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Inverse design of mesoscopic models for compressible flow using the ChapmanEnskog analysis
Advances in Aerodynamics volume 3, Article number: 5 (2021)
Abstract
In this paper, based on simplified Boltzmann equation, we explore the inversedesign of mesoscopic models for compressible flow using the ChapmanEnskog analysis. Starting from the singlerelaxationtime Boltzmann equation with an additional source term, two model Boltzmann equations for two reduced distribution functions are obtained, each then also having an additional undetermined source term. Under this general framework and using NavierStokesFourier (NSF) equations as constraints, the structures of the distribution functions are obtained by the leadingorder ChapmanEnskog analysis. Next, five basic constraints for the design of the two source terms are obtained in order to recover the NSF system in the continuum limit. These constraints allow for adjustable bulktoshear viscosity ratio, Prandtl number as well as a thermal energy source. The specific forms of the two source terms can be determined through proper physical considerations and numerical implementation requirements. By employing the truncated Hermite expansion, one design for the two source terms is proposed. Moreover, three wellknown mesoscopic models in the literature are shown to be compatible with these five constraints. In addition, the consistent implementation of boundary conditions is also explored by using the ChapmanEnskog expansion at the NSF order. Finally, based on the higherorder ChapmanEnskog expansion of the distribution functions, we derive the complete analytical expressions for the viscous stress tensor and the heat flux. Some underlying physics can be further explored using the DNS simulation data based on the proposed model.
1 Introduction
The Boltzmann equation is of vital importance in the kinetic theory of dilute gases [1]. It is noted that the construction of gas kinetic models has a long history. After the pioneering work of WangChang and Uhlenbeck [2] on the eigenvalues and eigenfunctions of the linearized Boltzmann equation, Gross and Jackon [3] established the relations between the linearized Boltzmann equation and some models for monatomic gas by comparing the eigenvalue spectra of the collision operators. Later, Hanson and Morse [4] obtained the kinetic model equations for polyatomic gases based on Wang ChangUhlenbeck equation. The original collision operator in the Boltzmann equation is a complex integral term, which makes direct numerical simulation of the system very costly. The simplest choice is to replace the original collision operator with the BhatnagerGrossKrook (BGK) model [5]. It should be noted that the original collision operator is directly based on the physical description of molecule interactions while the BGK model describes the fact that the distribution of the molecules relaxes to the local equilibrium state through particle collisions without considering the detailed molecule interactions. It has long been recognized that such an approximation works well beyond its theoretical limit as long as the relaxation time can be made to capture the relevant physics [6, 7].
By applying the ChapmanEnskog expansion to the Boltzmann equation with the BGK collision operator, the NSF system can be recovered, but with a unit Prandtl number which does not obey the physical reality. Hence, some improved models have been developed to overcome this limitation from different physical considerations, such as the Shakhov (S) model [8], the ellipsoidal statistical (ES) model [9], the internal energy doubledistributionfunction (IEDDF) model [10], the total energy doubledistributionfunction (TEDDF) model [11], and the Rykov (R) model [12–14]. Further, we notice that, for example, the ratio between the bulk to shear viscosity in S model is always less than 2/3. Therefore, for a fixed specific heat ratio, the S model cannot be used to investigate the physical effect of the bulktoshear viscosity ratio in compressible flows.
Some merits and drawbacks of these models are briefly discussed below. The S model may encounter a negative value of the particle distribution function because of the modified equilibrium distribution to accommodate arbitrary Prandtl number, while the ES model and the TEDDF model can avoid such unphysical deficiency. However, Chen et al. [15] showed that the S model may yield more accurate solutions than that from the ES model in the transition regime and they proposed a generalized model which combines the advantages of the S model and ES model. For both IEDDF and TEDDF models, two distribution functions are introduced with different relaxation times for the nonequilibrium part of the particle distribution function because the momentum and energy have different relaxation time scales during the collision process as suggested by Wood [16]. In the TEDDF model, spatial and time derivatives of the hydrodynamic velocity are not involved in the source terms while they are involved in the source terms of the IEDDF model which may introduce some numerical errors and lead to some unphysical phenomena in fluid systems containing large spatial gradients. In the R model, by considering the elastic and nonelastic particle collision processes, the hydrodynamic flow variables corresponding to the translational and rotational processes can be evaluated separately. Both the total internal energy and the total heat flux are the sum of the contributions from the two processes. The bulktoshear viscosity ratio can be modified in the R model through the ratio of the total number of translational and rotational collisions to that of rotational collisions.
During the past few decades, the BGK model has been widely used to simulate different flows such as homogeneous isotropic turbulence [17], turbulent channel flows [18] and multiphase flows [19], by different numerical approaches such as the lattice Boltzmann method (LBM) [20], the gas kinetic scheme (GKS) [21], the unified gas kinetic scheme (UGKS) [22], and the discrete unified gas kinetic scheme (DUGKS) [23, 24]. Recently, Liu et al. [25] claimed that the predictions based on the BGK model for highly nonequilibrium flows are only qualitatively correct in the transitional regime since the BGK model filters out the information of the detailed molecularinteraction processes. They compare the Boltzmann equation and its model equations through some test cases where the distribution functions are far from equilibrium. From these tests, they found that information contained in the nonequilibrium moments and the different relaxation rates of high and lowspeed molecules is essential in adjusting the behaviors of model collision terms. However, many existing works have shown that the BGK model is adequate in simulating many flows accurately for both the continuum and rarefied regimes [17–19, 26–28].
