieved on a given grid even if the overall scheme is only second order accurate; (iv) reconstruction methods based on slope-limiting techniques tend to generate artificial, high-frequency acoustic waves during the evolution of the flow; and (v) unlimited reconstruction methods introduce oscillations in the thermal stratification near the convective boundary, where the entropy gradient is steep.
In this paper, high order semi-implicit well-balanced and asymptotic preserving finite difference WENO schemes are proposed for the shallow water equations with a non-flat bottom topography. We consider the Froude number ranging from O(1) to 0, which in the zero Froude limit becomes the “lake equations” for balanced flow without gravity waves. We apply a well-balanced finite difference WENO reconstruction, coupled with a stiffly accurate implicit-explicit (IMEX) Runge-Kutta time discretization. The resulting semi-implicit scheme can be shown to be well-balanced, asymptotic preserving (AP) and asymptotically accurate (AA) at the same time. Both one- and two-dimensional numerical results are provided to demonstrate the high order accuracy, AP property and good performance of the proposed methods in capturing small perturbations of steady state solutions.
An effective and unified framework, termed anti-dissipation pressure correction, is established to overcome the deterioration in the accuracy observed in Godunov-type schemes in low-speed scenarios. Based on the scale analysis of the nonlinear characteristic line relation, the deteriorated-accuracy source term is associated with the speed of sound and velocity difference term. Subsequently, the anti-dissipation pressure correction term is constructed by using the scaling Mach number function to rescale the error source term to be of the magnitude of the convective velocity and ensure a satisfactory accuracy. This scaling function is a polynomial of the local Mach number and restrained by the shock indicator. Results of the asymptotic analysis demonstrate that the anti-dissipation pressure correction helps recover the correct Ma2 scaling of the pressure fluctuations and damps the checkerboard modes, and this aspect is verified considering low Mach number flows over the NACA0012 airfoil. This novel framework can remarkably enhance the low Mach number performance and is also applicable to scenarios involving moderate or high Mach numbers. The performance of the approach is validated through numerical investigations across a wide range of Mach numbers.
A method of second-order accuracy is described for integrating the equations of ideal compressible flow. The method is based on the integral conservation laws and is dissipative, so that it can be used across shocks. The heart of the method is a one-dimensional Lagrangean scheme that may be regarded as a second-order sequel to Godunov's method. The second-order accuracy is achieved by taking the distributions of the state quantities inside a gas slab to be linear, rather than uniform as in Godunov's method. The Lagrangean results are remapped with least-squares accuracy onto the desired Euler grid in a separate step. Several monotonicity algorithms are applied to ensure positivity, monotonicity and nonlinear stability. Higher dimensions are covered through time splitting. Numerical results for one-dimensional and two-dimensional flows are presented, demonstrating the efficiency of the method. The paper concludes with a summary of the results of the whole series “Towards the Ultimate Conservative Difference Scheme.”
This paper presents a new, simple low-dissipation numerical flux function of the AUSM-family for all speeds, called the simple low-dissipation AUSM. In contrast with existing all-speed schemes, the simple low-dissipation AUSM features low dissipation without any tunable parameters in a low Mach number regime while it keeps the robustness of the AUSM-family fluxes against shock-induced anomalies at high Mach numbers (e.g., carbuncle phenomena). Furthermore, the simple low-dissipation AUSM has a simpler formulation than the other all-speed schemes. These advantages of the present scheme are demonstrated in numerical examples of a wide spectrum of Mach numbers.
Shock-capturing finite volume schemes often give rise to anomalous results in hypersonic flow. We present a wide-ranging survey of numerical experiments from 12 different flux functions in one- and two-dimensional contexts. Included is a recently developed function that satisfies the second law of thermodynamics. It is found here that there are at least two kinds of shock instabilities: one is one-dimensional and the other is multidimensional. According to the results, the former does not appear if a flux function satisfies the second law of thermodynamics, and the latter is suppressed by an additional dissipation with a multidimensional character. However, such dissipation has no effect on the one-dimensional mode. Among the flux functions investigated, no universally stable schemes are found that are free from both one- and multidimensional shock instabilities. The appearance of these instabilities depends-on the relative positioning of the shock on the grid.
