Numerical treatment of the energy equation in compressible flows simulations

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Numerical treatment of the energy equation in compressible flows simulations
C. De Michelea, G. Coppolaa
aUniversità di Napoli “Federico II”, Dipartimento di Ingegneria Industriale, Napoli, Italy
Abstract
We analyze the conservation properties of various discretizations of the system of compressible Eu-
ler equations for shock-free flows, with special focus on the treatment of the energy equation and
on the induced discrete equations for other thermodynamic quantities. The analysis is conducted
both theoretically and numerically and considers two important factors characterizing the various
formulations, namely the choice of the energy equation and the splitting used in the discretization
of the convective terms. The energy equations analyzed are total and internal energy, total en-
thalpy, pressure, speed of sound and entropy. In all the cases examined the discretization of the
convective terms is made with locally conservative and kinetic-energy preserving schemes. Some
important relations between the various formulations are highlighted and the performances of the
various schemes are assessed by considering two widely used test cases. Together with some popular
formulations from the literature, also new and potentially useful ones are analyzed.
Keywords: Energy conservation, Compressible Navier-Stokes equations, Turbulence simulations
1. Introduction
The compressible Navier-Stokes equations are written as the balance equations for mass, mo-
mentum and an ‘energy’ variable specifying the thermodynamic state of the system, as total or
internal energy, or entropy. The choice of the ‘energy’ variable is usually made depending on some
physical or mathematical requirement and, assumed sufficient smoothness of the flow, the various
formulations are usually seen as equivalent, since one can pass from one equation to another through
the usual rules of calculus and the equation of state.
It is well known that, when turning to discrete formulations, this equivalence is typically lost,
since the classical rules of calculus, which are required to pass from one set of equations to another,
cannot be applied, in general, at a discrete level [1, 2]. As an example, the product and chain rules
do not hold in general for finite-difference operators [3], which implies that the steps required to pass
from the equations for the ‘primary’ variables (i.e. the balance equations directly discretized) to that
for the secondary or ‘induced’ ones cannot be reproduced at a discrete level. This circumstance can
have strong effects on the quality of the discrete solutions, since the derived, or induced, quantities
evolve satisfying discrete equations that are, in general, different from the discretized versions of
the continuous equations.
The effects of this discrepancy are evident when considering some symmetries of the continuous
system, which are typically lost in the discrete formulation, if discretization is not properly done.
Email address: gcoppola@unina.it (G. Coppola)
Preprint submitted to Computers and Fluids October 5, 2022
arXiv:2210.01251v1 [physics.flu-dyn] 3 Oct 2022
The most evident case is that of the conservation properties induced by the divergence structure
of the convective terms in the system of non-viscous equations. In compressible flow equations the
convective term is expressed as the divergence of a flux vector. Integration of each equation on the
whole domain and application of the Gauss divergence theorem easily shows that the convective
mechanisms do not influence the evolution of the integrated balanced quantities over the entire
domain, apart from boundary terms. The reproduction of this property at a discrete level is
usually considered an important quality of the discretization procedure.
In the case of primary variables, for which the evolution equations are directly discretized, the
divergence structure of the convective terms can be discretely enforced by using a Finite Volume
(FV) approach, which is based on the direct specification of the flux at cell boundaries. In this
case the convective term is expressed as difference of fluxes at adjacent nodes, which is the discrete
local representation of the divergence structure. We will term a discretization of this type a ‘locally
conservative’ discretization. The global conservation of the quantity on the whole domain follows
by virtue of the telescoping property. In the case of a Finite Difference (FD) discretization, the
divergence operator is approximated through a suitable derivative matrix, and the local conservation
form is not evident a priori. This is especially true when an equivalent ‘advective’ form of the
convective term (i.e. an expression of the divergence of the product of two or more variables as a
sum of products obtained by applying the product rule) is directly discretized. However, if one
limits to the case of central schemes on uniform Cartesian meshes, it is known that almost all the
forms in which the convective terms can be written (e.g. divergence, advective, split. . . ) admit a
‘difference of fluxes’ expression [4, 5, 6]. The extension of this and other conservation criteria to
a wider class of derivative schemes, even on nonuniform meshes, is discussed in a recent paper by
Coppola and Veldman [7].
In the case of induced, or secondary variables, the situation is less definite. A discretization
that is locally conservative for primary variables does not guarantee that the induced ones evolve
by satisfying a discrete equation in which the convective terms can be cast as a difference of fluxes.
As an example, the direct discretization of the system of equations for mass, momentum and total
energy through a locally conservative formulation, guarantees that these quantities are locally (and
globally) conserved, but the kinetic or internal energies, or the entropy, usually evolve satisfying
a discrete equation in which the convective terms cannot be cast as difference of fluxes, which
means that local (and global) preservation is spuriously affected by discrete convective terms, in a
potentially unbounded manner.
