Localized Artificial Diffusivity Research Articles

A high-order, localized-artificial-diffusivity method for Eulerian simulation of multi-material elastic-plastic deformation with strain hardening

A high-order method for Eulerian simulation of material undergoing large elastic–plastic deformation is developed. Thermodynamically consistent hyperelastic constitutive relations are assumed, facilitating the treatment of solids, liquids, and gases in a unified manner. The method enables the simulation of multi-material interactions using a diffuse interface approach. Numerical capturing of material interfaces, shock waves, contact surfaces, and elastic-plastic strain discontinuities using high-order compact-difference schemes is assisted by Localized Artificial Diffusivity (LAD). In the new setting involving elastic–plastic deformation, the previously established terms for the artificial properties are verified to effectively regularize normal shocks. Additional LAD terms are introduced to the elastic and plastic kinematic equations to regularize shear shocks and other strain discontinuities, improving solution stability. Other important features of the method that improve robustness include the numerical treatment of compatibility terms in the kinematic equations, and the treatment of rotation. Particular emphasis is focused toward new advancements of the methods for plastic-deformation integration and the associated strain hardening of the material, including rate-dependent plasticity. The method is demonstrated on a variety of test problems, including 1-D impacts, a variant of the Shu-Osher problem, a Taylor impact, and a Richtmyer-Meshkov instability between two elastic–plastic solids with strain hardening.

Comparison of high-order numerical methodologies for the simulation of the supersonic Taylor–Green vortex flow

This work presents a comparison of several high-order numerical methodologies for simulating shock/turbulence interactions based on the supersonic Taylor–Green vortex flow, considering a Reynolds number of 1600 and a Mach number of 1.25. The numerical schemes considered include high-order Finite Difference, Targeted Essentially Non-Oscillatory, Discontinuous Galerkin, and Spectral Difference schemes. The shock capturing methods include high-order filtering, localized artificial diffusivity, non-oscillatory numerical fluxes, and local low-order switching. The ability of the various high-order numerical methodologies to both capture shocks and represent accurately the development of turbulent vortices is assessed.

Parametric effects on Richtmyer–Meshkov instability of a V-shaped gaseous interface within linear stage

The Richtmyer–Meshkov instability of a V-shaped air/SF6 gaseous interface is numerically studied via a high-order finite difference scheme and a localized artificial diffusivity method. The oblique angle of the interface ranges from 20° to 75°, and the incident shock Mach number varies from 1.05 to 1.75. The wave patterns and the vortex structures are visualized during the interface evolution. A cavity is observed at the spike fingertip when the oblique angle decreases, which proves to be formed due to Mach reflection of the transmitted shock through velocity decomposition. By analyzing the linear growth rates of the interface, a modified empirical model for the reduction factor is suggested with model coefficients acquired by linear fitting for different Mach numbers. With shock polar analysis (SPA) method and visualization of the wave configuration, a criterion is proposed to explain the non-monotonic dependence of the linear growth rate on the oblique angle. In addition, Mach number effects on the linear growth rate are discussed by the SPA method, especially the anomalous behavior of the Mach 1.05 case.

Inverse asymptotic treatment: Capturing discontinuities in fluid flows via equation modification

A major challenge in developing accurate and robust numerical solutions to multi-physics problems is to correctly model evolving discontinuities in field quantities. These manifest themselves as interfaces between different phases in multi-phase flows, or as shock and contact discontinuities in (either single- or multi-phase) compressible flows. A plethora of bespoke discretization schemes have been developed to capture both types of discontinuities in physics-based simulations. However, when a quick response is required to rapidly emerging challenges such as the need to design novel hypersonic vehicles, the complexity of such implementations impedes a swift transition from problem formulation to computation. This is exacerbated by the need to compose multiple interacting physics, which may cause specialized schemes to break unexpectedly as governing equations need to be changed and/or added. We introduce “inverse asymptotic treatment” (IAT) as a unified framework for capturing discontinuities in fluid flows that enables building directly computable models based on standard, off-the-shelf numerics. By capturing discontinuities through modifications at the level of the governing equations, rather than via reliance on specialized discretization schemes, IAT can seemlessly handle additional physics and thus makes it easier for novice end users to quickly obtain numerical results for a variety of multi-physics scenarios. This strategy also facilitates the use of a “multi-physics” compiler that automates the conversion of the modified PDEs to numerical source code readable by popular computational frameworks like OpenFOAM. We outline IAT in the context of phase-field modeling of two-phase incompressible flows, and then demonstrate its generality by showing how localized artificial diffusivity (LAD) methods for single-phase compressible flows can be viewed as instances of IAT. Through the real-world example of a laminar hypersonic compression corner, we illustrate IAT’s ability to, within a span of just a few months, generate a directly computable model whose wall metrics predictions for sufficiently small corner angles come close to that of NASA’s state-of-the-art VULCAN-CFD solver. Finally, we propose a novel LAD approach via “reverse-engineered” PDE modifications, inspired by total variation diminishing (TVD) flux limiters, to eliminate the problem-dependent parameter tuning that plagues traditional LAD. Through canonical numerical tests, we demonstrate that this “limiter-inspired” LAD approach, when combined with second-order central differencing, can robustly and accurately model compressible flows.