Several numerical schemes for the solution of hyperbolic conservation laws are based on exploiting the information obtained by considering a sequence of Riemann problems. It is argued that in existing schemes much of this information is degraded, and that only certain features of the exact solution are worth striving for. It is shown that these features can be obtained by constructing a matrix with a certain “Property U.” Matrices having this property are exhibited for the equations of steady and unsteady gasdynamics. In order to construct thems it is found helpful to introduce “parameter vectors” which notably simplify the structure of the conservation laws.
Acceleration methods are presented for solving the steady state incompressible equations. These systems are preconditioned by introducing artificial time derivatives which allow for a faster convergence to the steady state. We also consider the compressible equations in conservation form with slow flow. Two arbitrary functions α and β are introduced in the general preconditioning. An analysis of this system is presented and an optimal value for β is determined given a constant α. It is further sown that the resultant incompressible equations form a symmetric hyperbolic system and so are well posed. Several generalizations to the compressible equations are presented which extend previous results.
A new multigrid relaxation scheme is developed for the steady-state solution of the Euler and Navier-Stokes equations. The lower-upper Symmetric-Gauss-Seidel method (LUSGS) does not require flux splitting for approximate Newton iteration. The present method, which is vectorizable and unconditionally stable, needs only scalar diagonal inversions. Application to transonic flow shows that the new method is efficient and robust.
With the increasing popularity of Computational Fluid Dynamics (CFD), the reliability of numerical scheme becomes prominent. The work presents a newly improved scheme more reliable in all Mach number regimes to circumvent some typical symptoms of the previous AUSM-family schemes observed in hypersonic and very low speeds. This scheme is facilitated by reconstructing pressure diffusion term in mass flux, velocity diffusion term in pressure flux and numerical sound speed. Then, a variety of benchmark test cases are selected to systematically assess the effects of these key ingredients and investigate the additional features in terms of robustness and accuracy. The proposed scheme attains stronger shock robustness against carbuncle instability, better low-speed accuracy and higher resolution of oblique shocks compared with many existing upwind schemes. Moreover, it can exactly resolve contact discontinuity, preserve positivity, damp numerical overshoots and avert the global cut-off strategy. Numerical results for a wide spectrum of Mach numbers indicate its potential and reliable application to all Mach number flows.
Aeroheating predictions play an important role in the heatshield design of Mars entry capsule. This paper numerically investigates the effects of sphere-cone angle on the aerothermodynamic performances of heatshield configurations. Three-dimensional Navier-Stokes equations with chemical non-equilibrium models are employed to simulate the flowfield around the capsule. The laminar and turbulent heating rates of different heatshield configurations are compared and analyzed in detail. A novel correlation for turbulent heating augmentation in terms of laminar momentum thickness Reynolds number is developed to provide a guidance for engineering design and application. The proposed correlation can be more accurate and applicable due to its consideration of the sphere-cone angle effects. Finally, the laminar and turbulent maximum heat flux and total heat load along the flight trajectory are investigated for all the configurations. The numerical study is expected to illustrate the aeroheating characteristics of different heatshield configurations and provide an insight into the rational configuration design for future Mars entry capsules.
Effective high-order solver based on the model of thermally perfect gas has been developed for hypersonic heat transfer computation. The technique of polynomial curve fit coupling to thermodynamics equation is suggested to establish the current model and particular attention has been paid to the design of proper numerical flux for thermally perfect gas. We present procedures that unify five-order WENO (Weighted Essentially Non-Oscillatory) scheme in the existing second-order finite volume framework and a line implicit method that improves the computational efficiency without increasing memory consumption. A variety of hypersonic viscous flows are performed to examine the capability of the resulted high order thermally perfect gas solver. Numerical results demonstrate its superior performance compared to low order calorically perfect gas method and indicate its potential application to hypersonic heating predictions for real-life problem.
