Artificial viscosity

W.D. Schultz, A Tensor Artificial Viscosity for Numerical Hydrodynamics, J. Math. Phys. 5, Na. 1 (1964). 

L.G. Margolin, A Centered Artificial Viscosity for Cells with Large Aspect Ratio, UCRL-53882, Lawrence Livermore National Laboratory, Livermore, CA, 1988. 

W.F. Noh, Errors for Calculations of Strong Shocks Using an Artificial Viscosity and an Artificial Heat Flux, J. Comput. Phys. 72 (1978). 

A drawback of the Lagrangean artificial-viscosity method is that, if sufficient artificial viscosity is added to produce an oscillation-free distribution, the solution becomes fairly inaccurate because wave amplitudes are damped, and sharp discontinuities are smeared over an increasing number of grid points during computation. To overcome these deficiencies a variety of new methods have been developed since 1970. Flux-corrected transport (FCT) is a popular exponent in this area of development in computational fluid dynamics. FCT is generally applicable to finite difference schemes to solve continuity equations, and, according to Boris and Book (1976), its principles may be represented as follows. 

In the earliest applications of numerical methods for the computation of blast waves, the burst of a pressurized sphere was computed. As the sphere s diameter is reduced and its initial pressure increased, the problem more closely approaches a point-source explosion problem. Brode (1955,1959) used the Lagrangean artificial-viscosity approach, which was the state of the art of that time. He analyzed blasts produced by both aforementioned sources. The decaying blast wave was simulated, and blast wave properties were registered as a function of distance. The code reproduced experimentally observed phenomena, such as overexpansion, subsequent recompression, and the formation of a secondary wave. It was found that the shape of the blast wave at some distance was independent of source properties. 

The code reproduced shock-jump conditions well, but many details in the solution were lost because of the smearing effect of artificial viscosity. 

Recently various computational hydro codes have been adapted to the determination of underwater shock parameters. A Lagrangian code (with artificial viscosity) augmented by a sharp shock routine was used by Sternberg Hurwitz (Ref 12) to generate the curves shown in Figs 22,23 24... 

The main reason why numerical waves have not been discussed much in the CFD community is that most RANS codes use excessive artificial viscosity and large turbulent viscosity levels (due to turbulence models) which kills all numerical waves. They also kill all acoustic waves and all hydro-dynamic modes and cannot be used for the present needs of combustion research. Methods which can compute accurately waves in reacting flows must use centered schemes and LES (or DNS) formulations in order to avoid damping all waves (physical and numerical). A convenient way to illustrate this point is to compare the various viscosities pla3ung a role in a CFD code ... 

Many CFD codes also add an artificial viscosity / . This viscosity can be explicit or it can be hidden, for example in the case of upwind schemes. An important dissipation is also introduced by large time steps and implicit schemes which are commonly used in RANS. 

Table 8.2 shows typical levels of physical viscosity, turbulent viscosity i/t and artificial viscosity i/a reached in a combustion chamber for a standard regime. All viscosities are scaled by the physical viscosity. Note that viscosity affects all scales and not only the small scales. For example, acoustic waves are very strongly dissipated in a RANS code because the turbulent viscosity acts on them too. This is a collateral effect of turbulence models formulated using turbulent viscosities but it implies that such methods cannot be used for the present objectives. 

A less pleasant implication of Table 8.2 is that, as soon as high-fidelity methods such as DNS or LES are developed, they have to avoid large values of turbulent and artificial viscosities. This requires small mesh sizes, high-order schemes, small time steps [268 362 340]. But even after all these improvements, these methods will remain sensitive to numerical waves [363]. In DNS or LES, numerical waves are intrinsic elements of the simulation and must be controlled by something other than viscosity. This usually means significant improvements of initial and boundary conditions and a careful... 

As is usual with centered schemes, artificial viscosity is used. This is done with great care to preserve accuracy, applying it very locally (using specific sensors) and with the minimum level of viscosity [346]. 

A Taylor series analysis on the ID transport equation shows that the transient artificial viscosity coefficients for explicit upwind differencing is given by [157, 158] ... 

Number of stations at which data is taken Artificial viscosity coefficient Grid mesh (inches)... 

Simulating shock phenomenon is not the strongest part of SPH. SPH relies on the use of artificial viscosity, in order to resolve shocks. Typical shock resolution is then around three smoothing lengths. State-of-the-art finite difference schemes, like the Piecewise Parabolic Method (PPM) [98], can achieve far better (at comparable resolution). Typically, SPH needs numbers of particles in excess of 100,000, when shocks are present. 

Experiments have not yet confirmed this model. However, computer simulations have been done that illustrate the behavior. These use a version of the SALE code developed at Los Alamos for two-dimensional, time-dependent compressible flow in either Eulerian or Lagrangian coordinates. It incorporates artificial viscosity to accommodate shock fronts. The equation of state used is the Van der Waals equation as generalized by Callen to include the internal energy. ... 

In fact, it is necessary to introduce artificial viscosity to reduce some "numerical noises". The term added to the second member of the conservative form is... 

Particular care must be paid to the choice of the differencing scheme used for the solution of the governing equations. The choice is not univocal, since a tradeoff between accuracy and computational cost exists. A first-order approximation for the convection term is the most stable approach, however if the target is the prediction of the mixing efficiency between ammonia and exhaust gas stream, then the solution will be affected by a significant amount of artificial viscosity, comparable to the turbulent one [33]. This issue can be overcome resorting to more... 

The modeling of detonation, which is a reactive shock, requires a number of tricks to make an assembly of finite elements behave acceptably when the parameters are evaluated and advanced with finite time steps. The two major variations involve artificial viscosity and rezoning, both discussed in the descriptions of KIVA (1) and its predecessor ALE (2). 

H. M. Sternberg and W. A. Walker, Artificial Viscosity Method Calculation of an Underwater Detonation , Fifth International Symposium on Detonation, ACR-184, 597 (1970). 

Numerical reactive hydrodynamic codes such as SIN, TDL or 2DE include the Forest Fire decomposition rate. For unresolved burns, it is necessary for the decomposition front of a detonation wave to occur over several computer meshes or cells so that the physics of the flow, the shock jump conditions, are properly described. Historically this was accomplished by adjusting the artificial viscosity so that the burn occurred over about 3 cells. If the mesh size changed, a new viscosity coefficient was determined empirically that would result in a realistic burn. As shown in the movie on the CD-ROM at /MOVIE/VISC.MVH, if there is insufficient viscosity, one obtains a reactive front with a peak that oscillates. If there was too much viscosity, a flat pressure profile occurs at the front. The problem is not unique to Forest Fire as other burn rates such as Arrhenius have the same numerical problems when numerically unresolved in reactive hydrodynamic codes. 

Bulk or Artificial Viscosity XX Viscosity Deviator XZ Viscosity Deviator Gas Constant ZZ Viscosity Deviator XX Elastic Stress Deviator XZ Elastic Stress Deviator ZZ Elastic Stress Deviator Temperature "K... 

---