Finite-scale equations for compressible fluid flow

L.G. Margolin

Abstract

Finite-scale equations (FSE) describe the evolution of finite volumes of fluid over time. We discuss the FSE for a one-dimensional compressible fluid, whose every point is governed by the Navier–Stokes equations. The FSE contain new momentum and internal energy transport terms. These are similar to terms added in numerical simulation for high-speed flows (e.g. artificial viscosity) and for turbulent flows (e.g. subgrid scale models). These similarities suggest that the FSE may provide new insight as a basis for computational fluid dynamics. Our analysis of the FS continuity equation leads to a physical interpretation of the new transport terms, and indicates the need to carefully distinguish between volume-averaged and mass-averaged velocities in numerical simulation. We make preliminary connections to the other recent work reformulating Navier–Stokes equations.

1. Introduction

Finite-scale equations (FSE) describe the evolution of finite volumes of fluid over time. Here, we introduce the FSE for parcels of fluid whose every point is governed by the compressible Navier–Stokes equations. This reformulation of classical fluid dynamics offers useful insights, especially in the context of numerical simulations of fluid flow.

In previous work, Margolin & Rider (2002) and Margolin et al. (2006; hereafter, MR 2002 and MRG 2006) described the derivation of FSE that are based on Burgers' equation and on incompressible Navier–Stokes equations. In both cases, the FSE consist of the underlying equations augmented (at lowest order) by new terms quadratic in the length scales over which the averaging is performed. In those papers, the analysis was used as a rationale to justify the numerical technique of implicit large eddy simulation (ILES); see Grinstein et al. (2007) for a detailed description and applications of this technique. Similarly, a principal goal of this paper is to provide a rationale for the ILES approach applied to compressible turbulence. However, we emphasize that the FSE themselves are an analytic result, independent of numerical considerations; in particular, the FSE are continuous equations, not discrete.

Our principal result in considering Burgers' and the incompressible flow equations was the appearance of a new momentum transport term. This term arises directly from the nonlinearity of the advective term after averaging in space. In MR (2002), we discussed the similarity of that term to truncation terms that are inherent in nonoscillatory finite-volume (NFV) approximations widely used in numerical simulations in a Eulerian framework, and to the artificial viscosity that is added explicitly to simulations in a Lagrangian framework. This led to the suggestion that these terms have a physical, rather than a numerical, origin. In MRG (2006), we discussed the further similarity of the new momentum transport term in multiple spatial dimensions to a class of subgrid scale models used in large eddy simulation (LES). Similar terms will arise in our consideration of the momentum and energy equations for compressible flow.

In this paper, we will be concerned with the analysis of the FSE as a model for the numerical simulation of compressible flows with high Reynolds number—e.g. flows with turbulence and/or shocks. In §2, we give a brief summary of the derivation and then present the averaged equations. In §3, we focus on the terms that arise from the nonlinearity of the advective terms. Consideration of the FS continuity equation will require us to generalize the concept of Lagrangian volumes and will lead to a physical explanation of the new transport terms for momentum and energy that consequently arise. In §4, we focus on the terms that arise from the nonlinearity of the pressure–velocity work terms. We also discuss the role of unresolved kinetic energy and its relationship to thermodynamics. In §5, we summarize the overlap between the FSE and the current state of the art of finite-volume Lagrangian/arbitrary Lagrangian–Eulerian (ALE) and Eulerian methodologies.

Many of the new terms in our FSE have their origin in distinguishing between two velocities that naturally appear in our analysis—the average or advective velocity and the momentum or Favre-averaged velocity. This distinction is also made in other recent reformulations of Navier–Stokes equations. In appendix A, we briefly discuss two of these—the volume transport concept and the Navier–Stokes α-model. We do not suggest the equivalence of these theories to FSE, but merely point out the similarity of consequences that follow from the distinction of these two velocities.

2. Coarse graining

We begin this section by offering a concise statement of the theme of this paper. We consider a fluid in one dimension whose every point is described by the compressible Navier–Stokes equation. In conservative form, these are as follows:Embedded Image(2.1)Here, ρ, u, E have their usual meanings of density, velocity and specific total energy. The pressure, p, is the sum of the thermodynamic pressure plus the viscous pressure, η(∂u/∂x), where η is the dynamic viscosity. Also, q is the heat flux. At this point, we make no particular assumptions about the equation of state or the heat flux, but we will return to this point in some detail in §4.