The reduced dissipation approach is applied to AUSM-family flux functions of SLAU2 (as well as its predecessor, SLAU) and AUSM+-up for high resolution simulations. In this approach, a dominant dissipation term (of the pressure flux) in each flux function is locally controlled (0 < γHR < 1, γHR: dissipation coefficient) if a cell geometry is of high quality (i.e., fully or nearly rectangular) and flow is smooth, and the original method is recovered otherwise (γHR = 1). Numerical tests demonstrate that the proposed HR (High-Resolution, or Hi-Res) -SLAU2 achieves better resolution (while maintaining robustness) for a wide-ranging Mach numbers (from Mach 6 × 10-4 to 8.1), compared with the original counterparts (γHR = 1) or an existing method (HR-Roe), whereas HR-AUSM+-up shows degraded resolution due to a large cutoff Mach number at low speeds and insufficient dissipation at super- and hypersonic speeds, although a smaller γHR is allowed. Furthermore, a new wiggle detector is proposed to improve both convergence and solution accuracy.
A modification of the Roe scheme called L2Roe for low dissipation low Mach Roe is presented. It reduces the dissipation of kinetic energy at the highest resolved wave numbers in a low Mach number test case of decaying isotropic turbulence. This is achieved by scaling the jumps in all discrete velocity components within the numerical flux function. An asymptotic analysis is used to show the correct pressure scaling at low Mach numbers and to identify the reduced numerical dissipation in that regime. Furthermore, the analysis allows a comparison with two other schemes that employ different scaling of discrete velocity jumps, namely LMRoe and a method of Thornber et al. To this end, we present for the first time an asymptotic analysis of the last method.Numerical tests on cases ranging from low Mach number (M∞= 0.001) to hypersonic (M∞ = 5) viscous flows are used to illustrate the differences between the methods and to show the correct behavior of L2Roe. No conflict is observed between the reduced numerical dissipation and the accuracy or stability of the scheme in any of the investigated test cases. This article is protected by copyright. All rights reserved.
In low speed flow computations, compressible finite-volume solvers are known to a) fail to converge in acceptable time and b) reach unphysical solutions. These problems are known to be cured by A) preconditioning on the time-derivative term, and B) control of numerical dissipation, respectively. There have been several methods of A) and B) proposed separately. However, it is unclear which combination is the most accurate, robust, and efficient for low speed flows. We carried out a comparative study of several well-known or recently-developed low-dissipation Euler fluxes coupled with a preconditioned LU-SGS (Lower-Upper Symmetric Gauss-Seidel) implicit time integration scheme to compute steady flows. Through a series of numerical experiments, accurate, efficient, and robust methods are suggested for low speed flow computations.
We present ideas and procedure to extend the AUSM-family schemes to solve flows at all speed regimes. To achieve this, we first focus on the theoretical development for the low Mach number limit. Specifically, we employ asymptotic analysis to formally derive proper scalings for the numerical fluxes in the limit of small Mach number. The resulting new scheme is shown to be simple and remarkably improved from previous schemes in robustness and accuracy. The convergence rate is shown to be independent of Mach number in the low Mach number regime up to M ∞ =0·5, and it is also essentially constant in the transonic and supersonic regimes. Contrary to previous findings, the solution remains stable, even if no local preconditioning matrix is included in the time derivative term, albeit a different convergence history may occur. Moreover, the new scheme is demonstrated to be accurate against analytical and experimental results. In summary, the new scheme, named AUSM+-up, improves over previous versions and eradicates fails found therein.
This paper presents an asymptotic analysis (in power of Mach number) of the flux difference splitting approximation of compressible Euler equations in the low Mach number limit. We prove that the solutions of discrete system contain pressure fluctuations of order of Mach number, while the continuous pressure scales with the square of Mach number. This explains in a rigorous manner why this approximation of compressible equations fails to compute subsonic flows. Finally, we show that a preconditioning of numerical dissipation tensor allows to recover a correct scaling of the pressure. These theoretical results are confirmed by numerical experiments.
