Artificial Viscosity Research Articles

Adaptive Total Lagrangian Eulerian SPH for high-velocity impacts

This paper presents a novel coupling between the Total Lagrangian and Eulerian formulations of the Smooth Particle Hydrodynamics (SPH) method. It is implemented such that a Total Lagrangian formulation can adaptively convert to an Eulerian formulation (referred to as AdapTLE in this work) and applied to high-velocity impact problems. The novel coupling makes use of the mixed kernel-and-gradient correction scheme, which is applied in order to improve consistency and conserve momentum. Additionally, the symmetrical terms, which are often found in SPH, are included in the derivation of the conservation equations to reduce the number of calculations required. A novel implementation of artificial viscosity is suggested for the Total Lagrangian formulation, which makes use of the kernel gradient in the undeformed state and therefore does not require neighbour lists to be updated. This form of artificial viscosity can be applied to a singularly Total Lagrangian formulation or a coupled one. Numerical examples including a patch test, and two-dimensional and three-dimensional high-velocity impact problems were simulated to evaluate this novel coupling method and demonstrate the benefits of AdapTLE, which was found to produce superior results to either a Total Lagrangian or Eulerian formulation independently. This was due to the retention of a Total Lagrangian formulation until severe distortions required the conversion to an Eulerian formulation, which significantly reduced the effect of the tensile instability in the latter.

Fully compressible simulations of waves and core convection in main-sequence stars

Context. Recent, nonlinear simulations of wave generation and propagation in full-star models have been carried out in the anelastic approximation using spectral methods. Although it makes long time steps possible, this approach excludes the physics of sound waves completely and requires rather high artificial viscosity and thermal diffusivity for numerical stability. A direct comparison with observations is thus limited. Aims. We explore the capabilities of our compressible multidimensional Seven-League Hydro (SLH) code to simulate stellar oscillations. Methods. We compare some fundamental properties of internal gravity and pressure waves in 2D SLH simulations to linear wave theory using two test cases: (1) an interval gravity wave packet in the Boussinesq limit and (2) a realistic 3 M⊙ stellar model with a convective core and a radiative envelope. Oscillation properties of the stellar model are also discussed in the context of observations. Results. Our tests show that specialized low-Mach techniques are necessary when simulating oscillations in stellar interiors. Basic properties of internal gravity and pressure waves in our simulations are in good agreement with linear wave theory. As compared to anelastic simulations of the same stellar model, we can follow internal gravity waves of much lower frequencies. The temporal frequency spectra of velocity and temperature are flat and compatible with the observed spectra of massive stars. Conclusion. The low-Mach compressible approach to hydrodynamical simulations of stellar oscillations is promising. Our simulations are less dissipative and require less luminosity boosting than comparable spectral simulations. The fully-compressible approach allows for the coupling of gravity and pressure waves in the outer convective envelopes of evolved stars to be studied in the future.

Global Weak Solutions for a Nonlinear Hyperbolic System

In this paper, we study the global existence of weak solutions for the Cauchy problem of the nonlinear hyperbolic system of three equations (1.1) with bounded initial data (1.2). When we fix the third variable s, the system about the variables ρ and u is the classical isentropic gas dynamics in Eulerian coordinates with the pressure function $$P(\rho ,s) = {{\rm{e}}^s}{{\rm{e}}^{ - {1 \over \rho }}}$$ , which, in general, does not form a bounded invariant region. We introduce a variant of the viscosity argument, and construct the approximate solutions of (1.1) and (1.2) by adding the artificial viscosity to the Riemann invariants system (2.1). When the amplitude of the first two Riemann invariants (w1(x, 0), w2(x, 0)) of system (1.1) is small, (w1(x, 0), w2(x, 0)) are nondecreasing and the third Riemann invariant s(x, 0) is of the bounded total variation, we obtained the necessary estimates and the pointwise convergence of the viscosity solutions by the compensated compactness theory. This is an extension of the results in [1].

