Flux-difference Splitting Research Articles

A computational study of leading-edge wall cooling effect on shock wave–boundary layer interaction in forward-facing step

A computational study has been done to understand the effect of leading-edge wall cooling on shock wave–boundary layer interaction. Shock wave–boundary layer interaction is studied over a forward-facing step at supersonic Mach 2.5. The study was carried out using Ansys. The work aims to explore the implementation of wall cooling at the leading edge as a separation control strategy for supersonic forward-facing step-induced flow separation. We use a finite-volume method based on upwind flux difference splitting and second-order upwind flow discretization. The simulation results are validated with the available experimental data. Furthermore, using numerical simulations, we found that the separation bubble size was reduced by 18.36% when the walls were marginally cooled to 150 K, while the separation was reduced by 32.65% when the walls were strongly cooled to 100 K.

Parallel unstructured finite volume lattice Boltzmann method for high-speed viscid compressible flows

Based on the double distribution function Boltzmann-BGK equations, a cell-centered finite volume lattice Boltzmann method on unstructured grids for high-speed viscid compressible flows is presented. In the equations, the particle distribution function is introduced on the basis of the D2Q17 circular function, and its corresponding total energy distribution function is adopted. In the proposed method, the advective term is evaluated by Roe’s flux-difference splitting scheme, and a limiter is used to prevent the generation of oscillations. The distribution functions on the interface are calculated by piecewise linear reconstruction, in which the gradient is computed by the least-squares approach. In order to do large-scale simulations, a parallel algorithm is illustrated. The present method is validated by a flow around the NACA0012 airfoil and a flow past a circular cylinder at high Mach numbers. The results agree well with the published results, which demonstrate that the present method is an efficient numerical method for high-speed viscid compressible flows. The parallel performance results show that the proposed parallel algorithm achieves 90% parallel efficiency on 4800 cores for a problem with [Formula: see text] unstructured triangle cells, which shows the potential to perform fast and high-fidelity simulations of large-scale high-speed viscid compressible flows in complicated computational domains.

Godunov-type schemes for the pressureless gas dynamics and related models

We study and develop new first-order Godunov-type schemes for the weakly hyperbolic pressureless gas dynamics equations and augmented Burgers’ equations. Each of these systems carries the information of propagation of waves with the same fluid velocity. The goal is achieved by first obtaining an Engquist-Osher (EO) type scheme for the pressureless system and then by enhancing the upwinding information present in the EO-type scheme to construct more accurate Godunov-type schemes. The resulting schemes present a lesser amount of numerical dissipation than existing Jordan decomposition-based Flux Difference Splitting (FDS) schemes recently proposed in [N. K. GARG, Numer. Algorithms, 83 (2020) 1091–1121] and are compact and robust. These schemes are tested on a number of numerical examples for one- and two-dimensional pressureless equations of gas dynamics and then to augmented Burgers’ equations.

Parallel finite-volume discrete Boltzmann method for inviscid compressible flows on unstructured grids.

In this paper, a finite-volume discrete Boltzmann method based on a cell-centered scheme for inviscid compressible flows on unstructured grids is presented. In the new method, the equilibrium distribution functions are obtained from the circle function in two-dimensions (2D) and the spherical function in three-dimensions (3D). Moreover, the advective fluxes are evaluated by Roe's flux-difference splitting scheme, the gradients of the density and total energy distribution functions are computed with a least-squares method, and the Venkatakrishnan limiter is employed to prevent oscillations. To parallelize the method we use a graph-based partitioning approach that also guarantees the load balancing. The method is validated by seven benchmark problems: (a) a 2D flow pasting a bump, (b) a 2D Riemann problem, (c) a 2D flow passing the RAE2822 airfoil, (d) flows passing the NACA0012 airfoil, (e) 2D supersonic flows around a cylinder, (f) an explosion in a 3D box, and (g) a 3D flow around the ONERA M6 wing. The benchmark tests show that the results obtained by the proposed method match well with the published results, and the parallel numerical experiments show that the proposed parallel implementation has close to linear strong scalability, and parallel efficiencies of 95.31% and 94.56% are achieved for 2D and 3D problems on a supercomputer with up to 4800 processor cores, respectively.

