Upstream-centered Scheme For Conservation Laws Research Articles

High-order discretization–based self-adaptive turbulence eddy simulation for supersonic base flow with PHengLEI software

PurposeThe purpose of the present study is to develop a new numerical framework that can predict the supersonic base flow more accurately, including the development of axisymmetrically separated shear layer and recompression shock. To this end, two aspects are improved and combined, i.e. a newly self-adaptive turbulence eddy simulation (SATES) turbulence modeling method and a high-order discretization numerical scheme. Furthermore, the performance of the new numerical framework within a general-purpose PHengLEI software is assessed in detail.Design/methodology/approachSatisfactory prediction of the supersonic separated shear layer with unsteady wake flow is quite challenging. By using a unified turbulence model called SATES combining high-order accurate discretization numerical schemes, the present study first assesses the performance of newly developed SATES for supersonic axisymmetric separation flows. A high-order finite differencing-based compressible computational fluid dynamics (CFD) code called PHengLEI is developed and several different numerical schemes are used to investigate the effects on shock-turbulence interactions, which include the monotonic upstream-centered scheme for conservation laws (MUSCL), weighted compact nonlinear scheme (WCNS) and hybrid cell-edge and cell-node dissipative compact scheme (HDCS).FindingsCompared with the available experimental data and the numerical predictions, the results of SATES by using high-order accurate WCNS or HDCS schemes agree better with the experiments than the results by using the MUSCL scheme. The WCNS and HDCS can also significantly improve the prediction of flow physics in terms of the instability of the annular shear layer and the evolution of the turbulent wake.Research limitations/implicationsThe small deviations in the recirculation region can be found between the present numerical results and experimental data, which could be caused by the inaccurate incoming boundary layer condition and compressible effects. Therefore, a proper incoming boundary layer condition with turbulent fluctuations and compressibility effects need to be considered to further improve the accuracy of simulations.Practical implicationsThe present study evaluates a high-order discretization-based SATES turbulence model for supersonic separation flows, which is quite valuable for improving the calculation accuracy of aeronautics applications, especially in supersonic conditions.Originality/valueFor the first time, the newly developed SATES turbulence modeling method combining the high-order accurate WCNS or HDCS numerical schemes is implemented on the PHengLEI software and successfully applied for the simulations of supersonic separation flows, and satisfactory results are obtained. The unsteady evolutions of the supersonic annular shear layer are analyzed, and the hairpin vortex structures are found in the simulation.

Improved hybrid approach of monotonic upstream-centered scheme for conservation laws and discontinuity sharpening technique for steady and unsteady flows

In finite-volume methods, monotonic upstream-centered schemes for conservation laws (MUSCL) offer second-order spatial accuracy but tend to produce highly dissipative solutions for density discontinuity and weak shock waves. To address this limitation within a second-order framework, a novel strategy for hybridizing MUSCL with the tangent of the hyperbola interface capturing technique for both steady and unsteady compressible flows is presented. This hybridization optimizes the process based on the degree of nonlinearity and discontinuity around the target cells, providing a novel method to sharply resolve weak shock waves and robustly compute strong shock waves within the hybrid scheme. The proposed scheme sharply captures exceedingly weak shock waves that conventional MUSCL fails to resolve accurately due to excessive numerical dissipation. Furthermore, for resolving small vortices induced by instability at slip lines, computational results demonstrate high-resolution surpassing fifth-order spatial accuracy schemes within this second-order spatial accuracy framework with less computational cost. Moreover, the scheme exhibits commendable convergence and robustness when applied to steady-state problems featuring strong shock waves. This scheme offers a more precise and high-resolution alternative to conventional MUSCL for compressible flow computations, as it requires no additional stencil for reconstruction, unlike conventional fifth-order schemes.

Numerical viscosity control in Godunov-like smoothed particle hydrodynamics for realistic flows modeling

In this study, we introduce a way to control the viscosity of the numerical approximation in the Godunov-like smoothed particle hydrodynamics (SPH) methods. This group of SPH methods includes momentum and energy fluxes in the right-hand sides of the equations, which are calculated by the solution of the Riemann problem between each pair of neighboring particles within the support radius of the smoothing kernel, which is similar to the procedure for the calculation of fluxes across cell boundaries in Godunov schemes. Such SPH methods do not require the use of artificial viscosity since the significant numerical viscosity is already introduced by a Riemann problem solution. We demonstrate that such a numerical viscosity may be measured and obtain the explicit expression for it depending on smoothed particle properties. In particular, we have found that Godunov-like SPH method with interparticle contact algorithms produces numerical viscosity several orders of magnitude higher than physical viscosity in materials. Modern approaches, such as SPH with monotonic upstream-centered scheme for conservation laws or weighted essentially non-oscillatory reconstruction techniques, have not only lower numerical viscosity but also too large for modeling real-world viscous flows. By constructing a correcting viscous stress tensor based on the analytical solution for discontinuous viscous flow, it is possible to reduce the viscous stresses of numerical origin. The use of such a correction makes it possible to improve the agreement with experiments in the simulation of viscous flows without using schemes of higher order reconstruction.

