12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

ULTIMATE HEAT TRANSFER IN TURBULENT THERMAL CONVECTION
BETWEEN POROUS WALLS

Fanyu Meng
Graduate School of Engineering Science
Osaka University
1-3 Machikaneyama, Toyonaka,
Osaka 560-8531, Japanmeng@tes.me.es.osaka-u.ac.jp

Shingo Motoki
Graduate School of Engineering Science
Osaka University
1-3 Machikaneyama, Toyonaka,
Osaka 560-8531, Japan
motoki@me.es.osaka-u.ac.jp

Genta Kawahara
Graduate School of Engineering Science
Osaka University
1-3 Machikaneyama, Toyonaka,
Osaka 560-8531, Japankawahara@me.es.osaka-u.ac.jp

ABSTRACT

instability analysis and similarity argument, respectively. In
contrast, Kraichnan (1962) predicted an asymptotic ultimate
regime as Nu ∼ Pr1/2 Ra1/2 with a logarithm correction stemming from turbulent boundary layers. The most notable implication is that the heat transport will be independent of thermal
conductivity or thermal diffusivity in the ultimate state. Several attempts had been done to achieve the ultimate scaling by
Calzavarini et al. (2005) , Pawar & Arakeri (2016) and Zhu
et al. (2019) numerically and experimentally.
According to previous studies, the effects induced by thewall has been recognized to prevent heat transfer thereon fromexhibiting the ultimate scaling. With the aid of wall permeability first introduced by Jiménez et al. (2001), Kawano et al.
(2021) showed that the ultimate scaling can be well realized
at high Rayleigh number in turbulent thermal convection between Darcy-type permeable walls on which the vertical velocity is given by w = ±β p/ρ. On the porous walls of fixed
geometry, the dimensionless permeability parameter βU f (U fbeing the buoyancy-induced terminal velocity) is considered toincrease as βU f ∼ Pr−1/2 Ra1/2 with increasing Ra, and thus
in a real situation the prefactor c in the scaling law Nu = cRaγ
increases with increasing Ra, implying deviation from thepower law, i.e. the ultimate scaling of γ = 1/2. To achieve
the ultimate scaling and verify the deviation from the ultimate
scaling in a more realistic model for the RB convection, we
performed direct numerical simulations based on high-ordercompact schemes and implemented the immersed boundary
method to mimic the interaction to let it be more realistic andrealizable.

A numerical study on the heat transfer and flow structure has been conducted for turbulent thermal convection between horizontal porous walls by using the immersed boundary method. In this research, we perform direct numerical simulation with high-order compact schemes for turbulent Rayleigh-Bérnard(RB) convection at the Rayleigh number
Ra = 4 × 108 –8 × 108 . The ultimate state represented by thescaling Nu ∼ Ra1/2 for this RB convection between porous
walls could be well achieved at a high Rayleigh number Ra,
where the appearance of the large-scale structures within thenear-wall region hold the existence of the ultimate state. Moreover, a transient state without a power law of Nu with Ra with
an distinct exceeded ultimate scaling Nu ∼ Ra could also be
observed at the lower Ra. The distinct increased vertical velocity over the terminal velocity has been recognized as the triggerof this novel scaling-like transitional process. The mechanismof transition to the ultimate state has been well investigatedand discussed based on the energy dissipation rate and vorticalstructures in this research.

Introduction
Rayleigh-Bénard (RB) convection is a canonical thermal
flow system induced by buoyancy in horizontal fluid layer
heated from below and cooled from above. The key parametersof this flow system are the Rayleigh number and the Prandtl
number, defined as Ra = gα∆T H 3 /(νκ) and Pr = ν/κ, whereg, α, κ and ν represent the acceleration of gravity, expansion coefficient, thermal diffusion coefficient and viscosity, respectively. The Rayleigh number Ra expresses the strength ofbuoyancy caused by gravity and temperature difference. The
Nusselt number Nu is defined to evaluate the intensity of aheat flux in comparison to thermal conduction. A power law
Nu ∼ Raγ has been long discussed, in which different values ofγ demonstrate distinct states in this system. Malkus (1954) andPriestley (1954) derived an Nu ∼ Ra1/3 law with a marginal