We present a low-Mach number fix for Roe’s approximate Riemann solver (LMRoe). As the Mach number Ma tends to zero, solutions to the Euler equations converge to solutions of the incompressible equations. Yet, standard upwind schemes do not reproduce this convergence: the artificial viscosity grows like 1/Ma, leading to a loss of accuracy as Ma→0. With a discrete asymptotic analysis of the Roe scheme we identify the responsible term: the jump in the normal velocity component ΔU of the Riemann problem. The remedy consists of reducing this term by one order of magnitude in terms of the Mach number. This is achieved by simply multiplying ΔU with the local Mach number. With an asymptotic analysis it is shown that all discrepancies between continuous and discrete asymptotics disappear, while, at the same time, checkerboard modes are suppressed. Low Mach number test cases show, first, that the accuracy of LMRoe is independent of the Mach number, second, that the solution converges to the incompressible limit for Ma→0 on a fixed mesh and, finally, that the new scheme does not produce pressure checkerboard modes. High speed test cases demonstrate the fall back of the new scheme to the classical Roe scheme at moderate and high Mach numbers.
The Harten, Lax, and van Leer with contact restoration (HLLC) scheme has been modified and extended in conjunction with time-derivative preconditioning to compute How problems at all speeds. It is found that a simple modification of signal velocities in the HLLC scheme based on the eigenvalues of the preconditioned system is only needed to reduce excessive numerical diffusion at the low Mach number. The modified scheme has been implemented and used to compute a variety of flow problems in both two and three dimensions on unstructured grids. Numerical results obtained indicate that the modified HLLC scheme is accurate, robust, and efficient for flow calculations across the Mach-number range.
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.
This paper considers a class of approximate Riemann solver devised by Harten, Lax, and van Leer (denoted HLL) for the Euler equations of inviscid gas dynamics. In their 1983 paper, Harten, Lax, and van Leer showed how, with a priori knowledge of the signal velocities, a single-state approximate Riemann solver could be constructed so as to automatically satisfy the entropy condition and yield exact resolution of isolated shock waves. Harten, Lax, and van Leer further showed that a two-state approximation could be devised, such that both shock and contact waves would be resolved exactly. However, the full implementation of this two-state approximation was never given. We show that with an appropriate choice of acoustic and contact wave velocities, the two-state so-called HLLC construction of Toro, Spruce, and Speares will yield this exact resolution of isolated shock and contact waves. We further demonstrate that the resulting scheme is positively conservative. This property, which cannot be guaranteed by any linearized approximate Riemann solver, forces the numerical method to preserve initially positive pressures and densities. Numerical examples are given to demonstrate that the solutions generated are comparable to those produced with an exact Riemann solver, only with a stronger enforcement of the entropy condition across expansion waves.
The missing contact surface in the approximate Riemann solver of Harten, Lax, and van Leer is restored. This is achieved following the same principles as in the original solver. We also present new ways of obtaining wave-speed estimates. The resulting solver is as accurate and robust as the exact Riemann solver, but it is simpler and computationally more efficient than the latter, particulaly for non-ideal gases. The improved Riemann solver is implemented in the second-order WAF method and tested for one-dimensional problems with exact solutions and for a two-dimensional problem with experimental results.
A new scheme, All-Speed-Roe scheme, was proposed for all speed flows. Compared with traditional preconditioned Roe scheme, All-Speed-Roe scheme changes non-linear eigenvalues in the numerical dissipation terms of Roe-type schemes. With an asymptotic analysis, the low Mach number behaviour of the scheme is studied theoretically in two ways. In one way, All-Speed-Roe scheme is regarded as finite magnification of Low-Speed-Roe scheme in the low Mach number limit. In the other way, a general form of All-Speed-Roe scheme is analyzed. Both ways demonstrate that All-Speed-Roe scheme has the same low Mach number behaviour as the original governing equation in the continuous case, which includes three important features: pressure variation scales with the square of the Mach number, the zero order velocity field is subject to a divergence constraint, and the second order pressure satisfies a Poisson-type equation in the case of constant-entropy. The analysis also leads to an unexpected conclusion that the velocity filed computed by traditional preconditioned Roe scheme does not satisfy the divergence constraint as the Mach number vanishes. Moreover, the analysis explains the reason of checkerboard decoupling and shows that momentum interpolation method provides a similar mechanism as traditional preconditioned Roe scheme inherently possesses to suppress checkerboard decoupling. In the end, general rulers for modifying non-linear eigenvalues are obtained. Finally, several numerical experiments are provided to support the theoretical analysis. All-Speed-Roe scheme has a sound foundation and is expected to be widely studied and applied to all speed flow calculations.
