Boundary Variation Diminishing Research Articles

A high-order limiter-free arbitrary Lagrangian–Eulerian discontinuous Galerkin scheme for compressible multi-material flows

In this paper, a high-order limiter-free arbitrary Lagrangian–Eulerian discontinuous Galerkin (ALE-DG) scheme is proposed for simulating one-dimensional compressible multi-material flows. The system is discretized by the DG method, and a kind of Taylor’s expansion basis functions in the general cell is used to construct the interpolation polynomials of the variables. The mesh velocity at the node recognized as the material interface is set to approximate fluid velocity, which makes our scheme can capture the material interface accurately and clearly. For controlling the oscillations and reducing the numerical dissipation, a reconstruction based on the boundary variation diminishing (BVD) algorithm and Tangent of Hyperbola for INterface Capturing (THINC) function is employed to minimize the jump between the values at the left and right sides of cell boundaries. Due to the essentially monotone and bounded properties of THINC function, the difficulties caused by solving discontinuous solutions and complexities of designing limiters can be avoided. Finally, several examples are presented to demonstrate the accuracy and good performance of our scheme such as the essentially non-oscillatory property and so on for simulating the compressible multi-material flows.

Shock wave induced by the collapse of a bubble cluster with each bubble distributed randomly

The phenomenon of bubble collapse is ubiquitous in both natural and industrial settings, often manifesting in the form of bubble clusters. Despite its prevalence, there has been limited research on the dynamics of bubble clusters. In order to gain a better understanding of shock wave emission, liquid jet propagation, and the resulting pressure loads during bubble cluster collapse, we have developed a high-fidelity numerical approach aimed at accurately capturing shock waves and phase interfaces. The interface compression technique is introduced into the compressible two-phase model to improve the sharpness of the phase interface. We solve the physical model using the BVD (Boundary Variation Diminishing) principle. Both the 5th order WENO (Weighted Essentially Non-Oscillatory) scheme and the interface capturing function THINC (Tangent of Hyperbola for INterface Capturing) are utilized to reconstruct primary variables at cell boundaries, effectively minimizing numerical dissipation. For further resolution improvement, we implement these numerical methods on the AMR (Adaptive Mesh Refinement) mesh. Our numerical results demonstrate that the distance between the wall and the bubble cluster, as well as the nearest bubble to the wall, significantly influence the wall peak pressure. However, they have a minimal impact on the overall evolution of the bubble cluster and the peak pressure induced in the flow field. Remarkably, the spatial distribution of bubbles and the total number of bubbles play a crucial role in shaping the dynamic behavior of the bubble cluster and the pressure loads exerted both on the solid wall and in the flow field. The crucial findings offer profound insights into a range of applications, including cavitation erosion, underwater explosions, and drug delivery.

A new high-fidelity Total Variation Diminishing scheme based on the Boundary Variation Diminishing principle for compressible flows

High-accuracy and stable simulations of compressible flows have been highly demanded in many engineering situations. However, it remains a challenging task due to the complicated fluid structures in compressible flows, which include both continuous and discontinuous solutions. The total variation diminishing (TVD) scheme is a mainstream numerical method because it has second-order accuracy for the smooth solution and suppresses numerical oscillations near the discontinuous solution. The properties of the TVD scheme depend on the design of the second-order slope-limiter function, Φ(r), for which minmod, superbee, and van Leer types are commonly adopted. However, the limiter function that provides both superior numerical accuracy and stability has not yet been established. In this paper, we provide an adaptive model for determining the suitable limiter function in the second-order TVD range, unlike the conventional methods, which use one type of limiter function for the entire calculation region. To designate the suitable limiter value, we use the boundary variation diminishing (BVD) principle, which suggests the most appropriate reconstruction method within the specific candidates. The numerical results of benchmark tests show that the proposed scheme has fewer numerical dissipation errors and provides a suitable limiter value that ensures both accuracy and stability.

