Monotonicity-preserving Scheme Research Articles

Gradient based reconstruction: Inviscid and viscous flux discretizations, shock capturing, and its application to single and multicomponent flows

This paper presents a gradient-based reconstruction approach for simulations of compressible single and multi-species Navier–Stokes equations. The novel feature of the proposed algorithm is the efficient reconstruction via derivative sharing between the inviscid and viscous schemes: highly accurate explicit and implicit gradients are used for the solution reconstruction expressed in terms of derivatives. The higher-order accurate gradients of the velocity components are reused to compute the viscous fluxes for efficiency and significantly improve the solution and gradient quality, as demonstrated by several viscous-flow test cases. The viscous schemes are fourth-order accurate and carefully designed with a high-frequency damping property, which has been identified as a critically important property for stable compressible-flow simulations with shock waves (Chamarthi et al., 2022). Shocks and material discontinuities are captured using a monotonicity-preserving (MP) scheme, which is also improved by reusing the gradients. For inviscid test cases, The proposed scheme is fourth-order for linear and second-order accurate for non-linear problems. Several numerical results obtained for simulations of complex viscous flows are presented to demonstrate the accuracy and robustness of the proposed methodology.

High-order numerical scheme for compressible multi-component real gas flows using an extension of the Roe approximate Riemann solver and specific Monotonicity-Preserving constraints

The purpose of this paper is to develop a high-order shock-capturing scheme capable of predicting flows where shock waves with high-temperature jumps interact with multi-component real gas mixtures, assuming a local thermodynamic equilibrium. We first propose a generalization of the Roe solver for distinct species with non-ideal thermodynamic properties that relies on the original method proposed by Vinokur & Montagné [1]. This method uses an approximation of compressibility factors to estimate a coherent value of the speed of sound at the Roe averaged state.This Roe averaged state is introduced in the One-Step Monotonicity-Preserving (OSMP) scheme, originally developed by Daru and Tenaud [2], to obtain an extension to the high-order with Lax-Wendroff procedure adequate for dealing with non-ideal gas flows. To avoid thermodynamic inconsistencies in the evolution of the Roe average state over a large stencil, we propose to reformulate the discrete total energy flux of the initial solver. This new formulation uses a combination of Riemann invariants related to the species mass fractions and avoids the influence of the independent values of the compressibility factors in the total energy flux computation. An additional M-P constraint on this new combination allows dealing with discontinuities. Based on the averaged speed of sound estimated by our proposed extension of the Vinokur & Montagné method, we demonstrate that this new formulation is equivalent to selecting a new combination of compressibility factors that completely fulfill the jump relationships of the Riemann problem.To properly capture discontinuities while optimizing the number of numerical cells, the new high-order OSMP scheme is combined with an Adaptive Multiresolution [3] procedure to automatically refine grid in regions where steep gradients occur and coarsen grid elsewhere. The order of the numerical method is evaluated on the convection of density and mass fraction waves. Its capability of capturing discontinuities is validated on a 1-D shock tube problem with a mixture of Nitrogen, Oxygen and dense refrigerant R22 gases. We show that smooth solutions, as well as discontinuities, are recovered with high accuracy. The 2-D interaction between a shock wave in Air with a cylindrical bubble initially filled with dense refrigerant R22 gas is also considered. Present results compare very well with both a recent fully resolved numerical solution of ideal gases and experimental results obtained with real gases. Compared to ideal gas solutions corresponding to calorically perfect gas, drastic changes are recorded on the predicted temperature and the bubble flow patterns that fully justify the use of relevant thermodynamics and the proposed numerical method to account for real gas properties.