The case of kinetic energy is of particular importance, and it has been the subject of several
studies in past years, for both incompressible and compressible flows and for temporal and spatial
discretizations [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. The reason for this interest lies in the fact
that for incompressible flows (global) kinetic energy is a norm of the solution vector. A procedure
that is able to bound the global kinetic energy gives also an important nonlinear stability criterion
for the discrete equations. The extension to (smooth) compressible flows has been pursued mainly
by analogy, and has shown great increases in the robustness of the simulations. General criteria
for the preservation of global kinetic energy in compressible flows equations have been recently
derived for both FD [7] and FV [17] methods. The details of these theories will be recalled in the
subsequent Section 3. For now, it is sufficient to mention that a globally kinetic-energy preserving
(KEP) discretization involves a coordinated treatment of the convective terms in the mass and
momentum equations, without any prescription on the discretization of the energy equation.
The situation regarding the energy equation is also interesting, although less studied. It is
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clear that a locally conservative discretization of the total-energy equation guarantees that total
energy is discretely preserved, both locally and globally, by convection. If a KEP discretization
has also been adopted for mass and momentum equations, global preservation of internal energy
follows as a reward, but entropy usually evolves satisfying a discrete equation which is not in
locally (nor globally) conservative form. This means that the complete discretization satisfies the
first principle of thermodynamics, but fails to satisfy the second. The reverse is true if one starts
directly by discretizing the equation for entropy. In this case a locally conservative formulation is
able to exactly preserve the entropy balance both locally and globally, but conservation of total
(and internal) energy is usually lost. The situation in the case of a direct discretization of one of the
other variables (e.g. internal energy, pressure, enthalpy, sound speed. . . ) is even more complicated,
since, in principle, neither total energy nor entropy are preserved by convection, if one does not
properly design the discretization details. Typically, and in absence of more suitable guidelines, the
‘energy’ equation (whichever one is considered among the mentioned ones) is discretized by using a
KEP formulation as it is done for momentum equation, which implies the exotic global preservation
by convection of quantities such as ρE2, ρe2or ρs2, in case the equation for total or internal energy,
or entropy, respectively, is directly discretized.
In the subsequent sections we will analyze some of the most common approaches used in the
literature in past years, together with some new formulations. Each formulation is characterized
by at least two factors. The first is the choice of the ‘energy’ equation to be directly discretized
among the various possibilities mentioned. The second is the particular splitting which is used to
discretize the various equations. It is known that both factors can strongly affect the robustness
of the simulation in different test cases, and a complete study assessing the advantages and disad-
vantages of the various options has not been made yet. In all the cases considered in this paper
we will always assume that a locally conservative and KEP discretization is performed, since these
two characteristics have been widely accepted as mandatory for a robust and reliable numerical
simulation of turbulent compressible flows. The analysis will be mainly developed by using a clas-
sical FD formalism based on (central) discretization of divergence, advective and split forms, as in
[6]. Since in all cases these formulations can be shown to be locally conservative, explicit numerical
fluxes (also of high order) will be derived. The preference for the FD formalism stems from the fact
that in this framework one can directly use the quite general necessary and sufficient condition for
kinetic energy preservation developed in [6], which is valid also in the high-order case. However, all
the numerical discretizations here analyzed can be reformulated in terms of numerical fluxes and
an equivalent treatment could have been developed starting from a FV perspective.
It is worth mentioning that, in the context of FV methods, the theory of entropy variables [19, 20]
allows the specification of the conditions for Entropy Conservative (EC) numerical fluxes, which
can be enforced together with the conditions for kinetic-energy preservation to construct explicit
centered numerical fluxes which are both entropy conservative and also preserve kinetic energy for
semi-discrete FV methods [21, 22, 23, 24]. This theory, however, is based on a specification of the
fluxes which typically uses a logarithmic mean value [22], which renders problematic the possibility
of recasting the method as a classical FD scheme based on the direct discretization of divergence
and advective forms as in [6]. Moreover, the EC schemes using the logarithmic mean have some
implementation issues (they need a treatment to avoid division by zero) and a non negligible increase
in computational cost when compared to classical FD schemes [25]. Since we mainly rely on FD
formulations in this work, which already produce many different alternatives, we will not consider
this approach here, leaving its analysis and a fairer comparison with standard FD approaches for
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future work. In Sec. 3 we recall some of the most important ingredients of the locally conservative
and KEP discretizations, whereas in Sec. 4 we will analyze the different formulations for the energy
equation. In Sec. 5 numerical tests on various formulations are reported for two test cases widely
used in the literature. Concluding remarks are given in Sec. 6.
2. Problem formulation
2.1. Euler equations
The compressible Euler equations can be written as
∂ρ
∂t =−∂ρuα
∂xα
,(1)
∂ρuβ
∂t =−∂ρuαuβ
∂xα−∂p
∂xβ
,(2)
∂ρE
∂t =−∂ρuαE
∂xα−∂puα
∂xα
(3)
where ρis the density, uαis the Cartesian velocity component, pis the pressure and Ethe total
energy per unit mass, which is the sum of internal and kinetic energies: E=e+uαuα/2. The
ideal gas law is assumed, which implies p=ρRT and e=cvT, where Tis the temperature, Rthe
gas constant and cvthe specific heat at constant volume. The ratio of specific heats at constant
pressure and volume γ=cp/cvis assumed to be 1.4. In Eq. (1)–(3) and in what follows we will
assume the convention that Greek subscripts refer to the components of Cartesian vectors; e.g. uα
is the component of the velocity vector along the α-direction with coordinate xα(α= 1,2,3).
