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

HEAT TRANSFER ENHANCEMENT BY TRAVELING WAVE-LIKE BODY
FORCE IN TURBULENT CHANNEL FLOWRyo Yamamoto, Junichi Morita, Hiroya Mamori, Takeshi Miyazaki
Department of Mechanical and Intelligent Systems Engineering
The University of Electro-Communications
1-5-1 Chofugaoka, Chofu, Tokyo, 182-8585, Japanmamori@uec.ac.jp

Shmpei Hara
Department of Energy and Mechanical Engineering
Doshisha University
1-3, Tatara Miyakodani, Kyotanabe-shi, Kyoto, JapanABSTRACT

fer. Kaithakkal et al. (2020) conﬁrmed the dissimilar effectby a traveling wave-like blowing and suction in the turbulent
channel ﬂow. Instead of the blowing and suction from the wall,
Murakami & Hara (2020) made an experimental study of thechannel ﬂow to investigate the dissimilar effect of the traveling wave. They employed a self-excited thin ﬁlm installedapart from the walls and conﬁrmed the dissimilarity, which
depended on the distance between the ﬁlm and the wall.
In this study, we reproduce the experiment of Murakami& Hara (2020) by direct numerical simulations. Instead of
the self-excited thin ﬁlm, we employed the traveling wave-likebody force control. The purpose is to investigate the effectof the control on the dissimilar control effect and clarify themechanism.

Direct numerical simulations of the turbulent channel
ﬂow controlled by the traveling wave are performed. We employed the traveling wave-like body force to mimic the selfexcited thin ﬁlm, which has a dissimilar effect on momentum
and heat transfer. The dissimilar effect is obtained when thewave travels in the downstream direction of which wavespeed
is slower than the bulk velocity. The three-component decomposition shows that the traveling wave generates velocity rotation, which increases the skin-friction drag and heat transfer. In addition, the coherent component of the Reynolds shear
stress decreases while that of the turbulent heat ﬂux increases,
which is responsible for the dissimilar effect.

INTRODUCTION
Energy consumption is increasing year by year, and population growth and improving energy efﬁciency is a common
issue globally. In ﬂuid engineering, the development of ﬂowcontrol methods to improve energy efﬁciency has been actively pursued. In particular, research on the reduction ofskin-friction drag and enhancement of heat transfer has beenfrequently conducted. It is known that there is a similarity
between heat and momentum transport. Especially in turbulent ﬂows, vortical structures promote the heat and momentum transports, and heat transfer and skin-friction drag are increased. However, increasing the heat transport and reducing(or increasing slightly) the skin-friction drag simultaneously
are required, and dissimilar controls have been investigated.
Hasegawa & Kasagi (2011) derived a control law basedon the sub-optimal theory using the intrinsic difference between velocity and temperature. The law was applied to thecontrol of blowing and suction from walls. Direct numericalsimulations (referred to as DNS) in the turbulent channel ﬂow
with the control were performed. They obtained tripled heattransfer with doubled skin-friction drag as compared with theuncontrolled case. Yamamoto et al. (2013) performed the DNSof the turbulent channel ﬂow controlled by the wall transpiration based on the optimal control theory. They obtained a 30 %reduction of the skin-friction drag and doubled the heat trans-

DIRECT NUMERICAL SIMULATIONThe DNS of the turbulent channel ﬂow with the traveling wave-like body force control is performed. The governing
equations are the continuity, the Navier-Stokes, and the energy
equations of the incompressible ﬂow, as

∂ ui
= 0,
∂ xi
∂ ui ∂ ui u j∂P1 ∂ 2 ui+=−++ fi ,∂t∂xj
∂ xi Reτ ∂ x j ∂ x j∂ T ∂ u jT
∂ 2T
1+= Q+,∂t∂xj
Reτ · Pr ∂ x j ∂ x j(1)(2)(3)

Here, t is time, P is pressure, T is temperature, Q is heat source
and fi is the body force term. The streamwise, wall-normal,and spanwise directions are deﬁned as x, y, and z, respectively,
and the corresponding velocity components are deﬁned as u,v, and w. We interchangeably use xi and ui (n = 1 ∼ 3) as thecoordinate and the velocity in this paper. The mean pressure
gradient in the streamwise direction is kept constant, and thefriction Reynolds number is set to 180. The Reynolds number
is deﬁned by the reference values of the channel half-width δ ∗1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Lz
body force equation
Δcy wave propagation

LyxhwzLx

Figure 1.