The Lagrangian hydrodynamics code magma2

ABSTRACTWe present the methodology and performance of the new Lagrangian hydrodynamics code magma2, a smoothed particle hydrodynamics (SPH) code that benefits from a number of non-standard enhancements. By default it uses high-order smoothing kernels and wherever gradients are needed, they are calculated via accurate matrix inversion techniques, but a more conventional formulation with kernel gradients has also been implemented for comparison purposes. We also explore a matrix inversion formulation of SPH with a symmetrization in the particle indices that is not frequently used. We find interesting advantages of this formulation in some of the tests, for example, a substantial reduction of surface tension effects for non-ideal particle setups and more accurate peak densities in Sedov blast waves. magma2 uses artificial viscosity, but enhanced by techniques that are commonly used in finite-volume schemes such as reconstruction and slope limiting. While simple to implement, this approach efficiently suppresses particle noise, but at the same time drastically reduces dissipation in locations where it is not needed and actually unwanted. We demonstrate the performance of the new code in a number of challenging benchmark tests including, for example, multidimensional vorticity creating Schulz–Rinne-type Riemann problems and more astrophysical tests such as a collision between two stars to demonstrate its robustness and excellent conservation properties.

Linear stability analysis of RELAP5 two-fluid model in nuclear reactor safety results

System thermal-hydraulic code RELAP5 is based on a two-fluid, non-equilibrium, and non-homogeneous hydrodynamic model for simulation of transient two-phase behavior. The code model includes six governing equations to incorporate the mass, energy, and momentum of the two fluids. In this paper, linear stability analysis is performed to check the ill-posedness of the RELAP5 specific two-fluid model (TFM) for all normal and accident conditions of a standard pressurized water reactor (PWR). The analysis gives information about the soundness of the model and identifies the range of parameters where the solutions obtained from the model will be numerically convergent. The linear two-phase fluid dynamic stability (by dispersion analysis) of the RELAP5 one-dimensional two-fluid model and numerical stability of the difference equation formulation (by Von Neumann method) is presented. The present analysis shows that the two-fluid model becomes ill-posed for some fluid conditions, where the results are less accurate, so sensitivity analysis plays an important role. The ill-posed nature implies that results thus obtained (by finite difference method) have to be interpreted carefully because of the sensitive nature of reactor safety analysis. It is also identified that the variation in various parameters (like slip ratio, system pressure, void fraction, and phasic velocities) can affect the error growth rates. It has been demonstrated that the basic system of one-dimensional two-phase flow equations, that possesses complex characteristics, exhibits unbounded instabilities in the short-wavelength limit and constitutes an improperly posed initial value problem. The semi-implicit numerical method, which is unconditionally stable for hyperbolic systems, becomes unstable for non-hyperbolic systems. For some of the fluid conditions, even after the introduction of artificial viscosity terms (in the difference equation formulation) that damp the high-frequency spatial components of the solution, are not sufficient for regularization of the two-fluid model. Thus, there is a need for the addition of newer terms, e.g. bubble collision, so that the existing incomplete model provides better results. It is also demonstrated that the basic TFM of RELAP5 with additional collision term makes the system unconditionally well-posed which are originally conditionally well-posed. Excellent agreement is obtained between the predicted and computed growth rates of harmonic disturbances. It is found that the instability associated with the two-fluid model discretized system of equations is related to the instability associated with the ill-posedness of the two-fluid model but is quantitatively different.

A Godunov-type tensor artificial viscosity for staggered Lagrangian hydrodynamics

This paper describes a tensor artificial viscosity for staggered Lagrangian hydrodynamics. Specifically, two viscous tensors are constructed using a discrete velocity gradient method. Under the condition of a piecewise constant distribution of velocity in each subcell, the Generalized Riemann Invariant relation, in the spirit of the Godunov methods, is applied to determine the coefficients of the viscous tensors. The artificial viscosity is found to be dissipative. Besides, the cylindrical symmetry of the developed artificial viscosity is demonstrated in an equi-angular polar grid. Typical numerical cases with strong shocks show that the tensor artificial viscosity is robust and performs well in different grid types.

