drag coe cient and lift coe cient are also provided, It is always shown that the one in classical limit is within the other quantum statistics.
5. In three dimensional lid driven cavity ow, the velocity vectors and the velocity of the centerlines of three slices are shown. The one in classical limit matches with previous works, it validates the semiclassical LB method in classical limit.
6.2 Future Work
Since the semiclassical LB methods have been derived, however, there exist some works to do in the future.
1. The experimental results for quantum hydrodynamics are rare and we only validate our results with the corresponding classical counterpart in classical limit. There are some experiments dealing with quantum uids in microchannel ow. For example, in [106], rst observation of the Knudsen minimum in normal liquid 3He was reported. Also, Knudsen minima has been found for phonons at dierent Knudsen numbers have been reported, see [107][108]. The present work should be compared with quantum uid experiments and also validated according to more experiments.
2. A new coupled thermal LB method based on double distribution func-tions with two relaxation times is derived for thermohydrodynamic Navier-Stokes equations in Appendix 2. The method is obtained by rst pro-jecting the governing two relaxation kinetic model equations onto the Hermite polynomial basis as pioneered by [79] and [16]. And the cou-pled thermal LB method should be possibly used for developing double
distribution function semiclassical LB method.
3. The present work is developed based on expanding the distribution func-tion (including equilibrium distribufunc-tion funcfunc-tion feq) onto Hermite poly-nomials. Although this strategy for building LB method attracts lots of attentions in recent years, there are still some criticisms concerning entropy and Galilean invariance on these procedures [115][116]. One of the disadvantages of expanding the equilibrium distribution function onto Hermite polynomials is that the roots of Hermite polynomials are irrational, the corresponding discrete velocities can not be tted into a regular space lling lattice. Such that, the resulting o-lattice models make the exact space discretization of the advection step which is the major advantage of the LB methods broken. Another drawback is that the solution of projecting the equilibrium distribution function in a -nite dimensional Hermite basis will lose the positivity of the distribution function in the truncation of Hermite polynomial expansions. One of the methods for solving these problems mentioned above is through the en-tropic formulation [117][118] which evaluates the Boltzmann H function at the nodes of the given quadrature instead of equilibrium distribution function. And in recent years, this idea is extended to derive a gener-alized Maxwell distribution function for LB method [119], in that work, all the previously introduced equilibria for LB are found as special cases of the generalized Maxwellian. Moreover, in [120] taking advantage of the closed-form generalized Maxwell distribution function and splitting the relaxation to the equilibrium in two steps, they proposed an entropic quasiequilibrium kinetic model for the simulation of low Mach number
ows. The future development of semiclassical LB method may follow
6.2. Future Work 105
the new trend of LB method.
Appendix A
Nomenclature
n number density
ρ density, Kg/m3
µ viscosity, or chemical potential c lattice velocity
cs speed of sound D dimension of system
ξi lattice streaming vector in i direction
e kinetic energy
F external force
f distribution function g acceleration due to gravity
H height, m
p pressure
P r Prandtl number
q heat ux vector
x position vector
Ra Rayleigh number
U maximum velocity at the inlet U0 top lid velocity
u uid velocity, ms−1
u, v, w x, y and z components of ow velocity
V discrete velocity space Πij momentum stress tensor
m mass, kg
t time, s
T temperature, K
δx lattice spacing, m δt time step, s
Subscripts
i, j, k Cartesian coordinate system Greek symbols x, y, z Cartesian coordinate system Greek symbols
a velocity component
Superscripts
eq equilibrium distribution function
(0) equilibrium distribution function after discretizing velocity space N order of the discretized equilibrium distribution function
Q semiclassical distribution function C classical distribution function
Abbreviations
LB Lattice Boltzmann
WKB Wentzel-Kramers-Brillouin UUB Uehling-Uhlenbeck-Boltzmann BGK Bhatnagar-Gross-Krook
FD Fermi-Dirac
BE Bose-Einstein
109
MB Maxwell-Boltzmann
DdQq d Dimension, q velocity
Appendix B
Chapman-Enskog Analysis of semiclassical LB method
B.1 Chapman-Enskog Analysis of Single Relax-ation Time semiclassical LB method
In this section, the generic evolution equation of single relaxation time semi-classical LB method is inspected by means of a Chapman-Enskog analysis.