In this paper, we focus on the inverse design of the source term in the model Boltzmann equation for compressible flows. Following the work done by Guo et al. [24], an adjustable parameter representing the internal degree of freedom of molecules is introduced to the Maxwellian equilibrium distribution function. We will demonstrate that the two source terms in the two reduced model Boltzmann equations can be redesigned to attain the following objectives. First, the NSF system can be recovered in the continuum limit by applying the ChapmanEnskog analysis. Second, the model Boltzmann system can have flexible Prandtl number as well as adjustable bulktoshear viscosity ratio. Third, an arbitrary thermal source/sink term can be added to the internal energy equation.
The rest of the paper is organized as follows. In Section 2, the model Boltzmann equation with an additional source term is introduced. By introducing two reduced distribution functions, two reduced model Boltzmann equations are obtained. Some notations and conventions are given in Section 3. In Section 4, the structures of the Boltzmann equations are obtained by applying the firstorder ChapmanEnskog expansion. Next, five requirements for the two reduced source terms are given in Section 5. In Section 6, we present one design for the two source terms by applying the Hermite expansion to the two source terms. In Section 7, we show that the S model, the TEDDF model as well as the R model are compatible with the five derived constraints. Then we discuss the derivation of the proper implementation of the hydrodynamic boundary conditions in Section 8. Next, we derive the complete analytical expression for the viscous stress and the heat flux based on the secondorder ChapmanEnskog expansion in the following three sections. Major conclusions are drawn in Section 12. In Appendix 13, we include the details on the Hermite polynomials and Hermite expansion. Appendix 14 contains the derivations of the requirements for the two reduced source terms. Appendix 15 documents some details on the Rykov model.
2 The reduced model Boltzmann system with source terms
The Boltzmann equation with an additional source term can be expressed as
where f(x,ξ,t) is the particle distribution function, x=(x_{1},...,x_{3}) is the spatial location, t is the time, ξ=(ξ_{1},...,ξ_{3}) is the particle velocity in three dimensional space, and ζ=(ζ_{1},...,ζ_{K}) represents Kdimensional internal degree of freedoms. a represents the body force per unit mass. The singlerelaxationtime BhatnagerGrossKrook (BGK) model [5] is used for the collision operator, i.e. Ω_{f}=(f^{eq}−f)/τ. τ=μ/p is the molecular relaxation time and μ is the shear viscosity which can be approximated by the hardsphere model [24, 29] or Sutherland’s law [30, 31]. p=ρRT is the pressure for ideal gas. S_{f} is a source term to be designed, which will allow for modification of both the Prandtl number Pr as well as the bulktoshear viscosity ratio χ=μ_{V}/μ with μ_{V} being the bulk viscosity.
By assuming that the particle motion in ζ subspace is at local equilibrium, the local Maxwellian equilibrium distribution function can be written as [24]
where ρ is the density of the fluid, R is the specific gas constant, T is the temperature, and c=ξ−u is the peculiar velocity with u being the hydrodynamic velocity.
The conservative variables are defined as the moments of the particle distribution function
where ε=C_{v}T is the internal energy per unit mass, C_{v} is the specific heat capacity at constant volume, and ρE is the total energy per unit volume which is the sum of the internal energy and the kinetic energy. All relations in Eq. (3) remain valid if f is replaced by f^{eq}. C_{v} and the specific heat at constant pressure C_{p} are determined by the number of the internal degrees of freedom, K, and the gas constant, R. By integrating the energy moment of the equilibrium distribution, we can obtain C_{v}=(K+3)R/2 and C_{p}=(K+5)R/2, which implies that the specific heat ratio and thus the Prandtl number are γ=C_{p}/C_{v}=(K+5)/(K+3) and Pr=μC_{p}/κ, where κ is the thermal conductivity.
In addition, by comparing the firstorder moment of the model Boltzmann equation with the NavierStokes equation, it can be shown that the viscous stress tensor σ is determined by the nonequilibrium part of the particle distribution function as
and, by comparing the energy moment of the Boltzmann equation with the macroscopic energy equation, the heat flux q can be determined as
The physical conservative requirements can be expressed through the moments of the collision operator Ω_{f}, which reads
Therefore, provided that the mass conservation and the momentum conservation laws are observed, we have the following basic requirements for the source term
where Λ represents a source term applied to the energy equation, an example of which is the thermal cooling function [30].
Physically, the evolution of the particle distribution function only depends on the particle velocity ξ. In order to remove the dependence of the passive variables and also reduce the computational cost in the practical implementation, two independent, reduced distribution functions g and h, residing in lower dimensional phase space, are introduced [24], namely, \(g=\int f d\boldsymbol {\zeta }\) and \(h=\int \zeta ^{2} fd\boldsymbol {\zeta }\). Therefore, the two model Boltzmann equations residing in lower dimensional space can be obtained
In Eq. (8), the collision operators and source terms are
where the equilibrium distribution functions g^{eq} and h^{eq} are
Based on Eq. (6), the conservation laws can be recasted in terms of the collision operators Ω_{g} and Ω_{h}, as follows,
From Eq. (7), the two reduced source terms must satisfy the following requirements
In addition, from Eq. (3), we find that the conservative variables can be evaluated as
Moreover, from Eqs. (4) and (5), the viscous stress σ and the heat flux q become
3 Notations and conventions
For convenience, two time derivatives are introduced
where D/Dt is the time derivative along the phasespace trajectory of a particle subjected to a body force a per unit mass and d/dt is the rate of change of a physical quantity along the path of a fluid element in the physical space. Three variables including the time t, the spacial location x, and the particle velocity ξ are assumed to be independent when these time derivatives act on the distribution functions g(x,ξ,t) and h(x,ξ,t).