Investigate Impact Force of Dam-Break Flow against Structures by Both 2D and 3D Numerical Simulations

The aim of this paper was to investigate the ability of some 2D and 3D numerical models to simulate flood waves in the presence of an isolated building or building array in an inundated area. Firstly, the proposed 2D numerical model was based on the finite-volume method (FVM) to solve 2D shallow-water equations (2D-SWEs) on structured mesh. The flux-difference splitting method (FDS) was utilized to obtain an exact mass balance while the Roe scheme was invoked to approximate Riemann problems. Secondly, the 3D commercially available CFD software package was selected, which contained a Flow 3D model with two turbulent models: Reynolds-averaged Navier-Stokes (RANs) with a renormalized group (RNG) and a large-eddy simulation (LES). The numerical results of an impact force on an obstruction due to a dam-break flow showed that a 3D solution was much better than a 2D one. By comparing the 3D numerical force results of an impact force acting on building arrays with the existence experimental data, the influence of velocity-induced force on a dynamic force was quantified by a function of the Froude number and the water depth of the incident wave. Furthermore, we investigated the effect of the initial water stage and dam-break width on the 3D-computed results of the peak value of force intensity.

Evaluation of breakup models for liquid side jets in supersonic cross flows

In this study, numerical simulations of the interaction between supersonic crossflow and a side liquid jet were performed. The processes of the liquid jet breakup, flow separation and shock waves resulting from the interaction of the gas flows and liquid droplets were predicted. In the numerical simulation, a two-way coupling Eulerian-Lagrangian method was used. The two-dimensional unsteady compressible Navier-Stokes Equations were solved based on the MUSCL type (Monotone Upstream-Centred Scheme for Conservation Laws) spatial differencing coupling with Roe-type flux-difference splitting and a third-order accurate Runge-Kutta method for time-splitting. The atomization of liquid droplets was simulated using the combined KH-RT breakup model. The modified breakup model considered both the interface instability and the sudden deformation of a liquid spray induced by the compressible flows. Shock waves, recirculation zones and breakup processes of droplet particles were captured. The predicted droplet distributions at various operating conditions were discussed. Satisfied agreement between simulations and measurements is shown in the current work.

Numerical Study about Aerodynamic Interaction for Coaxial Rotor Blades

In the present study, the interaction effect by inter-rotor spacing of coaxial rotor blades in hovering and forward flight is studied by using a Reynolds-averaged Navier–Stokes CFD flow solver based on mixed meshes. The meshes were composed of unstructured near-body region and Cartesian off-body region. The flux-difference splitting scheme of Roe was used for computing inviscid fluxes on the near-body region, whereas the 7th order WENO scheme was used for the computation of the inviscid fluxes on the off-body region to capture vortex with high resolution. The predicted results of the baseline inter-rotor spacing were compared with the reference results of other researchers for validation. The computed results of three different inter-rotor spacings were compared in terms of the thrust and torque coefficients for each flight condition. The cause of the different interaction effect by inter-rotor spacing between hovering and forward flight was also investigated.

Analytical Closure of the Inviscid Blade Force Model

A new formulation of the axisymmetric flow equations in turbomachinery is proposed, which inherently contains the additional blade force equations for simulation of the (de)swirling effect of the blade rows. Unlike in explicit blade force models, the blade force term does not need to be separately solved throughout the time-marching integration of the flow equations. The new formulation does not have an equation for the swirl momentum, but it describes instead a strictly two-dimensional flow with specific convective flux and source terms. The convective fluxes include the slope of the mean blade surface (i.e., the blade angle), whereas a blade load source term replaces the explicit blade force with the surface camber. Suitable leading/trailing edge boundary conditions are developed to match the new equations in bladed subdomains of the meridional flowpath with the conventional equations in those unbladed. These conditions are coupled to a discontinuous flux difference splitting technique for treatment of a range of transonic regimes. The numerical solution has capabilities to capture choking and passage shocks, which are first addressed theoretically and then shown by re-obtaining two simple exact solutions. The validation goes through comparison to experimental results, a full three-dimensional solution, and explicit blade force solutions.

Use of Jordan forms for convection-pressure split Euler solvers

In this study, convection-pressure split Euler flux functions which contain weakly hyperbolic convective subsystems are analyzed. A system of first-order partial differential equations (PDEs) is said to be weakly hyperbolic if the corresponding flux Jacobian does not contain a complete set of linearly independent (LI) eigenvectors. Thus, the application of existing flux difference splitting (FDS) based schemes, which depend heavily on both eigenvalues and eigenvectors, are non-trivial to such systems. In the case of weakly hyperbolic systems, a required set of LI eigenvectors can be constructed through the addition of generalized eigenvectors by utilizing the theory of Jordan canonical forms. Once this is achieved for a weakly hyperbolic convective subsystem, an upwind solver can be constructed in the splitting framework.In the present work, the above approach is used for developing two new schemes. The first scheme is based on the Zha–Bilgen type splitting while the second is based on the Toro–Vázquez splitting. Both the schemes are tested on various benchmark problems in one-dimension (1-D) and two-dimensions (2-D). The concept of generalized eigenvectors based on Jordan forms is found to be useful in dealing with the weakly hyperbolic parts of the considered Euler systems.