Governing equations and flow configuration
A staggered finite-difference code Incompact3D developed by Laizet & Lamballais (2009) and Laizet & Li (2011)
based on high-order compact schemes has been applied toensure a quasi-spectral accuracy. The immersed boundary
method was used to make the detailed effects induced by the1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
geometry structures in this flow system. The Boussinesq equations, which employs the Oberbeck-Boussinesq approximationin the Navier-Stokes equation are our governing equations,1∂u+ (uu · ∇)uu = − ∇p + ν∇2 u + gαT e z + f b
∂t
ρ

(1)

∇·u = 0(2)∂T+ (uu · ∇)T = κ∇2 T + qb∂t(3)
Figure 1: The geometry of porous patterns and flow con-in which, u (xx,t) = ueex + veey + weez and T (xx,t) denote thevelocity field and temperature field respectively. The velocity field and temperature field in this case follow the periodic boundary conditions in the horizontal x- and y-directions
with the same length as Lx = Ly = H. The f b and qb denotethe force field and heat source term on account of the implemented immersed boundary method, respectively. We use thebuoyancy-induced terminal velocity, U f = (gα∆T H)1/2 as areference velocity scale. Pr and Ra are control parameters inthe system and the Pr is set to unity in this study. Moreover, theaspect ratio i.e., the ratio of the horizontal extent to the heightis also a control parameter. The boundary conditions for computational domain of liquid parts have been set as blow,u(z = 0) = u(z = 3H) = w(z = 0) = w(z = 3H) = 0(4)

T (z = 0) = T (z = 3H) = 0(5)

figurations for simulations. The left slice shows theholes in porous media viewed from above. The right one
shows all the flow system viewed from the side. The redpart, blue part and grey part denote the heating porous
wall, cooling porous wall and fluid domain, respectively.

The temperature of the heated lower (or cooled upper) porous
wall is kept at T = ∆T > 0 (or T = 0). The boundary conditions of the surfaces on the porous media and the two baseplates at z = 0 and z = 3H overhead and underneath are no-slip
and isothermal. The width of square holes is set to 0.08H in all
simulations. There are plenum chambers of height 0.08H between the porous media and the base plates. The chamber is an
an important part in this simulation, which make the fluid getin and get out from the porous walls be more realizable. Moreover, according to several previous studies, the permeable wallwithout chamber can be considered as the roughness.

Scaling properties

and the boundary conditions for the solid parts are imposed as
no-slip and isothermal,
u=v=w=0(6)

T (z < H|solid ) = ∆T(7)

T (z > 2H|solid ) = 0(8)

The scaling relationship not only between the Ra and Nu
but also the statistical features of vertical velocity induced bybuoyancy term and temperature distributions will be studied
and considered. It can be easily observed from Figure 2, thatthe ultimate scaling Nu ∼ Ra1/2 is achieved by the introduction of the porous walls with chambers at the high Rayleighnumber Ra > 108 . The observed ultimate scaling in the caseof the porous walls is roughly consistent with that found by
Kawano et al. (2021). in the case of the permeable walls at
βU f = 3. At the lower Rayleigh number Ra < 108 in the caseof porous walls, a distinct scaling law which almost dominated
all the transition process can be observed, Nu ∼ Ra . This kindof scaling law exceeded the ultimate scaling and enhanced the
Nu intensely, so it can be called as a scaling-like transition process and this process is so novel when considered about several
studies. At the much lower Rayleigh number around Ra = 106 ,Nu in the conventional non-porous case is almost comparablewith that in the porous case, implying that the wall porosity
does not have significant effects on thermal convection.
From Figure 3, the relationship between the Ra and Nu
on different positions can be observed. Most notable thing isthat, as Ra > 108 the ultimate state is reached in almost every
position of the domain. Likewise, the same scaling law Nu ∼Ra is evident in every position of the domain when the global
state is in this scaling-like transition state.
The mean profile of temperature distribution has been

The intensity of the convection will be quantified in termsof the heat flux resulting from the temperature difference between the top and bottom walls given by Nusselt number Nu,

Nu =

−κ

d⟨T ⟩xyt
dz + ⟨wT ⟩xytκ∆T /H(9)

where < · >xyt denotes the horizontal and time average on thevariables at the center of domain.
The geometry and flow configuration are shown in Figure
1. The porous wall possesses a 10 × 10 matrix of square holes.

