Weighted Compact Nonlinear Scheme Research Articles

Unconditionally optimal high-order weighted compact nonlinear schemes with sharing function for Euler equations

Weighted compact nonlinear schemes (WCNS) are a type of high-order shock-capturing finite difference schemes commonly used in various applications. Their spatial discretizations involve a nonlinear interpolation step and a linear difference step. However, the nonlinear interpolation step requires significantly more computational resources compared to the linear difference step. Therefore, simplifying the interpolation step is an effective way to improve the efficiency of these schemes. In this paper, we propose a new approach to construct WCNS schemes based on the sharing function for Euler equations. This approach uses a set of common nonlinear interpolation weights for different components, resulting in a significant improvement in efficiency compared to the original WCNS schemes that use different sets of weights for each component. To ensure accuracy at critical points of any orders, we employ nonlinear weights that guarantee unconditionally optimal high order. Additionally, this new approach may reduce oscillations caused by non-characteristic interpolation. Furthermore, we also develop a new shock detector using the sharing function, enabling us to employ characteristic interpolation for nonsmooth regions and linear component-wise interpolation for the rest. We validate the proposed schemes based on the sharing function through numerical examples of one- and two-dimensional Euler equations, demonstrating their effectiveness in terms of shock-capturing capability and efficiency.

High-order discretization–based self-adaptive turbulence eddy simulation for supersonic base flow with PHengLEI software

PurposeThe purpose of the present study is to develop a new numerical framework that can predict the supersonic base flow more accurately, including the development of axisymmetrically separated shear layer and recompression shock. To this end, two aspects are improved and combined, i.e. a newly self-adaptive turbulence eddy simulation (SATES) turbulence modeling method and a high-order discretization numerical scheme. Furthermore, the performance of the new numerical framework within a general-purpose PHengLEI software is assessed in detail.Design/methodology/approachSatisfactory prediction of the supersonic separated shear layer with unsteady wake flow is quite challenging. By using a unified turbulence model called SATES combining high-order accurate discretization numerical schemes, the present study first assesses the performance of newly developed SATES for supersonic axisymmetric separation flows. A high-order finite differencing-based compressible computational fluid dynamics (CFD) code called PHengLEI is developed and several different numerical schemes are used to investigate the effects on shock-turbulence interactions, which include the monotonic upstream-centered scheme for conservation laws (MUSCL), weighted compact nonlinear scheme (WCNS) and hybrid cell-edge and cell-node dissipative compact scheme (HDCS).FindingsCompared with the available experimental data and the numerical predictions, the results of SATES by using high-order accurate WCNS or HDCS schemes agree better with the experiments than the results by using the MUSCL scheme. The WCNS and HDCS can also significantly improve the prediction of flow physics in terms of the instability of the annular shear layer and the evolution of the turbulent wake.Research limitations/implicationsThe small deviations in the recirculation region can be found between the present numerical results and experimental data, which could be caused by the inaccurate incoming boundary layer condition and compressible effects. Therefore, a proper incoming boundary layer condition with turbulent fluctuations and compressibility effects need to be considered to further improve the accuracy of simulations.Practical implicationsThe present study evaluates a high-order discretization-based SATES turbulence model for supersonic separation flows, which is quite valuable for improving the calculation accuracy of aeronautics applications, especially in supersonic conditions.Originality/valueFor the first time, the newly developed SATES turbulence modeling method combining the high-order accurate WCNS or HDCS numerical schemes is implemented on the PHengLEI software and successfully applied for the simulations of supersonic separation flows, and satisfactory results are obtained. The unsteady evolutions of the supersonic annular shear layer are analyzed, and the hairpin vortex structures are found in the simulation.

A multi-order candidates weighting framework for discontinuity capturing of hyperbolic conservation laws

Weighted compact nonlinear schemes (WCNSs) represent a popular family of high-order methods used for simulating compressible flows. Within these methods, the nonlinear interpolation process is crucial for the scheme's ability to capture shocks. In this study, we introduce a generalized weighting framework that utilizes candidates of varying orders, named multi-order candidates (MOC) weighting, to design nonlinear schemes capable of capturing discontinuities. Analyzing within the MOC framework reveals that the Z nonlinear weights are equivalent to the JS nonlinear weights, supplemented by an additional fifth-order candidate. The convergence accuracy of the MOC weighting is examined, indicating that higher-order candidates in the MOC weighting require smoothness indicators with higher-order leading terms for better accuracy. Based on this convergence analysis, consistently high-order smoothness indicators (CHOSI) are proposed for the MOC weighting, which are essential for achieving the designed high-order accuracy and high resolution. It is found that many existing nonlinear weighting methods can be integrated into the MOC framework. Three of these methods are analyzed and further improved within the MOC framework. Finally, the newly proposed schemes are tested across a range of benchmark problems to showcase their enhanced performance, also illustrating the MOC framework's capability in developing innovative schemes.