Numerical study of liquid jet and shock wave induced by two-bubble collapse in open field

The shock waves and high-speed jets induced by the violent collapse of bubbles can cause serious damage to nearby structures. To fully understand the shock wave and jet characteristics induced by bubble-bubble interaction, we developed a diffuse-interface method for the simulation of compressible two-phase flow, which combined interface compression technique with interface sharpening technique to keep the sharpness of the shock wave and phase interface. The thermodynamically consistent five-equation model was improved using interface compression technique to maintain a constant thickness of the phase interface. To further suppress numerical dissipation, the high-order scheme WENO (Weighted Essentially Non-Oscillatory) and interface sharpening function THINC (Tangent of Hyperbola for INterface Capturing) were used for the flux reconstruction at the grid boundary to ensure that the total variation at the cell boundary was minimized (Boundary Variation Diminishing, BVD), thus the spatial reconstruction scheme, named WENO-THINC-BVD, was developed. Moreover, we extended the present method to a block-structured adaptive mesh refinement framework to improve grid resolution and save computing resources. Numerical results of several benchmark tests fully demonstrated the advantages of the present method in simulating complex compressible flows containing shock-shock and shock-interface interactions. Based on the present high-fidelity numerical methods, we investigated the shock wave and liquid jet induced by the two-bubble interaction in an open space. A phase diagram of the liquid jet propagation direction as a function of the initial bubble pressure and the inter-bubble distance was summarized. Finally, according to the bubble-bubble interaction process, an empirical formula for the collapse strength and another for the attenuation of shock waves were developed.

A revised WENO‐THINC scheme for the general structured mesh and applications in the direct numerical simulation of compressible turbulent flows

AbstractThe Tangent of Hyperbola for INterface Capturing (THINC) scheme allows a jump‐like reconstruction and brings about a significant improvement in resolving the discontinuous part of the numerical solutions. However, it is found that the original THINC scheme loses accuracy when working on the stretched, curvilinear or highly‐skewed grid. In this study, we propose a simple strategy to determine the jump thickness parameter in the THINC function, so as to effectively suppress the unphysical oscillation. A Boundary Variation Diminishing (BVD) guideline is introduced to make options between the Weighted Essentially Non‐Oscillatory (WENO) scheme and the modified THINC scheme, thus both the smooth and discontinuous solutions can be reconstructed properly avoiding distortion of grids. Numerical validations indicate that the improved WENO‐THINC‐BVD approach maintains high resolution and is more robust on various types of non‐uniform meshes. The present method is further extended to validate the low‐dissipation property in resolving higher wave numbers portions by simulating an isotropic turbulence decay problem. Finally, we perform the direct numerical simulation (DNS) of a spatially evolving adiabatic flat plate boundary‐layer flow problem at a supersonic Mach number (). Numerical results show the predicted mean flow variables and the normalized shear stress agree well with the experimental data, significant improvements are found in the resolution of the small‐scale vortices, especially in the transition process.

Implicit Gradients Based Conservative Numerical Scheme for Compressible Flows

This paper introduces a novel approach to compute the numerical fluxes at the cell boundaries for a cell-centered conservative numerical scheme. Explicit gradients used in deriving the reconstruction polynomials are replaced by high-order gradients computed by compact finite differences, referred to as implicit gradients in this paper. A problem-independent shock capturing approach via Boundary Variation Diminishing (BVD) algorithm is used to suppress oscillations for the simulation of flows with shocks and material interfaces. Several numerical test cases are carried out to verify the proposed method’s capability using the implicit gradient method for compressible flows.

