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This paper is concerned with the sonic-supersonic structures for the two-dimensional steady compressible inviscid multiphase flow equations. We construct a local classical supersonic solution near a given smooth sonic curve. This problem is originated from the transonic channel multiphase flows, which are one kind of the most important problems in mathematical fluid dynamics. In order to overcome the difficulties caused by the parabolic degeneracy near sonic and the multivariable dependence of pressure, we adopt the mixed variables of the pressure and angle functions and derive the characteristic decompositions of these quantities. In terms of the angle coordinate system, the multiphase flow equations can be transformed into a new degenerate hyperbolic system with an explicit singularity-regularity structure. We verify the convergence of the iterative sequence generated by the new system and then return the solution to the original physical variables. As a by-product, we obtain the existence of sonic-supersonic solutions for the steady full Euler equations with non-polytropic gases. [ABSTRACT FROM AUTHOR]

In this paper, we study the steady flows to the compressible Euler-Poisson system for semiconductors with the nonzero angular velocity in a radially symmetric way in an annulus. The main purpose here is to elucidate the effect of the angular velocity in the structure of the steady flows. We show the well-posedness of all kinds of types of radially symmetric spiral flows including radial subsonic/supersonic/transonic flows, and further give a specific classification of the flow patterns under the assumption of various boundary conditions at the inner and the outer circle. Additionally, different from the purely radial case, the uniqueness of radial subsonic flow can not be obtained due to the nonlocal effect caused by the angular velocity, consequently we prove the uniqueness of the radial subsonic solution in the case without the semiconductor effect or with a small current assumption. Moreover, some new patterns of spiral flows with or without shock are observed, such as a smooth transonic flow and a supersonic-supersonic shock flow for a large relaxation time parameter. [ABSTRACT FROM AUTHOR]

The inviscid limit for the two-dimensional compressible viscoelastic equations in the half plane is considered under the no-slip boundary condition. When the initial deformation tensor is a perturbation of the identity matrix and the initial density is near a positive constant, we establish the uniform estimates of solutions to the compressible viscoelastic flows in the conormal Sobolev spaces. It is well-known that for the corresponding inviscid limit of the compressible Navier-Stokes equations with the no-slip boundary condition, one does not expect the uniform energy estimates of solutions due to the appearance of strong boundary layers. However, when the deformation tensor effect is taken into account, our results show that the deformation tensor plays an important role in the vanishing viscosity process and can surprisingly prevent the formation of strong boundary layers. As a result we are able to justify the inviscid limit of solutions for the compressible viscous flows under the no-slip boundary condition governed by the viscoelastic equations, based on the uniform conormal regularity estimates achieved in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we consider the one-dimensional isentropic compressible Euler equations with linear damping β (t , x) ρ u in a bounded domain, which can be used to describe the process of compressible flows through a porous medium. And the model is imposed a dissipative subsonic time-periodic boundary condition. Our main results reveal that the time-periodic boundary can trigger a unique subsonic time-periodic smooth solution which is stable under small perturbations on initial data. Moreover, the time-periodic solution possesses higher regularity and stability provided a higher regular boundary condition. [ABSTRACT FROM AUTHOR]

In this paper, we study the global existence and large time behavior of the classical solution for the non-isentropic compressible Navier-Stokes-Allen-Cahn (NSAC) system describing the mixture motion of a two-phase immiscible viscous compressible heat-conducting flow in R 3. We first establish the global well-posedness of the Cauchy problem to the non-isentropic compressible NSAC system when the initial data is a small regular perturbation near the constant state. Then by a delicate spectral analysis and the damping structure for the perturbation equation of the phase field, we prove that the phase field decays to the phase separation state at an exponential rate while the density, velocity and the temperature converge to the constant state at the optimal time rate (1 + t) − 3 / 4 in L 2 norm. Furthermore, for well-chosen initial data, we also establish the lower bounds of time decay rate. [ABSTRACT FROM AUTHOR]

We propose high-resolution numerical schemes for two-phase compressible flow simulations to reproduce dynamically created gas/vapor-liquid interfaces with phase change. In the Godunov-type finite volume framework, suppressing numerical dissipation errors in numerical schemes is crucial for capturing the discontinuous solutions. The MUSCL scheme has second-order accuracy for smooth solutions and non-oscillatory behavior near discontinuous solutions. However, the MUSCL scheme introduces excessive numerical dissipation and diffuses the discontinuities nonphysically, leading to difficulties in distinguishing gas/vapor and liquid phases, and thus causing a blur in the interfaces during the multi-phase flow simulations. The hybrid-type boundary variation diminishing (BVD) scheme in this paper combines the MUSCL scheme and the THINC scheme to reduce the numerical dissipation errors near the discontinuities. The MUSCL-THINC-BVD scheme applies the MUSCL scheme for smooth solutions and the THINC scheme for discontinuous solutions, resulting in the successful capture of the discontinuities including the dynamically created gas/vapor-liquid interfaces. The Adaptive THINC-BVD scheme, which switches two types of THINC schemes with different values of gradient parameter, also captures the discontinuities clearly. The numerical results of the benchmark tests show that the proposed BVD schemes can lucidly reproduce the vapor-liquid interfaces newly created during the dynamical process of phase change. • Hybrid BVD schemes are presented to compute gas/vapor interfaces in 6-equation model for multi-phase flows. • Interfaces newly generated between liquid and vapor during the evaporation process are well-reproduced for the first time. • Basic benchmark validations are conducted for two-phase flows associated with phase changes. • The numerical methods presented are well-suited for simulating multi-phase flows involving phase changes. [ABSTRACT FROM AUTHOR]

In this work, a bound- and positivity-preserving quasi-conservative discontinuous Galerkin (DG) method is proposed for the γ -based model of compressible two-medium flows. The contribution of this paper mainly includes three parts. On one hand, the DG method with the extended Harten-Lax-van Leer contact flux is proposed to solve the γ -based model, and satisfies the equilibrium-preserving property which preserves uniform velocity and pressure fields at an isolated material interface. On the other hand, an affine-invariant weighted essentially non-oscillatory (Ai-WENO) limiter is adopted to suppress oscillations near the discontinuities. The limiter with the Ai-WENO reconstruction method to the conservative variables not only is able to maintain the equilibrium property, but also generates sharper results around the locations of shock waves in contrast to that applying to the primitive variables. Last but not least, a flux-based bound- and positivity-preserving limiting strategy is introduced and analyzed, which preserves the physical bounds for auxiliary variables in the non-conservative governing equations, and the positivity for density and internal energy. Extensive numerical experiments in both one and two space dimensions show that the proposed method performs well in simulating compressible two-medium flows with high-order accuracy, equilibrium-preserving and bound-preserving properties. [ABSTRACT FROM AUTHOR]

We are interested in the multi-dimensional compressible viscoelastic flows of Oldroyd type, which is one of non-Newtonian fluids exhibiting the elastic behavior. In order to capture the damping effect of the additional deformation tensor, to the best of our knowledge, the "div-curl" structural condition plays a key role in previous efforts. Our aim of this paper is to remove the structural condition and prove a global existence of strong solutions to compressible viscoelastic flows in critical spaces. In absence of compatible conditions, the new effective flux is introduced, which enables us to capture the dissipation arising from combination of density and deformation tensor. The partial dissipation in non-Newtonian compressible fluids, is weaker than that of classical Navier-Stokes equations. [ABSTRACT FROM AUTHOR]

