Periodically Modulated Thermal Convection

Rui Yang ,

1,2,†

Kai Leong Chong ,

1,†

Qi Wang ,

1,3

Roberto Verzicco ,

1,4,5

Olga Shishkina ,

2

and Detlef Lohse

1,2,*

1

Physics of Fluids Group, Max Planck Center for Complex Fluid Dynamics, MESA+ Institute and J.M.Burgers Center for Fluid

Dynamics, University of Twente, P.O. Box 217, 7500 AE Enschede, Netherlands

2

Max Planck Institute for Dynamics and Self-Organisation, Am Fassberg 17, 37077 Göttingen, Germany

3

Department of Modern Mechanics, University of Scie nce and Tech nology of China, Hefei 230027, China

4

Dipartimento di Ingegneria Industriale, University of Rome ’Tor Vergata’, Via del Politecnico 1, Roma 00133, Italy

5

Gran Sasso Science Institute–Viale F. Crispi, 7, 67100 L’Aquila, Italy

(Received 29 April 2020; accepted 9 September 2020; published 9 October 2020)

Many natural and industrial turbulent flows are subjected to time-dependent boundary conditions .

Despite being ubiquitous, the influence of temporal modulations (with frequency f) on global transport

properties has hardly been studied. Here, we perform numerical simulations of Rayleigh-B´enard

convection with time periodic modulation in the temperature boundary condition and report how this

modulation can lead to a significant heat flux (Nusselt number Nu) enhancement. Using the concept of

Stokes thermal boundary layer, we can explain the onset frequency of the Nu enhancement and the optimal

frequency at which Nu is maximal, and how they depend on the Rayleigh number Ra and Prandtl number

Pr. From this, we construct a phase diagram in the 3D parameter space (f, Ra, Pr) and identify the

following: (i) a regime where the modulation is too fast to affect Nu; (ii) a moderate modulation regime,

where Nu increases with decreasing f, and (iii) slow modulation regime, where Nu decreases with further

decreasing f. Our findings provide a framework to study other types of turbulent flows with time-

dependent forcing.

DOI: 10.1103/PhysRevLett.125.154502

Turbulent flows driven by time-dependent forcing are

common in nature and industrial applications [1,2].For

example, Earth’s atmosphere circulation is driven by

periodical heating from solar radiation, the ocean tidal

current by periodical gravitational attractions from both the

Moon and the Sun, and the blood circulation by the

beating heart.

In periodically driven turbulence in shear flows, a mean-

field theory has been used to analyze the resonance maxima

of the Reynolds number [3,4]. Periodic forcing in other

turbulent systems, for example, in the homogeneous

isotropic turbulence [5–8], pipe flow [9–12], channel flow

[13,14], Taylor-Couette flow [15,16], and Rayleigh-B´enard

(RB) convection [17–19], is also shown to have highly

nontrivial response properties.

Here we picked turbulent Rayleigh-B´enard convection as

a model system to study how time periodic modulation of

temperature boundary condition influences global heat

transport. The RB system, consisting of a fluid layer heated

from below and cooled from above, has been extensively

studied as the paradigmatic and well-defined system for

convective thermal turbulence [20–22]. Also, several

modulation methods have been studied for that system,

such as bottom temperature modulation [17,19,23], rotation

modulation [18,24], and gravity modulation [25,26].

Intuitively, one may expect that the modulation effect on

time-averaged global quantities is limited because the net

force averaged over a cycle vanishes. Indeed, with bottom

temperature modulation in experiments, only a small

enhancement (≈7%) of the heat flux has been observed

so far [17,19]. However, in those experiments, the effects of

modulation in temperature have not yet been fully explored

because of the experimental challenge in having a broad

range of modulation frequency due to thermal inertia of the

plates. Note that also in numerical simulations thermal

inertial can straightforwardly be treated [27,28], but in this

study, for conceptional clarity, we keep the problems of

thermal transport in the RB cell and in the plates disen-

tangled and assume perfect conductivity of the plates.