A very high order Flux Reconstruction (FR) method for the numerical simulation of 1-D compositional fluid flow model in petroleum reservoirs

Compositional reservoir simulation is an extremely important tool for modeling the fluid flow in complex petroleum reservoirs, as in cases where the reservoir fluid is volatile and composed of several pseudo-components with distinct characteristics, or reservoirs that require the use of enhanced oil recovery process. For these cases, simpler black-oil models are not suitable. The compositional model involves the solution of a large system of partial differential equations, that comprises the mass conservation, the Darcy’s law and the fugacity constraints. However, the high number of equations and constraints render the compositional problems extremely complex and computationally demanding solutions. To improve accuracy and reduce the high computational costs, one can use higher than first or even second-order methods to approximate the advective flux terms in the hyperbolic conservation laws that describe the multicomponent transport in the reservoir. Those very high-order schemes can replace low-order approaches, such as the First Order Upwind (FOU) method, which is traditionally used in commercial petroleum reservoir simulators (CMG, 2019; ECLIPSE, 2022), obtaining more accurate solutions with reduced computational cost. In this work, we consider a three-phase and isothermal fluid flow of water, oil and gas and assume that there is no mass transfer between the water and the hydrocarbon phases. Physical dispersion and capillary effects are neglected. To solve the system of Partial Differential Equations (PDEs), we used the classical IMPEC (Implicit Pressure Explicit Composition) approach and, to model the complex phase behavior, we used the Peng–Robinson equation of state (PR-EOS). The classical Two Point Flux Approximation (TPFA) method is used to spatially discretize the diffusion terms in the pressure equation. For the first time in literature the very high order Flux Reconstruction (FR) method is adapted for the numerical simulation of the compositional model in petroleum reservoirs. The FR method is a numerical scheme employed to discretize the hyperbolic conservation laws using general grids. To avoid spurious oscillations in the vicinity of solution discontinuities, the h-MLP (hierarchical Multi-dimensional Limiting Process) is applied in the reconstruction stage. For the time integration, we have used the third-order Runge–Kutta approach. To appraise the robustness of our formulation, we have solved some one-dimensional benchmark problems found in literature and compared the FR solution with the FOU method and with the second-order Monotonic Upstream-Centered Scheme for Conservation Laws (MUSCL) method. From our results, the FR method has shown improved accuracy, efficiency and capability of detecting sharper shocks when compared to the other methods that we have considered for the numerical modeling of the compositional flow in the present paper.

Spectral difference method with a posteriori limiting: application to the Euler equations in one and two space dimensions

ABSTRACTWe present a new numerical scheme which combines the spectral difference (SD) method up to arbitrary high order with a-posteriori limiting using the classical MUSCL-Hancock scheme as fallback scheme. It delivers very accurate solutions in smooth regions of the flow while capturing sharp discontinuities without spurious oscillations. We exploit the strict equivalence between the SD scheme and a finite-volume scheme based on the SD control volumes to enable a straightforward limiting strategy. At the end of each stage of our high-order time-integration ADER (Arbitrary high order using Derivatives) scheme, we check if the high-order solution is admissible under a number of numerical and physical criteria. If not, we replace the high-order fluxes of the troubled cells by fluxes from our robust second-order MUSCL (Monotonic Upstream-centered Scheme for Conservation Laws) fallback scheme. We apply our method to a suite of test problems for the one-dimensional and two-dimensional Euler equations. We demonstrate that this combination of SD and ADER provides a virtually arbitrary high order of accuracy while at the same time preserving good sub-element shock capturing capabilities.