Latin subscripts as i, j or kare used to denote the values of the discretized variable on a nodal
point xi. When the Greek subscript is omitted (e.g. for quantities as uor x) it is assumed that the
relations hold for a generic value of it. In all cases, unless otherwise explicitly stated, the summation
convention over repeated Greek indices is assumed.
Equations (1)–(3) constitute a set of three partial differential equations (the second one being
vectorial) expressing the balance of mass, momentum and total energy. Together with the equation
of state, they describe the evolution of both kinematic and thermodynamic variables for an inviscid
compressible flow. In what follows, we will consider also the induced balance equations for various
quantities related to the primary variables ρ,ρuαand ρE. These equations are termed induced
because they are derived through Eqs. (1)–(3) and don’t constitute additional independent balance
equations. Examples of kinematic and/or thermodynamic quantities of interest are the kinetic
energy (per unit volume) ρκ =ρuαuα/2, the internal energy ρe, the pressure p, the total enthalpy
ρH =ρE+p, the sound speed c=√γRT and the entropy ρs =ρcvln(p/ργ). The balance equations
for these quantities are easily derived by combining Eqs. (1)–(3), together with the equation of state,
and by applying the usual rules of calculus (assumed valid for smooth solutions), namely the classical
chain and product rules of differentiation, with respect to both temporal and space variables. They
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can be written as
∂ρκ
∂t =−∂ρuακ
∂xα−uα
∂p
∂xα
,(4)
∂ρe
∂t =−∂ρuαe
∂xα−p∂uα
∂xα
,(5)
∂p
∂t =−∂puα
∂xα−(γ−1) p∂uα
∂xα
,(6)
∂ρH
∂t =−∂ρuαH
∂xα−∂puα
∂xα−(γ−1) p∂uα
∂xα
,(7)
∂ρc
∂t =−∂ρuαc
∂xα−(γ−1)
2ρc∂uα
∂xα
,(8)
∂ρs
∂t =−∂ρuαs
∂xα
.(9)
On a continuous ground, Eqs. (4)–(9) are always satisfied by the variables obtained as a combi-
nation of the solutions to Eqs. (1)–(3), once a sufficient smoothness has been assumed. In principle,
any of the Eqs. (5)–(9) can be used in place of Eq. (3), to describe the evolution of the system (note
that Eq. (4), being obtained by combining only Eqs. (1) and (2), is independent of the equation for
total energy, and cannot be used in place of it). To each choice of the ‘energy’ equation corresponds
a set of ‘primary’ variables, and the values of the other ‘induced’ ones can be obtained by algebraic
manipulations and through the equation of state.
2.2. Discrete approximations
In this paper we will assume that the equations of motion are discretized with a FD method
over a uniform Cartesian mesh of width h(with a colocated approach). We will also assume that
integration is performed through a semi-discretized approach, in which a spatial discretization step
is firstly performed, and the resulting system of Ordinary Differential Equations (ODE) is integrated
in time by using a standard solver. Since we focus on the space discretization step, we will assume
that all the manipulations involving time derivatives can be carried out at the continuous level. The
effects of time integration errors will be assumed to be negligible at sufficiently small time steps.
Spatial discretization is made by using central difference schemes which, among various important
properties, assure that the discrete counterpart of the integration by parts rule (i.e. the summation
by parts (SBP) rule) holds, for periodic boundary conditions [26]. Of course, SBP operators can
be derived also for non-periodic boundary conditions. In this case, all the reasonings which are
based on the SBP rule hold in the general case. In the derivation of the various properties of the
discrete equations, manipulation of spatial terms will be done by using only algebraic relations and
the SBP rule, whereas the product and chain rules of derivative will not be allowed, since they are
not valid, in general, for discrete operators. Under these assumptions, all the equations derived
from the primary ones will be valid at discrete level.
To distinguish between continuous and discrete operators, we use the symbol δfor discrete
derivatives, in contrast to the usual symbol ∂for partial derivatives. According to the previous
discussion, for discrete operators we will assume all the usual algebraic operations valid for deriva-
tive operators, including the SBP rule, but the product rule will not be allowed. The result of
manipulations with δoperators will hold also on a continuous ground, but the opposite, of course,
5
摘要：

NumericaltreatmentoftheenergyequationincompressibleowssimulationsC.DeMichelea,G.CoppolaaaUniversitàdiNapoliFedericoII,DipartimentodiIngegneriaIndustriale,Napoli,ItalyAbstractWeanalyzetheconservationpropertiesofvariousdiscretizationsofthesystemofcompressibleEu-lerequationsforshock-freeows,withspe...

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