Application of a high-order MP scheme to computation of multi-phase flows with heat and mass transfer

Reliable prediction of unsteady multiphase flows with mass changes, large density ratios, and complex flow features is very challenging due to the need to accurately resolve both complex interface deformations and steep gradients associated with material interfaces. This paper describes an application of our recently developed high-order monotonicity preserving (MP) scheme – a reliable and robust scheme for long-term computations of flows contain complex smooth structures interspersed with discontinuities (Ha and Lee, 2020) − to the solution of unsteady incompressible two-phase flows in the presence of heat and mass changes. The flow is governed by the multiphase Navier–Stokes equations, which comprises of the mixture continuity equation, the mixture momentum equations, the mixture energy equation, and the continuity equation for the individual phase. The numerical discretization is implemented on generalized curvilinear grids for facilitating flow simulations of the complex geometries. Our high-order MP scheme is applied to the calculations of the convective and surface tension terms. To obtain an effectively simultaneous solution of all equations, a novel dual-time stepping approach in conjunction with an implicit Runge–Kutta three-stage scheme is proposed. The reliability of the proposed numerical algorithm is then examined for a 1D Stefan condensation problem and two condensing flows including single bubble and steam-jet condensation problems. Particularly, for the single bubble condensation test, the bubble characteristics at different time instants are evaluated both qualitatively and quantitatively. For the steam-jet condensation test, the capability for capturing the evolution of interface deformations under different thermal and hydrodynamic flow conditions is highlighted. Finally, the effects of subcooling degree and steam mass flow rate on heat and mass transfer characteristics are investigated and discussed.

Monotonicity-preserving finite element schemes with adaptive mesh refinement for hyperbolic problems

This work is focused on the extension and assessment of the monotonicity-preserving scheme in [3] and the local bounds preserving scheme in [5] to hierarchical octree adaptive mesh refinement (AMR). Whereas the former can readily be used on this kind of meshes, the latter requires some modifications. A key question that we want to answer in this work is whether to move from a linear to a nonlinear stabilization mechanism pays the price when combined with shock-adapted meshes. Whereas nonlinear (or shock-capturing) stabilization leads to improved accuracy compared to linear schemes, it also negatively hinders nonlinear convergence, increasing computational cost. We compare linear and nonlinear schemes in terms of the required computational time versus accuracy for several steady benchmark problems. Numerical results indicate that, in general, nonlinear schemes can be cost-effective for sufficiently refined meshes. Besides, it is also observed that it is better to refine further around shocks rather than use sharper shock capturing terms, which usually yield stiffer nonlinear problems. In addition, a new refinement criterion has been proposed. The proposed criterion is based on the graph Laplacian used in the definition of the stabilization method. Numerical results show that this shock detector performs better than the well-known Kelly estimator for problems with shocks or discontinuities.

Modified Monotonicity Preserving Constraints for High-Resolution Optimized Compact Scheme

The monotonicity-preserving (MP) scheme is an accurate shock-capturing scheme. However, its performance is still inefficient for resolving high-frequency waves. In this paper, to improve the resolution characteristics, an upwind compact interpolation is proposed as a substitute to the original one in the MP scheme, and the coefficients of that were analytically optimized to minimize the dispersion and dissipation errors. Moreover, it was found that the limiting part of the original MP scheme degenerates the accuracy in a high-wavenumber region due to unnecessarily activation. This limitation is improved by applying a new indicator and criterion. The results of the nonlinear wave (N-wave) propagation demonstrate that the proposed scheme guarantees the robustness at the sharp discontinuity. At the same time, the solutions of linear wave propagation prove the excellent resolution of the proposed scheme. We intensively evaluated the performance for the standard and long-time situations of the shock-entropy wave interaction problems. The results prove that the usefulness of proposed scheme is more pronounced in the flow fields involving both of shock and waves.