A new paradigm of dissipation-adjustable, multi-scale resolving schemes for compressible flows

The scale-resolving simulation of high speed compressible flow through direct numerical simulation (DNS) or large eddy simulation (LES) requires shock-capturing schemes to be more accurate for resolving broadband turbulence and robust for capturing strong shock waves. In this work, we develop a new paradigm of dissipation-adjustable, shock capturing scheme to resolve multi-scale flow structures in high speed compressible flow. The new scheme employs a polynomial of n-degree and non-polynomial THINC (Tangent of Hyperbola for INterface Capturing) functions of m-level steepness as reconstruction candidates. These reconstruction candidates are denoted as PnTm. From these candidates, the piecewise reconstruction function is selected through the boundary variation diminishing (BVD) algorithm. Unlike other shock-capturing techniques, the BVD algorithm effectively suppresses numerical oscillations without introducing excess numerical dissipation. Then, an adjustable dissipation (AD) algorithm is designed for scale-resolving simulations. This novel paradigm of shock-capturing scheme is named as PnTm−BVD−AD. The proposed PnTm−BVD−AD scheme has following desirable properties. First, it can capture large-scale discontinuous structures such as strong shock waves without obvious non-physical oscillations while resolving sharp contact, material interface and shear layer. Secondly, the numerical dissipation property of PnTm−BVD−AD can be effectively adjusted between n+1 order upwind-biased scheme and non-dissipative n+2 order central scheme through a simple tunable parameter λ. Thirdly, with λ=0.5 the scheme can recover to n+2 order non-dissipative central interpolation for smooth solution over all wavenumber, which is preferable for solving small-scale structures in DNS as well as resolvable-scale in explicit LES. Finally, the under-resolved small-scale can be solved with the dissipation adjustable algorithm through the so-called implicit LES (ILES) approach. Through simulating benchmark tests involving multi-scale flow structures and comparing with other central-upwind schemes, the superiority of the proposed scheme is evident. For instance, the simulation results of the supersonic planar jet show that PnTm−BVD−AD schemes can achieve competitive results as Pn+2Tm−BVD schemes which utilize a higher degree of reconstruction polynomial. Thus, in comparison with the previous work, the proposed PnTm−BVD−AD schemes have the benefit of a more compact stencil and lower cost. In summary, this work provides an alternative scheme for solving multi-scale problems in high speed compressible flows.

A new 3D OpenFoam solver with improved resolution for hyperbolic systems on hybrid unstructured grids

Although the accuracy and the robustness of discretized schemes solving hyperbolic systems have been greatly improved over the last two decades, devising a low-dissipative and non-oscillatory method remains a challenge. Especially, it is algorithmically complicated and computationally expensive to extend and apply high-resolution non-oscillatory schemes to 3D hybrid unstructured grids. Therefore, in this work we develop accurate, practical, and robust 3D hybrid unstructured OpenFoam solvers which can be applied in wide industrial applications. Different from existing 3D high order numerical solvers of which reconstruction schemes are based on polynomials of high degree, the proposed 3D scheme employs a polynomial function and a hyperbolic tangent basis function as two candidates for reconstruction functions. The MUSCL (Monotone Upstream-centered Schemes for Conservation law) scheme with the MLP (Multi-dimensional Limiting Process) slope limiter is adopted as the polynomial function. The multi-dimensional THINC (Tangent of Hyperbola for INterface Capturing) function with quadratic surface representation and Gaussian quadrature, so-called THINC/QQ, is used as the non-polynomial reconstruction candidate. With these reconstruction candidates, a multi-stage boundary variation diminishing (BVD) algorithm which significantly minimizes numerical dissipation is designed on 3D unstructured grids to select the final reconstruction function. With this accurate and robust reconstruction scheme, then we developed new solvers for simulating inviscid high speed compressible single-phase flow and two-phase flow via the implementation of the proposed scheme and algorithm into the open-source computational fluid dynamics (CFD) software, OpenFOAM. The solver with the proposed low-dissipative scheme, named as EulerEquationFoam-M-TQ-BVD, is constructed with the density-based solver for inviscid high speed compressible flow simulations. To simulate compressible multi-material flow with moving material interfaces, a solver named as FiveEquationFoam-M-TQ-BVD is developed based on mechanical-equilibrium diffusive interface model and MieGrüneisen equation of state. The performance of new solvers is evaluated through wide range benchmark tests and are compared with existing high order schemes such as the third-order WENO (Weighted Essentially Non-Oscillatory) scheme. The new solvers produce superior solution across critical points, contact discontinuities and material interfaces. Moreover, in comparison with WENO schemes, the proposed solver enjoys algorithmic simplicity and greater flexibility in 3D hybrid unstructured grids. Thus, this work provides accurate, practical and robust numerical solvers for high speed compressible single- and multi-material flow simulations.