We define an averaging operator Embedded Image for any variable Χ(x, t)Embedded Image(2.2)and pose the question: what equations govern the evolution of Embedded Image, Embedded Image and Embedded Image? An answer to this question was given in MR (2002) for Burgers' equation in one dimension and also in MRG (2006) for incompressible Navier–Stokes in two dimensions. We termed these FSE. Each of these derivations followed a similar path, which we outline here.

The derivation begins by considering scales of length (L0) for which the flow is smooth, by which we mean the dependent variables can be expanded in a convergent Taylor series. Such scales always exist, due to the presence of molecular viscosity. In this case, the integrals over the variables, including nonlinear terms, can be evaluated directly, leading to the equations for Embedded Image, etc.; here, we explicitly indicate dependence on the averaging scale.

This derivation is ultimately not justified for large enough scales in the case of a flow with shocks or turbulence. To proceed, we note that the averaging process itself produces smoother fields. We quantify this by assuming that the variables averaged over any scale of (L) are smooth enough to be expanded in convergent Taylor series on scales of (2L). Next, we proceed by induction. We assume that the equations written for variables averaged over (L0) continue to hold for variables averaged over (L) when L0L, and then we prove that this implies that the equations for the variables averaged over (2L) have the same form when L→2L.

The net result of this derivation (MR 2002) is that for any quantities A and BEmbedded Image(2.3)where Ax≡(∂A/∂x) and HOT are higher order terms, e.g. Embedded Image, etc.1

We choose as our primary variables averaged density Embedded Image, momentum density Embedded Image, energy density Embedded Image and pressure Embedded Image. The FSE to order Embedded Image are as follows (from this point on, we refrain from writing HOT, implicitly recognizing that all of our equations are a second-order truncation of an infinite set of terms):Embedded Image(2.4)Embedded Image(2.5)Embedded Image(2.6)In these equations, we recognize the appearance of new terms of order Embedded Image that describe the transport of mass, momentum and energy. These terms originate specifically due to the difference of u and Embedded Image. Note that an evaluation of equation (2.2) in smooth regions of the flow where a Taylor expansion of the velocity is valid yieldsEmbedded Image(2.7)where we have replaced Embedded Image to Embedded Image.

We close this section with the observation that the FSE are equivalent in physical content to Navier–Stokes; they are derived from Navier–Stokes and, conversely, Navier–Stokes can be recovered in the limit L→0.

3. Analysis of the FSE

(a) The continuity equation

We begin our analysis with the continuity equation (2.4) and recognize an immediate issue—the appearance of Embedded Image and its derivatives. Because we have chosen momentum as a primary variable, Embedded Image cannot be a prognostic variable. However, we note that by equation (2.3)Embedded Image(3.1)We define a ‘momentum’ velocity Embedded Image asEmbedded Image(3.2)and rewrite the continuity equation in more familiar form asEmbedded Image(3.3)The meaning of this equation is as follows: when the boundaries of a parcel of fluid are moved with the momentum velocity Embedded Image, then the total mass within the parcel remains constant. This does not preclude the flux of mass across the boundary, but rather means that the integral of the flux around the boundary is zero. It is simple to estimate the exchange mass flux in one dimension as Embedded Image. The flux of mass implies a concomitant flux of momentum and energy, and the integrals of these fluxes do not in general vanish. That is, the net zero exchange of mass by a volume with its neighbours will lead to a non-zero exchange of momentum and energy.

(b) The momentum and total energy equations

We now rewrite the momentum equation (2.5) in terms of Embedded Image asEmbedded Image(3.4)where we have used Embedded Image in collecting the second-order terms. The new term, proportional to L2, is the net exchange of momentum referred to in §3a. Its origin lies in two sources, the first being the difference between the ‘true’ advective velocity u and the averaged advective velocity Embedded Image, equation (2.7), and the second being the difference between advective velocity Embedded Image and the momentum velocity Embedded Image, equation (3.2). We note that the first difference is present in both compressible and incompressible flows. However, the second difference is proportional to the gradient of density and so vanishes in incompressible flow.

Similarly, we rewrite the energy equation in terms of Embedded Image asEmbedded Image(3.5)where Embedded Image.2 Here also, we identify a new term representing energy transport (kinetic and internal) associated with the exchange of mass within the volume with its neighbours. The fact that it is the same exchange mass flux that transports both momentum and energy leads to a simple relationship between the respective coefficients of Embedded Image and Embedded Image—they are the same! This relationship is a special case of an Onsager reciprocity relationship.