A modified monotonicity-preserving high-order scheme with application to computation of multi-phase flows

The monotonicity preserving (MP) polynomial-based reconstruction scheme with Runge-Kutta time stepping by Suresh and Huynh (1997) has been applied extensively in computational fluid dynamics for solving hyperbolic partial differential equations and other convection-dominated problems. However, despite of its algorithmic simplicity in achieving a desired high-order of accuracy, the MP scheme is shown to lose the accuracy and robustness for a long-term computation of flows containing complex smooth structures interspersed with discontinuities. In this paper, the original MP scheme is first reviewed with a focus on the potential causes that lead to its failure. We then propose an improved MP scheme that can overcome the weakness in the MP scheme. The proposed MP scheme is constructed so that it can bypass smooth extrema without loss of accuracy while preserving monotonicity and properly reducing the order of accuracy near discontinuities. Various challenging scalar advection test cases are considered to demonstrate the performance of the presented method. Numerical results reveal that it achieves the highly accurate solutions in smooth parts of the flow and yields a significant improvement in accuracy and shape-preserving property. Moreover, the successful application to several classes of two- and three-dimensional unsteady multi-phase flows highlights the potential of the present method for resolving both sharp and complex interface deformations.

Hybrid Flux Method in Monotonicity-Preserving Scheme for Accurate and Robust Simulation in Supersonic Flow

The fifth-order monotonicity-preserving (MP5) scheme is an accurate and low dissipative numerical method. As a finite-volume method, MP5 adopts the Roe-flux scheme for solving the numerical flux in the compressible Euler equation. However, due to the deficiency of the MP limiter and Roe-flux in maintaining positive density and pressure, the calculation could fail in cases of extreme flow involving small values of density and pressure. In this study, to overcome such a limitation but still to achieve a high-accuracy of MP5, we propose a hybrid flux method: the Roe-flux is used in the global computational domain, but the first-order Lax-Friedrich (LF)-flux is adopted only for trouble grids. The numerical results of shock-tube and complicated interaction problems indicate that the present scheme is more accurate at discontinuities and local extrema compared to the previous scheme, maintaining positive density and pressure values. For two-dimensional applications, a supersonic jet is explored with different Mach numbers and temperature conditions. As a result, small vortices induced by the shear layer can be clearly captured by the proposed scheme. Furthermore, a simulation was successfully conducted without blow-up of calculation even in the extreme jet flow condition.

Multidimensional Vlasov–Poisson Simulations with High-order Monotonicity- and Positivity-preserving Schemes

Abstract We develop new numerical schemes for Vlasov–Poisson equations with high-order accuracy. Our methods are based on a spatially monotonicity-preserving (MP) scheme and are modified suitably so that the positivity of the distribution function is also preserved. We adopt an efficient semi-Lagrangian time integration scheme that is more accurate and computationally less expensive than the three-stage TVD Runge–Kutta integration. We apply our spatially fifth- and seventh-order schemes to a suite of simulations of collisionless self-gravitating systems and electrostatic plasma simulations, including linear and nonlinear Landau damping in one dimension and Vlasov–Poisson simulations in a six-dimensional phase space. The high-order schemes achieve a significantly improved accuracy in comparison with the third-order positive-flux-conserved scheme adopted in our previous study. With the semi-Lagrangian time integration, the computational cost of our high-order schemes does not significantly increase, but remains roughly the same as that of the third-order scheme. Vlasov–Poisson simulations on mesh grids have been successfully performed on a massively parallel computer.

Hybrid monotonicity-preserving piecewise parabolic method for compressible Euler equations

In this article, we present a high-accuracy high-resolution hybrid scheme with strong robustness for solutions of compressible Euler equations, particularly those relating to strong discontinuity. The fifth-order monotonicity-preserving (MP5) scheme of Suresh and Huynh has high accuracy; however, its robustness is relatively weak. To improve the robustness and resolution, the MP5 scheme is conjugated with a fourth-order piecewise parabolic method (PPM) in a complementary approach. An adaptive constraint is used to detect which scheme is applied at the current position to guarantee the strong robustness and high accuracy. Through numerical experiments, our hybrid scheme demonstrates a stronger robustness and higher resolution than the traditional scheme and the hybrid scheme MP5-R by He without loss of accuracy.

