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Investigation of Multiscale Non-equilibrium Flow Dynamics Under External Force

Field

Tianbai Xiao1, ∗and Kun Xu2, †

1Department of Mechanics and Engineering Science,

College of Engineering, Peking University, Beijing 100871, China

2Department of Mathematics, Department of Mechanical and Aerospace Engineering,

Hong Kong University of Science and Technology, Hong Kong

The multiple scale non-equilibrium gaseous ﬂow behavior under external force ﬁeld is investigated.

Both theoretical analysis based on the kinetic model equation and numerical study are presented

to demonstrate the dynamic eﬀect of external force on the ﬂow evolution, especially on the non-

equilibrium heat ﬂux. The current numerical experiment is based on the well-balanced uniﬁed

gas-kinetic scheme (UGKS), which presents accurate solutions in the whole ﬂow regime from the

continuum Navier-Stokes solution to the transition and free molecular ones. The heat conduction

in the non-equilibrium regime due to the external forcing term is quantitatively investigated. In the

lid-driven cavity ﬂow study, due to the external force ﬁeld the density distribution inside cavity gets

stratiﬁed and a multiscale non-equilibrium ﬂow transport appears in a single gas dynamic system.

With the increment of external forcing term, the ﬂow topological structure changes dramatically, and

the temperature gradient, shearing stress, and external force play diﬀerent roles in determination

of the heat ﬂux in diﬀerent layers corresponding to diﬀerent ﬂow regime. Besides the non-Fourier’s

heat eﬀect in the transition regime, such as the heat ﬂux from cold to hot region in the absence

of external ﬁeld, the additional external force enhances the heat ﬂux signiﬁcantly along the forcing

direction, which could trigger the gravity-thermal instability through the heat ﬂow from the high

potential cold region to the low potential hot region. At the same time, the wave propagation

phenomena for the temperature ﬁeld in the transition regime have been observed, which may be

helpful for the modeling of hyperbolic heat evolution equation. Through the numerical experiment,

it is clear that the external force plays an important role in the dynamic process of non-equilibrium

heat transfer.

PACS numbers: 05.20.Dd, 47.70.Nd, 44.05.+e

Key Words: multiscale ﬂow, non-equilibrium phenomena, external force ﬁeld, uniﬁed gas-kinetic scheme, ﬂow trans-

port, heat transfer

I. INTRODUCTION

The gas dynamic system under external force ﬁeld is essentially associated with multiple scale nature due to the

possible large variation of gas density, and the corresponding local Knudsen number. From the kinetic theory, as an

example we can use the Bhatnagar-Gross-Krook (BGK) [1] equation to illustrate the eﬀect of the external force on

the variation of distribution function. The evolution of particle distribution function f(xi, t, ui, ξ) in space and time

is described by

∂f

∂t +ui

∂f

∂xi

+φi

∂f

∂ui

=g−f

τ.(1)

Here ξis the internal variable for the rotation and vibration and φiis the external forcing term. The Eq. (1) can be

rewritten in the following form,

f=g−τ(ft+uifxi+φifui).(2)

Even with an initial Maxwellian distribution function, the free transport of particles during the traveling time between

two successive collisions evolves the system away from equilibrium state. Under external force ﬁeld, the particle

acceleration or deceleration during this time interval results in a distortion of the distribution function in the velocity

space simultaneously. Consequently, the particle collision takes eﬀect to drive the system back to equilibrium state.

∗Email:xiaotianbai@pku.edu.cn

†Email:makxu@ust.hk

arXiv:1610.05544v2 [physics.flu-dyn] 19 Oct 2016

2

However, in rareﬁed regime, the particle free transport and collision are loosely coupled due to a large particle collision

time. Much complicated nonlinear dynamics due to particle transport, collision, and external force acceleration, will

appear and presents a peculiar non-equilibrium ﬂow behavior. Usually the non-equilibrium eﬀects are expected to

emerge in the highly dissipative regions, such as the shock and boundary layers. However, with the existence of

external force ﬁeld non-equilibrium gas evolution may spread to the whole ﬂow system in a large scale.

It is noted that there is only limited study on non-equilibrium ﬂow under external force ﬁeld. Theoretically, the

existence of external force ﬁeld, such as gravity, introduces a characteristic length scale H∼kBT/mφ [2], where kBis

the Boltzmann constant and mis the particle mass. It denotes a vertical distance over which the force ﬁeld produces

a signiﬁcant eﬀect on the gas evolution. For a gravitational system under laboratory condition or a micro-electro-

mechanical system (MEMS) with geometric characteristic length L, the relation H >> L holds naturally, and thus it

is reasonable to omit the inﬂuence of external force. However, in the case where His comparable to L, the eﬀect of

external force will appear. For example, let us consider a large scale cavity full of gases under gravitational ﬁeld, such

as the atmospheric environment. The external force will result in an observable variance of density in the cavity, and

so is the variation of the particle mean free path and the local Knudsen number. Similar cases appear in small scale,

but with large acceleration, such as material interface with shock impingement. Therefore, it is interesting to study

the multiple scale non-equilibrium gas eﬀect due to external force.

Diﬀerent gas dynamic equations are used to describe the ﬂow in diﬀerent modeling scale. In hydrodynamic scale

modeling, in 1822 Fourier [3] proposed a well-known phenomenological heat conduction law. Based on the Fourier’s law

and Newton’s stress and strain relationship, the Navier-Stokes-Fourier (NSF) equations were constructed to describe

the ﬂuid motion and heat transfer in macroscopic scale. In the Navier-Stokes modeling, the ﬂuid element picture

is used and the intensive particle collisions prevent particle penetration between elements. In such a modeling, the

diﬀusion formulation for heat ﬂux introduces inﬁnite propagation speed for the temperature ﬁeld, which prevents

its accurate description for the non-equilibrium heat transfer in rareﬁed ﬂow [4]. The NS equations have no a clear

modeling scale theoretically, and the boundary for its validation is not clearly deﬁned. On the contrary, the Boltzmann

equation is well deﬁned on the modeling scales of particle mean free path and collision time, where the particle free

transport and the collision can be described eﬃciently in an operator splitting way. It also limits the application of

the Boltzmann equation to other scales, except resolving other scales all the way to the mean free path and particle

collision time.