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. • Seventh-order accuracy using first two moments of Legendre basis. • Genuinely high-order accuracy using discontinuity detector where smooth solution regions apply flux reconstruction. • Efficient reconstruction as the gradients are shared between viscous and inviscid fluxes. [ABSTRACT FROM AUTHOR]

In the previous studies, the high-order gas-kinetic schemes (HGKS) have achieved successes for unsteady flows on three-dimensional unstructured meshes. In this paper, to accelerate the rate of convergence for steady flows, the implicit non-compact and compact HGKSs are developed. For non-compact scheme, the simple weighted essentially non-oscillatory (WENO) reconstruction is used to achieve the spatial accuracy, where the stencils for reconstruction contain two levels of neighboring cells. Incorporate with the nonlinear generalized minimal residual (GMRES) method, the implicit non-compact HGKS is developed. In order to improve the resolution and parallelism of non-compact HGKS, the implicit compact HGKS is developed with Hermite WENO (HWENO) reconstruction, in which the reconstruction stencils only contain one level of neighboring cells. The cell averaged conservative variable is also updated with GMRES method. Simultaneously, a simple strategy is used to update the cell averaged gradient by the time evolution of spatial-temporal coupled gas distribution function. To accelerate the computation, the implicit non-compact and compact HGKSs are implemented with the graphics processing unit (GPU) using compute unified device architecture (CUDA). A variety of numerical examples, from the subsonic to supersonic flows, are presented to validate the accuracy, robustness and efficiency of both inviscid and viscous flows. • To accelerate the convergence for steady flows, the implicit non-compact and compact HGKSs are developed for compressible flows. • For non-compact HGKS, the third-order WENO method is adopted, and the GMRES method with Jacobian matrix is used for temporal evolution. • For the compact HGKS, the third-order HWENO method is used, a simple strategy is used to update the cell averaged gradient with GMRES method. • To accelerate the computation, the current schemes are implemented with GPU. [ABSTRACT FROM AUTHOR]

An artificial viscosity-based shock capturing scheme is extended to the context of high-order triangular Spectral Difference method for solving gas dynamics problems featuring discontinuities and shock waves. The equations are regularised thanks to an artificial diffusivity method combined with a shock sensor based on dilatation and vorticity fields valid for any polynomial order of approximation. The methodological novelty of this paper is an L 2 projection based iterative Restriction Prolongation filtering operation introduced for high-order triangular elements. This filtering operation applied to the artificial bulk viscosity and thermal conductivity stabilises the simulations and improves the quality of the shock capturing approach. The methodology is validated on various canonical relevant compressible test cases. Numerical results demonstrate the performance and the flexibility of the triangular high-order Spectral Difference method for solving simultaneously compressible flows and shock waves. [ABSTRACT FROM AUTHOR]

This paper presents a model for simulating a two-phase flow involving cavitation and condensation, focusing on its application to turbopump flows. The model is derived by extending Kapila's approach, which incorporates thermal and free energy relaxation terms, to compressible multiphase flow equations in a rotating reference frame. The numerical method employed is a finite volume approach based on the Godunov scheme, ensuring pressure equilibrium and total energy conservation through a relaxation step. The model utilizes separate Equations of State (EOS) for the liquid and vapor phases, with the experimental saturation curve, expressed as p sat (T) , being the only parameter needed for cavitation modeling. To broaden its applicability beyond axial pumps, a specific Riemann solver is developed to couple rotating and non-rotating regions within the flow domain. The thermodynamic phase change process is validated using simple cases, and the model is further utilized to simulate water flows in a turbopump inducer. The characteristic curve of the pump is obtained in the non-cavitating regime for several mass flow rates conditions, and the model is successfully assessed in severe cavitation conditions, demonstrating the performance breakdown caused by vapor pockets. The modeling also provides detailed flow descriptions, including analysis of vapor pockets and their impact on blade pressure loading. • Modeling multiphase flow in rotating reference frame. • A new Riemann problem to connect fixed and rotating flow regions. • Phase change modeling by a relaxation method from pure phase to mixture. • Cavitating flow in turbopumps without parameter for the growth of the cavitation pockets. [ABSTRACT FROM AUTHOR]

The paper deals with stochastic homogenization of a system modeling immiscible compressible two-phase, such as water and gas, flow in random porous media. The problem is written in terms of the phase formulation, i.e. the saturation of one phase and the pressure of the second phase are primary unknowns. This formulation leads to a coupled system consisting of a nonlinear degenerate parabolic equation for the gas pressure and a nonlinear degenerate parabolic diffusion-convection equation for the liquid saturation, subject to appropriate boundary and initial conditions. We consider the behavior of compressible two-phase flow in heterogeneous reservoirs with permeability and porosity being realizations of given statistically homogeneous random fields. We derive the effective (macroscopic) problem and prove the convergence of solutions. Our approach relies on stochastic two-scale convergence techniques, the realization-wise notion of stochastic two-scale convergence being used. Also, we exploit various a priori estimates as well as monotonicity and compactness arguments. To our best knowledge, this is the first stochastic homogenization result in the case of compressible two-phase flow in random porous media. [ABSTRACT FROM AUTHOR]

This paper presents under-resolved simulations of compressible flows in complex structures like buildings. The mean pressure field is the primary focus as it is of high importance in many cases, such as the overpressure generated by the detonation of an explosive charge. In such high-risk situations, the mean pressure in the different building rooms has to be determined as quickly as possible, in order to help first responders evaluating the risk level of entering the building right after an explosion for instance. Conventional computational fluid dynamics methods are obviously able to deliver those requested pressure fields. However, they also require a long and tedious meshing process due to the presence of small openings in the computational domain (doors, windows or stairwells). In turn, these small openings need spatial and temporal resolution having significant consequences on computation time. But when dealing with pressing situations, a long pre-processing is not acceptable. The present paper proposes a two-step solution method. First, a very rough mesh is constructed which does not consider small openings. The mesh is constructed in an easy and particular way such that simplifications can be made when computing the fluid flux through these openings. Second, a specific Riemann solver dealing with geometric discontinuities is developed to estimate the fluid flow. The Riemann solver takes advantage of the simplifications provided by the meshing method, and accounts for both unchoked and choked flows through openings. The overall method then consists of both a specific meshing process that requires very little pre-processing, which considerably reduces the time needed to produce a complete simulation, and a Riemann solver. The proposed method is named MUZO in reference to its "MUlti-ZOne" flow solver. The MUZO solver is validated against resolved computations and provides accurate simulated mean pressure fields within a realistic building in a few seconds. • 3D simulations of compressible flow discharge in buildings are addressed through a very fast numerical framework. • Only 2D footprints without geometric details (doors, windows, etc.) are required to generate 3D meshes. • The previously omitted geometric restrictions are considered through a specific Riemann solver. [ABSTRACT FROM AUTHOR]