On the use of high order central difference schemes for differential equation based wall distance computations

A computationally efficient high-order solver is developed to compute the wall distances by solving the relevant partial differential equations, namely: Eikonal, Hamilton–Jacobi (HJ) and Poisson equations. In contrast to the upwind schemes widely used in the literature, we explore the suitability of high-order central difference schemes (explicit/compact) for the wall-distance computation. While solving the Hamilton–Jacobi equation, the high-order central difference schemes performed approximately 1.4–2.8 times faster than the upwind schemes with a marginal improvement in the solution accuracy. A new pseudo HJ formulation based on the localized artificial diffusivity (LAD) approach has been proposed. It is demonstrated to predict results with an accuracy comparable to that of the Eikonal equation and the simulations are ≈ 1.5 times faster than the baseline HJ solver using upwind schemes. A curvature correction is also incorporated in the HJ equation to correct for the near-wall errors due to concave/convex wall curvatures. We demonstrate the efficacy of the proposed methods on both the steady and unsteady test cases and exploit the unsteady wall-distance solver to estimate the instantaneous shape and burning surface area of a dendrite propellant grain in a solid propellant rocket motor.

A diffuse-interface compact difference method for compressible multimaterial elastic–plastic flows

In this paper, a novel high-order numerical method is proposed for the simulation of compressible multimaterial solid–fluid problems involving elastic–plastic solid behaviors. The solid, liquid, and gas behaviors are described by an Eulerian interface-capturing model derived from hyperelasticity theory. The deformations in each component of the solids are tracked using a single deviatoric strain tensor. The numerical algorithm employs a tenth-order compact finite difference scheme and a fourth-order Runge–Kutta time-stepping scheme. To prevent numerical oscillations near the material interface, a mechanical equilibrium assumption is introduced for the elastic–plastic problems, and a localized artificial diffusivity method satisfying this assumption is designed and applied in the numerical algorithm. Numerical tests in one and two dimensions are shown to demonstrate the accuracy and feasibility of the proposed approach. In particular, the accuracy and numerical resolution of the method are verified using problems with analytical solutions. Numerical tests of interface advection problems demonstrate that our method preserves the velocity, pressure, and elastic stress equilibria at the material interface and prevents overshoots in the rest of the domain. The two-dimensional Richtmyer–Meshkov instabilities between two elastic or two elastic–plastic solids are simulated to verify the suitability of the numerical method for solving problems involving large deformations. Impact of a 2D Taylor bar to a rigid wall is also simulated to test the robustness of the numerical method.

On a robust and accurate localized artificial diffusivity scheme for the high-order flux-reconstruction method