Physically, the valid application of distinguishable gas dynamic equations, such as NS and the Boltzmann, depends

on their clear scale separation. However, for a system under external force ﬁeld, the ﬂow physics may vary continuously

from the kinetic Boltzmann modeling in the upper rareﬁed layer to the hydrodynamic one in the lower dense region.

Theoretically, in order to study the cross-scale ﬂow physics, a valid modeling here should have a smooth scale variation

between the kinetic and hydrodynamic ones. A uniﬁed approach is preferable to simulate such a system to capture

a continuum spectrum of ﬂow dynamics from rareﬁed to continuum one. The uniﬁed gas-kinetic scheme (UGKS)

provides such a choice [5, 6]. The corresponding well-balanced scheme under external force ﬁeld has been developed

recently [7]. Through a coupled treatment of particle transport, collision, and external forcing eﬀect in the evaluation

of ﬂux transport across a cell interface and inner cell ﬂow evolution, a continuous spectrum of gas dynamic equations

can been recovered [8]. In this paper, the well-balanced UGKS is employed to investigate the multiscale gas evolution

under external force ﬁeld.

Even under simple geometric condition, the cavity ﬂow displays complex ﬂuid mechanical phenomena with multiple

scales, including shearing layers, eddies, secondary ﬂows, heat transfer, hydrodynamic instabilities, and laminar-

turbulence transition, etc [9]. Great eﬀorts have been devoted to the study of the ﬂow physics in diﬀerent ﬂow

regimes as well. In the continuum regime, the cavity problem is a typical benchmark case for the validation of

numerical algorithms for the NS solutions [9–13]. In rareﬁed regime, the direct simulation Monte Carlo (DSMC) [14]

and kinetic Boltzmann solvers [15, 16] provide the benchmark solutions. Naris et al. [17] discretized a linearized BGK

equation to investigate the rarefaction eﬀect on the ﬂow pattern and physical quantities over the whole range of the

Knudsen number. Mizzi et al. [18] compared the simulation results from the Navier-Stokes-Fourier equations (NSF)

with slip boundary conditions and the DSMC results in a lid-driven micro cavity case. John et al. [19] applied the

DSMC, discovered counter-gradient heat transport in the transition regime, and investigated the dynamic eﬀect from

the expansion cooling and viscous dissipation on the heat transport mechanism. In all previous work, there is few

study about the cavity ﬂow under external force ﬁeld. Due to the external force eﬀect, the cavity ﬂow becomes even

more complicated with its non-equilibrium multiple scale nature. A few new phenomena, including the connection

between the heat transfer and external force, and wave propagation of the temperature ﬁeld, have been observed

through this study.

This paper is organized as follows. The basic kinetic theory and the analysis of the inﬂuence of external force on

the macroscopic ﬂow transport are presented in Section 2. The well-balanced uniﬁed gas-kinetic algorithm under

external force ﬁeld is introduced in Section 3. Section 4 presents the numerical experiments and discussion on the

non-equilibrium ﬂow transport and heat transfer in diﬀerent ﬂow regimes. The last section is the conclusion.

3

II. KINETIC ANALYSIS ON THE EXTERNAL FORCE

The macroscopic ﬂow transport and heat transfer have a close relationship with the particle motion in the kinetic

scale. In this section, we use the BGK equation to illustrate the inﬂuence of external force on a near-equilibrium gas

dynamic system. For simplicity, the one-dimensional case is considered ﬁrst.

Notice that Eq. (1) can be rewritten in a successive form:

f=g−τDf

Dt =g−τD

Dt g−τDf

Dt =··· ,(3)

where D/Dt is the material derivatives. The τ=µ/p is the collision time, and the Maxwellian distribution gwrites,

g=ρλ

πK+1

2

e−λ[(u−U)2+ξ2],(4)

where λ=m/2kBTand Kis the internal degree of freedom. If we consider the ﬁrst order approximation of Eq. (3)

with respect to the collision time τ, the distribution function fhas the corresponding expansion,

f=g−τ(gt+ugx)−τφgu.(5)

The ﬁrst two terms in Eq. (5) describe the free transport of particles during the traveling time between two successive

collisions. This expression is consistent with the Chapman-Enskog expansion for the Navier-Stokes solutions, and

a dynamic viscous coeﬃcient µ=τp can be obtained [20, 21]. At the same time, the force acceleration distorts

the distribution function in the velocity space with the following contributions to the macroscopic ﬂow variables.

Macroscopic variables are related with particle distribution function through velocity moments,

W=

ρ

ρU

ρE

=Zψf dΞ,

p=1

3Z(u−U)2+ξ2fdΞ,

q=1

2Z(u−U)(u−U)2+ξ2fdΞ,

where dΞ = dudξ and ψ=1, u, 1

2(u2+ξ2)Tis the vector of moments for collision invariants. For the moments of

Maxwellian distribution RuαξβgdΞ = ρ<uαξβ>, it has the property that

< uαξβ>=< uα>< ξβ> .

For simplicity, let us consider a gas system with zero macroscopic velocity. In this case, the moments of Maxwellian

distribution function are

< u0>= 1,

< u1>= 0,

< u2>=1

2λ,

< u3>= 0,

< u4>=3

4λ2,

< ξ2>=K

2λ.

(6)

4

The net contribution of macroscopic variables from the external forcing term can be written as

∆W=

∆ρ

∆ρU

∆ρE

=Zψ(−τφgu)dΞ,

∆p=1

3Zu2+ξ2(−τφgu)dΞ,

∆q=1

2Zuu2+ξ2(−τφgu)dΞ.

If the forcing term φand the collision time τare viewed as local constants, and the exact Maxwellian distribution in

Eq. (4) is used to evaluate its velocity derivative, we have the following relations,

∆W=

∆ρ

∆ρU

∆ρE

= 2τ φλ ZψugdΞ=2τ φλρ

< u1>

< u2>

1

2< u3>+< uξ2>

,

∆p=2

3τ φλ Zuu2+ξ2gdΞ = 2

3τ φλρ < u3>+< uξ2>,

∆q=τ φλ Zu2u2+ξ2gdΞ = τ φλρ < u4>+< u2ξ2>.

