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The effects induced by numerical schemes and mesh geometry on the solution of two-dimensional supersonic inviscid flows are investigated in the context of the compressible Euler equations. Five different finite difference schemes are considered: the Beam and Warming implicit approximate factorization algorithm, the original Steger and Warming flux vector splitting algorithm, the van Leer approach on performing the flux vector splitting and two different novel finite difference interpretations of the Liou AUSM + scheme. Special focus is given to the shock wave resolution capabilities of each scheme for the solution of an external supersonic inviscid flows around a blunt body. Significant changes in the shock structure are observed, mainly due to special properties of the scheme in use and the influence of the domain transformation procedure. Perturbations in the supersonic flow upstream of the shock are also seen in the solution, which is a non-physical behavior. Freestream subtraction, flux limiting and the explicit addition of artificial dissipation are employed in order to circumvent these problems. One of the AUSM + formulations presented here is seen to be particularly more robust in avoiding the appearance of some of these numerically induced disturbances and non-physical characteristics in the solution. Good agreement is achieved with both numerical and experimental results available in the literature.

In this paper, we propose a novel Local Macroscopic Conservative (LoMaC) low rank tensor method for simulating the Vlasov-Poisson (VP) system. The LoMaC property refers to the exact local conservation of macroscopic mass, momentum and energy at the discrete level. This is a follow-up work of our previous development of a conservative low rank tensor approach for Vlasov dynamics ( arXiv:2201.10397 ). In that work, we applied a low rank tensor method with a conservative singular value decomposition to the high dimensional VP system to mitigate the curse of dimensionality, while maintaining the local conservation of mass and momentum. However, energy conservation is not guaranteed, which is a critical property to avoid unphysical plasma self-heating or cooling. The new ingredient in the LoMaC low rank tensor algorithm is that we simultaneously evolve the macroscopic conservation laws of mass, momentum and energy using a flux-difference form with kinetic flux vector splitting; then the LoMaC property is realized by projecting the low rank kinetic solution onto a subspace that shares the same macroscopic observables by a conservative orthogonal projection. The algorithm is extended to the high dimensional problems by hierarchical Tuck decomposition of solution tensors and a corresponding conservative projection algorithm. Extensive numerical tests on the VP system are showcased for the algorithm’s efficacy.

This article presents the development of a fifth-order multi-resolution finite volume weighted essentially non-oscillatory (WENO) scheme combined with the advection upstream splitting method based on flux vector splitting (AUSMV) numerical flux for analyzing two-phase flow in both horizontal and vertical pipelines. The drift flux flow model comprises of two separate mass conservation equations for each phase for liquid and gas and one momentum equation for mixture and submodels for thermodynamics and hydrodynamics. The two mass conservation equations describe the behavior of each phase in the flow. The mixture-momentum equation takes into account the frictional and gravitational forces acting on the mixture of both phases. The thermodynamic and hydrodynamic submodels provide additional information to fully describe the flow and close the drift flux model. In the presence of these source terms and submodels, it is a challenging task to develop a high order efficient and accurate numerical schemes. The proposed numerical technique captures the peaks of pressure wave, suppresses the erroneous oscillations at the transition zones and resolves the discontinuities more efficiently and accurately. The accuracy of proposed numerical technique is verified by solving the various test problems. Furthermore, the solution obtained by developed numerical technique are compared to those attained with the high-resolution improved CUP and simple finite volume WENO numerical schemes.

Supersonic flow over a half-angle wedge (θ = 15°) with an upstream Mach number of 2.0 was investigated using 2D Euler equations where sea level conditions were considered. The investigation employed the Steger–Warming flux vector splitting (FVS) method executed in MATLAB 9.13.0 (R2022b) software. The study involved a meticulous comparison between theoretical calculations and numerical results. Particularly, the research emphasized the angle of oblique shock and downstream flow properties. A substantial iteration count of 2000 iteratively refined the outcomes, underscoring the role of advanced computational resources. Validation and comparative assessment were conducted to elucidate the superiority of the Steger–Warming flux vector splitting (FVS) scheme over existing methodologies. This research serves as a link between theoretical rigor and practical applications in high-speed aerospace design, enhancing the efficiency of aircraft components.