Improved self-adaptive turbulence eddy simulation for complex flows and stall prediction using high-order schemes

It is challenging to apply numerical simulations to accurately predict the stall behavior of aircraft equipped with high-lift devices. Simulations with Reynolds-Averaged NavierStokes (RANS) models suffer from lack of the reliability at high angles of attack with separated and reattached boundary layers, whereas wall-resolved Large Eddy Simulations (LES) of wall-bounded flows at high Reynolds numbers currently costs too much computational resources. A new unified hybrid turbulence modeling approach, denoted the Self-Adaptive Turbulence Eddy Simulation (SATES), is proposed and applied for complex turbulent flows combining with high-order numerical scheme of the Weighted Compact Nonlinear Scheme (WCNS) in the present study. It enables a seamless evolution from unsteady RANS to LES and finally approaches Direct Numerical Simulation (DNS) depending on the turbulent scales. In the framework of SATES, a new SATES-σ model with an adaptive model coefficient is developed by extending the underlying LES mode based on an enhanced sub-grid-scale model of the σ-model. The new SATES-σ is first examined in two benchmark cases of channel flow and flow past a square cylinder. Then, it is validated in supercritical flow past a circular cylinder to assess the performance of turbulent models. The results show significant improvements over the previous SATES and IDDES in the predictions of boundary layer flow. Finally, successful application is achieved in the accurate prediction of the stall of the MD-30P30N airfoil at a Reynolds number of 9×106 with wide angles of attack. The simulation results show good agreement with experimental results for surface pressure even for the challenging cases of 21 and 23 deg angles of attack. Again, the SATES-σ shows better results than the previous SATES and IDDES. The presented method has considerable potential for the challenging stall predictions.

Two classes of third-order weighted compact nonlinear schemes for Hamilton-Jacobi equations

The solutions of Hamilton-Jacobi (HJ) equations may contain discontinuous derivatives, which brings numerical difficulties in capturing sharply these derivatives. This paper presents a third-order weighted compact nonlinear scheme (WCNS) and a third-order perturbed WCNS (WCNS-P) for solving HJ equations. A linear combination of hybrid cell-edge and cell-node values of the function is applied to approximate the spatial derivatives of the function. For the WCNS, the upwind-biased nonlinear weighted interpolation is used for the unknown cell-edge values, while for the WCNS-P perturbations with a free parameter is introduced to the upwind-biased linear interpolation. The free parameter can be optimized to reduce numerical errors in smooth regions. To enhance the numerical stability, a monotone polynomial interpolation method is designed to switch the WCNS-P to the WCNS. Several numerical experiments are performed to test the order of accuracy and the discontinuity-capturing ability of the two schemes.

Self-Adaptive Turbulence Eddy Simulation with a high-order finite differencing method for high Reynolds number complex flows

Accurate prediction of high Reynolds number wall-bounded flows is a challenge for turbulence modeling and numerical discretization. Moreover, complex engineering applications increase the difficulties in implementing robust and low-dissipation numerical schemes. A newly developed unified turbulence approach, Self-Adaptive Turbulence Eddy Simulation (SATES), is assessed in the simulation of a multi-element 30P30N airfoil at a high Reynolds number of 1.71 × 106. To reduce the numerical error, high-order finite differencing-based weighted compact nonlinear schemes (WCNSs) are successfully applied. The effects of grid resolution, numerical schemes and turbulence model are investigated by grouping studies. The SATES shows low sensitivity to grid resolution and predicts satisfactory results even on a coarse grid containing approximately 7.2 million cells. Compared with second-order accurate central differencing, the high-order accurate WCNS has less numerical dissipation and significantly improves the resolution of turbulence structures and the accuracy of numerical results. Z-type nonlinear weighted averaging interpolation, designed for shock capturing property in the WCNS scheme, leads to extra numerical dissipation and affects the accuracy of the results. The numerical results of the SATES are more accurate than those of the Smagorinsky Large Eddy Simulation (LES) and the Improved Delayed Detached Eddy Simulation (IDDES). The SATES provides sufficient modeling of the boundary layer on the coarse grid and a fast ‘transition’ from the RANS to the LES mode for accurate prediction of the separated shear layer and flap boundary layer separation. These results confirm that the SATES turbulence modeling approach is efficient for high Reynolds number flows and that the high-order accurate WCNS schemes show sufficient robustness and accuracy for engineering applications.