For a monatomic gas with internal degree of freedom K= 2 for the random particle motion in the yand zdirections,

based on the moments in Eq. (6), we get

∆W=

∆ρ

∆ρU

∆ρE

=

0

τφρ

0

,(7)

∆p= 0,(8)

∆q= 1.25τ φρ

λ.(9)

The Eq. (7), (8) and (9) present a quantitative contribution of external force ﬁeld on the macroscopic ﬂow variables,

which isn’t included in the traditional Navier-Stokes equations. It is clear that there exists contribution to the

macroscopic ﬂow velocity and the heat ﬂux from the external forcing term. Between two successive collisions, the

particles get acceleration and enhanced velocity, resulting in a macroscopic mass and energy transport along the

direction of external force. In the continuum ﬂow regime with intensive particle collisions, the collision time τtends

to zero, and the inﬂuence of external force ﬁeld can be omitted. However, at a limited particle collision time τand

non-vanishing local Knudsen number, the external forcing term does eﬀect the transport, especially in the transition

regime. It is noted that the above analysis is based on the near-equilibrium assumption. With the increment of degree

of rarefaction, even strong non-equilibrium eﬀect is expected to appear.

5

III. WELL-BALANCED UNIFIED GAS-KINETIC SCHEME

The well-balanced uniﬁed gas-kinetic scheme is a direct modeling on the length scale of cell size and time scale of

local time step for the construction of corresponding governing equations in such scales. With the notation of cell

averaged distribution function

fxi,yj,tn,uk,vl=fn

i,j,k,l =1

Ωi,j (~x)Ωk,l(~u)ZΩi,j ZΩk ,l

f(x, y, tn, u, v)d~xd~u,

the update of macroscopic conservative variables and the particle distribution function are coupled in the following

way,

Wn+1

i,j =Wn

i,j +1

Ωi,j Ztn+1

tnX

r=1

∆Sr·Frdt +1

Ωi,j Ztn+1

tn

Gi,j dt, (10)

fn+1

i,j,k,l =fn

i,j,k,l +1

Ωi,j Ztn+1

tnX

r=1

urˆ

fr(t)∆Srdt

+1

Ωi,j Ztn+1

tnZΩi,j

Q(f)d~xdt +1

Ωi,j Ztn+1

tnZΩi,j

G(f)d~xdt,

(11)

Gi,j =ZΩk,l −φx∆t∂

∂u fi,j,k,l −φy∆t∂

∂v fi,j,k,l ψdudvdξ, (12)

Q(f) = f+

i,j,k,l −fn+1/2

i,j,k,l

τ,

G(f) = −φx

∂

∂u fn+1

i,j,k,l −φy

∂

∂v fn+1

i,j,k,l.

(13)

where ψ=1, u, v, 1

2(u2+v2+ξ2)Tis the vector of moments for collision invariants, and ~

φ=φx

~

i+φy~

jis the

external force acceleration. The implementation of the full Boltzmann collision term can be done as well when the

time step is on the order of particle collision time [8]. However, if the time step is a few times of the local particle

collision time, the use of kinetic model equation is accurately enough because the accumulating physical eﬀect in a

multiple particle collision time scale is not sensitive to the individual particle collision anymore.

The conservative variables are updated ﬁrst, with Frbeing the ﬂuxes of conservative ﬂow variables and Gi,j the

source term from external force. The updated macroscopic variables can be used for the construction of the equilibrium

state in Q(f) at tn+1 time step, and the derivatives of particle velocity in G(f) are evaluated via implicit upwind

ﬁnite diﬀerence method in the discretized velocity space.

The BGK-type kinetic model is used to evaluate the interface distribution function fr. The model equation with

external force term in the two-dimensional Cartesian coordinate system is

ft+ufx+vfy+φxfu+φyfv=g−f

τ,(14)

where τ=µ/p is the particle collision time.

In the uniﬁed scheme, at the center of a cell interface (xi+1/2, yj) the solution fi+1/2,j,k,l is constructed from

the integral solution of Eq. (14). With the notations xi+1/2= 0, yj= 0 at tn= 0, the time-dependent interface

distribution function writes

f(0,0, t, uk, vl, ξ) = 1

τZt

0

g(x0, y0, t0, u0

k, v0

l, ξ)e−(t−t0)/τ dt0

+e−t/τ f0(x0, y0,0, u0

k, v0

l, ξ),

(15)

where x0=−u0

k(t−t0)−1

2φx(t−t0)2, y0=−v0

l(t−t0)−1

2φy(t−t0)2, u0

k=uk−φx(t−t0), and v0

l=vl−φy(t−t0) are

the trajectories in physical and velocity space, and (x0, y0, u0

k, v0

l) = (−(uk−φxt)t−1

2φxt2,−(vl−φyt)t−1

2φyt2, uk−

6

φxt, vl−φyt) is the initial location in physical and velocity space for the particle which passes through the cell

interface at time t. The time accumulating eﬀect from the external forcing term on the time evolution of the particle

distribution function is explicitly taken into consideration.

To the second order accuracy, the initial gas distribution function f0is reconstructed as

f0(x, y, 0, uk, vl, ξ) = (fL

i+1/2,j,k,l +σi,j,k,lx+θi,j,k,ly, x ≤0,

fR

i+1/2,j,k,l +σi+1,j,k,lx+θi+1,j,k,ly, x > 0,

where fL

i+1/2,j,k,l and fR

i+1/2,j,k,l are the reconstructed initial distribution functions at the left and right hand sides of

a cell interface. The van Leer limiter is used in the reconstruction.

The equilibrium distribution function around a cell interface is constructed as

g=g01 + (1 −H[x])aLx+H[x]aRx+by +At,

where g0is the Maxwellian distribution at (x= 0, t = 0). Here the coeﬃcients aL, aR, and Aare from the Taylor

expansion of a Maxwellian,

aL,R =aL,R

1+aL,R

2u+aL,R

3v+aL,R

4

1

2(u2+v2+ξ2) = aL,R

αψα,

b=b1+b2u+b3v+b4

1

2(u2+v2+ξ2) = bαψα,

A=A1+A2u+A3v+A4

1

2(u2+v2+ξ2) = Aαψα.