In addition, S=(∇u^{T}+∇u)/2 is the strain rate tensor and Ω=(∇u^{T}−∇u)/2 is the rotation tensor. 𝜗=∇·u is the velocity divergence (also known as dilatation).
The Newtonian constitutive relation for the viscous stress σ^{(NS)} and the Fourier’s law for the heat flux q^{(NS)} are, respectively,
4 The structure of the particle distribution functions
In the continuum limit, the relaxation time τ, when normalized by the acoustic time scale l_{0}/c_{0}, is proportional to the Knudsen number, where l_{0} is a system length scale and c_{0} is the speed of the sound at a reference temperature T_{0}. Therefore, τ may be taken as a small parameter in the Boltzmann equation. At the level of NSF equations, terms higher than O(τ) in the distribution functions can be neglected.
The derivatives of the equilibrium distribution function g^{eq} will be multiplied by τ to form the O(τ) terms in the distribution functions. Therefore, we only need to evaluate them to O(1). Direct evaluation yields the derivatives of the equilibrium distribution function g^{eq} as
The three coefficients in three derivatives are found to be polynomials of the peculiar velocity c and are related to the time t and spatial location x through the relation c=ξ−u.
By employing the Euler equations in Appendix 14 to replace the time derivatives of the hydrodynamic variables with spatial derivatives in ∂g^{eq}/∂t, we obtain the expression for Dg^{eq}/Dt and Dh^{eq}/Dt to the leading order, as
where G=G_{1}+G_{2}+G_{3}. The coefficients are given explicitly as
Therefore, up to the order O(τ) in the ChapmanEnskog expansion [1], the structure of the distribution functions g and h can be obtained and they are
5 Five basic requirements for the two source terms
Based on the structure of the distribution function, we shall now propose five basic requirements for the two source terms, S_{g} and S_{h}. The requirements are given as follows and the details for their derivations are included in Appendix 14. If these five requirements are satisfied up to the order of O(τ), then the NSF system can be recovered by applying the ChapmanEnskog expansion to the two model Boltzmann equations.
The first and second requirements come from the continuity and the momentum equations
The third requirement is used to modify the bulk viscosity in the viscous stress term, and it is
The fourth requirement follows from the energy equation and it is
The fifth requirement is expressed as
which is needed to modify the heat flux and thus the resulting Prandtl number.
As a result of the design constraints, Eqs. (21a)–(21d), the model Boltzmann equation will yield the following NSF system
6 A possible design of the two reduced source terms
There are many possible ways to design the specific form of the two source terms. By applying the Hermite expansion to the two source terms, a new mesoscopic model with both adjustable Prandtl number and bulk viscosity is proposed next. Any reasonable design of the two source terms should satisfy the five basic requirements presented in Eqs. (21a) to (21d) in the continuum limit.
Due to the desire to keep the order of GaussHermite quadrature as low as feasible in the numerical implementation, we further require \(\int \boldsymbol {\xi \xi \xi } S_{g} d\boldsymbol {\xi }=\boldsymbol {0}\). Then, using Eq. (21a), we have \(\int S_{g}\mathscr {H}^{(3)}\left (\boldsymbol {\xi },T_{0}\right)d\boldsymbol {\xi }=\boldsymbol {0}\). The Eqs. (21a)–(21b) can also be written as
where T_{0} is a reference temperature and the velocity ξ are scaled with \(\sqrt {RT_{0}}\).
Therefore, by using Eq. (23) and keeping the Hermite expansion (see Appendix 13) of the source term S_{g} up to the thirdorder, we obtain
From Eqs. (21a)–(21b), we can obtain
Combination of Eqs. (21c), (21d) and (25) yields
Therefore, we obtain
Eqs. (21b) and (21c) together imply that
Combination of Eqs. (27) and (28) yields one design for the source term S_{h}
In low Mach number thermal flows, the secondorder central finite difference scheme is adequate for calculating ∇·u in Eqs. (24) and (29). In addition, two different methods can be used to evaluate the heat flux q in Eq. (29). One way is to first obtain ∇T by the secondorder central finite difference scheme, then it follows that q=−κ∇T. Another way is to evaluate the heat flux through the velocity moment of distribution functions. For example, under the framework of DUGKS proposed by Guo et al. [24], the heat flux term at the cell center and the cell interface can be calculated by \(\boldsymbol {q}=\frac {2\tau }{2\tau +\Delta t Pr}\frac {1}{2}\int \boldsymbol {c}\left (c^{2} \tilde g+\tilde h\right) d\boldsymbol {\xi }\) and \(\boldsymbol {q}=\frac {2\tau }{2\tau +(\Delta t/2) Pr}\frac {1}{2}\int \boldsymbol {c}\left (c^{2} \bar {g}+\bar {h}\right) d\boldsymbol {\xi }\) after the distribution functions \(\tilde g, \tilde h\) and \(\bar g, \bar h\) are updated. No obvious difference can be observed for global turbulence statistics when the velocity divergence in the source terms is evaluated by either a secondorder or fourthorder finite difference scheme in compressible isotropic turbulence [32, 33]. Some improved shock capturing schemes with low dissipation and highorder accuracy could also be tried to evaluate ∇·u when the turbulent Mach number is further increased.