A shock-stable HLLEM scheme with improved contact resolving capability for compressible Euler flows

Due to its high resolution for discontinuities and high efficiency, the HLLEM scheme is a popular numerical method to calculate the conservative hyperbolic systems. When capturing multidimensional strong shock waves, however, it will encounter different forms of numerical oscillations, such as the odd-even decoupling and carbuncle phenomenon, which is a great hindrance to its application in hypersonic flow problems. In this paper the numerical shock instability of the HLLEM scheme is investigated from a mathematical perspective and a corresponding cure is proposed. A special linearized small perturbation analysis shows that perturbations evolved in the streamwise direction will be attenuated while the density and shear velocity perturbations evolved in the transverse direction will not be attenuated and thus induce the occurrence of instability. In order to suppress the numerical oscillation in capturing strong shock waves, a function based on the pressure ratio at the cell interface is defined to control the dissipation of the HLLEM flux. And a BVD (boundary variation diminishing) algorithm is implemented to pick the smallest density jump from all combinations among reconstructed density values to reduce the numerical dissipation, thus the contact resolving capability of the HLLEM scheme is further improved. The higher resolution for contact discontinuities and better stability for shock waves brought by the current strategies are demonstrated through solving a series of one- and two-dimensional benchmark test problems.

A simple FORCE-type centred scheme accurate for contact discontinuities: Application to compressible Euler flows

The FORCE-type centred schemes are simple and efficient and do not explicitly require the wave propagation information of the system to calculate the numerical flux. However, their poor resolution for contact discontinuities seriously affects their applications. In the present work, a simple FORCE-type centred scheme accurate for contact discontinuities is proposed and applied to calculations of compressible Euler flows. The missing contact wave of the original FORCE centred scheme is restored with an algebraic method and the resolution for contact discontinuities is further improved by using the boundary variation diminishing (BVD) algorithm to minimize the density jump in the numerical diffusion term of the original FORCE centred scheme. Numerical results of several one- and two-dimensional benchmark test problems fully demonstrate that the proposed centred scheme is capable of capturing contact discontinuities more sharply than the complete-wave HLLC upwind scheme. In addition, another advantage of the proposed centred scheme is that it is free from the carbuncle phenomenon which afflicts many contact-capturing upwind schemes (e.g. Roe and HLLC) in calculations of multidimensional flow problems involving strong shock waves.

Hybrid Discontinuous Galerkin/Finite Volume Method with Subcell Resolution for Shocked Flows

A hybrid discontinuous Galerkin (DG)/finite volume (FV) method under the multistage boundary variation diminishing (BVD) reconstruction paradigm is developed for shocked inviscid and viscous flows. The methodology uses high-order unlimited polynomials and the THINC (which stands for tangent of hyperbola for interface capturing) function as the candidate reconstruction functions to perform FV computation on embedded subcells. The reconstruction scheme on the FV subcells can be designed in a way to keep the resulting approach as compact as the underlying DG method. The reconstruction information based on the modified BVD selection algorithm has been applied as an indicating strategy so that no additional discontinuity indicator is required for the hybrid DG/FV computation. The designed approach has been investigated through the coupling with the explicit and implicit time discretization scheme for both inviscid and viscous flows. Numerical results demonstrate the high-fidelity characteristics of the method by using a much coarser grid to capture both the discontinuity and the smooth flow structure with high resolution.