The coeﬃcients above can be evaluated from the spatial distribution of conservative variables on both sides of the

cell interface and the compatibility condition. After all the coeﬃcients are determined, the time dependent interface

distribution function becomes

f(0,0, t, uk, vl, ξ) = 1−e−t/τ g0

+τ(−1 + e−t/τ ) + te−t/τ aL,Rukg0

−ττ(−1 + e−t/τ ) + te−t/τ +1

2t2e−t/τ aL,Rφxg0

+τ(−1 + e−t/τ ) + te−t/τ bvlg0−ττ(−1 + e−t/τ ) + te−t/τ +1

2t2e−t/τ bφyg0

+τt/τ −1 + e−t/τ Ag0

+e−t/τ fL

i+1/2,k0,l0+−(uk−φxt)t−1

2φxt2σi,k0,l0

+−(vl−φyt)t−1

2φyt2θi,k0,l0Huk−1

2φxt

+fR

i+1/2,k0,l0+−(uk−φxt)t−1

2φxt2σi+1,k0,l0

+−(vl−φyt)t−1

2φyt2θi+1,k0,l0(1 −Huk−1

2φxt)

=egi+1/2,j,k,l +e

fi+1/2,j,k,l,

(16)

where egi+1/2,j,k,l is related to equilibrium state integration and e

fi+1/2,j,k,l is related to the initial non-equilibrium

distribution. The ﬂux of conservative variables can be constructed as

Fi+1/2,j =ZΩk,l

ukf(0,0, t, uk, vl, ξ)ψdΞ.

IV. NUMERICAL EXPERIMENTS ON THE NON-EQUILIBRIUM FLOW EVOLUTION

In this section, we are going to present and discuss several numerical experiments to illustrate the non-equilibrium

ﬂow behavior under external force ﬁeld. The hard sphere (HS) monatomic perfect gas is employed in all test cases. Di-

7

mensionless quantities are introduced with ˆx=x/L∞, ˆρ=ρ/ρ∞,ˆ

T=T/T∞, ˆu=u/(2RT∞)1/2,ˆ

t=t(2RT∞)1/2/L∞,

ˆp=p/(2ρ∞RT∞), ˆ

φ=φL∞/(2RT∞). For simplicity, we will drop the hat notation henceforth.

A. Static heat conduction

Consider a vertical column of gas in which a temperature gradient is maintained through the top and bottom

boundary temperature condition. Instead of studying the Rayleigh-B´enard convection [22, 23], we conﬁne us to a

static heat conduction problem to evaluate the correlation between heat ﬂux and external force ﬁeld. With the

external force along the negative y-direction, the NSF equations indicate the following relations,

∂

∂y p=−ρg, (17)

∂

∂y κ(y)∂

∂y T= 0.(18)

As is analyzed in Section 2, the external forcing term will inﬂuence the heat evolution process, resulting in a deviation

of the temperature and heat ﬂux proﬁle away from the above theoretical solution in Eq. (18).

In the simulation, unit Prandtl number P r = 1.0 of a monatomic gas is assumed. The reference Knudsen number

is set as Knr ef = 0.001,0.075,1.0, which deﬁnes the dynamic viscosity in the reference state via variable hard sphere

model (VHS),

µref =5(α+ 1)(α+ 2)√π

4α(5 −2ω)(7 −2ω)Knr ef .(19)

We choose α= 1.0 and ω= 0.5 to recover a hard sphere monatomic gas. The viscosity for the hard-sphere model is,

µ=µref T

Tref θ

,(20)

where Tref is the reference temperature and θis the index related to HS model. In this case we adopt the value

θ= 0.72. The local collision time is evaluated with the relation τ=µ/p. The temperature ratio of the bottom hot

wall to the top cold one is set up with r=Th/Tc= 1.2. An external acceleration φyis imposed along the direction of

temperature gradient to the system with diﬀerent values. The heat ﬂux distribution along yaxis is presented in Fig.

1.

Y

QY

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

7.5E-05

8E-05

8.5E-05

9E-05

9.5E-05

0.0001

0.000105

0.00011

0.000115

g=0

g=0.05

g=0.1

g=0.2

(a)Kn=0.001

Y

QY

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.005

0.0052

0.0054

0.0056

0.0058

0.006

0.0062

0.0064

0.0066

0.0068

0.007

g=0

g=0.05

g=0.1

g=0.2

(b)Kn=0.075

Y

QY

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.005

0.01

0.015

0.02

0.025

0.03

0.035

g=0

g=0.05

g=0.1

g=0.2

(c)Kn=1.0

FIG. 1: Heat ﬂux distribution in y-direction in diﬀerent regimes.

It is observed that the heat ﬂux is enhanced in the external force direction. In the near-equilibrium region with

weak external force, the ﬂux modiﬁcation is proportional to the magnitude of the external force. The simulation

results agree well with the predictive value in Eq. (9). In the convergent state, the heat ﬂux is determined from

Q=Qfourier +Qf orce.

With the increment of reference Knudsen number and the magnitude of the external force, the contribution of heat

ﬂux from the forcing term is much more signiﬁcant. Due to the non-equilibrium eﬀect and large variation of the local

8

Knudsen number, the additional heat ﬂux is no longer proportional to the magnitude of the external force. Now the

spatial inhomogeneous heat ﬂux is balanced by the macroscopic ﬂow transport. As analyzed in Section 2, the gas

under external force cannot be absolutely stationary. In the continuum limit, the velocity increment caused by the

external force follows the relationship in Eq. (7), which is aligned with the force direction. However, in the transition

regime, as shown in Fig. 2, the non-equilibrium ﬂow transport can present a reversed velocity.

Y

V

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-0.00014

-0.00012

-0.0001

-8E-05

-6E-05

-4E-05

-2E-05

0

2E-05

g=0

g=0.05

g=0.1

g=0.2

(a)Kn=0.001

Y

V

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-5E-05

0

5E-05

0.0001

0.00015

0.0002

0.00025

0.0003

0.00035

g=0

g=0.05

g=0.1

g=0.2

(b)Kn=0.075

Y

V

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-0.005

0

0.005

0.01

0.015

0.02

g=0

g=0.05

g=0.1

g=0.2

(c)Kn=1.0

FIG. 2: V-velocity distribution in y-direction in diﬀerent regimes.

The external forcing term may play an equivalent role as the temperature gradient in the determination of the heat

ﬂux. Fig. 3 shows the simulation result at a reference Knudsen number Knref = 0.1, and external forcing acceleration

φy=−1.0. With the variation of local Knudsen number, the ﬂow physics changes accordingly. In the region near the

bottom wall, the temperature gradient contributes more to heat ﬂux in comparison with the external forcing term,

and the ﬁnal heat ﬂux in this region is still along the negative temperature gradient direction. However, at the upper

rareﬁed region, the external forcing term dominates the heat transfer process, resulting the heat ﬂux in the opposite

direction. In the transition regime, the heat ﬂux is associated with complicated non-equilibrium process.

