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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).
References
[1] Prabhu Lal Bhatnagar, Eugene P Gross, and Max Krook. A model for collision processes in gases. i. small amplitude
processes in charged and neutral one-component systems. Physical review, 94(3):511, 1954.
[2] M Tij, V Garz´o, and A Santos. On the influence of gravity on the thermal conductivity. arXiv preprint cond-mat/0002397,
2000.
[3] Joseph Fourier. Theorie analytique de la chaleur, par M. Fourier. Chez Firmin Didot, p`ere et fils, 1822.
[4] Mohamed Gad-el Hak. The fluid mechanics of microdevicesthe freeman scholar lecture. Journal of Fluids Engineering,
121(1):5–33, 1999.
[5] Kun Xu. Direct modeling for computational fluid dynamics: construction and application of unified gas-kinetic schemes.
World Scientific, 2015.
[6] Kun Xu and Juan-Chen Huang. A unified gas-kinetic scheme for continuum and rarefied flows. Journal of Computational
Physics, 229(20):7747–7764, 2010.
12
[7] Tianbai Xiao, Qingdong Cai, and Kun Xu. A well-balanced unified gas-kinetic scheme for multiscale flow transport under
gravitational field. arXiv preprint arXiv:1608.08730, 2016.
[8] Chang Liu, Kun Xu, Quanhua Sun, and Qingdong Cai. A unified gas-kinetic scheme for continuum and rarefied flows iv:
Full boltzmann and model equations. Journal of Computational Physics, 314:305–340, 2016.
[9] PN Shankar and MD Deshpande. Fluid mechanics in the driven cavity. Annual Review of Fluid Mechanics, 32(1):93–136,
2000.
[10] UKNG Ghia, Kirti N Ghia, and CT Shin. High-re solutions for incompressible flow using the navier-stokes equations and
a multigrid method. Journal of computational physics, 48(3):387–411, 1982.
[11] R Schreiber and H.B Keller. Driven cavity flows by efficient numerical techniques. Journal of Computational Physics,
49(2):310 – 333, 1983.
[12] S Pratap Vanka. Block-implicit multigrid solution of navier-stokes equations in primitive variables. Journal of Computa-
tional Physics, 65(1):138–158, 1986.
[13] Shuling Hou, Qisu Zou, Shiyi Chen, Gary D Doolen, and Allen C Cogley. Simulation of cavity flow by the lattice boltzmann
method. arXiv preprint comp-gas/9401003, 1994.
[14] Graeme Austin Bird. Molecular gas dynamics and the direct simulation of gas flows. 1994.
[15] Carlo Cercignani. The Boltzmann Equation and Its Applications. Springer, 1988.
[16] Carlo Cercignani. Rarefied gas dynamics: from basic concepts to actual calculations, volume 21. Cambridge University
Press, 2000.
[17] Stergios Naris and Dimitris Valougeorgis. The driven cavity flow over the whole range of the knudsen number. Physics of
Fluids (1994-present), 17(9):097106, 2005.
[18] Simon Mizzi, David R Emerson, Stefan K Stefanov, Robert W Barber, and Jason M Reese. Effects of rarefaction on cavity
flow in the slip regime. Journal of computational and theoretical nanoscience, 4(4):817–822, 2007.
[19] Benzi John, Xiao-Jun Gu, and David R Emerson. Investigation of heat and mass transfer in a lid-driven cavity under
nonequilibrium flow conditions. Numerical Heat Transfer, Part B: Fundamentals, 58(5):287–303, 2010.
[20] Walter Guido Vincenti and Charles H Kruger. Introduction to physical gas dynamics. Introduction to physical gas dynamics,
by Vincenti, Walter Guido; Kruger, Charles H. New York, Wiley [1965], 1, 1965.
[21] Kun Xu. A gas-kinetic bgk scheme for the navier–stokes equations and its connection with artificial dissipation and godunov
method. Journal of Computational Physics, 171(1):289–335, 2001.
[22] Henri B´enard. Les tourbillons cellulaires dans une nappe liquide propageant de la chaleur par convection: en r´egime
permanent. Gauthier-Villars, 1901.
[23] Lord Rayleigh. Lix. on convection currents in a horizontal layer of fluid, when the higher temperature is on the under side.
The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 32(192):529–546, 1916.
[24] Carlo Cattaneo. Sulla conduzione del calore. In Some Aspects of Diffusion Theory. 1948.
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.