In this Letter, we numerically study modulated RB

convection within a wide range (more than 4 orders of

magnitude) of modulation frequency at the bottom plate

temperature and observe a significant (≈25%) enhance-

ment in heat transport. To explain our findings, we show the

relevance of the Stokes thermal boundary layer (BL), which

is analogous to the classical one for an oscillating plate

[29], in determining the transitional frequency for the heat

transport enhancement and the optimal frequency for the

maximal heat transport. In particular, we calculate the

transition between the different regimes in phase space and

show how they depend on the Rayleigh and Prandtl

numbers, which represent the ratios between buoyancy

and viscosity and between momentum diffusivity and

thermal diffusivity, respectively. Our modulation method

PHYSICAL REVIEW LETTERS 125, 154502 (2020)

0031-9007=20=125(15)=154502(6) 154502-1 © 2020 American Physical Society

is complementary to hitherto used concepts of using

additional body force or modifying the spatial structure

of the system to enhance heat transport, for example,

adding surface roughness [30–32], shaking the convection

cell [33], including additional stabilizing forces through

geometrical modification [34–36], rotation [37], inclination

[38,39], or a second stabilizing scalar field [40].

Next to the aspect ratio of the horizontal and vertical

extensions of the container, the dimensionless control

parameters are the Rayleigh number Ra ¼ αgH

3

Δ=ðνκÞ

and the Prandtl number Pr ¼ ν=κ, with α, ν, and κ being,

respectively, the thermal expansion coefficient, kinematic

viscosity and thermal diffusivity of the fluid, g the

gravitational acceleration, and Δ the temperature difference

between the bottom and top boundaries. The time, length,

and temperature are made dimensionless by the free-fall

time τ ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

H=αgΔ

p

, the height H of the container, and the

temperature difference Δ, respectively. In the following, all

quantities are dimensionless, if not otherwise explicitly

stated. In the periodically modulated RB, we give a

sinusoidal modulation signal to the bottom temperature as

θ

bot

¼ 1 þ A cosð2πftÞ: ð1Þ

For modulated RB, two more parameters have to be

introduced, namely the modulation frequency f and its

amplitude A, which is kept fixed in this study, A ¼ 1. The

efficiency of the heat transport and flow strength in the

system are represented in terms of the Nusselt number Nu

(the dimensionless heat flux) and the Reynolds number

Re (the ratio between inertia and viscous forces). Direct

numerical simulation (DNS) for incompressible Oberbeck-

Bousinesq flow are employed [41]; the numerical details

are provided in the Supplemental Material [42]. The DNS

are conducted in a two-dimensional square box with no-slip

and impermeable boundary conditions (BCs) for all walls.

The explored parameter range spans 10

7

≤ Ra ≤ 10

9

,

1 ≤ Pr ≤ 8, and 10

−4

≤ f ≤ 4. We are aware of the

limitation of the two-dimensionality of the system on

which we focus, but in particular for Pr ≥ 1 two- and

three-dimensional RB convections show very close simi-

larities and features [44]. To support that our results are also

relevant for 3D RB, we conduct a set of three-dimensional

DNS in a cubic box at Ra ¼ 10

8

and Pr ¼ 4.3 with various

frequencies.

Figure 1(a) shows how the global convective heat

flux Nu depends on the modulation frequency f at fixed

Pr ¼ 4.3 (corresponding to water). The dependence of Nu

on f exhibits a universal trend for both, two- and three-

dimensional results, which is independent of Ra: When f is

large enough, Nu is not sensitive to the modulation

frequency, and the value is close to the value Nu

0

for

the case without modulation. However, when f is below a

certain onset frequency (denoted as f

onset

), there exists an

intermediate regime with significantly enhanced heat flux

as compared to Nu

0

. With f decreasing further, one

observes an optimal frequency f

opt

at which Nu is maximal

with an enhancement of approximately 25%. Such a

large enhancement of Nu is highly nontrivial because

the time-averaged temperature of the bottom plate is still

fixed at 1, and we only have changed the bottom

temperature from a steady value to a time periodic signal.

In Fig. 1(b), we further examine the NuðfÞ dependence for

different Pr, with Ra fixed at 10

8

. One can see that

both f

onset

and f

opt

are much more sensitive to Pr than

to Ra.

We first examine whether the transition is related to the

strength of the large-scale circulation (LSC). Figure 1(c)

shows the global Reynolds number Re as function of f for

various Ra, from which we can see that Re is maximized at

a Ra-dependent frequency f

opt;Re

(see Reynolds resonance

in Supplemental Material [42] for further analysis of

f

opt;Re

). However, when comparing the Nu and Re behav-

ior, one observes that the position of the strongest LSC does

not correspond to that of the maximum heat transport

(a)

(b)

(c)

FIG. 1. (a) Modulated frequency dependence of the Nusselt

number NuðfÞ, normalized by Nu

0

¼ Nuðf ¼ 0Þ, for different

Rayleigh numbers and fixed Pr ¼ 4.3. (b) NuðfÞ=Nu

0

for differ-

ent Prandtl numbers and fixed Ra ¼ 10

8

. (c) Global ReðfÞ

normalized by the Re

0

¼ Reðf ¼ 0Þ for different Rayleigh

numbers and fixed Pr ¼ 4.3.