212th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022(a)Non-porous case
Cha anne et al. , 2005(Pr ≥ 0.7≥
Porous case
Kawano et al. , 2021(ββf = 3≥
Nu ∼ Ra 1∼3
Nu ∼ Ra 1∼2
Nu ∼ Ra

Ra
Ra
Ra
Ra
Ra

0.8

/Δ

Nu

1.0

T xyt T103

= Δ × 10
= 2 × 10
= 1 × 10
= 9 × 10

Ra
Ra
Ra
Ra
Ra

8

8

8

8

7

= 8 × 10
= 7 × 10
= 5 × 10
= 4 × 10
= 1 × 10

7

7

7

7

7

0.6

>

102

= 5 × 10

<

0.4

101

0.2

100 4
10

107

106

105

Ra

108

109

1010

0.0
0.0

1011

0.2

0.4

zH

0.6

0.8

1.0

0.8

1.0

/

(b)

Figure 2: Nusselt number Nu as a function of Rayleighnumber Ra. The blue line and red lines represent theclassical scaling Nu ∼ Ra1/3 and the ultimate scaling
Nu ∼ Ra1/2 respectively. The black dashed line denotes
the distinct scaling-like transition process with scaling
law Nu ∼ Ra. The blue circles represent the results
obtained from the non-porous RB convection and thesky-blue squares denote the experimental data in a nonporous cylindrical cell done by Chavanne et al. (2001).
The red circles represent the results obtained from the
RB convection between porous walls. The orange
squares represent the results obtained from Kawano
et al. (2021) for the RB between permeable walls at
βU f = 3

0.30

Ra
Ra
Ra
Ra
Ra

0.25

= 3 × 10
= 2 × 10
= 1 × 10
= 9 × 10

Ra
Ra
Ra
Ra
Ra

8

8

8

8

7

= 8 × 10
= 7 × 10
= 5 × 10
= 4 × 10
= 1 × 10

7

7

7

7

7

/

wrms Uf

0.20= 5 × 10

0.15

0.10

0.05

0.00
0.0

0.2

0.4

zH

0.6

/

(c)
1.0

Ra
Ra
Ra
Ra
Ra
Ra
Ra
Ra
Ra

= 0.20= 0.00= 2.80== 2.92= 0.08

/Δ

z
H
z
H
z H
z
H

z
H
z
H
Nu Ra
= 1.75
= 1.5
∼

1∼2

0.6

>

H
H
H
H

= 3.0

= 1.25= 1 × 10
= 2 × 10
= 3 × 10
= 4 × 10
= 5 × 10
= Δ × 10
= 8 × 10
= 1 × 10
= 2 × 10

Δ

Δ

Δ

Δ

Δ

Δ

Δ

8

8

0.4

<

z
z
z
z

T xyt T

0.8

2
0.2

0.0
0.0

Nu

10

10

0.5

1.0

zH
1.5

2.0

2.5

3.0

/

1

(d)
Ra
Ra
Ra
Ra
Ra
Ra
Ra

0.25

Ra

10

8

0.20

/

10

7

wrms Uf

10

0

0.15

= 8 × 10
= 3 × 10
= 2 × 108

8

8

= 1.8 × 10
= 1.5 × 10
= 1 × 10
= 8 × 10

8

8

8

7

Ra
Ra
Ra
Ra
Ra
Ra
Ra

= 7 × 10
= 5 × 10
= 4 × 10
= 3 × 10
= 2 × 10
= 1 × 10
= 1 × 10

7

7

7

7

7

7

6

0.10

Figure 3: Nusselt number Nu as a function of Rayleighnumber Ra for porous cases with different positions.
The red line represents the ultimate scaling Nu ∼ Ra1/2 .