Low-dissipation BVD schemes for single and multi-phase compressible flows on unstructured grids

Solving compressible flows containing both smooth and discontinuous flow structures still remains a big challenge for finite volume methods, especially on unstructured grids where one faces more difficulties in building high-order polynomial reconstruction and limiting projection to suppress numerical oscillations in comparison with the case of structured grids. As a result, most of the current finite volume schemes on unstructured grids are of second order and too dissipative to resolve fine structures of complex flows. In this paper, we report two novel hybrid schemes to resolve vortical and discontinuous solutions on unstructured grids by reducing numerical dissipation. Different from conventional shock capturing schemes that use polynomials and limiting projections for reconstruction, the proposed schemes employ two second-order schemes, i.e. a polynomial and a sigmoid function as candidate reconstruction functions to approximate smooth and discontinuous solutions respectively. As the polynomial function, the MUSCL (Monotone Upstream-centered Schemes for Conservation law) scheme with the MLP (Multi-dimensional Limiting Process) slope limiter is adopted, while being a sigmoid function, the multi-dimensional THINC (Tangent of Hyperbola for INterface Capturing) function with quadratic surface representation and Gaussian quadrature, so-called THINC/QQ, is used to mimic the discontinuous solution structure. With these candidates for reconstruction, a single-step boundary variation diminishing (BVD) algorithm, which aims to minimize numerical dissipation, is designed on unstructured grids to select the final reconstruction function. The resulting two variant schemes, MUSCL-THINC/QQ-BVD schemes with two and three candidates respectively, are algorithmically simple and show great superiority to other existing schemes in capturing discontinuous and vortical flow structures for single and multiphase compressible flows on unstructured grids. The performance of the proposed schemes has been extensively verified through benchmark tests of single and multi-phase compressible flows, where discontinuous and vortical flow structures, like shock waves, contact discontinuities and material interfaces, as well as vortices and shear instabilities of different scales, coexist simultaneously. The numerical results show that the proposed schemes that hybrids two second-order schemes are capable of capturing sharp discontinuous profiles without numerical oscillations and resolving vortical structures along shear layers and material interfaces with significantly improved solution quality superior to other schemes of even higher order reconstructions.

High-order central-upwind shock capturing scheme using a Boundary Variation Diminishing (BVD) algorithm

In this paper, we present a novel hybrid nonlinear explicit-compact scheme for shock-capturing based on a boundary variation diminishing (BVD) reconstruction. In our approach, we combine a non-dissipative sixth-order central compact interpolation and a fifth-order monotonicity preserving scheme (MP5) through the BVD algorithm. For a smooth solution, the BVD reconstruction chooses the highest order possible interpolation, which is central, i.e. non-dissipative in the current approach and for the discontinuities, the algorithm selects the monotone scheme. This method provides an alternative to the existing adaptive upwind-central schemes in the literature. Several numerical examples are conducted with the present approach, which suggests that the current method is capable of resolving small scale flow features and has the same ability to capture sharp discontinuities as the MP5 scheme.

A flux split based finite-difference two-stage boundary variation diminishing scheme with application to the Euler equations

In this paper, we present a new finite difference (FD) two stage boundary variation diminishing (BVD) P4T2 (fourth-degree polynomial (P4) and Tangent of Hyperbola for Interface Capturing (THINC) function with a two-level steepness (T2)) scheme for the compressible Euler equations. First, after splitting the flux vector into positive and negative flux vectors using an appropriate flux vector splitting methods, we use the P4 and THINC functions with a two-level steepness to obtain the upwind numerical fluxes at cell boundaries for both the positive and negative flux vectors. Then, we adopt the corresponding downwind version of the P4 and THINC functions with a two-level steepness to obtain the downwind numerical fluxes at cell boundaries for both the positive and negative flux vectors. The final upwind numerical flux is determined for the positive/negative flux respectively through the two-stage BVD principle, which minimizes the jumps of the reconstructed values at the cell boundaries. The final scheme, called FD-P4T2-BVD, exhibits a better resolution, compared with WENO5 scheme, for small-scale structures, particularly the contact discontinuities in various one-dimensional and two-dimensional cases.