In this paper, we propose a modification of an acoustic-transport operator splitting Lagrange-projection method for simulating compressible flows with gravity. The original method involves two steps that respectively account for acoustic and transport effects. Our work proposes a simple modification of the transport step, and the resulting modified scheme turns out to be a flux-splitting method. This new numerical method is less computationally expensive in the low-Mach regime, more memory efficient, and easier to implement than the original one. We prove stability properties for this new scheme by showing that under classical CFL conditions, the method is positivity preserving for mass, energy and entropy satisfying. The flexible flux-splitting structure of the method enables straightforward extensions of the method to multi-dimensional problems (with respect to space) and high-order discretizations that are presented in this work. We also propose an interpretation of the flux-splitting solver as a relaxation approximation. Both the stability and the accuracy of the new method are tested against one-dimensional and two-dimensional numerical experiments that involve highly compressible flows and low-Mach regimes. • The all-regime acoustic/transport splitting solver proposed in [16] can be modified into a flux-splitting, one-step version. • The resulting method is less computationally expensive, more memory efficient, and easier to implement than the original one. • Stability properties are derived by interpreting the scheme as a convex combination of the advection/pressure steps. • This recasting allows an easy high-order extension as the new method is a standard flux-based solver. • Following the lines of [32] , we equip our scheme with the well-balanced property for gravity source terms. [ABSTRACT FROM AUTHOR]

In this paper, we construct a high order weighted essentially non-oscillatory (WENO) finite difference discretization for compressible Navier-Stokes (NS) equations, which is rendered positivity-preserving of density and internal energy by a positivity-preserving flux splitting and a scaling positivity-preserving limiter. The novelty of this paper is WENO reconstruction performed on variables from a positivity-preserving convection diffusion flux splitting, which is different from conventional WENO schemes solving compressible NS equations. The core advantages of our proposed method are robustness and efficiency, which especially are suitable for solving tough demanding problems of both compressible Euler and NS equation including low density and low pressure flow regime. Moreover, in terms of computational cost, it is more efficient and easier to implement and extend to multi-dimensional problems than the positivity-preserving high order discontinuous Galerkin schemes and finite volume WENO scheme for solving compressible NS equations on rectangle domain. Benchmark tests demonstrate that the proposed positivity-preserving WENO schemes are high order accuracy, efficient and robust without excessive artificial viscosity for demanding problems involving with low density, low pressure, and fine structure. • A positivity-preserving high order accurate finite difference WENO scheme for compressible N-S equations. • The method is the high order accuracy and positivity-preserving. • WENO reconstruction is performed on variables from a positivity-preserving convection diffusion flux splitting. • The method is robust and suitable for solving the compressible flow with low density and low pressure. [ABSTRACT FROM AUTHOR]

This is the first of two companion papers that introduce a new class of positivity-preserving, entropy stable schemes for the 1-D compressible Navier-Stokes equations. In this paper, we develop and analyze a novel first–order finite volume scheme that is discretized on Legendre-Gauss-Lobatto grids used for high-order spectral collocation methods. The scheme is constructed by regularizing the Navier-Stokes equations by adding artificial dissipation in the form of the Brenner-Navier-Stokes diffusion operator. The key distinctive property of the proposed scheme is that it is proven to satisfy the discrete entropy inequality and to guarantee the pointwise positivity of density, temperature, pressure, and internal energy for 1–D compressible viscous flows, while using high–order spectral collocation summation-by-parts operators for discretizing the Navier–Stokes viscous terms. To eliminate time step stiffness caused by the high-order approximation of the viscous terms, we develop two new limiters that bound the magnitude of velocity and temperature gradients and preserve the entropy stability and positivity properties of the baseline scheme. Numerical results demonstrating positivity-preserving and discontinuity-capturing capabilities of the new scheme are presented for viscous and inviscid flows with nearly vacuum regions and very strong shocks and contact discontinuities. • Novel first-order artificial dissipation based on the Brenner diffusion operator. • New 1st-order positivity-preserving L 2 stable scheme for 1D Navier-Stokes equations. • New positivity-preserving entropy stable velocity and temperature limiters. [ABSTRACT FROM AUTHOR]

This is the second of two companion papers, in which we develop and analyze a new class of positivity-preserving, entropy stable spectral collocation schemes of arbitrary order of accuracy for the 1-D compressible Navier-Stokes equations. The key distinctive property of the proposed methodology is that it is proven to guarantee the pointwise positivity of thermodynamic variables for compressible viscous flows. The new schemes are constructed by combining a positivity-violating entropy stable method of arbitrary order of accuracy and a novel first-order positivity-preserving entropy stable method developed in the companion paper [1]. This first-order scheme is discretized on the same Legendre-Gauss-Lobatto points used for the high-order counterpart. As a result, no interpolation or restriction is required between high- and low-order elements which are coupled by using the simultaneous approximation term (SAT) penalty method. To provide positivity preservation and excellent discontinuity-capturing properties, the Navier-Stokes equations are regularized by adding artificial dissipation in the form of the Brenner-Navier-Stokes diffusion operator. The high- and low-order schemes are combined by using a limiting procedure, so that the resultant scheme provides conservation, guarantees pointwise positivity of thermodynamic variables, preserves the design–order of accuracy of the corresponding high-order baseline spectral collocation method for smooth solutions, and satisfies the discrete entropy inequality, thus facilitating a rigorous L 2 -stability proof for the symmetric form of the discretized Navier-Stokes equations. The proposed methodology is very general and can be extended to multiple dimensions and other summation-by-parts SAT-type schemes. • A new physics-based artificial viscosity built as a function of entropy residual. • A novel high-order entropy stable pointwise positivity-preserving flux limiter. • New high-order positivity-preserving L 2 stable schemes for Navier-Stokes equations. [ABSTRACT FROM AUTHOR]

In this paper, a fully conservative sharp-interface method is developed for compressible multiphase flows with phase change. To enforce conservation, the coupling between two different phases is achieved via the interfacial fluxes, which are determined by solving a general Riemann problem with phase change. In addition, thermodynamical consistency is guaranteed by taking account of latent heat in the phase-change Riemann problem. This is efficiently and robustly solved by a novel approximate Riemann solver, where the complex eight-dimensional root-finding procedure required in the exact Riemann solver is simplified to solely iterating the mass flux. Compared to the existing methods, the present approximate Riemann solution consists of four waves, at which the jump conditions are imposed strictly to enforce the conservation of mass, momentum, and energy. Furthermore, for the evaluation of the mass flux, through numerical simulations and theoretical analysis, we have demonstrated that the adjacent states of a phase interface should be used to ensure numerical consistency, which, to our knowledge, has not been reported in the literature. Various numerical examples, including the phase-change Riemann problem, the oscillating droplet problem, and the shock-droplet interaction problem, are simulated to validate the present method. The numerical results are in good agreement with the exact solutions and the previous simulations. It is shown that, with the present method, the phase change effects in compressible multiphase flows are well resolved with conservation strictly preserved. • A fully conservative sharp-interface method is developed for compressible multiphase flows with phase change. • A novel approximate Riemann solver is developed to solve the two-phase phase-change Riemann problem. • The present Riemann solver employs a 4-wave model with all the jump conditions imposed strictly, which is more physical. • We ensure numerical consistency by using the adjacent states of the phase interface to compute the mass flux. [ABSTRACT FROM AUTHOR]