0.05

0.00
0.0

0.5

1.0

zH
1.5

2.0

2.5

3.0

/

Figure 4: Mean temperature as a function of z/H for(a) conventional cases and (c) porous cases. The rootmean-square(RMS) vertical velocity normalized by U ffor (b) conventional cases and (d) porous cases. The
1/2
RMS vertical velocity is defined as wrms = ⟨w2 ⟩xyt , inwhich ⟨·⟩xyt denotes the time and horizontal averaged.

been checked in Figure 4a and Figure 4c for convectional casesand porous cases respectively. It can be seen that in the nearwall region, the temperature becomes steeper and steeper inconvectional cases with increasing Ra. At higher Ra, thermal boundary thickness becomes thinner. Meanwhile, for the
porous cases, in the holes on the porous surfaces we havethe turbulent heat flux in addition to the thermal conduction
heat flux, and thus the near-wall temperature gradient does notchange much.
The vertical root-mean-square velocities are shown in
Figure 4b and Figure 4d for different configurations, wrms inconventional cases cannot scaled with the U f in the bulk region. In the porous cases, where a significant increase in thevertical velocity fluctuation can be observed, with respect toU f at Ra < 108 below the ultimate regime in which the ve-locity fluctuation can scale with U f . Moreover, in the porous
media region(0 ≤ z ≤ H and 2H ≤ z ≤ 3H), an increase wrms
normalized by U f in the porous can also be observed, which
may play a key role for the transition process.

3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
(a)

Heat-flux structures, Energy dissipation andVortical structures
The mean thickness of the thermal boundary layer is givenbyδT = −∆T

d < T >xytdz
z=0 or H

−1
=

H
,
2Nu

(10)

in which z = 0 and z = H imply the non-porous cases andporous cases, respectively. The turbulent heat flux inducedby the buoyancy term can describe the strength and structuresof the convection. The turbulent heat flux normalized by U f
and ∆T could be obtained as wT /(U f ∆T ). It can be seenfrom the Figure 5 that the small scale plume structures dominate the near-wall region even at high Ra for the conventionalcase. However, the structures of the turbulent heat flux obtained from porous cases are so distinct and the large scale
structures dominate the near-wall region, which have been observed in the ultimate state(Kawano et al., 2021)).
In order to figure out the mechanism for transient behaviour towards the 1/2 power law, the local energy dissipation rate

ϵ=

ν
2



∂ ui ∂ u j+∂ x j ∂ xi

2

(b)
,

(11)

is examined. In Figure 6,the horizontal and time averaged energy dissipation rate is shown as a function of z/H.
From Figure 6, we confirm that at the low Rayleigh number Ra = 106 , although the weak large scale structures appear,
the state in the porous media is quiescent, as represented bysmall energy dissipation therein. At higher Ra, the stronger
large scale structures appear to enhance thermal convection as
discussed before. In the transient state at Ra < 108 , however,
the energy dissipation in the porous media is still low withrespect to U f3 /H, the normalized dissipation there increases
with increasing Ra. This less dissipation in the porous media
in spite of the appearance of large scale structures enhancing
thermal convection might lead to the excess of the velocity
fluctuation over the buoyancy-induced terminal velocity U f .
Figure 7 shows the dependence of the local dissipation on Ra
at the two positions of the porous media, z = 0.5H and 0.
We see from this figure that in contrast with the abovementioned behaviour of the local dissipation in the porous media in Figure 7a, just on the surface of the porous media in
Figure 7b, the dissipation decreases with Ra at lower Ra. It
is usual laminar behaviour of the normalized dissipation. At
higher Ra, the dissipation in Figure 7b also increases with increasing Ra. This change of the behaviour would be attributed
to the appearance of dissipative vortical structures around theporous surface. Around Ra = 108 we have enough energy dissipation in the system, so that the velocity fluctuation does notexceed U f but scale U f , leading to the ultimate scaling.
As the variation of the energy dissipation rate not only
near the wall but also in the holes, the vortical structures’ appearance, distribution and stretching can be considered as theevidences for these three states, classical, transition and ultimate. As shown in Figure 8, when the exceeded ultimate
scaling Nu ∼ Ra occurs and continues, i.e. when the Ra is at
1 × 107 and 7 × 107 , it can be seen that the energy dissipation occupies a certain distribution inside the porous medium,
while the vortical structure does not enter too much, i.e. it does
not provide enough dissipation for the whole system, resulting