Numerical Approach of a Coupled Pressure-Saturation Model Describing Oil-Water Flow in Porous Media

Two-phase flow in porous media is a very active field of research, due to its important applications in groundwater pollution, {text {CO}}_2 sequestration, or oil and gas production from petroleum reservoirs, just to name a few of them. Fractional flow equations, which make use of Darcy’s law, for describing the movement of two immiscible fluids in a porous medium, are among the most relevant mathematical models in reservoir simulation. This work aims to solve a fractional flow model formed by an elliptic equation, representing the spatial distribution of the pressure, and a hyperbolic equation describing the space-time evolution of water saturation. The numerical solution of the elliptic part is obtained using a finite-element (FE) scheme, while the hyperbolic equation is solved by means of two different numerical approaches, both in the finite-volume (FV) framework. One is based on a monotonic upstream-centered scheme for conservation laws (MUSCL)-Hancock scheme, whereas the other makes use of a weighted essentially non-oscillatory (ENO) reconstruction. In both cases, a first-order centered (FORCE)-alpha numerical scheme is applied for intercell flux reconstruction, which constitutes a new contribution in the field of fractional flow models describing oil-water movement. A relevant feature of this work is the study of the effect of the parameter alpha on the numerical solution of the models considered. We also show that, in the FORCE-alpha method, when the parameter alpha increases, the errors diminish and the order of accuracy is more properly attained, as verified using a manufactured solution technique.

Lax-Wendroff method for incompressible flow

The Lax-Wendroff scheme is extended to incompressible fluid flow problems within the framework of artificial compressibility method (ACM) utilizing a “cell-centered finite-volume” Δ-approximation on a non-orthogonal non-staggered grid. An ACM induces a transformation between conservative and primitive variables; the physical relevance of ACM is to bring about “matrix preconditionings”, provoking density perturbations. The coupled algorithm is pressure-based; it benefits from enhanced accuracy and greater flexibility using the “monotone upstream-centered schemes for conservation laws (MUSCL)” approach. Numerical experiments in reference to “buoyancy-driven” flows with strong sources illustrate that the entire strategy augments overall damping capability and robustness adhering to the factored pseudo-time integration method. Conclusively, the associated limiter function has little influence on the high resolution for selected test cases.

Numerical study of the thermodynamics and supercavitating flow around an underwater high-speed projectile using a fully compressible multiphase flow model

In this study, the thermodynamic behavior and supercavitating flow around projectiles of high-subsonic to supersonic speeds were numerically analyzed using a fully compressible multiphase flow with phase change. The mathematical model is formulated based on the typical conservation laws of mixtures, including mass, momentum, and energy balance, and by maximizing the thermodynamic properties of water and vapor to enable the analysis of the characteristics of high-speed flows. The model is solved on a body-fitted grid using a finite-volume Riemann solver combined with a monotonic upstream-centered scheme for conservation laws. The numerical method was validated by comparing the supercavitating flow around an underwater projectile with the experimental results available in the literature. The temperature, shock, velocity, and supercavity around the high-speed projectile and the effects of the moving velocities on the physical aspects were investigated. It was observed that the temperature around the stagnation point increased significantly with the increase in the stream velocity, and such high temperatures may soften the steel encasing of the projectile in a manner that the excessive pressure in the area could damage the projectile. The physical context and thermodynamics around the transonic projectile were further observed through numerical investigation.

Application of High-Order WENO Scheme in the CFD/FW–H Method to Predict Helicopter Rotor Blade–Vortex Interaction Tonal Noise

The accurate prediction of helicopter rotor blade–vortex interaction (BVI) noise is challenging. This paper presents an implementation of the seventh-order improved weighted essentially non-oscillatory (WENO-Z) scheme for predicting rotor BVI noise using a high-resolution numerical method based on the Reynolds-averaged Navier–Stokes and the Ffowcs Williams–Hawkings equations. The seventh-order improved WENO-Z scheme is utilized to minimize the inherent numerical dissipation of the reconstruction method in the monotone upstream-centered scheme for conservation laws (MUSCL), thereby improving the rotor wake resolution and the BVI noise-prediction accuracy. The three-layer dummy cell method is used to ensure that the flux at the boundary maintains seventh-order accuracy. The effectiveness of the flow solver and the acoustic solver is validated using the Helishape-7A rotor and the UH-1H rotor, respectively. The flow field and BVI noise characteristics of the OLS rotor obtained from the fifth- and seventh-order WENO-Z schemes are compared with that of the third-order MUSCL for coarse and fine background grids. The wake resolution, noise-prediction accuracy, and computational cost of the three schemes are compared. The results show that the high-order WENO scheme provides higher accuracy for flow field simulation and BVI noise prediction than the MUSCL, but the computational cost of the WENO scheme increases substantially as the grid resolution increases. However, the WENO scheme can predict BVI using a coarser grid than the MUSCL. The computational cost of the WENO scheme is relatively low under the same flow field simulation resolution.