7 An examination of three existing mesoscopic models in our design framework
7.1 The Shakhov model
In this section, we will prove that the wellknown Shakhov model [8, 24] is compatible with our inversedesign here. Through the ChapmanEnskog analysis, it can be verified that the ratio of bulk viscosity to shear viscosity in the Shakhov model is μ_{V}/μ=2K/[3(K+3)]. In addition, no cooling function is considered, namely, Λ=0. Therefore, the five general requirements (Eqs. (21a) to (21d)) for the two source terms can be reduced as
By using Eq. (30), the source term S_{g} can be assumed as
where ω(c,T) is the peculiarvelocitybased weighting function, a^{(3)}(x,t) is the coefficient and \(\mathscr {H}^{(3)}(\boldsymbol {c}, T)\) is the thirdorder Hermite polynomial.
Note that \(\boldsymbol {a}^{(3)}(\boldsymbol {x},t)\equiv \int S_{g}\mathscr {H}^{(3)}(\boldsymbol {c}, T)d\boldsymbol {\xi }\) is also symmetrical with respect to the components of c because \(\mathscr {H}^{(3)}(\boldsymbol {c}, T)\) remains unchanged under the permutation operation, the simplest way is to assume that the coefficient a^{(3)} takes the form \(a_{ijk}^{(3)}=A_{i} \delta _{jk}+A_{j} \delta _{ki}+A_{k}\delta _{ij}\), where A(x,t)=(A_{1},A_{2},A_{3}) is a vector coefficient to be determined. Then, Eq. (31) gives
Moreover, it can be shown that \(\int \boldsymbol {c} c^{2}S_{g}d\boldsymbol {c}=5\boldsymbol {A} (RT)^{3/2}\).
Similarly, the source term S_{h} can be designed as
Therefore, the coefficient A and B should satisfy the following relation,
and C is a vector coefficient to be determined.
If the coefficients A,B and C are chosen specifically as
then Eq. (34) is satisfied and the two source terms S_{g} and S_{h} are given as
In the Shakhov model, the source term S_{f} corresponding to the original distribution function f is given by
Substitution of Eq. (37) into Eq. (9) yields the same results given in Eq. (36). Therefore, the Shakhov model is consistent with five constraints here.
7.2 The total energy doubledistributionfunction model
The total energy doubledistributionfunction model (TEDDF) is originally proposed by Guo et al. [11] and then generalized by Liu et al. [34] to simulate thermal compressible flows. The TEDDF model is briefly introduced as follows. From the original distribution function f, two new distribution functions g and b are introduced, namely, \(g=\int f d\boldsymbol {\zeta }\) and \(b=(1/2)\int \left (\xi ^{2}+\zeta ^{2}\right)f d\boldsymbol {\zeta }\). Therefore, this kinetic system can be expressed as
where the relaxation times are τ=τ_{g}=μ/p, τ_{b}=τ_{g}/Pr and τ_{bg}=τ_{b}τ_{g}/(τ_{g}−τ_{b}). The bulk viscosity of this model is shown to be 2Kμ/[3(K+3)]. The equilibrium b^{eq} can be written as b^{eq}=(ξ^{2}g^{eq}+h^{eq})/2. From Eq. (38), we find that the relation 2b=ξ^{2}g+h holds. The expressions for the hydrodynamic variables are the same as those given in Eqs. (13) and (14). Using g and h, the kinetic system Eqs. (38) can be rewritten as
Therefore, the two source terms are
Eq. (40) indicates that the source terms in the TEDDF model can be expressed in terms of the collision operators.
By noticing that the conservation law for the internal energy, \(\int \left (c^{2}\Omega _{g}+\Omega _{h}\right)d\boldsymbol {\xi }=0\), and the heat flux can be expressed as the moments of the collision operators, \(\boldsymbol {q}=(1/2)\tau \int \boldsymbol {c}\left (c^{2}\Omega _{g}+\Omega _{h}\right)d\boldsymbol {\xi }\), we find that the five general conditions given by Eqs. (21a) – (21d) are satisfied.
Therefore, we conclude that the TEDDF model is also a special design of the two source terms. Although two relaxation times are used to modify the Prandtl number, the TEDDF model is equivalent to a mesoscopic model with a single relaxation time.
7.3 The Rykov model
The wellknown Rykov model for diatomic gases with rotational degrees of freedom is originally obtained by Rykov [12, 13]. Recently, Wu et al. [14] has generalized this model to polyatomic gases. The elastic and nonelastic collision processes are considered respectively in this model. By integrating the particle distribution function f with respect to the rotational energy e, the following twoequation kinetic system can be established.
where the equilibrium distribution functions corresponding to the elastic and nonelastic processes are
Here f_{0} is the velocity distribution function and f_{1} is the distribution for rotational energy. \(f_{0}^{t}\) and \(f_{0}^{r}\) denote the equilibrium distributions of the elastic and nonelastic collision processes for f_{0}, respectively. Similarly, \(f_{1}^{t}\) and \(f_{1}^{r}\) denote the equilibrium distributions of the elastic and nonelastic collision processes for f_{1}, respectively. f_{M} is the Maxwellian equilibrium distribution function. T_{t} is the translational temperature corresponding to the translational degrees of freedom of particles, T_{r} is the rotational temperature corresponding to the rotational degrees of freedom, and T is the total temperature in the local equilibrium state. q^{t} is the translational heat flux and q^{r} is the rotational heat flux. The total heat flux is decomposed as q=q^{t}+q^{r}.
