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Numerical simulations of high-speed compressible flows remain challenging in engineering as the appearance of shock waves poses difficulties for high-resolution schemes as well as explicit large eddy simulation methods. Therefore, in this work, we propose a practical numerical solver for simulations of shock-vortex and shock-turbulence problems with the implicit large eddy simulation approach and newly developed low-dissipative schemes. In order to handle the multi-scale raised in the shock-vortex and shock-turbulence problems, it is required that the flow solver can simultaneously solve the large-scale flow structures such as shock waves with numerical oscillation-free, the resolvable flow structures above the grid scale with low-dissipation, and the under-resolve isotropic sub-grid scale with numerical stability. To this end, a low-dissipative, structure-preserving scheme with rigorously adjusted numerical dissipation is employed in the proposed solver. The structure-preserving property of this scheme can ensure that the large-scale structure is solved without numerical oscillations and the low-dissipative property of this scheme can produce high-resolution results for the resolvable flow scales. Moreover, the sub-grid scale is solved and stabilized by the inherent numerical dissipation in the shock-capturing scheme. The proposed numerical solver is then applied to simulate a wide range of shock-vortex and shock-turbulence interaction problems including supersonic planar jets, transonic flows past a deep cavity and impingement of a supersonic jet on a cone mounted on a flat plate. A comparison is also made with the solver using conventional total variation diminishing schemes. The numerical results of the supersonic planar jet have demonstrated that the proposed solver can resolve the small-scale structure such as Kelvin–Helmholtz instability involved in shock and shear layer with high-resolution. Through the results of transonic flow past a deep cavity and comparisons with the experimental data, it is verified that the proposed solver can reproduce the turbulence statistical data. Furthermore, the proposed solver resolves the complex turbulent fluid mechanical phenomena in the impingement of a supersonic jet on a cone, which demonstrates the proposed solver can robustly handle irregular geometry. The proposed solver also enjoys simplicity without using the explicit sub-grid scale model and involving the complexity of very high-order schemes. Thus, this work provides an accurate and practical numerical solver for shock-vortex and shock-turbulence problems in high-speed flows. • An accurate and practical solver for shock-vortex and shock-turbulence problems. • Resolvable flow scale is solved by structure-preserving ROUND schemes. • Sub-grid scale is modeled with inherent numerical dissipation. • Challenging case such as impingement of a supersonic jet on a cone is simulated. • The proposed solver enjoys simplicity and can handle irregular geometry. [ABSTRACT FROM AUTHOR]

• Accurate, practical and robust numerical solvers for high-speed single- and multi- material flow simulations. • New 3D advection schemes with improved resolution across discontinuities. • Implementation with OpenFoam for wide industrial applications. 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. [ABSTRACT FROM AUTHOR]

• 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. [ABSTRACT FROM AUTHOR]

We proposed a novel multi-moment finite volume method (MMFVM) to compute compressible viscous flows on arbitrary unstructured grids. Unlike the previous volume-average/point-value multi-moment (VPM) method (Xie et al. 2014 [60]), the present method, so-called VPM-Ex (VPM with extended-stencil reconstruction), formulates novel spatial reconstructions with substantially improved robustness and accuracy for both smooth and discontinuous solutions. Our efforts in this work have brought at least the following new features to the numerical formulation: 1) a more robust and accurate spatial reconstruction based on the stencil of all immediately adjacent cells that supports higher-order limiting projection; 2) two different options for both explicit and implicit reconstructions; 3) a simplified and more accurate formulation for point-value. In addition, we have proposed a smoothness-adaptive fitting (SAF) method for limiting projection to improve the solution quality for discontinuities. The resulting reconstruction method that combines VPM-Ex scheme with SAF-based limiting projection schemes is able to attain 3rd-order accuracy for smooth solution, and non-oscillatory and less-dissipative numerical results for discontinuous solution on unstructured grids of different types of elements. The numerical method has been extensively verified by various benchmark tests for scalar conservation equation, Euler equations and Navier–Stokes equations in 2 and 3 dimensions. The numerical results substantiate that the present method provides a novel numerical formulation of great practical significance to accurately resolve both continuous and discontinuous solutions. • Novel high-order VPM (volume-average/point-value multi-moment) method for arbitrary unstructured grids. • Sophisticated WENO (weighted essentially non-oscillatory) limiting process for suppressing the numerical oscillations. • Low-dissipative SAF (smoothness adaptive fitting) scheme for high-fidelity reconstruction of discontinuous solutions. • Verified solution quality on smooth and non-smooth solutions for various flow problems. [ABSTRACT FROM AUTHOR]