Y

Kn

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.05

0.1

0.15

0.2

0.25

0.3

(a)Local Knudsen number

Y

QY

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-0.01

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

(b)Heat ﬂux

Y

V

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

(c)V-velocity

FIG. 3: Distribution of macroscopic quantities in y-direction with φy=−1.0 and Knref = 0.1.

B. Lid-driven cavity ﬂow under external forcing ﬁeld

Diﬀerent from the previous case with the same direction for the temperature gradient and external force, the

horizontal moving upper surface of the cavity will induce dissipative shear eﬀect into the dynamic system under

vertical external force ﬁeld. The square cavity has four walls with length L= 1. The upper wall moves in tangential

direction with a velocity Uw= 0.15. The external forcing acceleration is set up with φy= 0.0,−0.1,−0.3,−0.5,−1.0

respectively in the negative y−direction. The magnitude of gravity φyis denoted by g. Non-dimensional Froude

number can be deﬁned in this system to quantify the relative importance of the upper wall’s driving velocity and the

eﬀect of external force,

F r =Uw

√gL .(21)

9

The initial density and pressure are deﬁned as

ρ(x, y, t = 0) = exp(φyy), p(x, y, t = 0) = exp(φyy),

and wall temperature is Tw= 2. Maxwell’s diﬀusive reﬂecting boundary condition is used in the simulation. The

Prandtl number of the gas is Pr = 0.67. The reference Knudsen number is selected as Knr ef = 0.001,0.075, which

is deﬁned by reference state at bottom of the cavity ρref = 1.0 and pref = 1.0. The computational domain is divided

into 45 ×45 uniform cells, and 28 ×28 Gaussian points in velocity space.

1. Near equilibrium regime

In this case, the movement of upper surface and the external forcing term are two driving sources for the ﬂuid motion.

The initial hydrostatic distribution of density and pressure is perturbed by the upper wall’s movement. Besides the

viscous dissipation and heat conduction, the external force ﬁeld participates in the ﬂow and heat transport inside

the cavity. Diﬀerent from the non-equilibrium ﬂow phenomena in the absence of external force, here the simple one

large eddy topological structure covering the whole cavity domain may not necessarily appear due to the large density

variation and diﬀerent transport mechanism for diﬀerent local Knudsen number ﬂow. Fig. 4 presents the temperature

contour along with the heat ﬂux under diﬀerent external forcing terms at Knref = 0.001, and Fig. 5 shows the velocity

distribution along the center lines. At a small magnitude of external force, there exists a single large eddy running

through the whole cavity domain, and the distribution of U-velocity along the vertical center line is a monotonic curve.

However, with the increment of the magnitude of external force, the ﬂow pattern changes signiﬁcantly. As presented

in Fig. 6, with the relatively large external force, there is an obvious density variation along the vertical center line.

In spite of the driving eﬀect at the upper surface, the high density region around the bottom forms a ”death ﬂow

region”, and the eddy is restricted to the upper half domain of the cavity. Under this situation, an inﬂexion point

appears in the U-velocity curve, leaving the lower part ﬂow almost stationary. In fact, Fig. 6 shows that the ﬂow in

the upper region of the cavity stays in the transition regime, where profound non-equilibrium ﬂow phenomena with

a variation of local Knudsen number appear in such a gas dynamic system.

The heat transfer inside the cavity is closely coupled with ﬂow transport. In the absence of external force, particle

collisions at the top right corner result in a viscous heating at the macroscopic level, as shown in Fig. 4a. Due to

intensive particle collisions, the expansion cooling at the top left corner is not obvious in this case, and the temperature

around other three boundaries is almost uniform. This is consistent with the NS solutions in the continuum regime

[8]. With an increment of external forcing term, the localized hot and cold spots move away from the corner regions,

and propagate into the cavity. The penetration of the spots is related to the scale of the main eddy. From the results

in Fig. 4 and 5, the center of the hot spot is located around the place where the negative U-velocity approaches

to its maximum value, and the center of the cold spot locates a little bit higher than the hot one. At the current

Knr ef = 0.001, the particle distribution function will not deviate far from the Maxwellian equilibrium state. As

analyzed in Section 2, the correlation between the heat ﬂux and the external force is proportional to the magnitude of

the forcing term. Although the heat ﬂux is still aligned with the temperature gradient in the upper domain, the heat

ﬂux in the lower static region lines up with the direction of the force ﬁeld. The adjustment of the particle distribution

function due to the external forcing term provides the dominant mechanism for the non-equilibrium heat transport.

2. Transition regime

Now let us turn our attention to the case of fully transition regime at Knref = 0.075. As shown in Fig. 7 and

Fig. 8, the particle penetration and eﬃcient mixing generates one large eddy in all cases. The stabilizing eﬀect due

to external force ﬁeld is to reduce the rotating speed of the vortex. With the increment of external force, the velocity

proﬁle in Fig. 8 is ﬂattened, indicating a weaker vortex motion.

Even with the similar main vortex structure, in the transition regime the external force ﬁeld exerts a greater impact

on the heat transfer process. As presented in Fig. 7, in the absence of external force ﬁeld, the expansion cooling and

viscous heating both have distinguishable contribution to the heat ﬂux, which presents a phenomena for the heat ﬂow

from the cold to hot region. This observation is consistent with the DSMC simulation and uniﬁed scheme solution

[8, 19]. With the increment of the external force, the heat transfer gradually turns into the vertical direction along

with the forcing ﬁeld. The hot spot moves downwards, while the cold region expands along the horizontal direction.

As demonstrated, even with the viscous heating from the isothermal upper wall, the temperature decreases there

due to the energy exchange among kinetic, internal, and potential one. The cooling of the upper zone may have the

similar mechanism as the dynamic cooling in the atmosphere. In the case with φy=−1.0, the heat ﬂux is almost

10

parallel to the external force direction, and the heat transport from the upper cold region to the bottom hot region.

This clearly indicates the gravity-thermal instability. In the transition regime, the external force plays an important

role in the determination of non-equilibrium heat transport.

C. Wave-type heat transfer

Based on the diﬀusion process in Fourier’s law, the NSF equations can be used to describe the temperature evolution.