Figure 5: Turbulent heat flux wT normalized by U f ∆Tat (a) z = 2δT (Lz = H) in the non-porous case at Ra =5 × 108 and (b) z − H = 5δT (Lz = 3H) in the porous
case at Ra = 1.8 × 108 , where δT denotes the thicknessof a thermal conduction layer.

in an actual dissipation rate higher than the Taylor’s dissipation
law, ϵ ∼ U f3 /H. When the Ra is at 1.8 × 108 , it can be seen in
Figure 8c that the smoother vortical structures start to enter the
porous medium to provide sufficient energy dissipation for thesystem, i.e. the energy dissipation rate of whole the system
will be scaled by Taylor’s dissipation rate as shown in Figure
6. As the Ra increases to 8 × 108 , the appearance of the energy

412th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

100

Ra = 1
Ra = 13<  > xyǫ /(Uf /H)

Ra = 2

10

-1

10

-2

10

-3

10

-4

0.0

0.2

0.4

× 106
× 107
× 107

Ra = 4
Ra = 5Ra = 7

0.6

× 107
× 107
× 107

0.8

Ra = 1

(a)

(b)

(c)

(d)

× 108
× 108
× 108

Ra = 1.8
Ra = 3

1.0

1.2

1.4

z/H

Figure 6: The horizontal and time averaged energy dissi-

pation rate normalized by U f3 /H in the lower half of thedomain in the porous case at several values of Ra.

(a)

0.004

|

3 −1
<  > xyǫ /(Uf H ) z = 0.5H

0.005

Figure 8: Instantaneous energy dissipation rate and vor-

0.003

tical structures for porous cases at (a) Ra = 1 × 107 ,(b) Ra = 7 × 107 , (c) Ra = 1.8 × 108 and (d) Ra =8 × 108 . The red and yellow objects, respectively, represent the isosurfaces of the energy dissipation rate andof the second invariant of the velocity gradient tensor,(a) ϵ/(U f2 /H 2 ) = 5 × 10, Q/(U f2 /H 2 ) = 1.8 × 10, (b)ϵ/(U f2 /H 2 ) = 1.2 × 103 , Q/(U f2 /H 2 ) = 1.5 × 102 , (c)ϵ/(U f2 /H 2 ) = 1.8 × 103 , Q/(U f2 /H 2 ) = 2.5 × 102 and(d) ϵ/(U f2 /H 2 ) = 4 × 103 , Q/(U f2 /H 2 ) = 5 × 102

0.002

0.001

0.000
10

6

10

7

10

8

Ra

(b)

<  > xyǫ / Uf Hz

( 3 −1 )| =

H

0.8

0.6

Summary and outlook
It can clearly be observed from Figure 2, that the ultimate scaling Nu ∼ Ra1/2 is achieved by the introductionof the porous walls with chambers at high Rayleigh number
Ra ≥ 108 . The pre-factor in the observed ultimate scaling in
the case of the porous walls is roughly consistent with thatfound by Kawano et al. (2021) in the case of the permeable
walls at βU f = 3. At the lower Rayleigh number Ra < 108 in
the case of the porous walls, a exceeded ultimate scaling law
Nu ∼ Ra has been well observed and confirmed, which is sonovel for this kind of research and can be recognized as thedistinct results induced by the porous structures. At the much
lower Rayleigh number around Ra = 106 , Nu in the conventional non-porous case is almost comparable with that in the
porous case, implying that the wall porosity does not have significant effects on thermal convection.
The geometry of the plume structures near the walls also
shows differences between the porous cases and non-porous
cases in Figure 5. It can be seen that the small-scale plume
structures dominate the near-wall region even at high Ra forthe conventional case. However, the structures of the turbulent heat flux found in the porous cases are so distinct, i.e, thelarge-scale structures dominate the near-wall region, which areessentially the same as those observed on the permeable walls

0.4

0.2

0.0
10

6

10

7

10

8

Ra

Figure 7: Local energy dissipation normalized with
U f3 /H as a function of the Rayleigh number at (a) z =0.5H and (b) z = H.

dissipation and vortical structures in the same time and spacedemonstrate that the vortical structures’ stretching becomes toprovide the most dissipation rate for the whole system, wherethe vortical structures become to be dominated within the areaof the energy dissipation in the porous holes. This phenom isalso consistent as the result in Figure 7, i.e. the dissipationof the whole system is in a saturated state, as the transition
process to ultimate state finished.