Fully compressible multiphase model for computation of compressible fluid flows with large density ratio and the presence of shock waves

We propose a fully compressible multiphase model for simulating compressible interfacial flows. The mathematical model is formulated on the basis of the typical conservation laws for mixtures, including mass, momentum, and energy balance. The mass balance equation in each phase is considered to discretely conserve the mass of each phase; therefore, it is always mass conservative for each phase. The numerical solution procedure for this model is developed using a high-resolution shock-capturing finite-volume method based on the Harten–Lax–van Leer-contact (HLLC) Riemann solver and total variation diminishing (TVD) time-integration schemes for simulating unsteady multiphase flows and capturing strong shocks. The finite-volume Riemann solver is combined with a monotonic upstream-centered scheme for conservation laws (MUSCL) and compressive limiters to maintain a sharp interface. The numerical framework yields a conceptually simple but robust and versatile numerical procedure. Numerical results of benchmark Riemann problems demonstrate the ability of the proposed method to obtain sharp interfaces, free of spurious oscillations, high accuracy, conservation properties, and thermodynamic consistency over various Mach numbers.

Numerical simulation of compressible multiphase hydrodynamic problems using reduced five-equation model on body-fitted grids

This paper focuses on the numerical simulation of compressible multiphase hydrodynamic problems on body-fitted mapped grids. Assuming the mixture of water and gas with homogeneous mixture properties, the reduced five-equation model (Kapila et al., 2001) is utilized to simulate the flow. Meanwhile, to circumvent the difficulties during computation such as preserving the volume fraction positivity, the pressure relaxation method (Saurel et al., 2009) is adopted to improve the robustness. The method is extended by considering the gravity effect and axisymmetric flows which are absent in the original model. A Godunov method based on the monotone upstream-centered scheme for conservation laws (MUSCL) is utilized to discretize the governing equations and the HLLC approximate Riemann solver is used to compute the numerical flux across a mesh cell face. Source terms representing the effect of gravitational and axisymmetric flow are calculated by the Runge–Kutta method. Several test cases including water–air shock tubes, dam break, underwater explosion, water entry of a hemisphere and an oblique cylinder are presented in the paper. The results obtained agree well with the exact solutions, experimental results, and other numerical computations, demonstrating the capability of the current method to deal with multiphase hydrodynamic problems involving complex geometries.

Research on Computational Method of Supersonic Inlet/Isolator Internal Flow

In this research, a CFD solver is developed for solving the 2D/3D compressible flow problem: the finite volume method based on multi-block structural grids is used to solve the compressible Reynolds averaged Navier–Stokes equations (RANS). Included in the methodology are multiple high-order reconstruction schemes, such as the 3rd-order MUSCL (Monotone Upstreamcentered Schemes for Conservation Laws), 5th-order WENO (Weight Essentially Non-Oscillatory), and 5th-order MP (Monotonicity-Preserving) schemes. Of the variety of turbulence models that are embedded, this solver is mainly based on the shear stress transport model (SST), which is compatible with OpenMP/MPI parallel algorithms. This research uses the CFD solver to conduct steady-state flow simulation for a two-dimensional supersonic inlet/isolator, incorporating these high-precision reconstruction schemes to accurately capture the shock wave/expansion wave interaction and shock wave/turbulent boundary layer interaction (SWTBLI), among other effects. By comparing the 2D/3D computation results of the same inlet configuration, it is found that the 3D effects of the side wall cannot be ignored due to the existing strong lateral flow near the corner. To obtain a more refined turbulence simulation, the commercial software ANSYS Fluent 18.0 is used to carry out the detached eddy simulation (DES) and the large eddy simulation (LES) of the same supersonic inlet, so as to reveal the flow details near the separation area and boundary layers.

Development of artificial compressibility method with elliptic relaxation approach

An artificial compressibility (AC) method resorting to the pressure-based algorithm is developed on a non-orthogonal non-staggered grid using a cell-centered finite-volume Δ-approximation for incompressible fluid flow problems. A pressure time-derivative is used to perturb the equation of continuity. This artifact provokes a density preconditioning which transforms conservative variables to primitive ones; the physical relevance of density preconditioning signifies to convoke compatible linearizations of residuals in collaboration with an AC parameter. The avoidance of pressure-velocity decoupling is enhanced by a consistently formulated cell-face dissipation approach, preserving an increased accuracy with a greater flexibility analogous to the MUSCL (monotone upstream-centered schemes for conservation laws) approach. An elliptic-relaxation scheme having accountability to promoting anisotropic diffusion coefficients is applied to smooth out the pressure residual. Numerical experiments dictate that pertaining to the pseudo-time integration method, the overall contrivance is benefited with an enhanced robustness having sensible oscillation damping characteristics.