A semi‐implicit stabilized particle Galerkin method for incompressible free surface flow simulations

SummaryA new particle Galerkin method is introduced to solve the Naiver‐Stokes equations in a Lagrangian fashion. The present method aims to suppress key numerical instabilities observed in the strong form Lagrangian particle methods such as smoothed particle hydrodynamics (SPH), incompressible SPH, and moving particle semi‐implicit for incompressible free surface flow simulations.It is well‐known that strong form Lagrangian particle methods usually rely on ad hoc particle stabilization techniques based on particle shifting, artificial viscosity, or density‐invariant condition due to some formulation inconsistency issues. In the present method, we introduce a momentum‐consistent velocity smoothing algorithm which is used to combine with the second‐order rotational incremental pressure‐correction scheme to stabilize the pressure field as well as to enforce the consistency of Neumann boundary condition. To further impose slip‐free or nonslip boundary conditions for the fluid flow, a penalty method which is free of ghost or dummy particles is developed. Finally, a particle insertion‐deletion adaptive scheme is proposed when the violent fluid flow is considered. Four numerical examples are studied to validate the accuracy and stability of the present method.

A Simple, Entropy-based Dissipation Trigger for SPH

Abstract Smoothed particle hydrodynamics (SPH) schemes need to be enhanced by dissipation mechanisms to handle shocks. Most SPH formulations rely on artificial viscosity and, while this works well in pure shocks, attention must be paid to avoid dissipation where it is not wanted. Commonly used approaches include limiters and time-dependent dissipation parameters. The former try to distinguish shocks from other types of flows that do not require dissipation while in the latter approach the dissipation parameters are steered by some source term (“trigger”) and, if not triggered, they decay to a predescribed floor value. The commonly used source terms trigger on either compression, , or its time derivative. Here we explore a novel way to trigger SPH-dissipation: since an ideal fluid conserves entropy exactly, its numerical nonconservation can be used to identify “troubled particles” that need dissipation because they either pass through a shock or become noisy for other reasons. Our new scheme is implemented into the Lagrangian hydrodynamics code MAGMA2 and is scrutinized in a number of shock and fluid instability tests. We find excellent results in shocks and only a very moderate (and desired) switch-on in instability tests. The new scheme is robust, trivial to implement into existing SPH codes, and does not add any computational overhead.

Experimental Analysis and Numerical Simulation of Hydraulic Jump

Saint Venant equations are numerically solved to simulate the formation of hydraulic jump in a rectangular channel having a small bed slope. The MacCormack’s scheme is used for the solution by applying specified initial and boundary conditions until a steady state flow is reached. The location of the hydraulic jump is determined as a part of these computations. The artificial viscosity technique should be used in the computations to dampen the superior oscillations near the steep gradient of the simulated hydraulic jump. Twenty laboratory experiments were carried out for verification of the numerical model. The upstream Froude number for these experiments ranged from 2.17 to 7.0 in three different bed slopes 0, 0.02174, 0.0475. The simulated hydraulic jump profiles using the MacCormack’s scheme shows a good agreement with the experimental data. An empirical equation was developed to determine the location of hydraulic jump using regression analysis based on simulated data. Software based on Computational Fluid Dynamics (CFD) was also used to simulate two of these experiments. The results obtained from CFD analysis matched fairly with the experimental results.

Features of High-Velocity Pulsating Liquid Jets

Processes proceeding in a pulsed liquid-jet generator were investigated in the acoustic and gasdynamic approximations. In the acoustic approximation, the problem on the one-dimensional movement of the liquid in the generator with no movement of its piston and no liquid outflow from it was considered. In the gasdynamic approximation, the problems on one-dimensional and axisymmetric liquid flows in the generator were solved numerically with regard for the movement of the generator piston, the liquid outflow from the generator, and the cavitation of the liquid in it. A one-dimensional liquid flow in the generator was calculated using the potential method, a scheme with introduction of artificial viscosity, the Godunov method, and modifications of this method. Results of calculations of the movement of the liquid in the generator by different models were compared, and corresponding conclusions have been made.

Numerical and Experimental Study of Classical Hydraulic Jump

The present work is an effort to simulate numerically a classical hydraulic jump in a horizontal open channel with a rectangular cross-section, as far as the jump location and free surface elevation is concerned, and compare the results to experiments with Froude numbers in the range 2.44 to 5.38. The governing equations describing the unsteady one-dimensional rapidly varied flow have been solved with the assumption of non-hydrostatic pressure distribution. Two finite difference schemes were used for the discretization of the mass and momentum conservation equations, along with the appropriate initial and boundary conditions. The method of specified intervals has been employed for the calculation of the velocity at the downstream boundary node. Artificial viscosity was required for damping the oscillations near the steep gradients of the jump. An iterative algorithm was used to minimize the difference of flow depth between two successive iterations that must be less than a threshold value, for achieving steady state solution. The time interval varied in each iteration as a function of the Courant number for stability reasons. Comparison of the numerical results with experiments showed the validity of the computations. The numerical codes have been implemented in house using a Matlab® environment.