Abstract A novel 5th-order shock capturing scheme is presented in this paper. The scheme, so-called P 4 T 2 − 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 P 4 T 2 − 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 P 4 T 2 − 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. P 4 T 2 − 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. Highlights • An innovative fifth-order shock capturing scheme is proposed. • Low-dissipation linear scheme is retrieved for smooth solution over all wave numbers. • Discontinuous and smooth solution of wide-band scales are simultaneously solved with high accuracy. • The free-mode solutions are faithfully maintained in long term computation. [ABSTRACT FROM AUTHOR]

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 P n T m. 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 P n T m − BVD − AD. The proposed P n T m − 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 P n T m − 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 P n T m − BVD − AD schemes can achieve competitive results as P n + 2 T m − BVD schemes which utilize a higher degree of reconstruction polynomial. Thus, in comparison with the previous work, the proposed P n T m − 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. • Novel dissipation-adjustable schemes for scale-resolving simulation. • Numerical dissipation is effectively adjusted between high order upwind and central scheme. • Capture discontinuous large-scale structures such as strong shocks without oscillations, and contact with less dissipation. • Small-scale is resolved by non-dissipative central interpolation which can be retrieved over all wavenumber. • Under-resolved scale can be solved with dissipation-adjustable algorithm through ILES approach. [ABSTRACT FROM AUTHOR]

• Novel hybrid schemes of less numerical dissipation based on BVD algorithm for unstructured grids. • Applicable to single and multiple phase compressible flows involving shocks and material interfaces. • Superior solution quality for both vortical and discontinuous flow structures in comparison with conventional schemes. • Easy to implement on unstructured grids and of significant practical impact. 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. [ABSTRACT FROM AUTHOR]

In this article, we developed an unstructured fluid solver based on finite volume framework for the low-Mach number compressible flows. The present method, so-called FVMS3 (Finite Volume method based on Merged Stencil with 3rd-order reconstruction) formulates two different numerical procedures for spatial reconstructions based on the quadratic polynomial which is performed by using least-square approximations on a merged stencil. In order to improve the reconstruction for discontinuities, we propose the limiting projection approach and smoothness adaptive fitting (SAF) scheme to suppress the numerical oscillation and limit the numerical dissipation. The resulting discretization algorithm that combines FVMS3 with SAF-based limiting projection scheme has third-order accuracy and resolves both smooth and non-smooth solutions with excellent quality. Additionally, a novel numerical model has been proposed by introducing the advection upstream splitting method (AUSM) flux into the pressure projection formulation which results in a unified scheme that works uniformly up to the incompressible limit. The fluid solver that integrates all above new efforts provides high-fidelity solutions for compressible viscous flows particularly for the low Mach regime. The performance of this new solver is verified by numerous benchmark tests. Our numerical results show that the proposed scheme gives accurate and robust solutions for a wide spectrum of test problems. • Third-order, non-oscillatory finite volume method is proposed for polyhedral grids. • Sophisticated SAF algorithm is used to reduce the numerical dissipation. • A new Mach uniform model is developed for compressible flows of various Mach numbers. • High-fidelity solution quality on numerical solutions is verified by benchmark tests. [ABSTRACT FROM AUTHOR]