When solving compressible multi-material flow problems, an unresolved challenge is the computation of advective fluxes across material interfaces that separate drastically different thermodynamic states and relations. A popular idea in this regard is to locally construct bimaterial Riemann problems, and to apply their exact solutions in flux computation. For general equations of state, however, finding the exact solution of a Riemann problem is expensive as it requires nested loops. Multiplied by the large number of Riemann problems constructed during a simulation, the computational cost often becomes prohibitive. The work presented in this paper aims to accelerate the solution of bimaterial Riemann problems without introducing approximations or offline precomputation tasks. The basic idea is to exploit some special properties of the Riemann problem equations, and to recycle previous solutions as much as possible. Following this idea, four acceleration methods are developed, including (1) a change of integration variable through rarefaction fans, (2) storing and reusing integration trajectory data, (3) step size adaptation, and (4) constructing an R-tree on the fly to generate initial guesses. The performance of these acceleration methods is assessed using four example problems in underwater explosion, laser-induced cavitation, and hypervelocity impact. These problems exhibit strong shock waves, large interface deformation, contact of multiple (>2) interfaces, and interaction between gases and condensed matters. For all the problems, the acceleration methods are able to significantly reduce the computational cost without affecting solver robustness or solution accuracy. In different cases, the solution of bimaterial Riemann problems is accelerated by 37 to 87 times. As a result, the total cost of advective flux computation, which includes the exact Riemann problem solution at material interfaces and the numerical flux calculation over the entire computational domain, is accelerated by 18 to 81 times. • Bimaterial Riemann problems can be constructed to compute interfacial fluxes. • Computational cost is often prohibitive, if complex equations of state are involved. • An efficient exact Riemann solver is proposed, with four algorithmic improvements. • Significant cost reduction (37-87X) is achieved for four challenging examples. • Does not introduce approximations or require offline precomputation. [ABSTRACT FROM AUTHOR]

In this paper, we present a detailed numerical scheme for a single-phase compressible flow without diffusion of a multi-component mixture in porous media with the higher-order approximation in both space and time. The mathematical model consists of Darcy velocity, transport equations for each component of a mixture, pressure equation and associated relations for physical quantities such as viscosity or equation of state. The discrete problem is obtained using a combination of the discontinuous Galerkin method for the transport equations and the mixed-hybrid finite element method for the Darcy velocity and the pressure equation. In both methods the higher-order approximation is used. The resulting nonlinear problem for concentrations is solved with the fully mass-conservative iterative IMPEC method. Experimental order of convergence analysis (EOC) and some numerical experiments of a 2D flow are carried out. • A new numerical scheme is presented for simulation of single-phase flow in a porous medium. • The second-order approximation is attained in both space and time. • The second-order approximation in both time and space is verified using the experimental order of convergence. • Several computational examples illustrate advantageous features of our approach. [ABSTRACT FROM AUTHOR]

This paper proposes a hierarchy of numerical fluxes for the compressible flow equations which are kinetic-energy and pressure equilibrium preserving and asymptotically entropy conservative, i.e., they are able to arbitrarily reduce the numerical error on entropy production due to the spatial discretization. The fluxes are based on the use of the harmonic mean for internal energy and only use algebraic operations, making them less computationally expensive than the entropy-conserving fluxes based on the logarithmic mean. The use of the geometric mean is also explored and identified to be well-suited to reduce errors on entropy evolution. Results of numerical tests confirmed the theoretical predictions and the entropy-conserving capabilities of a selection of schemes have been compared. [ABSTRACT FROM AUTHOR]

The discrete-equations method (DEM) [1] provides a universal approach to solve multi-phase-flow equations as it combines the solutions of pairwise Riemann problems. Although very robust, the original DEM with piecewise-constant volume fractions suffers from strong diffusion preventing accurate interface capturing. High-order interface reconstruction, however, introduces a restrictive time-step limit. This paper presents RDEMIC, a robust extension of DEM for accurate interface capturing on Cartesian meshes. By a modified partitioning of the Riemann solutions and a specific combination of fluxes and non-conservative terms, the time-step restriction is effectively prevented, which is critical for making the method practically applicable. Moreover, the accuracy of interface and shock-wave propagation is maintained. RDEMIC is not limited to two-phase flow but defined for an arbitrary number of phases. The method is combined with a THINC scheme [2] to reconstruct volume fractions. The reconstruction is enhanced by a positivity-preserving averaging procedure, which is consistent with the underlying multi-stage Runge–Kutta scheme of the flow solver. The resulting scheme consisting of RDEMIC and the positivity-preserving THINC reconstruction is very robust and captures the interface with high accuracy. We demonstrate its performance for various cases of shock-interface interactions, which show very good agreement with reference results from literature. • Robust high-resolution numerical model and algorithm for compressible multi-phase flow with arbitrary number of phases. • No further time-step restrictions implied by high-order interface reconstruction. • Accurate capturing and advection of multi-phase interfaces with positivity preservation. • No numerical artifacts for wave propagation across interfaces. [ABSTRACT FROM AUTHOR]

Aircraft aerodynamic design often involves evaluating flow conditions that span low subsonic to transonic or even supersonic Mach numbers. Compressible flow solvers are a natural choice for such design problems, but these solvers suffer from reduced accuracy and efficiency at low Mach numbers. In addition, simulations with supersonic conditions can be challenging to converge because of large gradients in the flow field. This paper presents three contributions to address these issues in the context of an approximate Newton–Krylov solver for the Reynolds-averaged Navier–Stokes equations. First, we propose a method for scaling the artificial dissipation terms in the Jameson–Schmidt–Turkel scheme to improve its accuracy at low Mach numbers while retaining the simplicity of the original scalar dissipation formulation. Second, we show that characteristic time-stepping combined with an approximate Newton method can accelerate convergence for low Mach number flows by reducing the stiffness in the linear system for each Newton iteration. Third, we introduce a dissipation-based continuation method for flows with shocks that improves robustness and accelerates convergence without sacrificing accuracy. These methods can make compressible flow solvers more accurate and efficient across low and high Mach number regimes. • A simple change to a scalar dissipation scheme improves accuracy at low Mach numbers. • Characteristic time-stepping reduces linear system stiffness for Newton-type solvers. • Scaling artificial dissipation is needed to achieve speedups at low Mach numbers. • Dissipation-based continuation improves robustness for flows with shocks. [ABSTRACT FROM AUTHOR]

In this paper, a new type of hybrid weighted essentially non-oscillatory (WENO) schemes with third-order and fourth-order accuracy is developed to simulate compressible flow problems on unstructured meshes. First, an improved discontinuity sensor is designed on unstructured meshes, which can automatically and accurately identify the troubled cells without manually adjusting the parameters related to the problem. Second, a hybrid unequal-sized WENO (US-WENO) scheme with different accuracy is developed based on the new hybridization strategy and the existing US-WENO scheme to reduce the computational overhead caused by the computation of nonlinear weights and local characteristic decompositions in WENO procedures. Finally, some benchmark examples are employed to verify the performance of the proposed method in terms of numerical dissipation, shock capture ability, and computational efficiency. Numerical experiments show that the proposed hybrid US-WENO schemes can inherit the excellent characteristics of the original US-WENO schemes while improving computational efficiency. • An improved, simple and artificial parameter free discontinuity sensor is presented on triangular meshes. • An efficient hybrid unequal-sized WENO scheme is designed to solve compressible problems on triangular meshes. • These new hybrid WENO schemes are more efficient than the existing unequal-sized WENO scheme. [ABSTRACT FROM AUTHOR]

• A positivity-preserving high order accurate finite volume hybrid HWENO scheme for compressible N-S equations. • The method is the high order accuracy and positivity-preserving. • The method is more efficient and more robust than the conventional HWENO methods. • The method is especially applicable for solving gas dynamics equations in low density and low pressure regime. In this paper, we construct a positivity-preserving high order accurate finite volume hybrid Hermite Weighted Essentially Non-oscillatory (HWENO) scheme for compressible Navier-Stokes equations, by incorporating a nonlinear flux and a positivity-preserving limiter. HWENO schemes have more compact stencils than WENO schemes but with higher computational cost due to the auxiliary variables. The hybrid HWENO schemes use linear reconstructions in smooth region thus are more efficient than conventional HWENO schemes. However, the hybrid HWENO is not robust for many demanding problems. The positivity-preserving hybrid HWENO scheme in this paper is not only more efficient but also much more robust than the conventional HWENO method for both compressible Euler and compressible Navier-Stokes equations, especially for solving gas dynamics equations in low density and low pressure regime. Numerical tests on low density and low pressure problems are performed to demonstrate the robustness and the efficiency of the positivity-preserving hybrid HWENO scheme. [ABSTRACT FROM AUTHOR]