5

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
in the ultimate state (Kawano et al., 2021). Meanwhile, thevertical velocity normalized by U f not only in the bulk region
but also in the near-wall region is found to be invariant in thepermeable ultimate state (Kawano et al., 2021), and the sameis true of the porous case at high Rayleigh number Ra > 108 .
At the Nu ∼ Ra Rayleigh number Ra = 107 –108 in the porous
case, however, the vertical velocity fluctuation increases relatively to U f .
When the porous parts are full-filled with the laminar
flow, βU f ∼ Ra1/2 can be achieved as the system in the stateof scaling-like transition and this kind of scaling law can leadthe deviant of the Nu based on the different pre-factor c(Ra) in
Nu ∼ c(Ra)Ra1 /2. The variation of the pre-factor leads whole
the system to enter the Nu ∼ Ra scaling law. As the vorticalstructures has been induced by the intensive turbulent structures from the bulk region, the vertical velocity in the porous
and on the inter-surface between the porous part and bulk region should be well considered due to the turbulent separated
vortical structures in the porous and on the near-wall region.
This kind of structures could lead the vertical structures with adifferent scaling relationship as follows,∆p∼ w2 ∼ U f2 .
ρwalls. We have also found that the large-scale structures become dominant even in the near-wall region for the porous
cases, leading to the increase of the vertical velocity normalized by U f . Moreover, the appearance of the large scale could
be recognized as a signal of the onset of the transition process. The energy dissipation rate normalized by U f3 /H increases with increasing Ra in the transition at low Ra, but it
is invariant at high Ra, implying the Taylor’s dissipation law,ϵ ∼ U f3 /H. Furthermore, the relative increase in ϵ to U f3 /H
and thus the larger velocity fluctuation than U f to enhance theenergy dissipation are the mechanisms of the transitional behavior between the classical and the ultimate scaling. Whenthe system has enough energy dissipation in the sense of the
Taylor’s dissipation, thermal convection is saturated down tothe ultimate state.

REFERENCES
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Ultimate heat transfer in ‘wall-bounded’ convective turbulence. Journal of Fluid Mechanics, 914 .
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Pawar, S. S. & Arakeri, J. H. 2016 Kinetic energy andscalar spectra in high Rayleigh number axially homogeneous buoyancy driven turbulence. Physics of Fluids, 28 6,065103.
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(12)

As the vortical structures enter the porous holes and try to fullfill the holes, whole the system will saturate back to the ultimate scaling from the scaling-like transition state as follows,
∆p
ρ

(13)

∆p −1
w ∼ Uf .
ρ

(14)

w∼β

β −1 ∼

Based on the relationship between the permeability β and U f ,βU f ∼ Ra0 can be realized, which applies to the same configuration in (Kawano et al., 2021) and the scaling law saturates
back to the Nu ∼ Ra1/2 from the scaling-like transition state.
We have confirmed that at higher Ra the stronger largescale structures appear to enhance thermal convection as discussed before. In the scaling-like transition state at Ra = 107 –
108 , however, as shown in Figure 6 the energy dissipation in
the porous media is still smaller than U f3 /H, so that the normalized dissipation therein increases with increasing Ra. Thisless dissipation in the porous media leads to the excess of thevelocity fluctuation over the buoyancy-induced terminal velocity U f .
In this work, we have observed the ultimate state at highRayleigh number as well as the transition with a distinct scaling law Nu ∼ Ra from the classical scaling to the ultimate
scaling at Ra = 107 –108 in RB convection between porous

6