In this section, we have analysed the properties of the FSE from the point of view of conservation and the convective fluxes. Next, we turn our attention to thermodynamic issues.

4. Energetics

(a) The internal energy equation

In continuum Navier–Stokes theory, one constructs an equation for internal energy in two steps: one manipulates the momentum equation to form an equation for kinetic energy, which one then subtracts from the total energy equation. The internal energy equation is redundant, i.e. internal energy can always be found as the difference of the total and kinetic energies. However, in many numerical algorithms for simulating high-speed flow, there are advantages to solving the internal energy equation instead of the total energy equation (Margolin & Shashkov 2004).

Directly averaging the internal energy equation for Embedded Image yieldsEmbedded Image(4.1)where Embedded Image.

We derive a FS equation for macroscopic kinetic energy Embedded Image by multiplying the momentum equation (3.4) by Embedded Image. Using the identity Embedded Image, we findEmbedded Image(4.2)From equations (3.5), (4.1) and (4.2), we see Embedded Image. Since at the continuum level Embedded Image, we must have Embedded Image. The issue is that Embedded Image.

(b) Resolved and unresolved kinetic energy

From a macroscopic point of view (i.e. from the point of view of an observer whose instruments can only resolve motion on scales greater than L), the cascade of fluid motions to scales smaller than L in the FSE appears to dissipate macroscopic kinetic energy. However, from the continuum viewpoint, this unresolved kinetic energy must continue its cascade down in scale until it is small enough to be dissipated by viscosity, and so does not affect the thermodynamic pressure immediately. This situation has been long recognized in the turbulence simulation community where an additional partition of the total energy, turbulence kinetic energy or TKE, is recognized, and a transport equation for turbulent kinetic energy is often used to represent its evolution; see, for example, the discussion in Pope (2000, pp. 124–128).

To proceed, we formally recognize the existence of an additional partition Embedded Image of the total energy, writing Embedded Image. (We write Embedded Image to emphasize that our discussion is not restricted to turbulent flows.) We can form the equation for the unresolved kinetic energy Embedded Image by subtracting the sum of equations (4.1) and (4.2) from equation (3.5):Embedded Image(4.3)In equation (4.3), the work term associated with the new momentum transport is seen to be a source term to the unresolved kinetic energy partition.

From its definition, Embedded Image is a quantity of Embedded Image, and in factEmbedded Image(4.4)The significance of equation (4.4) is that it is not necessary to solve an additional prognostic equation for Embedded Image, in contrast to the practice in turbulence transport theory.

(c) Enhanced momentum and heat fluxes

If we consider the density, momentum and total energy equations as fundamental, we see that an alternative interpretation of the new momentum exchange term is as a nonthermodynamic contribution to the pressure. That is, in the momentum equation (3.4), we could define an enhanced pressureEmbedded Image(4.5)In the energy equation, we substitute Embedded Image (ignoring Embedded Imagex as higher order)Embedded Image(4.6)Indeed, this is exactly how the artificial viscosity is implemented in high-speed flow codes. This reformulation simplifies the appearance of the FS momentum and energy equations, but does not simplify the internal energy equation. Instead, we now define a new variable Embedded Image. It is readily seen that Embedded Image obeys the equationEmbedded Image(4.7)In other words, in high-speed flow codes, it is Embedded Image that is actually calculated, not Embedded Image. This has important implications for how the thermodynamic equation of state should be evaluated in those codes.

Before proceeding to our final step of defining an enhanced heat flux, we consider the nature of the pressure–velocity average Embedded Image by invoking the assumption of a specific thermodynamic equation of state, the ideal gas. Here,Embedded ImageIt is straightforward to showEmbedded Image(4.8)That is, the average of the product of pressure and velocity (e.g. work) is the product of the averages plus a term identical to the heat exchange term except for dimensionless coefficient. This specific result for an ideal gas leads us to interpret Embedded Image more generally as another contribution to the heat flux.3

We define an enhanced heat flux byEmbedded Image(4.9)to derive our final result for the FSE as follows:Embedded Image(4.10)Embedded Image(4.11)Embedded Image(4.12)Embedded Image(4.13)Equations (4.12) and (4.13) may be replaced byEmbedded Image(4.14)

Embedded Image(4.15)

5. Discussion

(a) FSE results

Here, we summarize the major differences between the continuum Navier–Stokes equations and their FS counterparts. The principal difference lies in the presence of new transport terms for momentum and energy. Some of the new transport terms arise from the nonlinearity of the advective terms, as was the case in the incompressible equations (MRG 2006). However, additional terms also arise from the nonlinearity of the pressure–velocity terms (thermodynamic work). Furthermore, the FSE predict definite relationships among all these transport coefficients analogous to Onsager reciprocity relationships. The FSE also allow an estimate of the unresolved kinetic energy in terms of the volume-averaged variables, equation (4.4). This energy, dissipated from the resolved kinetic energy field, does not contribute to the thermodynamic pressure immediately; this effect is probably not important in shocks, but is important in turbulent flows.