This paper considers a coupling of slightly compressible Darcy-Brinkman-transport problem in fractured media with higher Reynolds numbers, involving Brinkman flow in the fractures with advection-diffusion transport in the whole considered media. A new two-layer reduced coupled model is introduced by treating the fracture as hyperplane. The finite difference method with second-order backward differentiation formula and modified upwind scheme by using the fewest nodal points is constructed to solve the new reduced coupled model on staggered nonuniform grids. Based on error derivation of the coupling term, the uniqueness and existence of solutions and second-order convergence rate of numerical method are derived in the mixed forms under the mass conservative transmission and Beavers-Joseph-Saffman conditions. Some experiments are provided to testify the accuracy and efficiency of the numerical method. The numerical analysis of new two-layer reduced model is illustrated to show the behavior of fluid flow and solute transport in the media with intersected, embedded and L-shaped fractures. [ABSTRACT FROM AUTHOR]

In this paper, we propose an adaptive mesh refinement method for 2D multi-material compressible non-viscous flows in semi-Lagrangian coordinates. The mesh adaptation procedure is local and relies on a discrete metric field evaluation. The remapping method is second-order accurate and we prove its stability. We propose a multi-material treatment using two ingredients: the local remeshing is performed in a way that reduces as much as possible the creation of mixed cells and an interface reconstruction method that can be used to avoid the diffusion of the material interfaces. The obtained method is almost Lagrangian and can be implemented in a parallel framework. We provide some numerical tests which attest the validity of the method and its robustness. • We propose a triangular mesh adaptation procedure adapted to lagrangian flows. • An interface reconstruction method is used to reduce numerical mixing propagation. • A second-order remapping procedure is described. Its stability is proven. • The efficiency of the method is illustrated through a set of numerical test cases. [ABSTRACT FROM AUTHOR]

This paper exploits the potential of a high-order implicit radial basis function-based differential quadrature-finite volume (IRBFDQ-FV) method for effective simulation of inviscid and viscous compressible flows using unstructured grids. The framework of IRBFDQ-FV method is based on finite volume discretization which can guarantee conservation of mass, momentum and energy. The flow field variables within each control cell are represented by a fourth-order approximation constructed by a Taylor series expansion to the cell center with spatial derivatives as the undetermined coefficients. All spatial derivatives are computed by the meshless and highly converged RBF-based differential quadrature (RBFDQ) technique. Regarding flux evaluation, the discrete gas-kinetic flux solver is applied to evaluate inviscid and viscous fluxes simultaneously for compressible viscous flows. Resultantly, the special mathematical treatment for the viscous flux, which is usually adopted in other high-order methods, can be avoided. A point extrema-based extended bounds (PEEB) shock-capturing limiter is introduced to eliminate spurious numerical oscillations near discontinuities. Due to the implicit nature of the proposed method, implicit time-stepping schemes compatible with the spatial accuracy are devised to efficiently solve the resulting discretized equations. Representative compressible flow problems are simulated to verify the high-order accuracy, excellent efficiency and robustness of the present method. Comparison with two other high-order finite volume methods further shows competitiveness of the present method in solving compressible flow problems on unstructured grids. • Compressible inviscid and viscous flow problems are solved by the present high-order method on unstructured grids. • RBFDQ meshless method is locally employed for derivative approximation. • Convective and viscous fluxes at the cell interface are simultaneously evaluated with high-order accuracy. • Implicit time integration methods are applied to obtain the steady-state and transient solutions with high-order accuracy. [ABSTRACT FROM AUTHOR]

The method of Objective Molecular Dynamics (OMD) provides exact solutions of the equations of molecular dynamics for atoms filling all of space in non equilibrium situations. The method can be used to simulate families of incompressible, compressible and unsteady flows, also with time-dependent vorticity. In this paper we develop OMD as an efficient computational tool and introduce some of its applications. We apply the method to the evolution of compressible heat-conducting monoatomic gas under general incompressible flow, and to flow in a nanochannel having realistic atomistic solid boundaries. The macroscopic flows of OMD are exact solutions of the Navier-Stokes equation with Newtonian and Fourier models, and we make comparisons with these solutions. We also report the simulation of homoenergetic dilatational flow exhibiting condensation using OMD techniques. • OMD for the time-dependent translation group is developed as a computational tool. • OMD provides a rigorous framework to perform non-equilibrium molecular dynamics. • The widely used velocity-Verlet algorithm is consistent with OMD. • In the regime of higher rates the Navier-Stokes-Fourier theory is no longer accurate. • OMD can also deal with spontaneous phase transition and boundary driven flow. [ABSTRACT FROM AUTHOR]

This paper presents a new framework for Aerodynamic Shape Optimization (ASO) implemented inside the open-source software SU2. A parametrized surface is iteratively morphed to improve a desired aerodynamic coefficient. At each loop, the computational grid must be updated. The radial basis function (RBF) mesh deformation technique is introduced to extend the capability to explore the design space. RBF is implemented with some state-of-the-art data reduction systems to lower the computational cost. The Discrete Adjoint is adopted to compute the sensitivity in combination with Automatic Differentiation to calculate the required Jacobians. RBF is differentiated as well, resulting in a method-dependent surface sensitivity. The gradient-based algorithm "Sequential Least Squares Programming" drives the research of the minimum. The study demonstrates that the presented combination is more robust than an ASO, including linear elasticity analogy (ELA) as mesh deformation method. It can handle complex geometries and apply larger displacements, making possible the optimization of a wing-winglet configuration and a rotating wind turbine. Results are presented in two and three-dimensions for compressible and incompressible flows, showing a stronger reduction of the drag without affecting the lift. • The optimization chain effectively reduces the drag in different velocity regimes. • RBF well preserves mesh quality, and it is computational cheap. • Different mesh deformation techniques bring to different sensitivity so a different final shape. • Complex and rotating geometries can be handled. • The optimization is cad-free and do not modify the grid connectivity. [ABSTRACT FROM AUTHOR]