A new type of weighted compact nonlinear scheme with minimum dispersion and adaptive dissipation for compressible flows

A new type of weighted compact nonlinear scheme (WCNS) with state-of-the-art spectral properties is developed based on an adaptive, nonlinear optimization strategy that considers the diversity of flow structures and the influence of nonlinearity. The optimization is realized by simultaneously adjusting the free parameters (η and ξ) and the nonlinear weighting function (C) at each local region. In this way, minimum dispersion and adaptive dissipation can be achieved for the scheme. The optimization contains two steps: (1) η is optimized to obtain minimum dispersion; (2) ξ and C are optimized depending on the local flow structure to obtain adaptive dissipation. Following these steps, a 6-point WCNS with excellent spectral properties is achieved. The optimized WCNS is then extended to multi-species flows by incorporating the double-flux algorithm. Numerical results of benchmark test cases show that the present WCNS exhibits better resolution and stronger robustness than recently-developed high-resolution schemes.

Efficient high-order gradient-based reconstruction for compressible flows

This paper extends the gradient-based reconstruction approach of Chamarthi [1] to genuine high-order accuracy for inviscid test cases involving smooth flows. A seventh-order accurate scheme is derived using the same stencil as of the explicit fourth-order scheme proposed in Ref. [1], which also has low dissipation properties. The proposed method is seventh-order accurate under the assumption that the variables at the cell centres are point values. A problem-independent discontinuity detector is used to obtain high-order accuracy. Accordingly, primitive or conservative variable reconstruction is performed around regions of discontinuities, whereas smooth solution regions apply flux reconstruction. The proposed approach can still share the derivatives between the inviscid and viscous fluxes, which is the main idea behind the gradient-based reconstruction. Several standard benchmark test cases are presented. The proposed method is more efficient than the seventh-order weighted compact nonlinear scheme (WCNS) for the test cases considered in this paper.

Nonlinear weights for shock capturing schemes with unconditionally optimal high order

We develop in this paper nonlinear weights for shock capturing schemes, such that the design optimal high order is achieved regardless of any order of the critical point. In particular, we validate the proposed nonlinear weights by testing their performance for a fifth-order weighted compact nonlinear scheme (WCNS), which has advantages in terms of implementing numerical fluxes and keeping the freestream property on curvilinear grids. Theoretical analysis is performed to demonstrate the unconditionally optimal high order of the proposed method. In addition, the spectral property of the proposed nonlinear weights is compared with some existing nonlinear ones. Several one- and two-dimensional benchmark examples are presented to validate the advantages of the proposed method. It is worth noting that the developed nonlinear weights can be applied straightforwardly for widely used weighted essentially non-oscillatory (WENO) schemes.

An efficient hybrid multi-resolution WCNS scheme for solving compressible flows

A hybrid finite difference fifth-order multi-resolution weighted compact nonlinear scheme (WCNS) is presented for simulating the inviscid and viscous compressible flows in this paper. This scheme borrows and improves the idea of the recently developed multi-resolution WENO scheme for node-to-midpoint nonlinear interpolation procedures of the WCNS scheme, which implements higher-order approximate interpolation at midpoints based on a series of nested stencils. This strategy can achieve the expected design accuracy under different linear weights, and can gradually decrease from high-order accuracy to first-order accuracy near the discontinuities. In addition, a hybridization strategy is proposed to further improve the computational efficiency of the presented multi-resolution WCNS scheme. The strategy does not depend on a specific problem to be solved, nor does it contain any problem-related artificial parameters. Finally, some benchmark examples are used to verify the performance of the presented scheme in terms of numerical accuracy, shock capture capability, robustness, computational efficiency, etc.