単調性保存スキームI : 定常移流・拡散方程式の局所厳密解に基づく有限可変差分法

単調性保存スキームI : 定常移流・拡散方程式の局所厳密解に基づく有限可変差分法

Characteristic-based and interface-sharpening algorithm for high-order simulations of immiscible compressible multi-material flows

The present work focuses on the simulation of immiscible compressible multi-material flows with the Mie–Grüneisen-type equation of state governed by the non-conservative five-equation model [1]. Although low-order single fluid schemes have already been adopted to provide some feasible results, the application of high-order schemes (introducing relatively small numerical dissipation) to these flows may lead to results with severe numerical oscillations. Consequently, attempts to apply any interface-sharpening techniques to stop the progressively more severe smearing interfaces for a longer simulation time may result in an overshoot increase and in some cases convergence to a non-physical solution occurs. This study proposes a characteristic-based interface-sharpening algorithm for performing high-order simulations of such flows by deriving a pressure-equilibrium-consistent intermediate state (augmented with approximations of pressure derivatives) for local characteristic variable reconstruction and constructing a general framework for interface sharpening. First, by imposing a weak form of the jump condition for the non-conservative five-equation model, we analytically derive an intermediate state with pressure derivatives treated as additional parameters of the linearization procedure. Based on this intermediate state, any well-established high-order reconstruction technique can be employed to provide the state at each cell edge. Second, by designing another state with only different reconstructed values of the interface function at each cell edge, the advection term in the equation of the interface function is discretized twice using any common algorithm. The difference between the two discretizations is employed consistently for interface compression, yielding a general framework for interface sharpening. Coupled with the fifth-order improved accurate monotonicity-preserving scheme [2] for local characteristic variable reconstruction and the tangent of hyperbola for the interface capturing scheme [3] for designing other reconstructed values of the interface function, the present algorithm is examined using some typical tests, with the Mie–Grüneisen-type equation of state used for characterizing the materials of interest in both one- and two-dimensional spaces. The results of these tests verify the effectiveness of the present algorithm: essentially non-oscillatory and interface-sharpened results are obtained.

A high-order monotonicity-preserving scheme for hyperbolic conservation laws

This work aims at designing a numerical model for discretizing non-linear hyperbolic systems of conservation laws, with high-order accuracy, while verifying a positivity property of the discretization. From a finite-volume discretization of the PDEs, we identify three positivity constraints for the resulting numerical scheme to be a convex combination of the stencil supporting the discretization. These constraints are then introduced into the MUSCL-like least-square interpolation procedure by using limiters conveniently sized. Lastly, a high-order stability-preserving time integration method (SSPRK) enables to generate the final algorithm. Extensive numerical tests on scalar problems help to assess the benefits and potentialities of the resulting algorithm. Then, upon using a local diagonalization, we extend the scope of the algorithm to non linear hyperbolic systems. To this aim, we propose two strategies for identifying local directions that promotes this diagonalization. Numerical tests on standard problems of gas dynamics, help to appreciate the advantages and drawbacks of each strategy. The resulting numerical scheme is a fourth-order least-square positive scheme (LSQP4) and it is designed and formulated for discretizing unsteady multi-dimensional hyperbolic conservation laws, over non-Cartesian meshes.

A low-dissipation monotonicity-preserving scheme for turbulent flows in hydraulic turbines

The objective of this work is to improve the inherent dissipation of the numerical schemes under the framework of a Reynolds-averaged Navier-Stokes (RANS) simulation. The governing equations are solved by the finite volume method with the k-ω SST turbulence model. Instead of the van Albada limiter, a novel eddy-preserving limiter is employed in the MUSCL reconstructions to minimize the dissipation of the vortex. The eddy-preserving procedure inactivates the van Albada limiter in the swirl plane and reduces the artificial dissipation to better preserve vortical flow structures. Steady and unsteady simulations of turbulent flows in a straight channel and a straight asymmetric diffuser are demonstrated. Profiles of velocity, Reynolds shear stress and turbulent kinetic energy are presented and compared against large eddy simulation (LES) and/or experimental data. Finally, comparisons are made to demonstrate the capability of the eddy-preserving limiter scheme.