Series of studies of substantial aerodynamic loss for highly-loaded turbine shows that due to the potential flow field induced by interaction of end-wall boundary layer, complex massive vortex system will be formed upstream of the blade leading edge; and the large-scale secondary flow is related to boundary layer separation of the incoming end wall. In this work, simulation for such secondary flow fields is performed by implicit time-integrated higher-order Discontinuous Galerkin method (DGM) for the numerical approximation of Shear-Stress-Transport background Improved Delayed Detached Eddy Simulation (SST-IDDES). The mean strain rate involved with mean rotation rate and shear rate is modified to preserve the anisotropic characteristics encountered in strong secondary flows. Control of stochastic oscillations in multi-scale vortex region is guaranteed by a local correction method based on streamline curvature which also extends model ability in predicting large adverse-pressure-gradient flows. To reduce the numerical oscillations caused by inviscid term in meshes with complex geometrical configuration, a modified entropy conserving flux-vector splitting is applied across cell edges, and state vector is corrected by penalization of gradient jumps. While focusing on the analysis of secondary-vortex-system structure, basic model validation is also provided: fully-developed turbulent flow on flat-plate is simulated to validate the basic accuracy of proposed methodology in computing the near-wall turbulence; the case of NACA4412 airfoil is performed to verify the model ability in predicting large scale flow separation. Reasonable results obtained by current model provide detailed microscale flow structure and sufficiently accurate calculation of turbulent characteristics.

This paper is concerned with the derivation of a well-balanced kinetic scheme to approximate a shallow flow model incorporating non-flat bottom topography and horizontal temperature gradients. The considered model equations, also called as Ripa system, are the non-homogeneous shallow water equations considering temperature gradients and non-uniform bottom topography. Due to the presence of temperature gradient terms, the steady state at rest is of primary interest from the physical point of view. However, capturing of this steady state is a challenging task for the applied numerical methods. The proposed well-balanced kinetic flux vector splitting (KFVS) scheme is non-oscillatory and second order accurate. The second order accuracy of the scheme is obtained by considering a MUSCL-type initial reconstruction and Runge-Kutta time stepping method. The scheme is applied to solve the model equations in one and two space dimensions. Several numerical case studies are carried out to validate the proposed numerical algorithm. The numerical results obtained are compared with those of staggered central NT scheme. The results obtained are also in good agreement with the recently published results in the literature, verifying the potential, efficiency, accuracy and robustness of the suggested numerical scheme.

Water is a weakly compressible fluid medium. Due to its low compressibility, it is usually assumed that water is an incompressible fluid. However, if there are high-pressure pulse waves in water, the compressibility of the water medium needs to be considered. Typical engineering applications include water hammer protection and pulse fracturing, both of which involve the problem of discontinuous pulse waves. Traditional calculation and simulation often use first-order or second-order precision finite difference methods, such as the MacCormark method. However, these methods have serious numerical dissipation or numerical dispersion, which hinders the accurate evaluation of the pulse peak pressure. In view of this, starting from the weakly compressible Navier–Stokes (N-S) equation, this paper establishes the control equations in the form of flux, derives the expressions of eigenvalues, eigenvectors, and flux vectors, and gives a new flux vector splitting (FVS) formula by considering the water equation of state. On this basis, the above flux vector formula is solved using the fifth-order weighted essentially non-oscillatory (WENO) method. Finally, the proposed FVS formula is verified by combining the typical engineering examples of water hammer and pulse fracturing. Compared with the traditional methods, it is proved that the FVS formula proposed in this paper is reliable and robust. As far as we know, the original work in this paper extends the flux vector splitting method commonly used in aerodynamics to hydrodynamics, and the developed model equation and method are expected to play a positive role in the simulation field of water hammer protection, pulse fracturing, and underwater explosion.

In order to solve Maxwell's equations so as to study the electromagnetic stealth characteristics of targets, a 3-D meshless method is developed. According to the meshless method, the spatial derivatives on clouds of points are computed by using the weighted least square approach. After that, the Steger-Warming flux vector splitting approach is used to calculate the physical flux of Maxwell's equations. By using the developed method, the calculated bistatic radar cross sections (RCS) of a 3-D sphere are obtained, which are agree with the series solutions. Finally, the electromagnetic scattering characteristics for a 3-D stealth aircraft model with different situations is given, which shows the developed method has the ability to accommodate complicated 3-D configurations with multi-element.