PHYSICAL REVIEW LETTERS 125, 154502 (2020)

154502-2

(f

opt

≠ f

opt;Re

). What physics then governs the transitions

between the regimes of heat flux?

To gain insight into this problem, we analyze how the

flow structure is changed under modulation. Figure 2(a)

shows the temperature fields at different phases of modu-

lation at f ¼ 10

−3

. During the heating phase (θ

bot

> 1),

frequent plume emissions are observed near the bottom

plate. On the contrary, during the cooling phase (θ

bot

< 1),

there are no plume emissions from the bottom plate because

of the stable stratification near that surface, and the

resulting weakening of the circulation.

We further calculate the conditional average of the

temperature profiles at different phases, and compare

these profiles for different modulation frequencies in

Figs. 2(b)–2(d). Without modulation, we recover traditional

RB with a mean bulk temperature of 0.5 [Fig. 2(b)].

When f ¼ 10

−1

as shown in Fig. 2(c), the temperature

adjacent to the bottom is significantly affected by modu-

lation, whereas the bulk value is still close to 0.5. However,

the overall influence of the modulation is limited because it

is too fast to be sensed by the system. With decreasing

modulation frequency, the bulk temperature is more and

more influenced by the modulation (see Fig. 1 of the

Supplemental Material [42]). This suggests that there

exists a certain length scale which characterizes how deep

the influence of the modulation can penetrate into the

convective flow.

To better understand this length scale, we recall the

classical Stokes problem. In this flow, a BL is created by

an oscillating solid surface with modulating velocity

U cosð2πftÞ. Likewise, in modulated RB, we can draw

the analogy between an oscillating velocity and the

oscillating temperature θ

0

, where θ

0

¼ θ −

¯

θðzÞ, with

¯

θðzÞ being the temporally averaged temperature at height

z. The governing equation and corresponding BCs are

∂θ

0

=∂t ¼ðRaPrÞ

−1=2

∂

2

θ

0

=∂z

2

;

θ

0

ð0;tÞ¼A cosð2π ftÞ; θ

0

ð∞;tÞ¼0: ð2Þ

The analytical solution of this PDE is an exponential

profile:

θ

0

ðz; tÞ¼Ae

−z=λ

S

cos ð2πft − z=λ

S

Þ; ð3Þ

with the so-called Stokes thermal BL thickness

λ

S

¼ π

−1=2

f

−1=2

Ra

−1=4

Pr

−1=4

; ð4Þ

which is the penetration depth of the disturbance created by

the oscillating temperature at the boundary. The distortion

[Eq. (3)] travels as a transverse wave through the fluid.

From Eq. (4) one can see that the thickness λ

S

of the

Stokes thermal BL decreases with increasing modulation

frequency. Depending on the relative thicknesses of λ

S

, that

of the thermal BL λ

θ

, and that of the momentum BL λ

u

,we

can obtain three regimes shown in Fig. 2(e). Here we have

restricted our discussion to 1 ≤ Pr ≤ 8, where λ

u

≥ λ

θ

.

Regime (i): for λ

S

< λ

θ

< λ

u

, the effect of modulation is

confined inside the thermal BL, which is also shown by the

temperature profiles in Fig. 2(c). In such case, the effect of

modulation is negligible and the heat transport is almost

unaffected.

Regime (ii): for λ

θ

≤ λ

S

< λ

u

, the plume emission, which

occurs at the edge of the thermal BL, can now be influenced

by the modulation [Fig. 2(e)], leading to the enhancement

of heat transport. We note that in thermal convection with a

rough plate, a Nu enhancement can also be observed when

the thermal BL is perturbed by roughness [45,46]. Here, we

understand the enhancement in Nu by the following

mechanism: In the heating phase (θ

bot

> 1), there is a

(a)

(b) (c) (d)

(e) (f)

FIG. 2. (a) Instantaneous temperature fields at different phases in one modulation period for Ra ¼ 10

8

,Pr¼ 4.3, f ¼ 10

−3

.