Higher-order aerodynamic numerical simulations in compressible RANS framework with inverse-ω scale variable

The use of higher-order methods to robustly compute turbulent flows governed by the multi-equation turbulence model in Reynolds-averaged Navier-Stokes (RANS) framework is a challenging topic in the computational fluid dynamics (CFD) community. In this paper, we propose two aspects of measures involving the physical model and the numerical scheme. Firstly, a λ-scale (=1/ωn) is derived instead of the well-known ω in SSG/LRR Reynolds-stress model (RSM) and SST eddy viscosity model (EVM) due to its natural boundary conditions for viscous walls and help to the positivity preservation of dissipation rate ε in turbulence equations when n=1/(2m),m=1,2,3.... Secondly, a geometric-conservation-preservation negativity-correction (GCPNC) method is proposed on the implicit time integration procedure with right-hand-side (RHS) term discretized by high-order weighted compact nonlinear schemes (WCNSs). It can guarantee the positivity of normal Reynolds stresses or turbulent kinetic energy. Numerical tests with real-life compressible flows are reported to demonstrate the robustness and accuracy of the present methods. Relevant cases include the supersonic square duct flow, the flows over delta wing, wing-body configuration and high-lift configuration.

Data-Driven Transition Models for Aeronautical Flows with a High-Order Numerical Method

Over the past years, there has been innovative ideas about data-driven turbulence modeling proposed by scholars all over the world. This paper is a continuity of these significant efforts, with the aim of offering a better representation for turbulence physics. Previous works mainly focus on turbulence viscosity or Reynolds stress, while there are few works for turbulence transition. In our work, two mapping functions between average flow parameters and transition intermittency, a virtual physical quantity describing the amount of turbulence at a given position, are refactored, respectively, with neuron networks and random forests. These two functions are then coupled with the Spalart–Allmaras (SA) model to reconstitute two models for transition prediction. To demonstrate that these two coupled models provide improved prediction accuracy on transition compared with previous SA models, we conduct test cases all under a high-order weighted compact nonlinear scheme (WCNS). The prediction results of both coupled models significantly improved the capture of natural transitions occurring in the flows. Furthermore, the interpolation generalisation and extrapolation generalisation abilities of the coupled models are also demonstrated in this paper. The results emphasize the potential for machine learning as a supplementary in turbulence transition modeling.

A New ϵ-Adaptive Algorithm for Improving Weighted Compact Nonlinear Scheme with Applications

To improve the resolution and accuracy of the high-order weighted compact nonlinear scheme (WCNS), a new ϵ-adaptive algorithm based on local smoothness indicators is proposed. The new algorithm introduces a high-order global smoothness indicator to adjust the value of ϵ according to the local flow characteristics. Specifically, the algorithm increases ϵ in smooth regions, which can help cover up the disparity in smoothness indicators of sub-stencils and make the nonlinear scheme approach the background linear scheme. As a result, optimal order accuracy can be achieved in smooth regions, including critical points. While near discontinuities, the algorithm decreases ϵ, thereby strengthening the stencil selection mechanism and further attenuating spurious oscillations. Meanwhile, the new algorithm makes nonlinear schemes scale-invariant of flow variables. The results of approximate dispersion relation (ADR) show that the new algorithm can greatly reduce spectral errors of nonlinear schemes in the medium and low wavenumber range without inducing instability. Numerical results indicate that the new algorithm can significantly improve resolution of small-scale structures and suppress numerical oscillations near discontinuities with only a minor increment in computational cost.

High order all-speed semi-implicit weighted compact nonlinear scheme for the isentropic Navier–Stokes equations

This paper presents a high order all-speed semi-implicit weighted compact nonlinear scheme (WCNS) for the isentropic Navier–Stokes system. To avoid the severe CFL stability restriction, the pressure and viscous terms are treated implicitly in time, while the other terms are treated explicitly in time. The third-order IMEX Runge–Kutta methods and the fifth-order WCNS are used for time discretization and spatial discretization, respectively. The generated linear equations of velocity components are solved by the GMRES iterative algorithm. Numerical results in one, two and three dimensions in both compressible and incompressible regimes are presented to show the performance of the designed scheme.

Improved weighted compact nonlinear scheme for implicit large-eddy simulations

Weighted compact nonlinear scheme (WCNS) is improved by replacing the nonlinear explicit interpolation with a nonlinear compact interpolation. The non-oscillatory behaviour of the weighting technique and the spectral resolution of the compact interpolation are maintained simultaneously. In order to investigate the performance of implicit large-eddy simulations (ILES) using WCNS with the new interpolation, several fundamental turbulent flows with relatively complex configurations are validated. Encouraging results are obtained in predicting mean flow profiles, unsteady vortex shedding and the frequency spectra, indicating that the improved WCNS is a promising algorithm for simulations of turbulent compressible flows.