The high-order flux reconstruction (FR) method on unstructured hexahedral grids is coupled with the localized artificial diffusivity (LAD) scheme for aiming at accurate simulation of shock-turbulence interaction. In order to overcome known robustness issues particularly with the high-order (r>0) derivative formulation, important properties of the artificial bulk viscosity (ABV) profile affecting the solution are investigated first. For the purpose of comparison, an approximated Gaussian filter and modified restriction-prolongation (RP) filters are developed and tested for the present FR-LAD approach. Then, we propose a multidimensional RP filter on unstructured quadrilateral and hexahedral grids to address the issues on non-Cartesian grids. It is shown that a simple extension of a one-dimensional RP filter for two-dimensional grids may result in insufficient smoothing of ABV with r=2, and that the proposed multidimensional RP filter can provide smooth ABV with both r=0 and r=2. The proposed FR-LAD scheme is tested for typical shock-related problems, including the 1D shock tube, 1D shock-entropy wave interaction, 2D steady shock flows, and 2D shock-vortex interaction. The FR-LAD scheme has favorable properties of subcell shock capturing with the length scale of O(h/p) and superior preservation of high-order accuracy for smooth flows. Finally, LES of an overexpanded supersonic jet is performed to demonstrate its capability for practical applications. The proposed FR-LAD scheme can be an attractive candidate for LES or DNS of compressible flows involving shocks, contact discontinuities, turbulence, and their interactions on unstructured meshes.

A localized artificial diffusivity method to simulate compressible multiphase flows using the stiffened gas equation of state

SummaryThe development of numerical approaches to perform direct numerical simulations of compressible multiphase flows has been an active field of research for several years. Proper treatment of fluid interfaces is crucial as important physics occur in this infinitesimally small region. Furthermore, the compressibility of the fluid requires proper treatment of discontinuities. Artificial diffusivity is among a number of methods widely used for compressible flows. This study develops a general form of consistent artificial diffusion fluxes and extends the localized artificial diffusivity method for high‐order central schemes to solve multiphase flows with an interface‐capturing method. These fluxes ensure an oscillation‐free interface for pressure, velocity, and temperature without employing a sharpening technique. Moreover, the high‐order representation of all scales in the flow helps capture the wide range of instabilities inherent in these flows. The goal is to develop an approach capable of performing high‐fidelity simulations supported by physics‐driven validation. This is achieved by solving the five‐equation model with the stiffened‐gas equation of state using the proposed method for multicomponent and multiphase flows on a variety of 1D and 2D problems.

A unified high-order Eulerian method for continuum simulations of fluid flow and of elastic–plastic deformations in solids

We develop a new high-order method for Eulerian simulations of solids undergoing large, elastic–plastic deformations. Thermodynamically consistent constitutive relations of classical hyperelasticity are used to describe the behavior of solids, liquids and gases in a unified manner. Two kinematic formulations, one based on the inverse deformation gradient tensor, and a second based on the symmetric Finger tensor, are used for tracking large deformations in solids. Simulations based on the Finger tensor are shown to be equivalent to those using the full inverse deformation gradient tensor at much lower computational expense. The numerical algorithm employs a 10th-order compact finite-difference scheme for spatial discretization and a 4th-order Runge–Kutta time-stepping scheme. An improved form of the Localized Artificial Diffusivity (LAD) method is used for numerical regularization of shocks and contact discontinuities. We show that this high-order numerical framework, previously used for simulations of fluid flows, is suitable for problems involving large deformations in elastic–plastic solids as well. Particular emphasis is laid on the choice of the artificial diffusivity parameters in order to sufficiently capture shocks and discontinuities in all the aforementioned continuum media with minimal added dissipation. Test cases in one and two dimensions are shown to demonstrate the feasibility and accuracy of the proposed approach. In particular, this choice of algorithms is shown to lead to excellent numerical resolution properties, and to preserve mass-consistency and curl/compatibility constraints with high order of accuracy. Potential extensions of this numerical framework include application to multi-material problems, involving compressible flow of fluids coupled to elastic–plastic deformations in solids, that are of significant engineering interest.

High-Order Eulerian Simulations of Multimaterial Elastic–Plastic Flow

We develop a new high-order numerical method for continuum simulations of multimaterial phenomena in solids exhibiting elastic–plastic behavior using the diffuse interface numerical approximation. This numerical method extends an earlier single material high-order formulation that uses a tenth-order high-resolution compact finite difference scheme in conjunction with a localized artificial diffusivity (LAD) method for shock and contact discontinuity capturing. The LAD method is extended here to the multimaterial formulation and is shown to perform well for problems involving shock waves, material interfaces and interactions between the two. Accuracy of the proposed approach in terms of formal order (eighth-order) and numerical resolution is demonstrated using a suite of test problems containing smooth solutions. Finally, the Richtmyer–Meshkov (RM) instability between copper and aluminum is simulated in two-dimensional (2D) and a parametric study is performed to assess the effect of initial perturbation amplitude and yield stress.