This paper proposes a simple modification of the variable reconstruction process within finite volume schemes to allow significantly improved resolution of low Mach number perturbations for use in mixed compressible/incompressible flows. The main advantage is that the numerical method locally adapts the variable reconstruction to allow minimum dissipation of low Mach number features whilst maintaining shock capturing ability, all without modifying the formulation of the governing equations. In addition, incompressible scaling of the pressure and density variations are recovered. Numerical tests using a Godunov-type method demonstrate that the new scheme captures shock waves well, significantly improves resolution of low Mach number features and greatly reduces high wave number dissipation in the case of homogeneous decaying turbulence and Richtmyer–Meshkov mixing. In the latter case, the turbulent spectra match theoretical predictions excellently. Additional computational expense due to the proposed modification is negligible.
When the energy of a flow is largely kinetic, many conservative differencing schemes may fail by predicting non-physical states with negative density or internal energy. We describe as positively conservative the subclass of schemes that by contrast always generate physical solutions from physical data and show that the Godunov method is positively conservative. It is also shown that no scheme whose interface flux derives from a linearised Riemann solution can be positively conservative. Classes of data that will bring about the failure of such schemes are described. However, the Harten-Lax-van Leer (HLLE) scheme is positively conservative under certain conditions on the numerical wavespeeds, and this observation allows the linearised schemes to be rescued by modifying the wavespeeds employed.
This paper deals with the development of an improved Roe scheme that is free from the shock instability and still preserves the accuracy and efficiency of the original Roe’s Flux Difference Splitting (FDS). Roe’s FDS is known to possess good accuracy but to suffer from the shock instability, such as the carbuncle phenomenon. As the first step towards a shock-stable scheme, Roe’s FDS is compared with the HLLE scheme to identify the source of the shock instability. Through a linear perturbation analysis on the odd–even decoupling problem, damping characteristic is examined and Mach number-based functions f and g are introduced to balance damping and feeding rates, which leads to a shock-stable Roe scheme. In order to satisfy the conservation of total enthalpy, which is crucial in predicting surface heat transfer rate in high-speed steady flows, an analysis of dissipation mechanism in the energy equation is carried out to find out the error source and to make the proposed scheme preserve total enthalpy. By modifying the maximum-minimum wave speed, the problem of expansion shock and numerical instability in the expansion region is also remedied without sacrificing the exact capturing of contact discontinuity. Various numerical tests concerned with the shock instability are performed to validate the robustness of the proposed scheme. Then, viscous flow test cases ranging from transonic to hypersonic regime are calculated to demonstrate the accuracy, robustness, and other essential features of the proposed scheme.
The aims of this paper are threefold: to increase the level of awareness within the shock capturing community to the fact that many Godunov-type methods contain subtle flaws that can cause spurious solutions to be computed; to identify one mechanism that might thwart attempts to produce very high resolution simulations; and to proffer a simple strategy for overcoming the specific failings of individual Riemann solvers.
The present time-derivative preconditioning algorithm is effective in flow conditions ranging from inviscid to very diffusive flows, as well as low subsonic to supersonic flow velocities. By means of a preconditioning matrix that (1) introduces well-conditioned eigenvalues and (2) avoids nonphysical time reversals for viscous flows, a mechanism is obtained which controls the inviscid and viscous time-step parameters at very diffusive flows. These capabilities are demonstrated for a variety of sample problems; convergence rates of solutions that are indistinguishable from those obtained without preconditioning are shown to be accelerated by as much as two orders of magnitude.