PurposeThe flux vector splitting (FVS) schemes are known for their higher resistance to shock instabilities and carbuncle phenomena in high-speed flow computations, which are generally accompanied by relatively large numerical diffusion. However, it is desirable to control the numerical diffusion of FVS schemes inside the boundary layer for improved accuracy in viscous flow computations. This study aims to develop a new methodology for controlling the numerical diffusion of FVS schemes for viscous flow computations with the help of a recently developed boundary layer sensor.Design/methodology/approachThe governing equations are solved using a cell-centered finite volume approach and Euler time integration. The gradients in the viscous fluxes are evaluated by applying the Green’s theorem. For the inviscid fluxes, a new approach is introduced, where the original upwind formulation of an FVS scheme is first cast into an equivalent central discretization along with a numerical diffusion term. Subsequently, the numerical diffusion is scaled down by using a novel scaling function that operates based on a boundary layer sensor. The effectiveness of the approach is demonstrated by applying the same on van Leer’s FVS and AUSM schemes. The resulting schemes are named as Diffusion-Regulated van Leer’s FVS-Viscous (DRvLFV) and Diffusion-Regulated AUSM-Viscous (DRAUSMV) schemes.FindingsThe numerical tests show that the DRvLFV scheme shows significant improvement over its parent scheme in resolving the skin friction and wall heat flux profiles. The DRAUSMV scheme is also found marginally more accurate than its parent scheme. However, stability requirements limit the scaling down of only the numerical diffusion term corresponding to the acoustic part of the AUSM scheme.Originality/valueTo the best of the authors’ knowledge, this is the first successful attempt to regulate the numerical diffusion of FVS schemes inside boundary layers by applying a novel scaling function to their artificial viscosity forms. The new methodology can reduce the erroneous smearing of boundary layers by FVS schemes in high-speed flow applications.

In this paper we propose a novel semi-implicit Discontinuous Galerkin (DG) finite element scheme on staggered meshes with a posteriori subcell finite volume limiting for the one and two dimensional Euler equations of compressible gasdynamics. We therefore extend the strategy adopted by Dumbser and Casulli (Appl Math Comput 272:479–497, 2016), where the Euler equations have been solved solved using a semi-implicit finite volume scheme based on the flux-vector splitting method recently proposed by Toro and Vázquez-Cendón (Comput Fluids 70:1–12, 2012). In our scheme, the nonlinear convective terms are discretized explicitly, while the pressure terms are discretized implicitly. As a consequence, the time step is restricted only by a mild CFL condition based on the fluid velocity, which makes this method particularly suitable for simulations in the low Mach number regime. However, the conservative formulation of the scheme, together with the novel subcell finite volume limiter allows also the numerical simulation of high Mach number flows with shock waves. Inserting the discrete momentum equation into the discrete total energy conservation law yields a mildly nonlinear system with the scalar pressure as the only unknown. Due to the use of staggered meshes, the resulting pressure system has the most compact stencil possible and can be efficiently solved with modern iterative methods. In order to deal with shock waves or steep gradients, the new semi-implicit DG scheme proposed in this paper includes an a posteriori subcell finite volume limiting technique. This strategy was first proposed by Dumbser et al. (J Comput Phys 278:47–75, 2014) for explicit DG schemes on collocated grids and is based on the a posteriori MOOD algorithm of Clain, Loubère and Diot. Recently, this methodology was also extended to semi-implicit DG schemes on staggered meshes for the shallow water equations in Ioriatti and Dumbser (Appl Numer Math 135:443–480, 2019). Within the MOOD approach, an unlimited DG scheme first produces a so-called candidate solution for the next time level t n + 1 . Later on, the control volumes with a non-admissible candidate solution are identified by using physical and numerical detection criteria, such as the positivity of the solution, the absence of floating point errors and the satisfaction of a relaxed discrete maximum principle (DMP). Then, in the detected troubled cells a more robust first order semi-implicit finite volume (FV) method is applied on a sub-grid composed of 2 P + 1 subcells, where P denotes the polynomial degree used in the DG scheme. For that purpose, the nonlinear convective terms are recomputed in the troubled cells using an explicit finite volume scheme on the subcell level. Also the linear system for the pressure needs to be assembled and solved again, but where now a low order semi-implicit finite volume scheme is used on the sub-cell level in all troubled DG elements, instead of the original high order DG method. Finally, the higher order DG polynomials are reconstructed from the piecewise constant subcell finite volume averages and the scheme proceeds to the next time step. In this paper we present, discuss and test this novel family of methods and simulate a set of classical numerical benchmark problems of compressible gasdynamics. Great attention is dedicated to 1D and 2D Riemann problems and we also show that for these test cases the scheme responds appropriately in the presence of shock waves and does not produce non-physical spurious numerical oscillations.