2D numerical modelling of dam break- applying for Namchien dam in Vietnam

The paper is dedicated to study a numerical model simulating dam-break based on two dimensional nonlinear shallow water equations (2D-NSWE). Finite Volume Method-Godunov type is applied to discretize this equation. Roe scheme is utilized to approximate Riemann problem, meanwhile method of flux difference splitting is implemented to construct numerical solvers of SWE. Besides, the semi implicit scheme is also invoked to solve friction term in case of high roughness coefficient. The proposed model is verified through a comparison between computed results and empirical data of two reference tests. A dam break flow over floodable area with different roughness coefficients is also researched. A total collapsed dam scenario of an arch dam-Nam Chien in Vietnam is simulated by the proposed model. Several hydraulic characteristics such as flood extent, arrival time and time histories of water depth at different gauges are estimated with different grid sizes.

Numerical experiments of flux difference splitting methods with high resolution scheme for supersonic flows

In this work, we have carried out the assessment of a high resolution scheme for unsteady compressible flow. For high order spatial accuracy, we have used fifth order weighted essentially non oscillatory (WENO) scheme. This scheme is applied to four flux difference splitting (FDS) methods: Harten-Lax-van-Leer (HLL), Roe solver, Harten-Lax-van Leer-Contact (HLLC), and Rusanov methods. We have compared results of these flux schemes with each other. WENO scheme is used for the reconstruction of left and right state variable across the cell interface for high resolution. The reconstruction procedure is performed in terms of primitive variables instead of conservative variable, in order to avoid spurious oscillation. We have considered two test cases: shock wave reflection and supersonic viscous flow over a flat plate, to access the performance of FDS schemes. An explicit third order TVD Runge-Kutta method is used for advancement of solution in time. The present results are compared with available numerical solutions. WENO-HLLC has good shock capturing capabilities as compare to WENO-Roe, WENO-HLL and WENO-Rusanov methods. It also provides best results inside and outside the boundary layer.

An accurate and robust AUSM-family scheme on two-dimensional triangular grids

Most numerical schemes for solving high-speed compressible flow problems exhibit an instability that usually occurs inside the numerical shock structure in low-dissipative shock-capturing finite volume methods. In examining several test cases, the flux-difference splitting and the AUSM family of schemes cannot satisfy the robustness requirement, which manifests as the carbuncle phenomenon on two-dimensional triangular grids. This paper presents an accurate and robust AUSM-family scheme ( $$\hbox {AUSMDV}^+$$ scheme) that is verified against shock-induced anomalies on two-dimensional triangular grids. The linearized discrete analysis of an odd–even decoupling problem is applied to investigate the perturbation damping mechanism of these schemes. The corresponding recursive equations show that the $$\hbox {AUSMDV}^+$$ scheme is less sensitive to these anomalies than are other schemes in the AUSM family. Finally, the presented scheme is extended to achieve second-order solution accuracy. Its robustness and efficiency are then evaluated on both structured and unstructured triangular grids. The $$\hbox {AUSMDV}^+$$ scheme yields a physically meaningful solution for all test cases without introducing an additional shock fix technique.