(b)–(d) Phase-averaged temperature profiles during one period for Ra ¼ 10

8

,Pr¼ 4.3 and different modulation frequencies, namely

(b) without modulation; (c) f ¼ 10

−1

; (d) f ¼ 10

−4

. The horizontal axis is the temperature and the vertical axis is the height. The

colorbar shows the bottom temperature (phase angle) from 0ð−π=2Þ to 2ðπ=2Þ. (e) Sketch of the relations between the three BLs [Stokes

thermal BL (λ

S

), thermal BL (λ

θ

), momentum BL (λ

u

)] for the three regimes (Pr ¼ 4.3): (i) λ

u

> λ

θ

> λ

S

; (ii) λ

u

> λ

S

> λ

θ

;

(iii) λ

S

> λ

u

> λ

θ

. Arrows represent the flow in the bulk. (b) Sket ch of two different phases during one period for regime ii: (a) heating

phase when θ

bot

> 1 and (b) cooling phase when θ

bot

< 1. (f) Phase-averaged center temperature for Ra ¼ 10

8

,Pr¼ 4.3. The red (blue)

curve represents the phase when the bottom temperature is maximal (minimal). The dashed lines (from right to left) correspond to f

onset

and f

opt

for Nu.

PHYSICAL REVIEW LETTERS 125, 154502 (2020)

154502-3

stronger convective flow and more energetic plumes, as

compared to the case without modulation. This can be seen

from the value of Nu at the top plate (see Supplementary

Material [42]), where it increases to the values above that

without the modulation during the heating phase. However,

in the cooling phase, Nu starts to decline but still remains

at values comparable to that without modulation, due to

the remaining convective flow. Therefore, there is a net

increase in Nu after one cycle, as compared to the value of

Nu without time-dependent modulation.

Regime (iii): for λ

θ

< λ

u

≤ λ

S

, the effect of temperature

modulation penetrates into the bulk region occupied by the

LSC. The role of the bulk flow is to efficiently bring the

injected hot/cold fluid near the plates during the heating/

cooling phase to the center of the system. Therefore, the

center temperature also varies with the phases as seen in

Fig. 2(f), in contrast to the situation in regimes (i) and (ii).

As a result, at the peak of the heating phase (θ

bot

¼ 2), the

temperature difference between the bottom plate and the

bulk cannot be maintained at Δθ ≃ 1.5. The thermal driving

in the heating phase becomes weaker for smaller f, and the

global Nu is expected to decrease for decreasing f. When

the frequency decreases further and goes to 0, the limiting

value of Nu should be higher than without modulation. This

is because the asymptotic value of Nu is the integral of the

NuðRaÞ (ranging from 0 to 2Ra). However, the relation

between Nu and Ra is nonlinear [47].

According to the physical picture of the three regimes, we

compare the relative BL thickness to obtain the boundaries

of the regimes, i.e., f

onset

ðRa; PrÞ and f

opt

ðRa; PrÞ.First,we

make use of the relations λ

θ

∼ Nu

−1

and λ

u

∼ Re

−1=2

for the

thermal and momentum BL thicknesses. Then we use the

Grossmann-Lohse model for the scaling of NuðRa; PrÞ and

ReðRa; PrÞ in the I

∞

regime (for large Pr) [47,48]:Nu∼

Pr

0

Ra

1=3

and Re ∼ Pr

−1

Ra

2=3

. The onset frequency f

onset

corresponds to the transition between regime i and regime ii

(λ

S

∼ λ

θ

), and we obtain

f

onset

∼ Ra

1=6

Pr

−1=2

: ð5Þ

The optimal frequency f

opt

corresponds to the transition

between regime ii and regime iii (λ

S

∼ λ

u

), and we have

f

opt

∼ Ra

1=6

Pr

−3=2

: ð6Þ

To check these predictions for f

onset

and f

opt

, we replot

NuðfÞ for various Ra but now versus the rescaled fre-

quency fRa

−1=6

; see Fig. 3(a) (Pr ¼ 4.3 fixed). Indeed, the

figure shows rather good collapses around the onset. Next,

we vary Pr for a fixed Ra ¼ 10

8

and plot Nu versus the

correspondingly rescaled frequencies, namely f Pr

1=2

for

the onset [Fig. 3(b)] and f Pr

3=2

for the optimum [Fig. 3(c)].

Indeed, one can see the rescaled frequencies (horizontal

axis) collapse well, indicating that equations (5) and (6)

correctly predict the onset frequency and the optimal

frequency for all Pr.

Finally, we present the phase diagram in the f vs Ra

and the f vs Pr parameter spaces in Figs. 3(d) and 3(e) .