Without considering the dissipation function from shear stress and the net work caused by the variation of pressure,

in the absence of heat source or sink the energy conservation equation is,

ρcp

DT

Dt =∇ · (κ∇T).(22)

The parabolic nature in the above equation implies an inﬁnite propagation speed for the heat. A ﬁrst attempt to

solve this problem was carried out by Cattaneo [24], who suggested Fourier’s heat ﬂux q=−κ∇Tshould be replaced

by a more general rule,

τ(T)∂q

∂t +q=−κ∇T , (23)

where τis the relaxation time which depends on the mechanism of heat transport. Although many other researchers

also developed new equations, a universal heat conduction law is still far from complete. Macroscopic equations alone

may not be possible to give a complete non-equilibrium description of transport process at all. The current UGKS

is based on both macroscopic and microscopic scale ﬂow evolution, where the degree of freedom for capturing non-

equilibrium state and transport is not ﬁxed and depends on the ﬂow regime. In fact, if temperature is regarded as an

internal energy, heat transfer could depend on a combination of many factors related to energy transformation, such as

temperature gradient, stress tensor, compressibility, degree of rarefaction, and the contribution from external forcing

term. In diﬀerent ﬂow regimes, these factors may play diﬀerent role, which explains the existence of heat transport

from cold to hot region in transition regime. In the kinetic scale, the particle transport and penetration take place for

the thermal energy transport and dissipation. In the Navier-Stokes limit, the ﬂow motion is approximately described

by the movement of ﬂuid elements, and energy exchange takes place only through the boundaries between the elements

by diﬀusion process without particle penetration. Due to the instant interaction between ﬂuid elements, the Fourier’s

law gives an arbitrary high propagating speed for the heat energy. Theoretically, all information propagated in a gas

system must go through the particle collision and transport, the particle random velocity Cc∼√RT determines the

physical speed. Hence, the propagation speed of thermal disturbance cannot be far away from the above speed in the

case with small ﬂuid bulk velocity. In the transition regime, with the variation of local Knudsen number a much more

complicated dynamic process will emerge in the determination of heat transfer.

Here we study the transient temperature evolution in two driven cavity cases to explore mechanism of heat transfer

in diﬀerent ﬂow regimes. Fig. 9 presents the time evolved temperature contour inside the cavity at Knref = 0.001

and φy=−0.1. Initially, both hot and cold spots generated at the upper surface permeate downwards through a

”transport” process even the generated ﬂuid velocity remains in the top region. When the localized hot and cool

spots reach the bottom at t= 0.92, complex heat transport process appears. As presented in Fig. 10, with the

stratiﬁed density distribution, the shearing generated ﬂow motion stays in the top region, the tiny ﬂow velocity at

the bottom contributes marginally to the thermal energy transport there. Even at t= 3.99, the main eddy is still

restricted in the upper half of the cavity, and the transverse ﬂow velocity at the bottom is very weak. Although the

heat diﬀusion smears the temperature gradient, the time variation of the temperature ﬁeld around bottom indicates

the wave-type thermal energy propagation. As shown in Fig. 9d to Fig. 9k, the thermal wave travels back and

forth between opposite solid walls in the bottom region, where the localized hot and cold spots exchange locations a

few times. Fig. 11 presents the time-series horizontal temperature distribution near the bottom wall, and the wave

pattern of temperature evolution is fully demonstrated. With the wave reﬂection and transmission between two solid

walls, the thermal wave is gradually dissipated. At end, two large temperature spots from the upper region move

slowly downward with the main ﬂow eddy.

For the case in the transition regime at Knref = 0.075 and φy=−0.5, the results are presented in Fig. 12. Here

the particle penetration speeds up the momentum and energy mixing, and the external force ﬁeld plays a much more

critical role. As shown in Fig. 13, the main eddy grows up quickly, and the particle transport dominates the heat

and mass transfer. In this case, the high-temperature gas moves downwards smoothly with the swirling ﬂow. After

reaching the downside wall, the hot gas spreads moderately along with the convection, and forms a hot spot near

the left corner in Fig. 12f. Although the dissipation process contributes much to the heat transfer now, the wave

pattern of temperature variation is still observed. As shown in Fig. 14, after reﬂected by the solid wall, the thermal

11

wave transmits reversely from t= 1.59 to t= 2.66. During this process, the localized hot spot is transporting against

the ﬂow velocity and moves to the right corner in Fig. 12g. It is noted that due to the increased collision time

and particle penetration at this Knudsen number, the thermal wave speed is much reduced in comparison with the

previous one. It takes about ∆t= 1.07 for the wave to travel across the bottom wall, while this process is ﬁnished in

∆t= 0.40 in the previous case. Actually, under external force ﬁeld the propagation of thermal wave may be closely

coupled with the gravitational wave, which needs further investigation. The amplitude of thermal wave here is one

quarter of the one at Knr ef = 0.001, indicating a much reduced wave energy. With increasing collision time and

external force, the thermal wave is dissipated rapidly, and the localized hot and cold spots exchange locations only

once. As a consequence, a stable temperature distribution is formed. So, it seems that the thermal wave propagating

phenomenon is a complex dynamic process related to non-equilibrium mechanism in diﬀerent ﬂow regimes. Diﬀerent

wave patterns are expected to appear due to the relative importance of particle transport over diﬀusion. The current

numerical experiments provide valuable observations for the theoretical modeling for the macroscopic thermal wave

propagating equations.

V. CONCLUSION

The gas dynamics under external force ﬁeld is intrinsically a multiple scale ﬂow problem due to large density

variation and a changeable local Knudsen number. In this paper, based on the well-balanced uniﬁed gas-kinetic

scheme we investigate the multiscale, non-equilibrium ﬂow dynamics under external forcing in diﬀerent ﬂow regimes.

For the near equilibrium ﬂow, the contribution of the external force to the heat ﬂux is analyzed based on the kinetic

model equation, and veriﬁed numerically as well. At the same time, a detailed investigation for lid-driven cavity

case has been conducted and the non-equilibrium ﬂow evolution has been quantitatively evaluated. The dynamic

eﬀect of the external force on the ﬂow pattern and heat transfer is presented. With the UGKS method, it is now

possible to explore the physics in the non-equilibrium transition ﬂow regime. The current study presents the following

observations. The external force has great contribution to the heat ﬂux and aligns the heat ﬂow in the forcing direction.