The relaxation time τ is related to the shear viscosity μ and pressure p through the relation τ=μ/p with p=ρRT. Physically, the relaxation time τ is related to the translational temperature T_{t} instead of the rotational temperature T_{r}. Therefore, in analogy to the case of a monatomic gas, the following assumption is used, τ=μ_{t}/p_{t}, μ_{t}=μ(T_{t}) and p_{t}=ρRT_{t}, p_{r}=ρRT_{r}. Z indicates the ratio of the total number of translational and rotational collisions to that of rotational collisions. When Z goes to infinity, the Rykov model can reduce to the Shakhov model for monatomic gas without energy exchange between translational and rotational motions. δ=(μ_{t}/ρ)D, where D is the gas selfdiffusion coefficient. ω_{0} and ω_{1} are two parameters which can be selected to achieve proper relaxation of the heat fluxes.
The hydrodynamic variables are defined by the following relationships.
In order to use our general results, we first notice K=2, Λ=0 in this case. Then we introduce two new distribution functions, g=f_{0} and h=2f_{1}.
Two new collision operators are defined as
Two new source terms are given by
The expressions for the hydrodynamic variables in Eq. (43) can be rewritten in terms of g and h, which are found to be the same as Eqs. (13) and (14). From Eqs. (43) and (44), we confirm that the newly defined collision operators, Ω_{g} and Ω_{h}, still satisfy the conservative requirements in Eq. (11). Furthermore, it can be shown that S_{g} and S_{h} indeed satisfy five basic requirements in Eqs. (21a) – (21d). The details of proof are provided in Appendix 15 in which we can confirm that the bulktoshear viscosity ratio χ is determined by the collision ratio Z through χ=4Z/15 in the continuum limit assuming that both τ and τZ are small in ChapmanEnskog analysis. By contrast, both the S model and TEDDF model give a constant bulktoshear viscosity ratio 4/15 for diatomic gas.
In addition, the Prandtl number can be identified as Pr=7Rμ/[2(κ^{t}+κ^{r})], where the tranlational and rotational thermal conductivity coefficients κ^{t} and κ^{r} as well as the total thermal conductivity coefficient are shown to be κ^{t}=15Rμ_{t}/(4A),κ^{r}=Rμ_{t}/B and κ=κ^{t}+κ^{r}, where A=1+0.5(1−ω_{0})/Z and B=δ+(1−δ)(1−ω_{1})/Z.
Therefore, we have proved that the Rykov model is compatible with our design in the continuum limit. The Rykov model is constructed from physical viewpoint with a broad range of bulktoshear viscosity ratios. In contrast, inverse design approach presented in this paper is directly based on the ChapmanEnskog analysis in order to obtain feasible model for compressible flows. The underlying assumptions for our model are (a) τ is small and (b) τχ(∼τZ) is small, which originates from the premise of the ChapmanEnskog analysis that the nonequilibrium part and the source terms only serve as small corrections to the equilibrium distribution function. As a result, the bulktoshear viscosity ratio should be limited such that the premise for the ChapmanEnskog expansion can be preserved.
8 Implementation of macroscopic hydrodynamic and thermodynamic boundary conditions
When numerically implementing mesoscopic methods based on a model Boltzmann equation, a challenge is to properly determine the unknown distribution functions near a solid boundary, such that the resulting scheme is fully consistent with the NSF system near the boundary. Since the NSF system is derived from the ChapmanEnskog expansion up to O(τ), it follows that the proper implementation of the boundary condition should be based on a consistent application of the ChapmanEnskog expansion up to O(τ). In the literature, this requirement is often not checked and thus not met rigorously, leading to degradation of the accuracy of a mesoscopic method. Furthermore, for thermal or compressible flows, as will be shown below, the implementations of velocity and temperature boundary conditions, at the level of the distribution functions, could be coupled. Source terms could also affect the implementation details. Such fine points are not fully realized in the literature. Below we shall explore the relations between the components of the distribution functions (typically the distribution functions between two opposite particle velocity directions after the particle velocity space is discretized).
By using the relation c=ξ−u, the expression for G_{1}(ξ),G_{2}(ξ),G_{3}(ξ) and Φ_{1}(ξ) in Eq. (19) can be rewritten in terms of the particle velocity ξ.
Obviously, we have G_{i1}(ξ)=−G_{i1}(−ξ),G_{i2}(ξ)=G_{i2}(−ξ) and Φ_{11}(ξ)=−Φ_{11}(−ξ),Φ_{12}(ξ)=Φ_{12}(−ξ),(i=1,2,3).
From Eq. (20), we have
where ϕ=g or h. Obviously, the coefficients satisfy the relations A_{ϕ}(ξ)=A_{ϕ}(−ξ) and B_{ϕ}(ξ)=−B_{ϕ}(−ξ). They can be expressed explicitly as follows,
If the particle distribution function ϕ(−ξ) is already known, then the particle distribution function ϕ(ξ) in the opposite direction can be obtained in the following way. From Eq. (50), we obtain a generalized bounce back scheme
where β is a coefficient to be determined. Specially, we can choose β=1 or β=−1 in real implementation. For this purpose, we have to evaluate the sum or difference of the equilibriums and source terms.
The sum and difference of the source terms depend on the specific form used. For S_{g} given in Eq. (24) and S_{h} given in Eq. (29), we have
Consider the threedimensional isothermal flow in the incompressible limit with constant temperature T_{0}. The internal degree of freedom is K=0 and the bulk viscosity is μ_{V}=0. No thermal cooling function is applied, i.e. Λ=0. The source term S_{g}=0. Then the Eq. (52) with ϕ=g and β=1 can be simplified as
In the lattice Boltzmann method [20], we first introduce a transformation as
where Δt is the time step.