Computational fluid dynamics is a direct modeling of physical laws in a discretized space. The basic physical laws include the mass, momentum and energy conservations, numerical scale-consistent transport process, and similar domain of dependence and influence in the physical and the numerical wave propagations. Therefore, a physically soundable numerical scheme should be a compact one which involves the closest neighboring cells within the domain of dependence for the solution update under a CFL number (∼1). In order to keep the compactness of the scheme and be high-order, subcell flow distributions or the equivalent degrees of freedom beyond the cell averaged flow variables are needed, such as the gradients of the flow variables inside each control volume. Under the above consideration, the direct use of Riemann solver as the evolution model doesn't provide such a mechanism. High-order dynamic evolution process should be used to update the time accurate cell-averaged flow variables and their gradients. The direct modeling of flow evolution under generalized initial condition will be developed. In order to provide reliable cell averaged flow variables and their gradients for the compact data reconstruction, the evolution process is constructed to provide possible discontinuous time-dependent flow variables across a cell interface. At the same time, the time accurate flux function at a cell interface can become a discontinuous function of time in its evolution process. Same as the spatial limiter in the conventional computational fluid dynamics methods, such as the total variation diminishing and weighted essentially non-oscillatory methods, the temporal limiter for the time accurate flux function is designed as well. The direct modeling method in this paper will update flow variables differently on both sides of a cell interface and nonlinearly limit the high-order time derivatives of the flux function in case of shock wave passing through a cell interface within a time step. The direct modeling method unifies the concepts of nonlinear limiters in spatial data reconstruction and temporal flux transport. The equivalent treatment of space and time makes the compact gas-kinetic scheme be super robust and accurate for the compressible flow simulation. At the same time, the 4th to 8th-order compact schemes can use a large CFL number (∼ 0.8) in flow computation, even for the hypersonic flow simulation. • A general principle for the development of high-order compact scheme is presented. • Same as the nonlinear limiter in space, the nonlinear limiter in time for the flux function is proposed. • According to the principle the high-order compact gas-kinetic scheme is developed. • Accurate solutions of compressible Euler and Navier-Stokes equations are obtained. [ABSTRACT FROM AUTHOR]

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. • We proposed a new fifth-order multi-resolution WCNS scheme. • The design process of this presented WCNS scheme is simpler than the original one. • A hybridization strategy is proposed to improve the computational efficiency of the presented scheme. • The linear weights of this new WCNS scheme can be taken arbitrarily. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel approximation strategy for time-dependent hyperbolic systems of conservation laws for the Euler system of gas dynamics that aims to represent the dynamics of strong interacting discontinuities. The goal of our method is to allow an approximation with a high-order of accuracy in smooth regions of the flow, while ensuring robustness and a non-oscillatory behaviour in the regions of steep gradients, in particular across shocks. Following the Multidimensional Optimal Order Detection (MOOD) ([15,17]) approach, a candidate solution is computed at a next time level via a high-order accurate explicit scheme ([3,5]). A so-called detector determines if the candidate solution reveals any spurious oscillations or numerical issue and, if so, only the troubled cells are locally recomputed via a more dissipative scheme. This allows to design a family of "a posteriori" limited, robust and positivity preserving, as well as high accurate, non-oscillatory and effective scheme. Among the detecting criteria of the novel MOOD strategy, two different approaches from literature, based on the work of [15,17] and on [35] , are investigated. Numerical examples in 1D and 2D, on structured and unstructured meshes, are proposed to assess the effective order of accuracy for smooth flows, the non-oscillatory behaviour on shocked flows, the robustness and positivity preservation on more extreme flows. • Presentation of a novel MOOD strategy for multidimensions. • MOOD on high order Residual Distribution schemes for compressible flows. • Application of the strategy to triangle and quad meshes. [ABSTRACT FROM AUTHOR]

Three asymptotic limits exist for the Euler equations at low Mach number - purely convective, purely acoustic, and mixed convective-acoustic. Standard collocated density-based numerical schemes for compressible flow are known to fail at low Mach number due to the incorrect asymptotic scaling of the artificial diffusion. Previous studies of this class of schemes have shown a variety of behaviours across the different limits and proposed guidelines for the design of low-Mach schemes. However, these studies have primarily focused on specific discretisations and/or only the convective limit. In this paper, we review the low-Mach behaviour using the modified equations - the continuous Euler equations augmented with artificial diffusion terms - which are representative of a wide range of schemes in this class. By considering both convective and acoustic effects, we show that three diffusion scalings naturally arise. Single- and multiple-scale asymptotic analysis of these scalings shows that many of the important low-Mach features of this class of schemes can be reproduced in a straightforward manner in the continuous setting. As an example, we show that many existing low-Mach Roe-type finite-volume schemes match one of these three scalings. Our analysis corroborates previous analysis of these schemes, and we are able to refine previous guidelines on the design of low-Mach schemes by including both convective and acoustic effects. Discrete analysis and numerical examples demonstrate the behaviour of minimal Roe-type schemes with each of the three scalings for convective, acoustic, and mixed flows. • Asymptotic analysis of numerical schemes at low Mach number. • Consider the convective, acoustic, and mixed convective-acoustic low Mach limits. • Derivation of required asymptotic scaling of artificial diffusion at each limit. • Use of modified equations applies to finite-volume or finite-difference schemes. • Application to Roe schemes shows excellent agreement with previous literature. [ABSTRACT FROM AUTHOR]

In this paper we develop a family of arbitrarily high-order non-oscillatory hybrid Discontinuous Galerkin(DG)-Finite Volume(FV) schemes for mixed-element unstructured meshes. Their key ingredient is a switch between a DG method and a FV method based on the CWENOZ scheme when invalid solutions are detected by a troubled cell indicator checking the unlimited DG solution. Therefore, the high order of accuracy offered by DG is preserved in smooth regions of the computational domain, while the robustness of FV is utilized in regions with strong gradients. The high-order CWENOZ variant used has the same spatial order of accuracy as the DG variant, while representing one of the most compact applications on unstructured meshes, therefore simplifying the implementation, reducing the computational overhead associated with large stencils of the original WENO reconstruction without sacrificing the desirable non-oscillatory properties of the schemes. We carefully investigate several parameters associated with the switching between DG and FV methods including the troubled cell indicators in a priori fashion. For the first time in the literature, we investigate the definition of the bounds for an admissible solution, the frequency by which we use the troubled cell indicators, and the evolution of the percentage of troubled cells for unsteady test problems. The 2D and 3D Euler equations are solved for well established test problems and compared with computational or experimental reference solutions. All the methods have been implemented and deployed within the UCNS3D open-source high-order unstructured Computational Fluid Dynamics (CFD) solver. The present coupling has the potential to improve the shortcomings of both FV-DG in a computational efficient manner. The improved accuracy and robustness provided is a characteristic of paramount importance for industrial-scale CFD applications, and favours the extension to other systems of governing equations. [ABSTRACT FROM AUTHOR]

A core aspect in the simulation of compressible fluid flow is the solution of Riemann problems. The solution is well understood for single-phase flows and is used to define numerical fluxes for finite volume and discontinuous Galerkin schemes. However, the solution of two-phase Riemann problems with phase transitions is still an active field of research. This is due to the strong coupling of complex thermodynamics and flow dynamics that leads to severe challenges in finding solutions. In this paper, we consider the Riemann problem based on the Godunov-Peshkov-Romenski formulation of the continuum flow equations and propose a novel strategy to model two-phase flows with phase transitions. We take into account heat transfer, which is modeled in this system of equations by hyperbolic terms. Our solver is applied to two-phase shock tubes with an evaporating fluid. It is validated against molecular dynamics simulations for the Lennard-Jones fluid with truncated and shifted potential. • Sharp-interface method for evaporating liquids. • Approximate solution of two-phase Riemann problem with evaporation based on the Godunov-Peshkov-Romenski equations. • Novel parameter free microscopic model (kinetic relation) of the phase boundary. • Validation against molecular dynamics data for evaporating LJTS fluid. • Better match to molecular dynamics compared with literature results. [ABSTRACT FROM AUTHOR]