The enhanced heat transport from the forcing term may overtake the contribution from the temperature diﬀusion

process, which determines the heat ﬂow from the upper cold high gravitational potential region to the lower hot low

potential region and triggers the gravity-thermal instability. In the transition regime, the thermal wave propagating

phenomena under external forcing ﬁeld have been observed and quantitatively evaluated. This provides a ﬁrst hand

data for the construction of generalized Fourier’s law to take account of the limited thermal propagating speed. For

a physical gas system under external force ﬁeld, it is naturally a multiple scale ﬂow problem with a large variation of

local Knudsen number. The understanding of the multiscale non-equilibrium ﬂow phenomena will have great help to

our understanding to the earth’s atmosphere environment.

Acknowledgement

The current research is supported by Hong Kong research grant council (16207715, 16211014, 620813), and National

Science Foundation of China (91330203,91530319).

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13

X

Y

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

T

2.011

2.01

2.009

2.008

2.007

2.006

2.005

2.004

2.003

2.002

2.001

2

(a)φy= 0

X

Y

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

T

2.024

2.022

2.02

2.018

2.016

2.014

2.012

2.01

2.008

2.006

2.004

2.002

2

1.998

1.996

1.994

1.992

1.99

(b)φy=−0.1, F r = 0.47

X

Y

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

T

2.04

2.035

2.03

2.025

2.02

2.015

2.01

2.005

2

1.995

1.99

1.985

1.98

1.975

(c)φy=−0.3,F r = 0.27

X

Y

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

T

2.05

2.045

2.04

2.035

2.03

2.025

2.02

2.015

2.01

2.005

2

1.995

1.99

1.985

1.98

1.975

1.97

(d)φy=−0.5,F r = 0.21

X

Y

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

T

2.05

2.045

2.04

2.035

2.03

2.025

2.02

2.015

2.01

2.005

2

1.995

1.99

1.985

1.98

1.975

1.97

1.965

1.96

(e)φy=−1.0,F r = 0.15

FIG. 4: Temperature contour and heat ﬂux with Knref = 0.001.

U/Uw

Y

-0.4 -0.2 0 0.2 0.4 0.6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

g=0

g=0.1

g=0.3

g=0.5

g=1.0

(a)U-velocity along the vertical center

line

X

V/Uw

0 0.2 0.4 0.6 0.8 1

-0.2

-0.1

0

0.1

0.2

g=0

g=0.1

g=0.3

g=0.5

g=1.0

(b)V-velocity along the horizontal

center line

FIG. 5: Velocity distribution along the center line with Knref = 0.001.

14

Y

Density

0 0.2 0.4 0.6 0.8 1

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

g=0

g=0.1

g=0.3

g=0.5

g=1.0

(a)Density

Y

Kn

0 0.2 0.4 0.6 0.8 1

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

0.0045

0.005

g=0

g=0.1

g=0.3

g=0.5

g=1.0

(b)Local Knudsen number

FIG. 6: Density and local Knudsen number distribution along the vertical center line with Knref = 0.001.

X

Y

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

T

2.014

2.012

2.01

2.008

2.006

2.004

2.002

2

1.998

1.996

1.994

1.992

(a)φy= 0

X

Y

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

T

2.012

2.01

2.008

2.006

2.004

2.002

2

1.998

1.996

1.994

1.992

1.99

1.988

(b)φy=−0.1,F r = 0.47

X

Y

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

T

2.006

2.004

2.002

2

1.998

1.996

1.994

1.992

1.99

1.988

1.986

1.984

1.982

1.98

(c)φy=−0.3,F r = 0.27

X

Y

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

T

1.998

1.996

1.994

1.992

1.99

1.988

1.986

1.984

1.982

1.98

1.978

1.976

1.974

1.972

1.97

1.968

(d)φy=−0.5,F r = 0.21

X

Y

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

T

1.98

1.975

1.97

1.965

1.96

1.955

1.95

1.945

1.94

1.935

1.93

1.925

1.92

(e)φy=−1.0,F r = 0.15

FIG. 7: Temperature contour and heat ﬂux with Knref = 0.075.

15

U/Uw

Y

-0.2 0 0.2 0.4 0.6

0

0.2

0.4

0.6

0.8

1

g=0

g=0.1

g=0.3

g=0.5

g=1.0

(a)U-velocity along the vertical center

line

X

V/Uw

0 0.2 0.4 0.6 0.8 1

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

g=0

g=0.1

g=0.3

g=0.5

g=1.0

(b)V-velocity along the horizontal

center line

FIG. 8: Velocity distribution along the center line with Knref = 0.075.