Next, in order to use the GaussHermite quadrature for the evaluation of the integrals, we introduce another transformation as,
where α denotes the directions of the discrete velocities ξ_{α} and W_{α} denotes the corresponding weight.
After some reorganization, the final result is
If we only keep the first term in Eq. (58), we can obtain
We note that the body force enters the implementation of the bounceback scheme, which is not well documented in the literature. Furthermore, it must be cautioned that Eq. (59) is not fully consistent with the ChapmanEnskog expansion of the NSF system as the O(τ) terms in Eq. (58) are not included. Luckily, in the special case of noslip boundary u_{wall}=0, the O(τ) terms in Eq. (58) will disappear. Note that the source term and velocity could all enter the implementation of the thermal boundary conditions.
9 Highorder structure of the distribution functions
The NSF equations, which are based on the continuum hypothesis, have been widely used in understanding flow behaviors in many natural and engineering problems. However, in some cases such as microchannel flows [35], compressible turbulence [30] and space vehicles in low earth orbits [36], the local Knudsen number may be finite such that the flow may lie in the continuumtransition regime locally. Therefore, the NSF equations are not adequate to capture the finite Knudsen number effect while the Boltzmann equation can describe the flows in all Kn number regimes.
In order to quantitatively estimate the departure from the local thermodynamic equilibrium and study the extended hydrodynamics, the secondorder ChapmanEnskog expansion of the particle distribution function is desired, which results in the socalled “Burnett equations” [37]. The Burnett equations have been derived from the original Boltzmann equation by applying the ChapmanEnskog expansion [1] or the Grad’s 13 moment equations [38] by the iteration approach [39]. However, these theoretical results are seldom compared with those using the singlerelaxationtime BGK model. In addition, detailed derivations are less reported or the final results are not presented in a general form. In this section, we will derive the structure of the distribution functions up to the order O(τ^{2}). Then, the complete analytical expressions for the viscous stress tensor and the heat flux are obtained in the subsequent sections. Furthermore, by comparing our results with those from Grad’s 13 moment equations, it is found that the mathematical form of the viscous stress tensor and the heat flux can be fully determined in the singlerelaxationtime BGK model. The difference from the literature in the coefficients could be attributed to different relaxation rates to the local equilibrium for different moments used in the literature.
By using the NSF equations, we obtain the expression for Dg^{eq}/Dt and Dh^{eq}/Dt to the order of O(τ),
where G and Φ_{1} have been given above, L and Φ_{2} are given as
By applying the ChapmanEnskog expansion, we can obtain the structure of the particle distribution function as
10 Viscous stress tensor up to O(τ ^{2})
When the local Knudsen number becomes finite, additional contributions from the nonequilibrium part of the distribution function result in the highorder components of the viscous stress tensor. Agarwal et al. [39] and Struchtrup [40] derived the viscous stress tensor up to the secondorder for Maxwell molecules from the Grad’s 13 moment equations by the series expansion in terms of the shear viscosity. Chen et al. [7] obtained the expression of viscous stress tensor up to the secondorder based on the singlerelaxationtime BGK model. However, they mainly focus on the incompressible limit and the terms proportional to the density gradient, the temperature gradient and the velocity divergence have been neglected in their derivation. By making an analogy between the turbulent fluctuations and microscale thermal fluctuations, they show that the Reynolds stress obtained by the BGKBoltzmann equation has model coefficients similar to some existing turbulence models. They also claimed that the turbulence phenomenon such as the secondary flow structures and rapid distortion processes [41] can be better understood according to the kinetic theory. As an extension of Chen’s work, the complete form of the viscous stress tensor will be derived up to O(τ^{2}) using the singlerelaxationtime BGK model considering the internal degree of freedom of molecules. Moreover, this new result will be compared to that obtained by Agarwal et al. [39] for Maxwell molecules.
The general expression of the viscous stress tensor is given as follows.
After some computation and simplification, we obtain the complete expression for the viscous stress tensor σ as
where the material derivative of the strain rate tensor dS/dt can be derived from the Euler equations, which reads
Furthermore, by setting K=0, χ=0, Λ=0 and neglecting all the terms proportional to the velocity divergence 𝜗, the density gradient ∇ρ and the temperature gradient ∇T, it is found that S_{g}=0 and the following approximate result obtained by Chen et al. [7] can be reproduced from the Eq. (64), namely,
For Maxwell molecules of which the shear viscosity is linearly proportional to the temperature, we have dμ/dT=μ/T. Therefore, the relations (1/τ)∇τ=−(1/ρ)∇ρ and (1/τ)dτ/dt=−(1/ρ)dρ/dt hold. Using the continuity equation, it follows that (1/τ)dτ/dt=𝜗. Then, using our notations, the results obtained by Agarwal et al. [39] can be rewritten as
Moreover, Eq. (64) can be simplified as
From Eqs. (67) and (68), we observe that these two expressions share identical mathematical form up to the order O(τ^{2}) except for values of some coefficients. It is observed that the nonlinear terms S·S,S·Ω−Ω·S and S𝜗 are exactly identical. Besides, the material derivative term d(S−(1/3)𝜗I)/dt is also the same and the terms related to the temperature gradient and temperature diffusion are also very close to each other. The sign of corresponding coefficients is also the same, which implies that the negative or positive contribution to the viscous stress tensor can be qualitatively determined based on the BGK collision model. Moreover, it is found that the viscous stress tensor can be changed by the body force effect included in the material derivative term d(S−(1/3)𝜗I)/dt at the secondorder expansion but not at the firstorder. Therefore, we conclude that although the BGK model only uses single relaxation time to characterize the relaxation process to the local equilibrium without considering rigorous collision interaction details, all the dominant terms in the viscous stress tensor can be recovered compared to those obtained from the Grad’s 13 moment equations. These expressions could potentially give guidelines to the functional form for Reynolds stress modeling in compressible turbulence.
