The reality of artificial viscosity

Abstract

Artificial viscosity is used in the computer simulation of high Reynolds number flows and is one of the oldest numerical artifices. In this paper, I will describe the origin and the interpretation of artificial viscosity as a physical phenomenon. The basis of this interpretation is the finite scale theory, which describes the evolution of integral averages of the fluid solution over finite (length) scales. I will outline the derivation of finite scale Navier–Stokes equations and highlight the particular properties of the equations that depend on the finite scales. Those properties include enslavement, inviscid dissipation, and a law concerning the partition of total flux of conserved quantities into advective and diffusive components.

Appendix: Historical threads and terminology

In this paper, I have tried to create a continuous thread from the formative paper of von Neumann and Richtmyer through the evolution of turbulence models, the development of nonoscillatory numerical methods and finally to their fusion in ILES and the finite scale theory. Here, I present some additional detail of a more historical nature.

1.1 Artificial viscosity

The terminology “artificial viscosity” is, in my opinion, unfortunately pejorative. However, the term artificial dissipation originates on page 1 of the von Neumann–Richtmyer paper [1]. I have used the alternate description “inviscid dissipation” to emphasize that in the finite scale equations, the dissipated energy does not depend on physical viscosity. This does not contradict the statement that in nature, it is the physical viscosity that is the mechanism of dissipation. A more mathematical characterization of inviscid dissipation and its origin in the work of Onsager can be found in [19].

There have been many improvements to the formulation of artificial viscosity and the finite scale theory can provide a new perspective on these. For example, one significant idea was put forward by Bill Noh [22] in 1987, namely that one should also implement an artificial heat conduction in the energy equation. The artificial heat conduction effectively deals with the well-known problem of wall heating [23]. In (27), the finite scale theory also predicts an artificial heat flux and indicates that the energy gradient should have the same coefficient as the velocity gradient in the momentum equation.

In [1], the form of (1) is not justified. However, in another Los Alamos report from 1948 [24], recently released, Robert Richtmyer (as the sole author) derives the quadratic form, arguing that both weak and strong shocks should have the same width on the mesh, e.g., roughly three or four computational cells. In the sense that the computational cell is the “observer,” I would like to think that Richtmyer had an early intuitive grasp of finite scale theory.

1.2 Turbulence modeling

In Sect. 5, I noted several overlaps of the properties of the finite scale theory with those of conventional LES. Indeed, Smagorinsky [3] describes the initial implementation of his eddy viscosity as a tensor version of the von Neumann–Richtmyer viscosity suggested by J. Charney and N. Phillips to control unphysical oscillations (“noodling”) in early weather simulations. Also, the averaging operator defined in (7) is the top hat filter of LES when the filter width is identified with the cell size.

One may ask then whether the FSNS equations are similar to some existing model in conventional LES. In fact, there is a substantial similarity to the tensor diffusivity model of Leonard [25]. However, it is found that the tensor diffusivity model is not sufficiently dissipative and must be stabilized by an additional Smagorinsky term [26]. By contrast, ILES simulation is stable by construction so long as appropriate time step constraints are employed. I conjecture that the essential difference is the extra dissipation in the NFV schemes that is proportional to the computational time step. This point of view has been explored by Ristorcelli [27] in an article concerned with the dynamics of interscale energy transfer in ILES.

1.3 Form invariance and enslavement

Form invariance implies that there is a relation between the resolved and the unresolved scales of motion, that the effects of the unresolved scales can be modeled in terms of the macroscopic variables. I have used the term enslavement as appears in the mathematical theory of the inertial manifold [28, 29] where it has the stronger sense that the large (resolved) scales of motion control the small (unresolved) scales. A similar concept of control is part of Haken’s theory of Synergetics [30].

This idea is fundamental to the idea of artificial viscosity and appears in [1], where it is explicitly noted that the speed of a shock, the jump conditions, and the production of entropy are all determined by conservation and are independent of the viscosity; it is only in determining the width and shape of the shock that viscosity plays a determining role. A similar result in turbulence theory is expressed by Kolmogorov’s 4 / 5 theorem [31] where it is proved that the energy dissipated in a turbulent flow is independent of the viscosity.

1.4 “Whither” shock theory

It was at a conference at Cornell University, convened by John Lumley in 1989 to define new directions for turbulence modeling, where Jay Boris first introduced his concept of implicit large eddy simulation which he termed MILES [32]. I believe a similar crossroad lies ahead in the near future for the computational simulation of shocks. It has been known for more than 40 years that the Navier–Stokes equations do not correctly describe the width and shape of gaseous shock waves as measured in physical experiments [33, 34]. Higher-order expansions of Chapman–Enskog theory and of Grad moment methods have been equally unsuccessful [35]. However, simulations of the more fundamental Boltzmann equation using Direct Simulation Monte Carlo (DSMC) have very accurately reproduced the widths and shape of the experiments [36].

These results would indicate that it is the transition from Boltzmann to Navier–Stokes, i.e., the Chapman–Enskog expansion, that is unjustified. Chapman–Enskog assumes that the collision integral is the dominant term in the Boltzmann equation, so implying a flow that is near equilibrium. However, the high Reynolds number of a shock would indicate that it is the advective terms which dominate and that is the regime of the finite scale theory. In [37], I have made a first attempt to apply the finite scale theory to the Boltzmann equation. New issues arise as the Boltzmann equation describes the evolution of a probability distribution rather than a field variable and the coarse-graining now must have the character of an ensemble average rather than a spatial average. However, those preliminary results indicate that a term quadratic in the velocity gradient will appear in the coarse-grained equations that will produce wider shocks.