The study on performances of kernel types in solid dynamic problems by smoothed particle hydrodynamics

As the earliest meshless method, smoothed particle hydrodynamics (SPH) has been applied in solid dynamics because of its great potentials in simulating extremely large deformations. In terms of the essence of SPH, it uses a kernel function for numerical approximations. Some studies demonstrate mathematically that the types of the kernel function directly influence the stability and overall accuracy of SPH simulation. However, less attention is paid to the influences derived from kernels, particularly on simulating dynamic solids problems by SPH. In addition, some researches introduce that there are strong relations between SPH kernels and tensile instability which is one of the shortcomings in SPH method, a kind of particles clumping phenomenon probably leading to some unphysical cracks when the solid material is largely stretched or compressed. At present, no method exists to completely avoid these instabilities, although a few corrections like artificial viscosity, artificial stress, corrective SPH or the Godunov-type SPH have been proposed.In this paper, the performances of SPH method with different kernels are studied, including classical types and several new ones proposed recently. Combined with some corrected techniques, the suitability of these kernels in SPH method is discussed in solid dynamic problems like bending deformation of elastic beam and some tests about elastic–plastic deformation in impact problems.

A new approach for numerical-diffusion control of flux-vector-splitting schemes for viscous-compressible flows

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.

Defect correcting extrapolation technique for Oseen and Navier–Stokes Flows

This report presents defect correcting extrapolation technique (DCE), an alternative approach for defect correction (DC). In contrast to conventional DC methods, which always solve for separate correction steps, DCE instantly assumes weighted linear combinations of several artificial viscosity approximations as a corrected solution. This way, it merely requires solving an identical problem with varying viscosities. The theory and computations show that DCE uncovers ”hidden turbulence” in artificially-diffused laminar solutions, and recovers their energy and turbulent characteristics.

Cell-centered Lagrangian Lax-Wendroff HLL hybrid scheme in cylindrical geometry

Lagrangian hydrodynamics as described by the Euler equations is treated by an improved version of the basic predictor-corrector Lax-Wendroff method that also has added HLL-type dissipative fluxes, including both artificial viscosity and energy dissipation. The method in Cartesian geometry is enhanced by a different weighting of the conservative variables in the predictor and a new treatment of material interfaces. The basic method is extended to cylindrical r,z geometry, where it satisfies the geometric conservation law and keeps exact spherical symmetry on equiangular polar meshes. The added viscosity does not preserve symmetry, so in order to achieve that we have added a symmetry correction. Numerical hydrodynamic tests, including the Noh, Sedov, spherical Sod, free expansion and Kidder problems show a reasonable performance of the method. In addition, the spherical Noh problem was simulated on an initially rectangular mesh with a very good result regarding its symmetry.

A new two-level defect-correction method for the steady Navier–Stokes equations

A new defect-correction method based on subgrid stabilization for the simulation of steady incompressible Navier–Stokes equations with high Reynolds numbers is proposed and studied. This method uses a two-grid finite element discretization strategy and consists of three steps: in the first step, a small nonlinear coarse mesh system is solved, and then, in the following two steps, two Newton-linearized fine mesh problems which have the same stiffness matrices with only different right-hand sides are solved. The nonlinear coarse mesh system incorporates an artificial viscosity term into the Navier–Stokes system as a stabilizing factor, making the nonlinear system easier to resolve. While the linear fine mesh problems are stabilized by a subgrid model defined by an elliptic projection into lower-order finite element spaces for the velocity. Error bounds of the approximate solutions are estimated. Algorithmic parameter scalings are derived from the analysis. Effectiveness of the proposed method is also illustrated by some numerical results.