The Rayleigh-Taylor instability is a classical hydrodynamic instability of great interest in various disciplines of science and engineering, including astrophysics, atmospheric sciences and climate, geophysics, and fusion energy. Analytical methods cannot be applied to explain the long-time behavior of the Rayleigh-Taylor instability, and therefore, numerical simulation of the full problem is required. However, in order to capture the growth of amplitude of perturbations accurately, both the spatial and temporal discretizations need to be extremely fine for traditional numerical methods, and long-time simulation may become prohibitively expensive. In this paper, we propose efficient reduced order model techniques to accelerate the simulation of the Rayleigh-Taylor instability in compressible gas dynamics. We introduce a general framework for decomposing the solution manifold to construct the temporal domain partition and temporally-local reduced order model construction with varying Atwood number. We propose two practical approaches in this framework, namely decomposition by physical time and by penetration distance. Numerical results are presented to examine the performance of the proposed approaches. • We develop nonlinear reduced order models for Rayleigh-Taylor instability in compressible gas with varying Atwood number. • We introduce a general indicator-based framework for decomposing the solution manifold and localizing reduced order models. • We compare two practical approaches, namely using time and penetration distance as physics-based indicator in the framework. • We exploit the Lagrangian frame for instantaneous evaluation of penetration distance to achieve good speed-up. • We present outstanding numerical examples which are reproducible with open-source softwares libROM and Laghos. [ABSTRACT FROM AUTHOR]

We present an a posteriori shock-capturing finite volume method algorithm called GP-MOOD that solves a compressible hyperbolic conservative system at high-order solution accuracy (e.g., third-, fifth-, and seventh-order) in multiple spatial dimensions. The GP-MOOD method combines two methodologies, the polynomial-free spatial reconstruction methods of GP (Gaussian Process) and the a posteriori detection algorithms of MOOD (Multidimensional Optimal Order Detection). The spatial approximation of our GP-MOOD method uses GP's unlimited spatial reconstruction that builds upon our previous studies on GP reported in Reyes et al. (2018) [20] and Reyes et al. (2019) [21]. This paper focuses on extending GP's flexible variability of spatial accuracy to an a posteriori detection formalism based on the MOOD approach. We show that GP's polynomial-free reconstruction provides a seamless pathway to the MOOD's order cascading formalism by utilizing GP's novel property of variable (2 R + 1) th-order spatial accuracy on a multidimensional GP stencil defined by the GP radius R , whose size is smaller than that of the standard polynomial MOOD methods. The resulting GP-MOOD method is a positivity-preserving method. We examine the numerical stability and accuracy of GP-MOOD on smooth and discontinuous flows in multiple spatial dimensions without resorting to any conventional, computationally expensive a priori nonlinear limiting mechanism to maintain numerical stability. • GP-MOOD is a posteriori scheme delivering solutions up to 7th-order accuracy. • GP-MOOD is a positivity-preserving scheme for highly compressible flow problems. • GP-MOOD uses smaller local stencils than the existing polynomial MOOD methods. • GP-MOOD maximizes the use of unlimited linear high-order spatial solutions. [ABSTRACT FROM AUTHOR]

The paper addresses a two-temperature model for simulating compressible two-phase flow taking into account diffusion processes related to the heat conduction and viscosity of the phases. This model is reduced from the two-phase Baer-Nunziato model in the limit of complete velocity relaxation and consists of the phase mass and energy balance equations, the mixture momentum equation, and a transport equation for the volume fraction. Terms describing effects of mechanical relaxation, temperature relaxation, and thermal conduction on volume fraction evolution are derived and demonstrated to be significant for heat conduction problems. The thermal conduction leads to instantaneous thermal relaxation so that the temperature equilibrium is always maintained in the interface region with meeting the entropy relations. A numerical method is developed to solve the model governing equations that ensures the pressure-velocity-temperature (PVT) equilibrium condition in its high-order extension. We solve the hyperbolic part of the governing equations with the Godunov method with the HLLC approximate Riemann solver. The non-linear parabolic part is solved with an efficient Chebyshev explicit iterative method without dealing with large sparse matrices. To verify the model and numerical methods proposed, we demonstrate numerical results of several numerical tests such as the multiphase shock tube problem, the multiphase impact problem, and the planar ablative Rayleigh–Taylor instability problem. • Two-temperature model for compressible two-phase flows with diffusion. • Consistency with thermodynamic laws and interface equilibrium conditions. • The efficient explicit local iteration method for solving nonlinear diffusion PDEs. • Application in the inertial confinement fusion. [ABSTRACT FROM AUTHOR]

• A novel framework of self-adjusting steepness-based schemes is proposed. • We theoretically prove that ln3 is the exact value that makes the THINC scheme second order. • The harmonic limiter is extended to a steepness-adjustable harmonic limiter. Although conventional total-variation-diminishing schemes can capture discontinuities with essentially non-oscillatory properties and are nominal of second-order accuracy in smooth regions, they cannot generally maintain sharp resolution for various discontinuities (especially contact discontinuity for long-time simulations) owing to excessive numerical dissipation. Designing schemes to overcome this disadvantage is a difficult problem. In this paper, we propose to solve this problem by introducing a novel framework of self-adjusting steepness-based schemes, which are mainly constructed in four steps: (1) design a slope limiter containing a steepness parameter β that provides a mechanism to enable the scheme to accurately solve both smooth and discontinuous problems with proper values of β ; (2) determine the infimum of β such that the scheme is order-optimized; (3) determine the supremum of β such that the scheme has a nonlinear stable anti-diffusion/compression effect; (4) calculate the steepness parameter β using an adaptive algorithm in terms of the infimum and supremum to ensure that the final scheme obtains essentially non-oscillatory and sharp resolutions for various discontinuities while maintaining the nominal second-order accuracy for smooth regions. Moreover, such a framework has been implemented on the tangent-of-hyperbola-for-interface-capturing scheme (Xiao et al. (2005) [27]) and the classical harmonic limiter (van Leer (1974) [37]), resulting in two specific self-adjusting steepness-based schemes. Finally, these two schemes were tested using a series of numerical examples in one and two dimensions. It was found that the schemes cannot only obtain second-order accuracy in smooth regions but also preserve discontinuous flow structures, especially contact discontinuities, even after long computation times. [ABSTRACT FROM AUTHOR]

• Numerical dissipation modification to SLAU2 flux is proposed to avoid negative pressure at gas-liquid interfaces. • The modification does not involve any iterative or tunable parameters in AUSM+-up(f). • The modified SLAU2 flux allows stable and high-resolution results in multiphase flows. This paper proposes a novel method for the computation of compressible multiphase flows under the assumption of pressure equilibrium based on a 6-equation model and the AUSM (Advection Upstream Splitting Method) family. In this study, we introduce a new numerical pressure flux dissipation term based on the relative velocities in the gas and liquid phases to develop an analogous carbuncle-suppression mechanism that is applicable to gas dynamics. We also propose a mass flux dissipation term based on the pressure ratio at the gas-liquid interface and incorporate both terms into SLAU2, an AUSM-family scheme, to achieve robustness against shock anomalies. [ABSTRACT FROM AUTHOR]

In this paper, we derive a dimensionless model for pure-component two-phase compressible flows with Van der Waals equation of state (EoS) and generalized Navier boundary condition (GNBC). We propose three energy-stable numerical schemes. One of them is based on the scalar auxiliary variable (SAV) approach for Helmholtz free energy for bulk and surface free energy, which leads to a modified energy and is proved to be unconditional stable. Another numerical scheme is based on the Lagrange multiplier approach for Helmholtz free energy for bulk and surface free energy, which leads to the original energy and is proved to be unconditional stable. Numerical results are presented to verify the effectiveness of the proposed methods. • A dimensionless model for two-phase compressible flows with GNBC is presented. • Three energy-stable numerical schemes are presented. • The numerical scheme leads to the original energy dissipation. [ABSTRACT FROM AUTHOR]