Localized artificial diffusivity scheme for deflagrations and detonation waves

The ability of a localized artificial diffusivity (LAD) method to capture reactive flow discontinuities, such as deflagrations and detonations, using high-order compact differencing scheme is evaluated in this paper. Different types of steady and unsteady problems under 1D and 2D conditions are tested. For the laminar premixed flame, the flame speed and numerical flame thickness are used when evaluating the accuracy of the LAD results. As the grid spacing increases, the physical behaviour of the flame is maintained using LAD. It is shown that the ability to obtain accurate flame structure are controlled by the artificial diffusivity terms for both the thermal conductivity and species diffusivity. The artificial species diffusivity term is found to affect the L2 error of the solution when the mesh is progressively refined. However, the error reduces according to the order of the discretization error for the schemes being used when the LAD terms are switched off. The choice of the model constants for the LAD terms, selected previously for non-reacting cases, are found appropriate as the flame thickness and chemical reaction rate are found to adjust such that the flame speed is well predicted. The finding is verified further using a 2D configuration involving vortex-flame interaction. The ability of LAD to retain the flame properties allows for the flame response to vortex interaction to be captured at larger grid sizes. For the CJ detonation wave, the structure of the detonation wave is well captured without spurious oscillations. The amplitude of wiggles near a shock or contact discontinuity and the number of grid points used to capture the discontinuity are largely independent of the mesh size. A 2D CJ detonation configuration is also investigated and shows the ability of the LAD method to capture the transverse variations of the detonation wave when subjected to an initial perturbation. The study demonstrates that the LAD method does not show any major detrimental effect on the discontinuity capturing under the conditions examined for both deflagration and detonation wave.

Modulation to compressible homogenous turbulence by heavy point particles. I. Effect of particles’ density

In this paper, two-way interactions between heavy point particles and forced compressible homogenous turbulence are simulated by using a localized artificial diffusivity scheme and a Eulerian–Lagrangian approach. The initial turbulent Mach number is around 1.0 and the Taylor Reynolds number is around 110. Seven different simulations of 106 particles with different particle densities (or initial Stokes numbers) are considered. The statistics of the compressible turbulence, such as the turbulence Mach number, kinetic energy, dilatation, and the kinetic energy spectra, from different simulations are compared with each other and with the one-way undisturbed case. Our results show that the turbulence is suppressed if the two-way coupling backward interactions are considered, and the suppression is more obvious if the density of particles is higher. The kinetic energy spectrum at larger initial Stokes number (higher density) exhibits a reduction at low wave numbers and an augmentation at high wave numbers, which is similar to those obtained in incompressible cases. The probability density functions of dilatation and normal upstream Mach number of shocklets also show that the modulation to the shocklet statistics is more apparent for particles with higher density.

A comparison of higher-order finite-difference shock capturing schemes

The efficiency of computational fluid dynamics simulations can be greatly enhanced by employing higher-order accurate numerical schemes which provide superior accuracy for a given cost. For unsteady turbulent flow simulations involving shocks, contacts, and/or material discontinuities, various higher-order shock capturing schemes are available in the literature. The desired numerical scheme should be free of spurious numerical oscillations across discontinuities and it should obtain higher-order accuracy in smooth flow regions in an efficient manner. Sufficient robustness is necessary for effectively utilizing these numerical methods in engineering and science applications. Three classes of higher-order shock capturing schemes are compared in this paper: (1) central finite-difference schemes with explicit artificial dissipation, (2) a compact centered finite-difference scheme with localized artificial diffusivity and (3) weighted essentially non-oscillatory schemes in both explicit and compact finite difference forms. Multiple variations of these methods were implemented and tested using a block-structured Cartesian mesh solver. The current paper assesses shock capturing capabilities as well as effects on the accuracy in smooth flow regions using a variety of test cases that range from canonical shock problems to homogeneous isotropic turbulence at a turbulent Mach number of 0.5 where shocklets are formed. Finally, a computational cost breakdown for each scheme is provided and the overall computational efficiency of the different schemes are compared to each other.