Effect of RANS Method on the Stall Onset Prediction by an Eigenvalue Approach

The eigenvalue approach is a recently developed compressor stability model used to predict stall onset. In this model, the flow field from a Reynolds-averaged Navier–Stokes (RANS) simulation provides the basic flow. This paper presents the effect of the RANS methods (including the computational grid, the turbulence model, and the spatial discretization scheme) on the eigenvalue and investigates the most influencing flow structures to the eigenvalue. The test compressor was the transonic compressor of NASA Rotor 37. Three individual meshes with different grid densities were used to validate the grid independence, and the results indicated that RANS simulation and eigenvalue calculation obtain grid independence at the same grid density. Then, the effect of four turbulence models (including Spalart–Allmaras (SA) turbulence model, two different k–ε models with the extended wall function model (EWFKE), and the Yang–Shih model (YSKE), and k–ω shear stress transport (SST) model), and three spatial discretization schemes (the central scheme, the flux difference splitting (FDS) scheme, and the symmetric total variation diminishing (STVD)) was also studied. Further investigation showed that the SA turbulence model combined with the STVD scheme provided the best stall point prediction, with a relative error of 0.05%. Detailed exploration of the three-dimensional flow field revealed that there were two flow patterns near the blade tip necessary for precisely predicting stall onset: the flow blockage generated by the shockwave-tip leakage vortex (TLV) interaction, and the trailing edge separation and corresponding wake flow. The effect of the blockage was greater than the effect of the trailing edge flow.

Development of a structured overset Navier–Stokes solver with a moving grid and full multigrid method

An unsteady Reynolds averaged Navier–Stokes solver with a structured overset grid method has been developed. Velocity–pressure coupling is achieved using an artificial compressibility approach, spatial discretization is based on a finite-volume method, and convective fluxes are evaluated by a third-order upwind scheme based on flux-difference splitting. Body motions are considered using the grid deformation technique and grid velocities in the convective term. Viscous fluxes are evaluated via second-order centered differencing. The full multigrid (FMG) method is applied to obtain fast convergence. The cell flag on a coarse grid level is determined using the cell flag on a fine grid level. In the coarse grid-level calculations at the FMG stage, the data are interpolated until the finest grid level is achieved at an overset update interval. Then, the data are updated based on the overset relations at the finest grid level and then transferred to a coarser grid level. For free surface treatments, a single-phase localized level-set method is employed. The computations for flows around a hull form, including an unsteady simulation with regular waves, are demonstrated.

Large time step HLL and HLLC schemes

We present Large Time Step (LTS) extensions of the Harten-Lax-van Leer (HLL) and Harten-Lax-van Leer-Contact (HLLC) schemes. Herein, LTS denotes a class of explicit methods stable for Courant numbers greater than one. The original LTS method (R.J. LeVeque, SIAM J. Numer. Anal. 22 (1985) 1051–1073) was constructed as an extension of the Godunov scheme, and successive versions have been developed in the framework of Roe's approximate Riemann solver. In this paper, we formulate the LTS extension of the HLL and HLLC schemes in conservation form. We provide explicit expressions for the flux-difference splitting coefficients and the numerical viscosity coefficients of the LTS-HLL scheme. We apply the new schemes to the one-dimensional Euler equations and compare them to their non-LTS counterparts. As test cases, we consider the classical Sod shock tube problem and the Woodward-Colella blast-wave problem. We numerically demonstrate that for the right choice of wave velocity estimates both schemes calculate entropy satisfying solutions.

Pitching effect on transonic wing stall of a blended flying wing with low aspect ratio

Numerical simulation of the pitching effect on transonic wing stall of a blended flying wing with low aspect ratio was performed using improved delayed detached eddy simulation (IDDES). To capture the discontinuity caused by shock wave, a second-order upwind scheme with Roe’s flux-difference splitting is introduced into the inviscid flux. The artificial dissipation is also turned off in the region where the upwind scheme is applied. To reveal the pitching effect, the implicit approximate-factorization method with sub-iterations and second-order temporal accuracy is employed to avoid the time integration of the unsteady Navier–Stokes equations solved by finite volume method at Arbitrary Lagrange–Euler (ALE) form. The leading edge vortex (LEV) development and LEV circulation of pitch-up wings at a free-stream Mach number M = 0.9 and a Reynolds number Re = [Formula: see text] is studied. The Q-criterion is used to capture the LEV structure from shear layer. The result shows that a shock wave/vortex interaction is responsible for the vortex breakdown which eventually causes the wing stall. The balance of the vortex strength and axial flow, and the shock strength, is examined to provide an explanation of the sensitivity of the breakdown location. Pitching motion has great influence on shock wave and shock wave/vortex interactions, which can significantly affect the vortex breakdown behavior and wing stall onset of low aspect ratio blended flying wing.