An improved accurate monotonicity-preserving scheme for the Euler equations

The accurate monotonicity-preserving (MP) scheme of Suresh and Huynh (1997) [5] is a high-order and high-resolution method for hyperbolic conservation laws. However, the robustness of the MP scheme is not very high. In this paper, a detailed analysis on this scheme is performed, and two potential causes which may account for the weak robustness are revealed. Furthermore, in order to enhance the robustness of the MP scheme, an improved version of the MP scheme is presented, in which a strict continuous total-variation-diminishing (TVD) numerical flux is used at a disturbed discontinuity so that oscillations cannot grow indefinitely without violating the TVD condition. Without destroying the very high resolution property, numerical tests show that the improved scheme shares a strong robustness in simulating extreme numerical tests.

Subgrid-scale turbulence in shock–boundary layer flows

Data generated by direct numerical simulation (DNS) for a Mach 2.75 zero-pressure gradient turbulent boundary layer interacting with shocks of different intensities are used for a priori analysis of subgrid-scale (SGS) turbulence and various terms in the compressible filtered Navier–Stokes equations. The numerical method used for DNS is based on a hybrid scheme that uses a non-dissipative central scheme in the shock-free turbulent regions and a robust monotonicity-preserving scheme in the shock regions. The behavior of SGS stresses and their components, namely Leonard, Cross and Reynolds components, is examined in various regions of the flow for different shock intensities and filter widths. The backscatter in various regions of the flow is found to be significant only instantaneously, while the ensemble-averaged statistics indicate no significant backscatter. The budgets for the SGS kinetic energy equation are examined for a better understanding of shock–tubulence interactions at the subgrid level and also with the aim of providing useful information for one-equation LES models. A term-by-term analysis of SGS terms in the filtered total energy equation indicate that while each term in this equation is significant by itself, the net contribution by all of them is relatively small. This observation is consistent with our a posteriori analysis.

Investigation of low-dissipation monotonicity-preserving scheme for direct numerical simulation of compressible turbulent flows

The influence of numerical dissipation on direct numerical simulation (DNS) of decaying isotropic turbulence and turbulent channel flow is investigated respectively by using the seventh-order low-dissipation monotonicity-preserving (MP7-LD) scheme with different levels of bandwidth dissipation. It is found that for both benchmark test cases, small-scale turbulence fluctuations can be largely suppressed by high level of scheme dissipation, while the appearance of numerical errors in terms of high-frequency oscillations could destabilize the computation if the dissipation is reduced to a very low level. Numerical studies show that reducing the bandwidth dissipation to 70% of the conventional seventh-order upwind scheme can maximize the efficiency of the MP7-LD scheme in resolving small-scale turbulence fluctuations and, in the meantime preventing the accumulation of non-physical numerical errors. By using the optimized MP7-LD scheme, DNS of an impinging oblique shock-wave interacting with a spatially-developing turbulent boundary layer is conducted at an incoming free-stream Mach number of 2.25 and the shock angle of 33.2°. Simulation results of mean velocity profiles, wall surface pressure, skin friction and Reynolds stresses are validated against available experimental data and other DNS predictions in both the undisturbed equilibrium boundary layer region and the interaction zone, and good agreements are achieved. The turbulence kinetic energy transport equation is also analyzed and the balance of the equation is well preserved in the interaction region. This study demonstrates the capability of the optimized MP7-LD scheme for DNS of complex flow problems of wall-bounded turbulent flow interacting with shock-waves.

Subgrid-Scale Models for Large-Eddy Simulations of Shock-Boundary-Layer Interactions

Direct numerical simulation and large-eddy simulation of a flat-plate supersonic turbulent boundary layer interacting with an oblique incident shock wave are conducted for a priori and a posteriori assessments of subgrid-scale models. The incident shock is strong enough to generate a marginal separation in the boundary layer near the interaction region providing the subgrid-scale models with a nontrivial challenge. The governing equations for direct numerical simulation and large-eddy simulation are solved by a new seventh-order monotonicity-preserving scheme for inviscid fluxes and a sixth-order compact finite-difference scheme for the viscous terms. The effect of subgrid-scale stress term on the resolved velocity field is shown to be significant and shock-dependent. The subgrid-scale models tested include the mixed-time-scale model, the dynamic Smagorinsky model, the dynamic mixed model and a new dynamic model termed the compressible serial-decomposition model. A priori analysis indicate that the new dynamic model is more accurate than other subgrid-scale closures. A posteriori tests also indicate better predictions of direct numerical simulation results by the large-eddy simulation employing the compressible serial-decomposition model.