On the use of the discontinuous Galerkin method for numerical simulation of two-dimensional compressible turbulence with shocks

In this paper, the discontinuous Galerkin (DG) method combined with localized artificial diffusivity is investigated in the context of numerical simulation of broadband compressible turbulent flows with shocks for under-resolved cases. Firstly, the spectral property of the DG method is analyzed using the approximate dispersion relation (ADR) method and compared with typical finite difference methods, which reveals quantitatively that significantly less grid points can be used with DG for comparable numerical error. Then several typical test cases relevant to problems of compressible turbulence are simulated, including one-dimensional shock/entropy wave interaction, two-dimensional decaying isotropic turbulence, and two-dimensional temporal mixing layers. Numerical results indicate that higher numerical accuracy can be achieved on the same number of degrees of freedom with DG than high order finite difference schemes. Furthermore, shocks are also well captured using the localized artificial diffusivity method. The results in this work can provide useful guidance for further applications of DG to direct and large eddy simulation of compressible turbulent flows.

Large Eddy Simulation of Pseudo-Shock Waves Using Wall Model

A supersonic turbulent flowfield involving the pseudo-shock waves in an isolator of a supersonic combustion ramjet is computed using two different LES codes which are a high-order upwind finite volume scheme, and a sixth order compact differencing scheme utilizing the localized artificial diffusivity method for stabilizing shock waves and employing a wall model to enable the use of coarse mesh. In the validation study where a supersonic turbulent boundary layer flow over a flat plate is examined, both LES codes are well validated using velocity profile in the boundary layer given by the hot-wire anemometry and normal stress given by the laser Doppler anemometry. In particular, the sixth order compact differencing scheme gives closer agreements with these experimental data. Then, the validated LES codes are applied to solve the Mach number 2.5 supersonic turbulent flowfield involving the pseudo-shock waves. It is shown that typical features of unsteady flowfield of the pseudo-shock waves are well obtained by both schemes. Again, it is indicated that the sixth order compact differencing scheme gives closer agreements with the existing velocity data obtained by particle image velocimetry and pressure fluctuation data on the wall surface. Besides, the computational cost of the compact differencing scheme is found to be 1/7 of that for the upwind finite volume scheme, even though a wall model is solved at each grid point on the wall surface. Therefore, the obtained results in the present study allow recommending the sixth order compact differencing scheme with a wall model for simulating supersonic turbulent flowfield in an isolator involving the pseudo-shock waves.

High-Resolution Numerical Method for Supercritical Flows with Large Density Variations

is introduced with the aim of simulating jet mixing under supercritical pressure environments. The nonlinear localized artificial diffusivity provides the stability to capture different types of discontinuity, such as shock wave, contact surface, and material interface, whereas the high-order compact difference scheme resolves broadband scales in the rest of the domain. The present method is tested on several one-dimensional discontinuity-related problems under super/transcritical conditions and a comparatively more illustrative two-dimensional lowtemperature planar jet problem under a supercritical pressure condition. The localized artificial diffusivity, especially artificial thermal conductivity for temperature gradients, effectively suppresses numerical wiggles near the interfaces. The effects of the artificial thermal conductivity on numerical stability and accuracy are examined. Comparisons between the present method and a conventional low-order scheme demonstrate the superior performance of the present method for resolving a wide range of flow scales while successfully capturing large density/temperature variations at interfaces.

Two-dimensional viscous flow simulation of a shock accelerated heavy gas cylinder

Numerical simulation of Richtmyer–Meshkov instability (RMI) is conducted using an improved localized artificial diffusivity (LAD) method, which is used to treat discontinuities in the form of material interfaces and shocks in the flow-field. The RMI occurs on a cylindrical interface between air and SF6 accelerated by a Mach 1.2 shock initially in air. Navier–Stokes simulation is conducted to accurately predict the mixing between the two fluids. The initial conditions for the two-dimensional simulations are matched to previous experimental work by C. Tomkins et al. [“An experimental investigation of mixing mechanisms in shock-accelerated flow,” J. Fluid Mech. 611, 131 (2008)] and good agreement is found between the experimental data and numerical results. The study on initial condition sensitivity indicates that the initial pressure and density gradient are critical parameters that determine the primary vortex generation responsible for the flow development. A grid convergence study is carried out and the relative contribution of the artificial properties introduced by the LAD method is characterized. Novel to this study is the exploration of the effect of the third species (acetone used as a tracer particle in the experiments to obtain contour fields using planar laser induced florescence). The effect of the presence of the third species on the evolution of the RMI and mixing is shown to be non-negligible and an estimate of the amount of the tracer species that was present in the initial experimental set-up is given.