(b) Numerical simulation

As stated in the introduction, we are interested in the FSE as an alternative model for numerical simulation. Here, we discuss the overlap of two modern approaches to simulating compressible flow with the FSE.

For compressible flow, especially for high Reynolds number flows with shocks and/or turbulence, most simulation codes are based on finite-volume approximations; this has dictated our choice of the fundamental variables for the FSE as the densities of the conserved variables—Embedded Image, Embedded Image and Embedded Image. For flows with multiple materials and strong shocks, and when vorticity does not play an important role, a Lagrangian or ALE formulation has considerable advantages (Margolin 1997). When vorticity becomes important, e.g. when the flow becomes turbulent, Lagrangian meshes tangle and a Eulerian formulation becomes preferable. Simulation codes may also be distinguished by the data structures of the computational variables. Lagrangian/ALE codes and many Eulerian codes are based on a staggered mesh where thermodynamic variables are placed at cell centres, but velocity is placed at the cell vertices. Other Eulerian codes, principally those based on Godunov-type methods, have all variables colocated at the cell centres. See Grinstein et al. (2007, ch. 4) for a detailed discussion of these.

Most modern Eulerian and ALE codes use nonoscillatory differencing to approximate the advective terms. Nonoscillatory approximation invokes mathematical notions such as the preservation of sign or of monotonicity to modify (limit) the advective fluxes. In MRG (2006), modified equation analysis was used to show that the combination of nonoscillatory differencing and finite-volume approximation (NFV) in the MPDATA advection scheme (Smolarkiewicz & Margolin 1998) effectively solves the FS momentum equation rather than the Navier–Stokes momentum equation. This ‘implicit’ appearance of the FS momentum exchange terms was offered as a rationale for the modelling technique of ILES. We expect the NFV differencing of the energy equation will produce similar terms to the FS energy equation (3.5). However, as noted above, not all of the new FSE transport terms of energy arise from the advective terms. Furthermore, there are still open questions in NFV methods as to the relationship of the limiting process in the separate conservation equations (Schar & Smolarkiewicz 1996), which our reciprocity relationships may help to clarify.

The situation in Lagrangian/ALE codes is more complicated. It was recognized from the earliest days of computing high-speed flows that an artificial viscosity was required to produce the requisite entropy production in shocks. Von Neumann & Richtmyer's (1950) viscosity is the momentum exchange term in the FS momentum equation (3.4), although it is typically only employed in compression. This basic viscosity with some generalization for multiple space dimensions is still in wide use today. Similarly, an artificial heat conduction was proposed by Noh (1978) to treat the numerical problem of wall heating. This term is not yet in common use. The relationship of artificial heat conduction to the artificial viscosity may be clarified by reciprocity of coefficients in the analogous terms predicted by the FSE.

The Lagrangian/ALE community has also partially confronted the thermodynamic issues discussed in §4. In these codes, velocity (more precisely, momentum velocity) is placed at the cell vertices, and so the new position of a cell and its volume can be calculated geometrically. However, Reynolds transport theorem (in integral form) offers an alternative equation to update volume. Problems associated with the lack of consistency of the geometric volume change and the discretization of the velocity divergence led to formulation of compatible differencing techniques (Bauer et al. 2006; Margolin & Shashkov 2008). However, thermodynamic issues have continued to affect these codes (Whalen 1996). One modern approach to these problems is the technique of subcells (Bauer et al. 2006). Although the subcell technique was originally developed to control null-space deformations (e.g. hourglassing), they also imply higher order approximations to the pressure and velocity distributions within a cell and so provide a more accurate approximation to the nonlinear pressure–velocity work term.