Numerical investigation on noise reduction of rotor blade-vortex interaction using blade surface jet blowing

The effects of blade surface jet blowing on the reduction of rotor blade-vortex interaction (BVI) noise are investigated using the compressible Reynolds-averaged Navier-Stokes equations and the Ffowcs Williams-Hawkings equations. A detailed analysis of the accuracy of the Weighted Essentially Non-Oscillatory (WENO) scheme and Monotone Upstream-Centered Scheme for Conservation Laws for simulating the flow field and predicting the BVI noise is performed using a two-bladed scaled AH-1 helicopter main rotor undergoing BVI as the baseline case. Additionally, a grid convergence study is conducted. The BVI noise is accurately predicted using the fifth-order WENO scheme and high-resolution grid system (the prediction errors of the peak sound pressure are less than 4%). Surface boundary conditions are introduced to model the effect of jet blowing on the blade surface on the flow control of the baseline case. Blade surface blowing is analyzed near the trailing edge and the leading edge near the rotor tip. The results show that blowing near the blade trailing edge is effective for BVI mitigation by altering the tip vortex strength and increasing the blade-vortex miss distance. The sound pressure level is reduced by 2.6 dB, and the peak amplitude is attenuated by more than 30%, with a 9.8% loss in rotor thrust. Therefore, blowing near the trailing edge may be considered a promising approach for active BVI noise control.

Numerical Simulating Open-Channel Flows with Regular and Irregular Cross-Section Shapes Based on Finite Volume Godunov-Type Scheme

A numerical model based on the Saint-Venant equations (one-dimensional shallow water equations) is proposed to simulate shallow flows in an open channel with regular and irregular cross-section shapes. The Saint-Venant equations are solved by the finite-volume method based on Godunov-type framework with a modified Harten, Lax, and van Leer (HLL) approximate Riemann solver. Cross-sectional area is replaced by water surface level as one of primitive variables. Two numerical integral algorithms, compound trapezoidal and Gauss–Legendre integrations, are used to compute the hydrostatic pressure thrust term for natural streams with arbitrary and irregular cross-sections. The Monotonic Upstream-Centered Scheme for Conservation Laws (MUSCL) and second-order Runge–Kutta methods is adopted to achieve second-order accuracy in space and time, respectively. The performance of the resulting scheme is evaluated by application in rectangular channels, trapezoidal channels, and a natural mountain river. The results are compared with analytical solutions and experimental or measured data. It is demonstrated that the numerical scheme can simulate shallow flows with arbitrary cross-section shapes in practical conditions.

Numerical simulations on propane/oxygen detonation in a narrow channel using a detailed chemical mechanism: formation and detailed structure of irregular cells

Numerical simulations of two-dimensional inviscid detonations for a stoichiometric propane/oxygen gas mixture are performed using a detailed chemical reaction model. The UC San Diego model which includes 57 chemical species and 268 elementary reactions is mainly used in the present study. It is shown that a grid size of 3 µm can capture important features such as the unburned gas pocket behind the detonation when compared to larger grid sizes. The effects of channel width show that the detonation propagates with the CJ (Chapman–Jouguet) velocity for all cases and for more than 100 times the channel width of 4.5 mm. Increasing the channel width results in an irregular detonation cell structure. A transverse detonation forms with cross-hatching marks on the maximum pressure history. The irregular detonation cell structure forms because both the reduced activation energy and the stability parameter have a value of approximately 10; however, the maximum thermicity in the detonation is one. The free radicals C3H7 and H2O2 play an important role in the propane oxidation under the high temperature in the detonation. The maximum concentration exists at a temperature of 2000–3000 K. The fifth-order WCNS (weighted compact nonlinear scheme) scheme can resolve the contact surface and complicated flow structure behind the detonation front compared to the second-order MUSCL (Monotonic Upstream-centered Scheme for Conservation Laws).