(a)

(d) (e)

(b)

(c)

FIG. 3. (a) Normalized Nu as a function of fRa

−1=6

, for different Ra and Pr ¼ 4.3, (b) f Pr

1=2

. (c) f Pr

3=2

for different Pr and

Ra ¼ 10

8

. Dashed lines show the onset frequency [where NuðfÞ starts to be affected, NuðfÞ=Nu

0

¼ 1.01] or optimal frequency [where

NuðfÞ reaches the maximum], averaged for different Ra or Pr. Phase diagram (a) in the f vs Ra and (b) in the f vs Pr parameter spaces. In

(a), the lower dashed line shows the optimal frequency f

opt

¼ 0.65Ra

−0.22

that corresponds to the maximal Nu. The upper dashed line

shows the onset frequency f

onset

¼ 0.015Ra

0.14

that corresponds to the onset of the heat flux enhancement. In (b), the lower dashed line

shows the optimal frequency f

opt

¼ 0.06 Pr

−1.35

, while the upper one shows the onset frequency f

onset

¼ 0.45 Pr

−0.65

. The prefactors

originate from fits to the DNS data for f

opt

and f

onset

[set to occur when NuðfÞ=Nu

0

¼ 1.01]; see Supplemental Material [42] for details

on the fitting.

PHYSICAL REVIEW LETTERS 125, 154502 (2020)

154502-4

We classify three regimes: classical RB regime (i), modu-

lation-enhancement regime (ii), and modulation-reduction

regime (iii). The boundary between the regimes is found

by fitting the numerically obtained f

onset

and f

opt

. The

fitting scaling relations for onset and optimum

(f

onset

∼ Ra

0.14

Pr

−0.65

, f

opt

∼ Ra

−0.22

Pr

−1.35

) show a good

agreement with the derived ones (f

onset

∼ Ra

1=6

Pr

−1=2

,

f

opt

∼ Ra

1=6

Pr

−3=2

) except f

opt

vs Ra, corresponding to

λ

S

∼ λ

u

. We notice that in our model, λ

S

is obtained based

on a diffusion equation. The neglected advection term can

become significant, particularly in regime iii where the

Stokes BL may penetrate into the bulk. It, therefore,

imposes uncertainty in estimating the weak Ra dependence

of λ

opt

. Our explored parameter range only spans 1 < Pr < 8

due to extreme costs to explore a wider range. But our

model is general for various Pr, as long as the boundary

layers exist and follow the given scaling relations. These

obviously no longer hold for extreme Pr values (i.e., very

large Pr when the flow becomes laminar and very small Pr,

when λ

u

< λ

θ

). Moreover, our model indicates the relation

of the magnitude of Nu enhancement with Ra and Pr. From

Figs. 1(a) and 1(b), the maximal Nu enhancement increases

as Pr increases while it is independent of Ra. This is

because f

onset

and f

opt

have the same scaling with Ra but

different scalings with Pr, as shown in Eqs. (5) and (6).As

Pr increases, the gap between f

onset

and f

opt

becomes

larger, and Nu keeps increasing in between. Therefore, the

maximal Nu increases with increasing Pr.

In conclusion, our results have substantial implications

for the investigation of modulated convection systems. For

a wide range of parameters in the three-dimensional

parameter space (modulation frequency f, Rayleigh num-

ber Ra, and Prandtl number Pr), we have demonstrated how

the global heat transport efficiency can be enhanced

through temperature modulation in both two- and three-

dimensional simulations. The high similarity between 2D

and 3D DNS results supports that our results are applicable

in both cases and robust. Based on the heat transfer

enhancement, we can identify three different regimes:

the classical RB regime for fast modulation, an inter-

mediate regime in which the modulation leads to increasing

Nu enhancement, and the slow modulation regime in which

it leads to decreasing Nu enhancement. The transitions

between the regimes are well predicted by the relative

thicknesses of thermal, momentum, and Stokes thermal

BLs. Our concept of explaining global transport properties

in modulated BL flows by the relative thicknesses of the

three relevant BLs can also be extended to the angular

velocity transfer in modulated turbulent Taylor-Couette

flow, or to the kinetic energy transfers in modulated

turbulent pipe flow.

This work was supported by the Priority Programme

SPP 1881 Turbulent Superstructures of the Deutsche

Forschungsgemeinschaft and by NWO via the

Zwaartekrachtprogramma MCEC and an ERC-Advanced

Grant under Project No. 740479. This work was partly

carried out on the national e-infrastructure of SURFsara.

We also gratefully acknowledge support by the Balzan

Foundation.

*

d.lohse@utwente.nl

†

These authors contributed equally to this work.

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