(c) Summary

We have presented and analysed the FSE that describe the evolution of finite parcels of fluid whose individual points are governed by the compressible Navier–Stokes equations. The volume-averaged variables are the fundamental variables of finite-volume numerical simulation codes when one identifies the averaging scale L with the computational cell size Δx. Direct comparison of the FSE with the discretized equations shows that many of the new features of the FSE are already incorporated into the codes on a less rigorous basis, and provides guidance for future improvement. The possibility of constructing NFV schemes to match new FSE is discussed in Margolin & Rider (2005).

A principal motivation for this work is to provide a framework for rationalizing the use of ILES (cf. Grinstein et al. 2007) for compressible flows, similar to our results for Burgers' fluids (MR 2002) and for incompressible flows (MRG 2006). We believe the simplicity of the ILES approach and the absence of ad hoc dimensional parameters make ILES an effective tool for practical applications that couples computational efficiency with predictiveness. Many documented implementations of ILES for compressible flows (Grinstein et al. 2007) serve to illustrate this promise. However, we recognize that the present lack of theoretical foundation impedes general acceptance of ILES by the turbulence community.

Acknowledgments

We appreciate many discussions with D. E. Burton, D. D. Holm, R. B. Lowrie, J. R. Ristorcelli and J. H. Schmidt. This work was performed under the auspices of the US Department of Energy's NNSA by the Los Alamos National Laboratory operated by Los Alamos National Security LLC under contract no. DE-AC52-06NA25396.

Appendix A

(a) Kinematics of volume transport

In a recent paper, Brenner (2005a) discusses the kinematics of volume transport—i.e. the transport and production of the extensive property of volume in a fluid (see also Brenner 2005b). Brenner's results have many features in common with FSE, although the derivation and context are quite different. In particular, Brenner remains in the context of the Navier–Stokes equations and does not introduce any additional length scale. All B-equations and page numbers in this subsection refer to Brenner (2005a).

Brenner begins with the recognition that the advective velocity and the momentum velocity are distinct, even in continuum Navier–Stokes theory. In contrast with our development, he chooses to maintain the framework of the advective velocity. Rewriting the continuity equation in terms of specific volume, v=1/ρ, he derives new terms representing the diffusion of v; cf. B-eqn (2.11). Furthermore, Brenner derives a relationship, B-eqn (3.15), between these velocities, which is similar to our equation (3.2) in its dependence on the logarithmic gradient of density. Lacking any additional length scale, Brenner invokes the length scales of molecular diffusion to construct the coefficient of the gradient—B-eqn (3.13).

Brenner acknowledges the redundancy of his volume transport B-eqn (2.11), but emphasizes that its consideration leads to new physical insights. In particular, he discusses the possibility that volume transport will play a dynamic role in transporting energy and momentum across material surfaces (B-page 43), and derives a pressure-dependent correction to the heat flux in B-eqn (5.20).

Brenner's theory is focused on explaining experimental results for low-speed flows and has not yet been connected to numerical simulation.

(b) Navier–Stokes α-models

The α-models (Holm et al. 1998) were introduced to describe the mean motion of ideal incompressible fluids. The equations are derived in a Euler–Poincaré formalism, which is the Lagrangian version of the Lie–Poisson Hamiltonian framework, and have been successfully used as a model for numerical simulation (e.g. Chen et al. 1999; Hecht et al. 2008). In the derivation of these equations, two velocities appear, the Lagrangian-averaged velocity and the Eulerian-averaged velocity, related by a Helmholtz operator. Comparison of eqn (1) of Hecht et al. (2008) with our equation (2.7) suggests that the two velocities of the α-model correspond to the unaveraged (u) and the averaged (Embedded Image) of our development. A further connection lies in the estimate of the total kinetic energy of the flow—compare eqn (8) of Holm et al. (1998) with our equation (4.4).

An important difference between the α-models and the FSE is that the spatial scale α is a flow-dependent quantity representing the small-scale fluctuations of the flow, whereas our L is a fixed scale representing the ‘observer’. However, as a model for numerical simulation, these may not be so dissimilar; i.e. α would appear to represent the size of a cell in a Lagrangian simulation, whereas L would represent the size of a Eulerian cell.

Footnotes

  • One contribution of 16 to a Discussion Meeting Issue ‘Applied large eddy simulation’.

  • This result is readily extended to three dimensions and to the time domain.

  • We endeavour to write our equations in terms of the primary variables chosen in §2. Note also that it is Embedded Image and Embedded Image that are calculated in finite-volume numerical codes, not Embedded Image and Embedded Image.

  • This would appear to be a result of dimensional analysis, not of the similarity of the underlying physical origins of these terms.

References

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