• A set of one-sided Riemann and GRP solver are described systematically. • One-sided GRP solver is applied for numerical boundary conditions. • Both 1-D and 2-D numerical simulations are tested for the resulting schemes. In the computation of compressible fluid flows, numerical boundary conditions are always necessary for all physical variables at computational boundaries while just partial physical variables are often prescribed as physical boundary conditions. Certain extrapolation technique or ghost cells are often employed traditionally for this issue but spurious wave reflections often arise to cause numerical instability. In this paper, we associate this issue with the one-sided generalized Riemann problem (GRP) solver motivated by the accelerated piston problem in gas dynamics so that the extrapolation technique can be actually avoided. In fact, the compatibility arguments naturally require to formulate the one-sided generalized Riemann problem and incorporate it into the numerical procedure of boundary conditions. Once the corresponding solver is available, the boundary control volume is treated in the same way as the usual interior control volume. As far as the interaction of nonlinear waves with physical boundaries, such a one-sided GRP solver shows significant benefits, as numerical experiments demonstrate, on avoiding spurious wave reflections at the computational boundaries. [ABSTRACT FROM AUTHOR]

In this paper we address the compressible Navier–Stokes equations written in the so-called conservative formulation. In particular, we focus on the possibility of uncoupling the computation of the problem unknowns, namely, density, linear momentum and total energy, a technique usually labeled as fractional step method, which allows to reduce the associated computational cost. The proposed methodology is a finite-element solver supplemented with a stabilization technique within the Variational Multi-Scale framework. In this regard, we consider orthogonal and dynamic definitions for the subscales. This discretization in space shows an adequate stability, permitting in particular the use of equal interpolation for all variables in play. However, we complement it with a shock-capturing operator in order to solve problems involving shocks. Several representative benchmark flow simulations are performed, which demonstrate the suitability of the proposed algorithm for a vast range of regimes. • Novel fractional step method for compressible flows in conservative variables. • Scheme (of first and second order) designed at the algebraic level. • A stabilized finite element method is employed. • A shock capturing technique is introduced to deal with the appearance of shocks. • Numerical experiments show the excellent performance of the formulations proposed. [ABSTRACT FROM AUTHOR]

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. • The shock instability of the HLLEM solver is studied in depth. • The shock instability is eliminated by the antidiffusion controlling technique with a shock indicator function. • The resolution of the HLLEM solver for contact discontinuities is improved by a BVD algorithm. • Numerical results demonstrate that the strategies adopted provide higher resolution and better shock stability. [ABSTRACT FROM AUTHOR]

• The macroscopic equations with actual numerical dissipative terms of the LBFS are derived. • The mechanisms of restraining pressure oscillation and stabilizing numerical computation in LBFS are revealed. • It proposes the reconstructed LBFS that recovers the numerical stability and accuracy of LBFS perfectly. • The reconstructed LBFS has higher computational efficiency than LBFS. The lattice Boltzmann flux solver (LBFS) uses the finite volume method (FVM) to update macroscopic variables while uses the local solution of lattice Boltzmann equation (LBE) to calculate fluxes at the cell interface. It overcomes the intrinsic drawbacks of the lattice Boltzmann method which include the limitation of uniform mesh, coupled time interval with mesh spacing, and extra memory requirement. The recovered macroscopic equations indicate that LBFS is a weakly compressible model. In general, directly solving weakly compressible models with the central difference scheme suffers numerical instability and additional treatments are needed to stabilize computation. The present paper firstly recovers the macroscopic equations of LBFS (MEs-LBFS) with actual numerical dissipative terms by approximating its actual computational process and then analyzes the mechanisms of the good performance of LBFS. Based on these mechanisms, the reconstructed LBFS (RLBFS), which directly uses macroscopic variables rather than LBE to calculate fluxes at the cell interface, is proposed. Numerical tests indicate that RLBFS preserves the mechanisms of restraining pressure oscillation and stabilizing numerical computation in LBFS perfectly. It is capable of simulating both steady and unsteady, 2D and 3D, viscous and inviscid flows as well, and has higher computational efficiency than LBFS. [ABSTRACT FROM AUTHOR]

• Generalization of the Roe approximate Riemann solver to Multicomponent Real Gas Flow. • High-order accuracy with One Step Monotonicity Preserving scheme. • Additional OSMP scheme on a combination of characteristic variables. • Accurate R 22 bubble/shock interaction test case with detailed thermodynamic model. 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. [ABSTRACT FROM AUTHOR]

For the compressible flows with broadband length scales, good dispersion and dissipation properties are crucial for the numerical schemes to realize high-fidelity simulations. It has been recognized that the minimization of dispersion error is an effective measure to improve the resolution of schemes. However, it is still an open problem for the schemes to control or adjust the dissipation. In this paper, a class of high-order finite difference schemes with minimized dispersion and adaptive dissipation is proposed. As the first step to automatically adjust the dissipation according to the flow structures, we devise a scale sensor to quantify the local length scale of the numerical solution as the effective scaled wavenumber. Then the dispersion-dissipation condition is used to construct the relationship between the dissipation parameter and the effective scaled wavenumber. Thus, we obtain a class of finite difference schemes with adaptive dissipation. To achieve the shock-capturing capability and maintain the superior spectral properties in the smooth regions, we combine the adaptive dissipation scheme with the corresponding weighted essentially non-oscillatory (WENO) scheme to construct a hybrid scheme. At each grid point, either of the sub-schemes is applied dynamically according to a new shock detector. The shock detection accuracy of the new detector has been effectively improved compared with classical ones. The approximate dispersion relation (ADR) shows that the new hybrid scheme is accurate and robust. Several benchmark test problems with broadband length scales as well as discontinuities are presented to verify the high resolution, high efficiency and the good shock-capturing capability of the proposed scheme. The performance of the proposed scheme in turbulence simulation is also demonstrated by the three-dimensional isotropic Taylor-Green vortex problem. [ABSTRACT FROM AUTHOR]

Usually, the temperature dependence of specific heats is neglected or the specific heats are frozen in interface computations for compressible two-material flows. In this paper, we present a practical interface treatment to faithfully capture the effect of high temperature on interface evolutions. A general technique for solving the Riemann problem equipped with a wide variety of equations of state (EOS) is established. In a unified framework for computing the interfacial states, it provides a convenient way to deal with the thermally perfect gas (PG) that considers the effect of temperature on specific heats. The algorithm of the complete and exact solution to Riemann problem with thermally PG is also designed in detail. Based on this technique, the modified ghost fluid method with an approximate Riemann solver is further extended to handle the interface of two-material flows involving thermally PG with chemical reactions. Several typical problems are selected to validate and test the present algorithm for the interaction between the thermally PG and other gases or liquids. The results indicate that the present algorithm enables an effective implementation for simulating various two-material multi-species flows with different types of EOS. As temperature increases, the behavior of the interfacial flows under the assumption of thermally PG EOS gradually differs from that under the assumption of calorically PG EOS. • Unified framework for simulating two-material multi-species flows. • Exact and approximate Riemann solvers for thermally perfect gases. • Modified ghost fluid method for high-temperature effects with chemical reactions. • Comparison of thermally/calorically perfect gases. [ABSTRACT FROM AUTHOR]