Hybrid Schemes of Shock Capturing Methods and Secondary Conservative Finite Difference

For compressible flow simulations, FDS, FVS, and AUSM are popular shock capturing schemes. However, these schemes do not work well for turbulent flow simulations because of their excessive numerical dissipation. On the other hand, the secondary conservative finite difference method is useful for unsteady turbulent flow simulations, although it does not work well for flows with discontinuities. In this study, we propose a computational technique that combine the shock capturing schemes and the secondary conservative finite difference. In addition, we try to combine the secondary conservative finite difference method and LAD (localized artificial diffusivity). These methods are combined properly with the aid of a shock detected sensor. In order to demonstrate the shock capturing performance and low numerical dissipation property of the hybrid schemes, numerical tests are done on a three-dimensional periodic inviscid flow, the shock tube problem (Sod problem), and a shock-vortex interaction problem. The computational results have shown that the proposed hybrid schemes based on numerical flux have desired conservation property, shock capturing performance, and low numerical dissipation in regions without discontinuity.

Large-Eddy Simulation of Jet Mixing in Supersonic Crossflows

Large-eddy simulation of an underexpanded sonic jet injection into supersonic crossflows is performed to obtain insights into key physics of the jet mixing. A high-order compact differencing scheme with a recently developed localized artificial diffusivity scheme for discontinuity-capturing is used. Progressive mesh refinement study is conducted to quantify the broadband range of scales of turbulence that are resolved in the simulations. The simulations aim to reproduce the flow conditions reported in the experiments of Santiago and Dutton [Santiago, J. G., and Dutton, J. C., Velocity Measurements of a Jet Injected into a Supersonic Crossflow, Journal of Propulsion and Power, Vol.132,1997, pp. 264―273] and elucidate the physics of the jet mixing. A detailed comparison with these data is shown. Statistics obtained by the large-eddy simulation with turbulent crossflow show good agreement with the experiment, and a series of mesh refinement studies shows reasonable grid convergence in the predicted mean and turbulent flow quantities. The present large-eddy simulation reproduces the large-scale dynamics of the flow and jet fluid entrainment into the boundary-layer separation regions upstream and downstream of the jet injection reported in previous experiments, but the richness of data provided by the large-eddy simulation allows a much deeper exploration of the flow physics. Key physics of the jet mixing in supersonic crossflows are highlighted by exploring the underlying unsteady phenomena. The effect of the approaching turbulent boundary layer on the jet mixing is investigated by comparing the results of jet injection into supersonic crossflows with turbulent and laminar crossflows.

Assessment of localized artificial diffusivity scheme for large-eddy simulation of compressible turbulent flows

The localized artificial diffusivity method is investigated in the context of large-eddy simulation of compressible turbulent flows. The performance of different artificial bulk viscosity models are evaluated through detailed results from the evolution of decaying compressible isotropic turbulence with eddy shocklets and supersonic turbulent boundary layer. Effects of subgrid-scale (SGS) models and implicit time-integration scheme/time-step size are also investigated within the framework of the numerical scheme used. The use of a shock sensor along with artificial bulk viscosity significantly improves the scheme for simulating turbulent flows involving shocks while retaining the shock-capturing capability. The proposed combination of Ducros-type sensor with a negative dilatation sensor removes unnecessary bulk viscosity within expansion and weakly compressible turbulence regions without shocks and allows it to localize near the shocks. It also eliminates the need for a wall-damping function for the bulk viscosity while simulating wall-bounded turbulent flows. For the numerical schemes used, better results are obtained without adding an explicit SGS model than with SGS model at moderate Reynolds number. Inclusion of a SGS model in addition to the low-pass filtering and artificial bulk viscosity results in additional damping of the resolved turbulence. However, investigations at higher Reynolds numbers suggest the need for an explicit SGS model. The flow statistics obtained using the second-order implicit time-integration scheme with three sub-iterations closely agrees with the explicit scheme if the maximum Courant–Friedrichs–Lewy is kept near unity.