16

2

2.018

1.984

1.992

1.998

2.008

2.002

2

X

Y

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

T

2.018

2.016

2.014

2.012

2.01

2.008

2.006

2.004

2.002

2

1.998

1.996

1.994

1.992

1.99

1.988

1.986

1.984

(a)t=0.13

1.992

1.99

2.012

2.01

2.006

2

1.996

2

2.002

X

Y

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

T

2.012

2.01

2.008

2.006

2.004

2.002

2

1.998

1.996

1.994

1.992

1.99

(b)t=0.40

1.993

2

1.995

1.997

1.999

2.009

2.007

2.005

2.003

2.001

2

2.002

2.003

X

Y

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

T

2.011

2.01

2.009

2.008

2.007

2.006

2.005

2.004

2.003

2.002

2.001

2

1.999

1.998

1.997

1.996

1.995

1.994

1.993

(c)t=0.66

1.994

2.007

2.005

1.995

1.997

2.003

2.001

1.999

2.004

2.005

2

2.001

1.999

2.001

2

1.999

1.997

2.002

X

Y

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

T

2.009

2.008

2.007

2.006

2.005

2.004

2.003

2.002

2.001

2

1.999

1.998

1.997

1.996

1.995

1.994

(d)t=0.92

2.006

1.998

1.994

1.996

1.998

2.006

2.004

2.002

2.004

2.002

2

2.001

1.999

1.998

1.998

2

2

X

Y

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

T

2.011

2.01

2.009

2.008

2.007

2.006

2.005

2.004

2.003

2.002

2.001

2

1.999

1.998

1.997

1.996

1.995

1.994

(e)t=1.33

2.01

1.998

2.006

2.002

2

1.994

1.998

2.006

2.002

2

1.998

X

Y

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

T

2.012

2.01

2.008

2.006

2.004

2.002

2

1.998

1.996

1.994

(f)t=1.59

2.008

2.006

2.004

2.004

2.002

2

1.997

1.996

2

1.999

2.003

X

Y

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

T

2.01

2.009

2.008

2.007

2.006

2.005

2.004

2.003

2.002

2.001

2

1.999

1.998

1.997

1.996

(g)t=1.99

2.009

1.996

2.003

2.001

1.998

2

1.997

1.999

2

2.001

2.003

2.005

1.999

X

Y

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

T

2.011

2.01

2.009

2.008

2.007

2.006

2.005

2.004

2.003

2.002

2.001

2

1.999

1.998

1.997

1.996

(h)t=2.66

2.007

2.003

2.001

2

1.997

1.999

1.999

2

2.004

2.002

2.001

X

Y

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

T

2.01

2.009

2.008

2.007

2.006

2.005

2.004

2.003

2.002

2.001

2

1.999

1.998

1.997

(i)t=3.06

2.009

2.005

2.002

2.001

1.999

2

2

2

2.003

2

X

Y

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

T

2.011

2.01

2.009

2.008

2.007

2.006

2.005

2.004

2.003

2.002

2.001

2

1.999

(j)t=3.32

2.01

2.007

2.005

2.003

2.002

1.998

1.997

1.999

2.001

2

X

Y

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

T

2.012

2.011

2.01

2.009

2.008

2.007

2.006

2.005

2.004

2.003

2.002

2.001

2

1.999

1.998

1.997

(k)t=3.59

2.012

2.008

2.004

2

1.994

1.998

2.002

X

Y

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

T

2.012

2.01

2.008

2.006

2.004

2.002

2

1.998

1.996

1.994

(l)t=3.99

2.01

2

1.995

1.997

1.999

2

2.008

2.006

2.004

2.002

X

Y

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

T

2.012

2.011

2.01

2.009

2.008

2.007

2.006

2.005

2.004

2.003

2.002

2.001

2

1.999

1.998

1.997

1.996

1.995

(m)t=6.64

1.99

1.994

1.998

2

2.004

2.008

2.016

2.012

2.01

X

Y

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

T

2.016

2.014

2.012

2.01

2.008

2.006

2.004

2.002

2

1.998

1.996

1.994

1.992

1.99

(n)t=13.29

FIG. 9: Evolution of temperature with K nref = 0.001 and φy=−0.1.

17

U/Uw

Y

-0.2 0 0.2 0.4 0.6

0

0.2

0.4

0.6

0.8

1

t=1.33

t=2.66

t=3.99

(a)U-velocity along the vertical center

line

X

V/Uw

0 0.2 0.4 0.6 0.8 1

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

t=1.33

t=2.66

t=3.99

(b)V-velocity along the horizontal

center line

FIG. 10: Transient velocity distribution along the center line with Knref = 0.001 and φy=−0.1.

X

T

0 0.2 0.4 0.6 0.8 1

1.99

1.995

2

2.005

2.01 t=0.93

t=1.33

t=1.99

t=2.66

FIG. 11: Transient temperature distribution along horizontal direction near the bottom with Knref = 0.001 and φy=−0.1.

2

2.015

2.005

2.01

1.995

1.99

1.985

1.98

1.975

X

Y

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

T

2.015

2.01

2.005

2

1.995

1.99

1.985

1.98

1.975

(a)t=0.13

2.01

2.005

1.97

1.98

1.99

1.995

2

1.995

1.975

X

Y

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

T

2.01

2.005

2

1.995

1.99

1.985

1.98

1.975

1.97

(b)t=0.26

2.004

2

1.966

1.97

1.974

1.978

1.982

1.986

1.99

1.994

1.998

1.968

2.002

X

Y

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

T

2.004

2.002

2

1.998

1.996

1.994

1.992

1.99

1.988

1.986

1.984

1.982

1.98

1.978

1.976

1.974

1.972

1.97

1.968

1.966

(c)t=0.66

1.97

1.996

1.994

1.974

1.978

1.982

1.986

1.99

1.994

X

Y

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

T

1.998

1.996

1.994

1.992

1.99

1.988

1.986

1.984

1.982

1.98

1.978

1.976

1.974

1.972

1.97

(d)t=1.06

1.97

1.996

1.992

1.992

1.988

1.984

1.98

1.976

X

Y

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

T

1.998

1.996

1.994

1.992

1.99

1.988

1.986

1.984

1.982

1.98

1.978

1.976

1.974

1.972

1.97

(e)t=1.20

1.97

1.998

1.992

1.988

1.984

1.98

1.976

1.974

1.99

1.992

X

Y

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

T

1.998

1.996

1.994

1.992

1.99

1.988

1.986

1.984

1.982

1.98

1.978

1.976

1.974

1.972

1.97

(f)t=1.59

1.968

1.998

1.994

1.99

1.986

1.982

1.978

1.974

1.97

1.992

X

Y

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

T

1.998

1.996

1.994

1.992

1.99

1.988

1.986

1.984

1.982

1.98

1.978

1.976

1.974

1.972

1.97

1.968

(g)t=2.66

1.968

1.998

1.994

1.99

1.986

1.982

1.978

1.974

1.972

X

Y

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

T

1.998

1.996

1.994

1.992

1.99

1.988

1.986

1.984

1.982

1.98

1.978

1.976

1.974

1.972

1.97

1.968

(h)t=3.99

FIG. 12: Evolution of temperature with K nref = 0.075 and φy=−0.5.

18

U/Uw

Y

-0.2 0 0.2 0.4

0

0.2

0.4

0.6

0.8

1

t=0.26

t=0.66

t=1.33

(a)U-velocity along the vertical center

line

X

V/Uw

0 0.2 0.4 0.6 0.8 1

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

t=0.26

t=0.66

t=1.33

(b)V-velocity along the horizontal

center line

FIG. 13: Transient velocity distribution along the center line with Knref = 0.075 and φy=−0.5.

X

T

0 0.2 0.4 0.6 0.8 1

1.989

1.9895

1.99

1.9905

1.991

1.9915

1.992

t=1.59

t=1.73

t=1.86

t=1.99

t=2.66

t=3.32

t=3.99

t=6.65

FIG. 14: Transient temperature distribution along horizontal direction near the bottom with Knref = 0.075 and φy=−0.5.