11 Heat flux up to O(τ ^{2})
Based on the secondorder ChapmanEnskog expansion of the distribution functions, the analytical expression for the heat flux q is given by
Noticing Eqs. (21a)–(21d) and Eqs. (19), (61), all the integrals in Eq. (69) can be evaluated term by term. After some reorganization, we obtain
where the time and spatial derivatives of the relaxation time are given by
The results in Eq. (70) are briefly discussed here. The first term is the Fourier’s law. The second term is determined by the divergence of the viscous stress tensor. The third term is caused by the variation of the particle relaxation time in both space and time. The fourth term is composed of the coupling terms between the strain rate, rotation rate, temperature gradient and density gradient as well as the divergence of the strain rate. The fifth term represents the contributions from the terms relevant to the thermal energy source. The last two integrals depend on the specific form of the source terms S_{g} and S_{h} used in different models.
Similar to what we have done for the viscous stress tensor, by setting K=0, χ=0, Λ=0 and S_{g}=0, a comparison would also be performed for heat flux for Maxwell molecules. The result obtained by Agarwal et al. [39] can be reformulated as
Correspondingly, Eq. (70) can be simplified as
Again, Eqs. (72) and (73) share the same mathematical form and the same sign for each contribution up to the order O(τ^{2}). In our model, S_{h} is mainly designed to modify the Prandtl number and thermal energy source. Note that we keep the term relevant to S_{h} in Eq. (73) but it can be evaluated once the specific form of S_{h} is given.
12 Conclusions
In this paper, an inverse design approach of mesoscopic models for compressible flows in continuum or nearcontinuum regime has been explored based on the ChapmanEnskog analysis. The design began with a model Boltzmann equation in a high dimensional phase space and with an undetermined source term. Then two reduced model Boltzmann equations in seven dimensional phase space are introduced, each containing a source term. First, it is found that there are many possible ways to design the source terms in order to recover the NSF system in the continuum limit, as long as five newlyderived requirements for the two source terms are met. These source terms allow for flexible Prandtl number, bulktoshear viscosity ratio, and a thermal energy source/sink term.
Second, based on the Hermite expansion, we have provided one design for the two source terms. This newly introduced model has been utilized to simulate decaying compressible isotropic turbulence [32] and forced compressible isotropic turbulence [33], achieving global turbulence statistics in excellent agreement with those based on solving the NSF system [29, 30]. Our model can be viewed as a Boltzmannequation based mesoscopic solver for compressible flows that solves the NSF system.
Third, three well accepted kinetic models, namely, the Shakhov model, the total energy doubledistributionfunction model, and the Rykov model, have been shown to be compatible with five basic constraints derived from the ChapmanEnskog analysis. For Rykov model, translational and rotational temperatures are introduced, which allows for larger bulktoshear viscosity ratio compared to other models. Similarly, in our model, the bulktoshear viscosity ratio should be limited such that the premise of ChapmanEnskog expansion is satisfied.
Furthermore, by applying the firstorder ChapmanEnskog expansion to the distribution functions, we discuss the structures of the distribution functions and the implementation of bounce back boundary conditions. These results can be used to improve the implementation of hydrodynamic boundary conditions in terms of the distribution functions, namely, constructing the missing distributions from the known distribution near a solid boundary, in both laminar and turbulent flows.
Finally, although present model is mainly designed for continuum or nearcontinuum compressible flows, with BGK approximation, we derive the complete analytical expressions for the viscous stress tensor and the heat flux based on the secondorder ChapmanEnskog expansion of the distribution functions, generalizing the previous results in the incompressible limit. These new results have been compared with those obtained from Grad’s 13 moment equations, which demonstrates that the final structure of the viscous stress tensor and heat flux can be fully determined by the singlerelaxationtime BGK model except for differences in some coefficients. We believe that highorder effects in compressible turbulence could be partially captured by the BGKBoltzmann equation, which certainly deserves further investigation. It would be desirable to explore underlying physics associated with the secondorder terms especially in compressible turbulence, in the future using DNS data. The secondorder terms may also provide a way to assess the difference between NSF flows and the flows governed by the model Boltzmann equation.
13 Hermite polynomials and hermite expansion
The nth order Hermite polynomial is defined by [42, 43],
where \(\boldsymbol {\nabla }_{\boldsymbol {\xi }}^{n}=\boldsymbol {\nabla }_{\boldsymbol {\xi }}\boldsymbol {\nabla }_{\boldsymbol {\xi }}\cdots \boldsymbol {\nabla }_{\boldsymbol {\xi }}\) implies that \(\mathscr {H}^{(n)}({\boldsymbol {\xi },T_{0}})\) is a symmetrical tensor of rankn. The weighting function is ω(ξ,T_{0})=(2πRT_{0})^{−D/2} exp(−ξ^{2}/2RT_{0}).
From the 4^{th}order Hermite expansion of g^{eq} and the 2^{nd}order Hermite expansion of h^{eq} [32], we obtain
where θ=T/T_{0} is the normalized temperature.