High order semi-implicit weighted compact nonlinear scheme for the full compressible Euler system at all Mach numbers

A high order semi-implicit weighted compact nonlinear scheme (WCNS) is presented to solve the full compressible Euler equations at all Mach numbers. To avoid the stringent acoustic CFL restriction for an explicit time discretization method, the pressure splitting methodology is introduced to split the Euler equations into nonstiff and stiff terms. The nonstiff and stiff terms are treated explicitly and implicitly, respectively, based on the high order IMEX Runge-Kutta time discretization method that is asymptotic preserving and asymptotically accurate in the zero Mach number limit. A fifth-order WCNS and fourth-order centered finite difference schemes with zero numerical viscosity are used for the spatial discretization. Numerical tests in one and two space dimensions are displayed to demonstrate the performance of the high order semi-implicit WCNS in compressible and incompressible regimes. Comparisons with the third-order explicit weighted essentially non-oscillatory (WENO) scheme are also made in order to better assess the proposed scheme.

A high-order weighted compact nonlinear scheme for compressible flows

This paper presents a weighted compact nonlinear scheme (WCNS) with high-order accuracy. The WCNS has the flexibility to select a numerical flux and is easily applied to general coordinates. In this study, the specific substencils of the original targeted essentially non-oscillatory (TENO) scheme, which is effective against discontinuities with a high-order interpolation, are combined with WCNS. The accuracy of the interpolation can be easily extended, and the expressions of the fifth, seventh, ninth, and eleventh-order accuracies are provided in this paper. For two types of weight coefficients (WCNS-JS and WCNS-T), the present scheme is investigated using a various benchmark-test problems including strong shock and high-frequency waves. Within substencils, the WCNS-JS can capture strong discontinuities; however, it cannot resolve small-scale fluctuations because of the dissipation effect. On the contrary, the WCNS-T with a TENO weighing coefficient can capture strong discontinuities and high-frequency waves because high-order accuracy works effectively on the premise that a suitable grid resolution is satisfied.

High-order weighted compact nonlinear scheme for one- and two-dimensional Hamilton-Jacobi equations

This paper designs a fifth-order weighted compact nonlinear scheme (WCNS) on a five-point stencil to solve one- and two-dimensional Hamilton-Jacobi equations. The five-point WCNS is used to compute the left and right limits of first-order spatial derivatives of the HJ equations in the Lax-Friedrichs monotone numerical Hamiltonian. The WENO-Z type interpolation for cell-edge values of the solutions is used to suppress numerical oscillations which may appear near discontinuities. Five- and seven-point WENO-Z schemes for Hamilton-Jacobi equations are also designed for comparisons. The performance of the WCNS and the two WENO-Z schemes is demonstrated by several numerical examples in one-dimensional and two-dimensional cases.

A positivity-preserving high-order weighted compact nonlinear scheme for compressible gas-liquid flows

We present a robust, highly accurate, and efficient positivity- and boundedness-preserving diffuse interface method for the simulations of compressible gas-liquid two-phase flows with the five-equation model by Allaire et al. using high-order finite difference weighted compact nonlinear scheme (WCNS) in the explicit form. The equation of states of gas and liquid are given by the ideal gas and stiffened gas laws respectively. Under a mild assumption on the relative magnitude between the ratios of specific heats of the gas and liquid, we can construct limiting procedures for the fifth order incremental-stencil WCNS (WCNS-IS) with the first order Harten–Lax–van Leer contact (HLLC) flux such that positive partial densities and squared speed of sound can be ensured in the solutions, together with bounded volume fractions and mass fractions. The limiting procedures are discretely conservative for all conservative equations in the five-equation model and can also be easily applied to any other conservative finite difference or finite volume scheme. Numerical tests with liquid water and air are reported to demonstrate the robustness and high accuracy of the WCNS-IS with the positivity- and boundedness-preserving limiters even under extreme conditions.

High order semi-implicit weighted compact nonlinear scheme for viscous Burgers’ equations

This paper develops a high order semi-implicit weighted compact nonlinear scheme (WCNS) for one- and two-dimensional viscous Burgers’ equations. This semi-implicit WCNS combines a fifth-order WCNS with a third-order implicit-explicit Runge–Kutta (IMEX-RK) time-stepping method. The advection terms of viscous Burgers’ equations are treated explicitly, while the viscosity terms are treated implicitly. Stability analysis shows that the CFL condition of the semi-implicit WCNS is controlled only by the advection terms. Compared to the explicit time-stepping method, the semi-implicit method has the advantage in terms of computational efficiency. Numerical results validate the accuracy and efficiency of the semi-implicit WCNS. This method can also solve viscous Burgers’ equations with large Reynolds numbers and has high-resolution shock-capturing ability.