A 3-D pseudo-arc-length moving-mesh method for numerical simulation of detonation wave propagation

In this paper, we propose a robust pseudo-arc-length moving-mesh method (PALM) which adopts the strategy of overall movement and block calculation for numerical simulation of detonation wave propagation in three dimensions. The pseudo-arc-length moving-mesh method involves governing equations’ evolution, mesh redistribution, and positivity-preserving analysis. Second-order finite-volume schemes and multistage total-variation-diminishing Runge–Kutta methods are used for governing equations’ evolution, while mesh redistribution is an iterative procedure that includes mesh point redistribution and cell average conservative interpolation. In addition, positivity-preserving analysis is discussed to avoid the density or pressure becoming negative in the process of numerical calculation. Finally, several numerical examples show that our method is feasible and effective. The advantage of the PALM scheme is that we can get similar results as the Monotonic Upstream-centered Scheme for Conservation Laws (MUSCL) which requires more cells and computational run time. It is demonstrated that the computational grids using the PALM scheme can capture the detonation front.

An interface-sharpening method with adaptive mesh refinement for volume-of-fluid simulations of two-phase compressible flows

We propose an interface-sharpening method for volume-of-fluid simulations of two-phase compressible flows. In the proposed method, sharpening of the interface between two phases is achieved by introducing a source term to the advection-reaction equation for the phase volume fraction. The inherent numerical diffusivity arising from the discretization of this equation is used to construct the source term. The resulting governing equations of the two-phase compressible flow are quasi-conservative and hyperbolic, with the phase volume fraction being bounded and free of spurious oscillations. The proposed method, called advection-reaction interface sharpening (ARIS), can be applied inline so that no post-processing of the phase volume fraction is required after each time step. The conservation equations for mass, momentum and energy are solved with a dimensional-splitting Monotone Upstream-centered Scheme for Conservation Laws (MUSCL). The interfacial resolution is improved by combining a fully-threaded-tree algorithm with adaptive mesh refinement. In validation tests performed on a variety of one- and two-dimensional benchmark flows, we find that ARIS can reduce the interface thickness to just three to four computational cells, improving the interfacial resolution without sacrificing computational efficiency. This study shows that ARIS can provide robust, quasi-conservative, oscillation-free and nearly sharp solutions of the phase volume fraction in various one- and two-dimensional interfacial flows, paving the way for extensions of this method to three-dimensional flows with more than two phases.

Direct numerical simulation of supersonic flow and acoustics over a compression ramp

We present direct numerical simulations of the shock wave boundary layer interaction (SBLI) at Mach number 2.9 over a 24° ramp. We study both the numerical accuracy and flow physics. Two classes of spatial reconstruction schemes are employed: the monotonic upstream-centered scheme for conservation laws and the Weighted Essentially Non-Oscillatory (WENO) scheme, of accuracy ranging from 2nd- to 11th-order. Using the canonical Taylor–Green vortex test-case, a simple and computationally inexpensive rescaling of the candidate stencil values—within the context of the high-order WENO scheme—is proposed for reducing the numerical dissipation, particularly in under-resolved simulations. For the compression ramp case, higher-order schemes are shown to capture the size of the SBLI separation zone more accurately, a consequence of resolving much finer turbulence structures. For second- and fifth-order schemes, the energy of the unresolved small scale turbulence shifts the cascade of the turbulence kinetic energy (TKE) spectrum, thus resulting in more energetic large scale turbulent structures. Consequently, the λ-shock foot shifts further downstream, leading to a smaller separation bubble size. Nonetheless, other statistical quantities, such as the turbulence anisotropy invariant map and the turbulence kinetic energy budget terms, show little dependence on the type and order of the spatial reconstruction scheme. Finally, using the more accurate ninth-order WENO results, it is reasoned that the interaction of the λ-shock with the post-shock relaxation region drives the low-frequency oscillation of the λ-shock.

3D cell-centered Lagrangian second order scheme for the numerical modeling of hyperelasticity system

The extension of the 3D cell-centered Finite Volume EUCCLHYD scheme to the hyperelasticity system is proposed here. This study is based on the left Cauchy-Green tensor B which enables to work in a fully updated Lagrangian formalism. The second order extension of this scheme is proposed using a MUSCL (Monotonic Upstream-centered Scheme for Conservation Laws) procedure combined with a GRP (Generalized Riemann Problem) approach. In particular, the limitation of the tensor fields is done in a component-wise manner. Moreover, the complete GRP procedure is proposed in the case of Neo-Hookean compressible solids. The scheme is validated on several test cases introducing small as well as large deformations. In particular, good results are found on the non trivial problems of oscillating and twisting beams.