Boundary variation diminishing algorithm for high‐order local polynomial‐based schemes

SummaryA novel approach for selecting appropriate reconstructions is implemented to the hyperbolic conservation laws in the high‐order local polynomial‐based framework, for example, the discontinuous Galerkin (DG) and flux reconstruction (FR) schemes. The high‐order polynomial approximation generally fails to correctly capture a strong discontinuity inside a cell due to the Gibbs phenomenon, which is replaced by more stable approximation on the basis of a troubled‐cell indicator such as that used with the total variation bounded (TVB) limiter. This paper examines the applicability of a new algorithm, so‐called boundary variation diminishing (BVD) reconstruction, to the weighted essentially nonoscillatory methodology in the FR framework including the nodal type DG method. The BVD reconstruction adaptively chooses a proper approximation for the solution function so as to minimize the jump between values at the left and right side of cell boundaries. The results of the BVD algorithm are comparable to those with the conventional TVB limiter in terms of oscillation suppression and numerical dissipation in one‐dimensional linear advection and nonlinear system equations, while the TVB limiter performs better in the case with strong discontinuities (the blast wave problem in the Euler equations). Overall, since the present BVD algorithm does not need any ad hoc constant such as the TVB parameter, it could be more reliable than the conventional TVB limiter that is often used in the DG and FR communities for shocks and other discontinuities. The proposed method would lead to a parameter‐free robust algorithm for the local polynomial‐based high‐order schemes.

A Higher Order Interpolation Scheme of Finite Volume Method for Compressible Flow on Curvilinear Grids

A higher order interpolation scheme based on a multi-stage BVD (Boundary Variation Diminishing) algorithm is developed for the FV (Finite Volume) method on non-uniform, curvilinear structured grids to simulate the compressible turbulent flows. The designed scheme utilizes two types of candidate interpolants including a higher order linear-weight polynomial as high as eleven and a THING (Tangent of Hyperbola for INterface Capturing) function with the adaptive steepness. We investigate not only the accuracy but also the efficiency of the methodology through the cost efficiency analysis in comparison with well-designed mapped WENO (Weighted Essentially Non-Oscillatory) scheme. Numerical experimentation including benchmark broadband turbulence problem as well as real-life wall-bounded turbulent flows has been carried out to demonstrate the potential implementation of the present higher order interpolation scheme especially in the ILES (Implicit Large Eddy Simulation) of compressible turbulence.

Constructing higher order discontinuity-capturing schemes with upwind-biased interpolations and boundary variation diminishing algorithm

Based on the fifth-order scheme in our previous work (Deng et. al (2019) [28]), a new framework of constructing very high order discontinuity-capturing schemes is proposed for finite volume method. These schemes, so-called PnTm−BVD (polynomial of n-degree and THINC function of m-level reconstruction based on BVD algorithm), are designed by employing high-order upwind-biased interpolations and THINC (Tangent of Hyperbola for INterface Capturing) functions with adaptive steepness as the reconstruction candidates. The final reconstruction function in each cell is determined with a multi-stage BVD (Boundary Variation Diminishing) algorithm so as to effectively control numerical oscillation and dissipation. We devise the new schemes up to eleventh order in an efficient way by directly increasing the order of the underlying upwind scheme using high order polynomials. The analysis of the spectral property and accuracy tests show that the new reconstruction strategy well preserves the low-dissipation property of the underlying upwind schemes with high-order polynomials for smooth solution over all wave numbers and realizes n+1 order convergence rate. The performance of new schemes is examined through widely used benchmark tests, which demonstrate that the proposed schemes are capable of simultaneously resolving small-scale flow features with high resolution and capturing discontinuities with low dissipation. With outperforming results and simplicity in algorithm, the new reconstruction strategy shows great potential as an alternative numerical framework for computing nonlinear hyperbolic conservation laws that have discontinuous and smooth solutions of different scales.