Level-set methods applied to the kinematic wave equation governing surface water flows

Critical issues arising from the governing nonlinear equations in surface water hydrodynamic include discontinuities in water surface levels, blow-up of water surface gradient, and treatment of dry beds or zero water depths, involving mathematical problems related to functional regularities of unknown variables such as the water depth. The level-set method is a powerful approach to relax requirements for functional regularities of unknowns in nonlinear partial differential equations of first order. In this study, the level-set method is applied to the one-dimensional kinematic wave equation, resulting in a linear level-set equation of the first order in a two-dimensional space to tackle dry beds. The zeros of the level-set function represent the water depths. Hypothesizing that the level-set function is continuous in the domain, it is numerically computed with a characteristic method. The development of overturning is regulated with singular viscosity regularization (SVR), whose effect is to relocate the zeros of the level-set function close to the exact positions of the shock fronts in dam-break problems. The method is firstly verified with the explicitly known exact solutions of primitive dam-break problems, optimizing a parameter of SVR. Then, abrupt water release from Chan Thnal Reservoir, Kampong Speu Province, Cambodia into an initially dry bed of its irrigation canal system is simulated as a practical demonstrative example. In contrast to most of the available software tools using either the shallow water equations with some artificial viscosity or the diffusion wave approximation, the proposed method turns out to be free from spurious diffusive deformation of water surfaces even if relatively coarse computational mesh is used.

Locally adaptive artificial viscosity strategies for Lagrangian hydrodynamics

To accurately model inviscid flow with shock waves using staggered-grid Lagrangian hydrodynamics, artificial viscosity is introduced to convert kinetic energy into internal energy, thereby providing a mechanism to generate the required entropy increase across shocks. In this paper, we propose a new method for constructing an adaptive, artificial viscosity in the context of one-dimensional, staggered-grid Lagrangian hydrodynamics. Our adaptive, artificial viscosity is defined in terms of two parameters that depend on density locally in a neighborhood around a shock, and hence, vary cell-by-cell. Our methodology is based on building a reference set of pre-computed optimal, globally constant artificial viscosity coefficients for a family of isolated shock test problems. For arbitrary flows, the evaluation of the unknown coefficients is automated by first estimating shock intensity locally, and second computing the corresponding adaptive parameter value via interpolation over the data from the pre-computed reference set of optimal, globally constant coefficient values. To illustrate the performance of our new approach, we compare our results against two existing methods. The first method is a limiter-based approach, which relies on estimating velocity gradients of the flow, and the second method is utilized in several commercial codes. We demonstrate that our new adaptive methodology produces more accurate results for a variety of tests with propagating shock waves, as well as for the aforementioned family of isolated shock problems.

A matrix-free hyperviscosity formulation for high-order ALE hydrodynamics

The numerical approximation of compressible hydrodynamics is at the core of high-energy density (HED) multiphysics simulations as shocks are the driving force in experiments like inertial confinement fusion (ICF). In this work, we describe our extension of the hyperviscosity technique, originally developed for shock treatment in finite difference simulations, for use in arbitrarily high-order finite element methods for Lagrangian hydrodynamics. Hyperviscosity enables shock capturing while preserving the high-order properties of the underlying discretization away from the shock region. Specifically, we compute a high-order term based on a product of the mesh length scale to a high power scaled by a hyper-Laplacian operator applied to a scalar field. We then form the total artificial viscosity by taking a non-linear blend of this term and a traditional artificial viscosity term. We also present a matrix-free formulation for computing the finite element based hyper-Laplacian operator. Such matrix-free methods have superior performance characteristics compared to traditional full matrix assembly approaches and offer advantages for GPU based HPC hardware. We demonstrate the numerical convergence of our method and its application to complex, multi-material ALE simulations on high-order (curved) meshes.

An analysis of a linearly extrapolated BDF2 subgrid artificial viscosity method for incompressible flows

This report extends the mathematical support of a subgrid artificial viscosity (SAV) method to simulate the incompressible Navier-Stokes equations to better performing a linearly extrapolated BDF2 (BDF2LE) time discretization. The method considers the viscous term as a combination of the vorticity and the grad-div stabilization term. SAV method introduces global stabilization by adding a term, then anti-diffuses through the extra mixed variables. We present a detailed analysis of conservation laws, including both energy and helicity balance of the method. We also show that the approximate solutions of the method are unconditionally stable and optimally convergent. Several numerical tests are presented for validating the support of the derived theoretical results.