14 Derivations of the requirements for the source terms
The Euler equations can be obtained by assuming that g=g^{eq}+O(τ) when evaluating the viscous stress and the heat flux. This leads to σ∼O(τ) and q∼O(τ). Therefore, the Euler equations are
Taking the firstorder moment of the Boltzmann equation for g, we have
By using the Eq. (20), we can make a closure of the viscous stress,
All the integrals in the RHS of Eq. (79) can be evaluated term by term,
Substitution of Eq. (80) into Eq. (79) yields
Hence, in order to recover the Newtonian constitutive law (see Eq. (16)), the Eq. (21b) must be satisfied.
Similarly, combining the secondorder moment of the Boltzmann equation for g and the zerothorder moment of the Boltzmann equation for h yields the following equation,
By using the Eqs. (20), we can make a closure of the heat flux term,
Because of \(\int \boldsymbol {c} c^{2} G_{1} g^{eq}d\boldsymbol {\xi }=5\rho (RT)^{2}\left ((1/T)\boldsymbol {\nabla } T\right)\) and \(\int \boldsymbol {c} c^{2} G_{2} g^{eq}d\boldsymbol {\xi }=\int \boldsymbol {c} c^{2} G_{3} g^{eq}d\boldsymbol {\xi }=\boldsymbol {0}\), we obtain
Because of \(\int \boldsymbol {c} G_{1} g^{eq}d\boldsymbol {\xi }=\int \boldsymbol {c} G_{2} g^{eq}d\boldsymbol {\xi }=\int \boldsymbol {c} G_{3} g^{eq}d\boldsymbol {\xi }=\boldsymbol {0}\), we have \(\tau \int \boldsymbol {c} G h^{eq}d\boldsymbol {\xi }=\boldsymbol {0}\). Further, we have
By substituting of Eqs. (84) and (85) into Eq. (83), we obtain
Therefore, we have derived the fifth requirement for the source term in Eq. (21d).
15 Details in the derivations of the Rykov model
Here we prove that the two source terms in Eq. (45) should satisfy the five general requirements. From the Euler equations for the Rykov model, the time derivative for the translational temperature T_{t} and rotational temperature T_{r} are
The first requirement for the source term is satisfied because of
The second requirement for the source term is satisfied because of
According to Eqs. (88) and (89), we obtain \(\int \boldsymbol {\xi \xi } S_{g} d\boldsymbol {\xi }=\int \boldsymbol {cc}S_{g} d\boldsymbol {\xi }\).
Since \(\int \boldsymbol {cc} f_{0}^{r} d\boldsymbol {\xi }=p\boldsymbol {I}\), \(\int \boldsymbol {cc} f_{0}^{t} d\boldsymbol {\xi }=p_{t} \boldsymbol {I}\) and \(\int \boldsymbol {c} \boldsymbol {c} f_{M}(T) d\boldsymbol {\xi }=p\boldsymbol {I}\), therefore,
From Eq. (87), we have
Therefore, we obtain
Substituting Eq. (92) into Eq. (90), we have
Therefore, the third requirement for the source term is proved. The ratio of bulktoshear viscosity χ=4Z/15 in ChapmanEnskog analysis.
From Eq. (90), we have
Using the relation \(pp_{r}=\frac {3}{2}\left (pp_{t}\right)\), we have
From Eqs. (94) and (95), the fourth requirement is satisfied.
We note that the following integrals can be carried out directly.
Hence, we have
By applying the ChapmanEnskog expansion, we can prove that the heat fluxes are given by
Combining Eqs. (97) and (98) gives
where the Prandtl number is Pr=μC_{p}/κ=7Rμ/2κ. Therefore, the fifth requirement is satisfied.
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Abbreviations
 NSF:

NavierStokesFourier
 CFD:

Computational fluid dynamics
 BGK:

BhatnagarGrossKrook
 S:

Shakhov
 ES:

Ellipsoidal statistical
 IEDDF:

Internal energy doubledistributionfunction
 TEDDF:

Total energy doubledistributionfunction
 R:

Rykov
 LBM:

Lattice Boltzmann method
 GKS:

Gas kinetic scheme
 UGKS:

Unified gas kinetic scheme
 DUGKS:

Discrete unified gas kinetic scheme
 EOS:

Equation of state
 DNS:

Direct numerical simulation
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Acknowledgements
Computing resources are provided by the Center for Computational Science and Engineering of Southern University of Science and Technology and by National Center for Atmospheric Research (CISLUDEL0001).
Funding
This work has been supported by the U.S. National Science Foundation (CNS1513031, CBET1706130), the National Natural Science Foundation of China (91852205, 91741101 & 11961131006), the National Numerical Wind Tunnel program, Guangdong Provincial Key Laboratory of Turbulence Research and Applications (2019B21203001), and Shenzhen Science & Technology Program (Grant No. KQTD20180411143441009).
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Tao Chen developed the derivations, drafted and edited the manuscript. LianPing Wang conceptualized the methodology and reviewed the derivations, edited the manuscript, provided suggestions for improving the manuscript, acquired funding for this research, and served as the corresponding author. Jun Lai checked the derivations. Shiyi Chen participated in funding acquisition and supervised this research. All authors read and approved the final manuscript.
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Chen, T., Wang, LP., Lai, J. et al. Inverse design of mesoscopic models for compressible flow using the ChapmanEnskog analysis. Adv. Aerodyn. 3, 5 (2021). https://doi.org/10.1186/s42774020000592
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DOI: https://doi.org/10.1186/s42774020000592