In LB method, the choice of lattice types is very important and it aects the accuracy and stability of LB simulation. The lattice must verify some sym-metry conditions to yield the desired asymptotic PDE. It requires appropriate lattice velocities ca and weights wa in developing LB method such that the following relations are veri ed:
(a)∑
a
wa= 1 (b)∑
a
wacai = 0
(c)∑
a
wacaicaj = c2sδij
(d)∑
a
wacaicajcak = 0 (e)∑
a
wacaicajcakcal= c4s(δijδkl+ δikδlj + δilδkj) (B.1)
Now, the LB-BGK equation (3.4) and semiclassical equilibria (3.38) up to third order are used and listed below.
fa(x + ζaδt, t + δt)− fa(x, t) =−(fa− fa(0))
First, computing all the following summations by the semiclassical equi-libria with the aids of (B.1) as:
∑
The above expressions of the derivatives are substituted into (B.2), and
B.1. Chapman-Enskog Analysis of Single Relaxation Time
semiclassical LB method 113
terms involving dierent orders of ε are separated as:
0 = −1 After summation with respect to a, the three dierent order mass conser-vation equations are recovered below:
−1 After multiplying with mζai and summing with respect to a, the three dierent order momentum conservation equations are recovered below:
−1 The local number density n and the local velocity u are
n =∑
From zeroth-order mass and momentum equations (B.16) and (B.19), we For deriving the Navier-Stokes equation, we add the rst and second order momentum equations (B.20) and (B.21).
∂t(ρui) + ∂jPij = 0 (B.24) where the momentum ux density tensor Pij(1) is given by
Pij = Pij0 + ε(2τ − 1 The Pij(1) is calculated as:
Pij1 = m∑
B.1. Chapman-Enskog Analysis of Single Relaxation Time
semiclassical LB method 115
After putting (B.29) and (B.30) into (B.28), we can get (B.27). Finally, put (B.25),(B.26) and (B.27) into (B.24), we can get the nal form of the momentum equation as
∂t(ρui) + ∂j(ρuiuj) =−∂ip + ν∂j[ρ∂iuj + ρ∂jui] (B.31) In symbolic notation the equation is
∂t(ρu) +∇ · (ρuu) = −∇p + ν∇ · [ρ∇u + ρ(∇u)T] (B.32) The Navier-Stokes equation is recovered from this equation in the incom-pressible limit(∂juj = 0) or (∇ · u = 0)
∂tui+ uj∂jui =−1
ρ∂ip + ν∇2ui (B.33) ν = (τ − 1
2)Tg5
2
g3
2
(B.34) In symbolic notation the equation is
∂tu + u· ∇u = −1
ρ∇p + ν∇2u (B.35)
Notice that ∇ · (∇)T =∇(∇ · u)
Appendix C
Derivations of Double Distribution Function LB method
In this chapter, we provides another possible way to solve the defects of single relaxation time semiclassical LB method from the ideas of developing thermal LB methods. Historically, the existing thermal LB methods can be classi ed into three categories. The rst category is the passive-scalar approach [121]
and the basic approach of this method is assuming that the viscous heat dis-sipation and the compression work done by the pressure can be neglected. As a consequence, the temperature can be simulated by a scalar density distri-bution function. The other two categories include the multi-speed distribu-tion funcdistribu-tion approach [122][123][124] and the double distribudistribu-tion funcdistribu-tions (DDF) approach [125][126][127][79]. The multi-speed approach is achieved by increasing the numbers of discrete velocities, then, the compressible Navier-Stokes equations could be recovered by those increasing degrees of freedom.
Although the multi-speed approach can reach a thermal LB method, but it suers from severe numerical instability and a narrow range of temperature variation. Moreover, the Prandtl number is usually xed at constant. On the other hand, the DDF approach utilizes two distribution functions, one is for describing the velocity eld, the other is for describing the internal energy or total energy. The DDF approach can achieve a better numerical stability than the multi-speed approach. We derive a thermal LB method which is
based on DDF method and should be suitable for developing DDF semiclas-sical LB method, in which both the Maxwell equilibrium velocity distribution function and a total energy equilibrium distribution function are expanded on tensor Hermite polynomials according to [79][16]. By choosing a proper reference velocity, the coupling of lattice velocities and the local temperature is avoided and the uncoupling process between velocity eld and temperature
eld as done in [79] is not necessary, thus, some of limitations mentioned such as valid only to small temperature variation and transport coe cients are independent of temperature can be overcome. We also apply the Chapman-Enskog method to the present coupled thermal LB-BGK equations to obtain the relations between the relaxation time and viscosity and thermal conduc-tivity. Hydrodynamics based on moments up to third order expansions are presented. Computational examples to illustrate the methods are given, and the results are carefully studied with the published result [114]. This new scheme provides a possible way for extending the current semiclassical LB method to double distribution functions semiclassical LB method.