Simulation of impacts on elastic\u2013viscoplastic solids with the flux-difference splitting finite volume method applied to non-uniform quadrilateral meshes

The flux-difference splitting finite volume method (Leveque in J Comput Phys 131:327–353, 1997; Leveque in Finite volume methods for hyperbolic problems. Cambridge: Cambridge University Press, 2002) is here employed to perform numerical simulation of impacts on elastic–viscoplastic solids on bidimensional non-uniform quadrilateral meshes. The formulation is second order accurate in space through flux limiters, embeds the corner transport upwind method, and uses a fractional-step method to compute the relaxation operator. Elastic–viscoplastic constitutive models falling within the framework of generalized standard materials (Halphen and Nguyen in J Mech 14:667–688, 1975) in small strains are considered. Many test cases are proposed and two particular viscoplastic constitutive models are studied, on which comparisons with finite element solutions show a very good accuracy of the finite volume solutions, both on stresses and viscoplastic strains.

Comparison among unstructured TVD, ENO and UNO schemes in two- and three-dimensions

This study focuses on unstructured TVD, ENO and UNO schemes applied to solve the Euler equations in two- and three-dimensions. They are implemented on a finite volume context and cell centered data base. The algorithms of Yee, Warming and Harten 1982; Harten; Yee and Kutler; Yee Warming and Harten 1985; Yee; Yee and Harten; Harten and Osher; Yang 1990, Hughson and Beran; Yang 1991; and Yang and Hsu are implemented to solve such system of equations in two- and three-dimensions. All schemes are flux difference splitting and good resolution is expected. This study deals with calorically perfect gas model and in so on the cold gas formulation has been employed. Two problems are studied, namely: the transonic convergent-divergent symmetrical nozzle, and the supersonic ramp. A spatially variable time step is implemented to accelerate the convergence process. The results highlights the excellent performance of the Yang 1990 TVD scheme, yielding an excellent pressure distribution at the two-dimensional nozzle wall, whereas the Harten and Osher scheme yields accurate values to the angle of the oblique shock wave and the best wall pressure distributions in the two-dimensional ramp problem. On the other hand, the excellent performance of the Harten scheme in the three-dimensional nozzle problem, yielding an excellent pressure distribution at the nozzle wall, and the Yee and Harten scheme yielding an accurate value to the angle of the oblique shock wave and the best wall pressure distribution in the three-dimensional ramp problem are of good quality.

Investigation of obstacle effect to improve conjugate heat transfer in backward facing step channel using fast simulation of incompressible flow

The current study investigates the conjugate heat transfer characteristics for laminar flow in backward facing step channel. All of the channel walls are insulated except the lower thick wall under a constant temperature. The upper wall includes a insulated obstacle perpendicular to flow direction. The effect of obstacle height and location on the fluid flow and heat transfer are numerically explored for the Reynolds number in the range of 10 ≤ Re ≤ 300. Incompressible Navier-Stokes and thermal energy equations are solved simultaneously in fluid region by the upwind compact finite difference scheme based on flux-difference splitting in conjunction with artificial compressibility method. In the thick wall, the energy equation is obtained by Laplace equation. A multi-block approach is used to perform parallel computing to reduce the CPU time. Each block is modeled separately by sharing boundary conditions with neighbors. The developed program for modeling was written in FORTRAN language with OpenMP API. The obtained results showed that using of the multi-block parallel computing method is a simple robust scheme with high performance and high-order accurate. Moreover, the obtained results demonstrated that the increment of Reynolds number and obstacle height as well as decrement of horizontal distance between the obstacle and the step improve the heat transfer.

Lax–Wendroff and TVD finite volume methods for unidimensional thermomechanical numerical simulations of impacts on elastic–plastic solids

We present in this work two finite volume methods for the simulation of unidimensional impact problems, both for bars and plane waves, on elastic–plastic solid media within the small strain framework. First, an extension of Lax–Wendroff to elastic–plastic constitutive models with linear and nonlinear hardenings is presented. Second, a high order TVD method based on flux-difference splitting [1] and Superbee flux limiter [2] is coupled with an approximate elastic–plastic Riemann solver for nonlinear hardenings, and follows that of Fogarty [3] for linear ones. Thermomechanical coupling is accounted for through dissipation heating and thermal softening, and adiabatic conditions are assumed. This paper essentially focuses on one-dimensional problems since analytical solutions exist or can easily be developed. Accordingly, these two numerical methods are compared to analytical solutions and to the explicit finite element method on test cases involving discontinuous and continuous solutions. This allows to study in more details their respective performance during the loading, unloading and reloading stages. Particular emphasis is also paid to the accuracy of the computed plastic strains, some differences being found according to the numerical method used. Lax–Wendoff two-dimensional discretization of a one-dimensional problem is also appended at the end to demonstrate the extensibility of such numerical scheme to multidimensional problems.