A new fuzzy criterion-referenced assessment with a fuzzy rule selection technique and a monotonicity-preserving similarity reasoning scheme

The use of a Fuzzy Inference System FIS as a part of Criterion-referenced Assessment CRA is not new. Nevertheless, there are several limitations in combining FIS models and CRA, as follows. i It is difficult to maintain the monotonicity property of the FIS-based CRA model; ii it is difficult and impractical to form a complete fuzzy rule base when the number of required rules is large, and iii reducing fuzzy rules may cause the “tomato classification” problem. In this paper, a practical solution to overcome these limitations is provided. We adopt the sufficient conditions i.e., a mathematical foundation as a set of governing equations for designing fuzzy membership functions and fuzzy rules to preserve the monotonicity property. In this paper, our works in [21] is extended and a new procedure that comprises of the sufficient conditions, fuzzy rule reduction and a monotonicity-preserving similarity reasoning SR is proposed. The new framework reduces the number of fuzzy rules gathered from an assessor i.e., selected rules with a proposed fuzzy rule selection approach. Selected rules are identified in such that the unselected rules can be deduced via a monotonicity-preserving SR technique. We formulate the process of minimizing the number of selected rules as a constrained optimization problem and a genetic algorithm GA technique is implemented. Unselected rules are predicted with a proposed monotonicity-preserving SR scheme. The proposed approach contributes to a solution to reduce the number of fuzzy rules to be gathered from an assessor while maintaining the monotonicity property. Besides, this paper also contributes to a new application of SR in education assessment. The proposed approach is evaluated with a case study relating to a lab assessment in Universiti Malaysia Sarawak UNIMAS.

An Optimized Low-Dissipation Monotonicity-Preserving Scheme for Numerical Simulations of High-Speed Turbulent Flows

This paper presents an optimized low-dissipation monotonicity-preserving (MP-LD) scheme for numerical simulations of high-speed turbulent flows with shock waves. By using the bandwidth dissipation optimization method (BDOM), the linear dissipation of the original MP scheme of Suresh and Huynh (J. Comput. Phys. 136, 83---99, 1997) is significantly reduced in the newly developed MP-LD scheme. Meanwhile, to reduce the nonlinear dissipation and errors, the shock sensor of Ducros et al. (J. Comput. Phys. 152, 517---549, 1999) is adopted to avoid the activation of the MP limiter in regions away from shock waves. Simulations of turbulent flows with and without shock waves indicate that, in comparison with the original MP scheme, the MP-LD scheme has the same capability in capturing shock waves but a better performance in resolving small-scale turbulence fluctuations without introducing excessive numerical dissipation, which implies the MP-LD scheme is a valuable tool for the direct numerical simulation and large eddy simulation of high-speed turbulent flows with shock waves.

Numerical simulation of the viscous shock tube problem by using a high resolution monotonicity-preserving scheme

This work deals with the flow generated in a shock tube after the shock wave has reflected at the end wall. For a viscous fluid, a complex unsteady interaction takes place between the incident boundary layer and reflected shock wave. The numerical simulation of this complex flow requires both robust and accurate numerical schemes. In this work, we rely on the one-step high-order scheme recently proposed by Daru and Tenaud [Daru V, Tenaud C. High order one-step monotonicity preserving schemes for unsteady flow calculations. J Comput Phys 2004;193:563–94]. With this scheme, converged results are obtained for Reynolds numbers in the range 200–1000. The interaction mechanisms are carefully analyzed as well as the flow dynamics.