A development of a sharp interface AUSMD scheme for an incompressible preconditiong multi-fluid model

In this study, the paper first proposes a sharp-interface cell interface variable reconstruction to solve the preconditioning incompressible multi-fluid flow model using the AUSMD scheme. Here, the preconditioning matrix modified from the Chorin's artificial compressibility makes the incompressible multi-fluid equations to be hyperbolic. Therefore, a simple AUSMD type flux-splitting with the MUSCL+THINC (Tangent of Hyperbola for Interface Capturing) type cell interface variable reconstruction is used to maintain the preservation of sharp interface evolutions of the multi-fluid simulation. A high-resolution interpolation MUSCL+THINC, the MUSCL method conjunction with the THINC decided under the boundary variation diminishing (BVD) concept, is used to determine the cell interfaces to evaluate the inviscid flux. The interface revolution of the Rayleigh Tayor instability, the dam breaking and the rising bubbles are shown and investigated.

Implicit large eddy simulation of compressible turbulence flow with PnTm−BVD scheme

• Implicit large eddy simulation of compressible turbulence is performed with newly proposed PnTm-BVD schemes. • More accurate predictions of the turbulence evolution are obtained compared with WENO schemes. • Faithful solutions to capture discontinuities are realized with BVD algorithm. • The superiority of PnTm-BVD is more notable in high wave-number region. Implicit large eddy simulation (ILES) of compressible turbulence with shock capturing schemes requires wide investigations and numerical experiments. In this study, a newly proposed P n T m − BVD (polynomial of n -degree and THINC function of m -level reconstruction based on BVD algorithm) shock capturing scheme is introduced to simulate compressible turbulence flow with ILES. The new scheme is designed by employing high-order linear-weight polynomials and THINC (Tangent of Hyperbola for INterface Capturing) functions with adaptive steepness as the reconstruction candidates. The final reconstruction function in each cell is determined with a multi-stage BVD (Boundary Variation Diminishing) algorithm so as to effectively control numerical oscillation and dissipation. Numerical tests involving shock waves and broadband turbulence are conducted in comparison with WENO (Weighted Essentially Non-oscillatory) schemes which are widely used in ILES. The results demonstrate performing ILES with P n T m − BVD scheme is able to obtain higher resolution and more faithful results than WENO does. Importantly, the superiority of P n T m − BVD becomes more notable in high wave-number region. Thus this paper provides and verifies a new scheme which is promising in providing high-resolution results for real-case ILES of compressible turbulence flow.

A fifth-order shock capturing scheme with two-stage boundary variation diminishing algorithm

A novel 5th-order shock capturing scheme is presented in this paper. The scheme, so-called P4T2−BVD (polynomial of 4-degree and THINC function of 2-level reconstruction based on BVD algorithm), is formulated as a two-stage spatial reconstruction scheme following the BVD (Boundary Variation Diminishing) principle that minimizes the jumps of the reconstructed values at cell boundaries. In the P4T2−BVD scheme, polynomial of degree four and THINC (Tangent of Hyperbola for INterface Capturing) functions with two-level steepness are used as the candidate reconstruction functions. The final reconstruction function is selected through the two-stage BVD algorithm so as to effectively control both numerical oscillation and dissipation. Spectral analysis and numerical verifications show that the P4T2−BVD scheme possesses the following desirable properties: 1) it effectively suppresses spurious numerical oscillation in the presence of strong shock or discontinuity; 2) it substantially reduces numerical dissipation errors; 3) it automatically retrieves the underlying linear 5th-order upwind scheme for smooth solution over all wave numbers; 4) it is able to resolve both smooth and discontinuous flow structures of all scales with substantially improved solution quality in comparison to other existing methods; and 5) it produces accurate solutions in long term computation. P4T2−BVD, as well as the underlying idea presented in this paper, provides an innovative and practical approach to design high-fidelity numerical schemes for compressible flows involving strong discontinuities and flow structures of wide range scales.