Schematics of the channel ﬂow with the traveling wave-like body force.

and the wall-friction velocity uτ∗ . To hold an analogy between
the momentum and heat transfer, we imposed a uniform heatgeneration condition and a Prandtl number of Pr = 1. The governing equations are discretized by using an energy-conservingsecond-order central difference method. As time advances, weemploy the second-order Crank-Nicolson method for the viscous term and the low storage third-order Runge-Kutta method
for the other terms.
Figure 1 shows the schematic of the channel ﬂow withthe traveling wave control. The computational domain is(Lx , Ly , Lz ) = (2π , 2, π ) and the grid points are (Nx , Ny , Nz ) =(256, 384, 128). The periodic boundary conditions are imposed in the streamwise and spanwise directions, and the noslip condition is imposed on the wall. The temperature on thewall is zero. We note that the body force control allows for
ﬂowing penetration of the ﬂow through the ﬁlm, unlike thecorresponding experiment by Murakami & Hara (2020). As
shown in Fig. 1, the control parameters are the amplitude A, thepenetration length ∆, the control position hw , wavelength λ ,and the wavespeed c. The paper shows the effect of wavespeedc while the others are ﬁxed at A = 500, ∆ = 0.1, hw = 0.5, andλ = π.

wave at 0 < c < 30 affects the cost functions. The skin-friction
coefﬁcient and the Nusselt number peak at c = 10. Because
the j/ f is positive and maximum, the increment of the heattransfer is larger than that of the skin-friction drag. It implies
the dissimilarity effect and we chose c = 10 as a reference casein the following. Figure 2(b) shows the dependency of hw on∆ = 0.1, A = 500, c = 11, and k = 2. The Nusselt numberand skin-friction coefﬁcient increase with an increase in hw ,whereas they slightly decrease at large values of hw . Moreover,
the j/ f factor increases, and the maximum value is obtained
at hw = 0.6.
Figure 3 compares the instantaneous streamwise velocity between the uncontrolled ﬂow and the reference case. The
white isosurface shows the vortical structure visualized by thesecond invariant of the velocity deformation tensor (so-called
the Q value), while the thresholds are 0.03 for the uncontrolled
ﬂow and 0.17 for the reference case. As a result, the turbulenceis well promoted by the traveling wave in the reference case. In
the uncontrolled case, the vortical structures are in the region
near the wall. On the other hand, the vortical structures clustered and distributed alternately in the streamwise direction in
the reference case.
We introduce a three-component decomposition method
to investigate the contribution of the traveling wave to the ﬂow.
Previous studies have distinguished contributions based on thedirect and indirect effects of traveling waves Mamori & Fukagata (2014). Accordingly, we deﬁne the three-component decomposition method for an arbitrary quantity f as

RESULT AND DISCUSSION
Figure 2 shows the control effect of the wavespeed c as afunction of a skin-friction drag coefﬁcient C f , a Nusselt number Nu, asf = f¯ + f ′τ∗Cf = 1 w ∗ ,∗2 ρ Ub(4)Nu =(5)2δ ∗ h∗,λ∗= ⟨ f ⟩ + f ′′ = f¯ + f˜ + f ′′ ,where f¯ and ⟨ f ⟩ are deﬁned as∫

1 2π(8)f¯(y) =
⟨ f ⟩d ϕx ,2π 0[]∫ T ∫ Lz11
⟨ f ⟩ (ϕx , y) =∑ T Lz 0 0 f (x, y, z,t)dzdt , (9)Nϕx x∈
ϕxHere, τw∗ is the wall shear stress, ρ ∗ is the ﬂuid density, Ub∗ isthe bulk velocity, h∗ is the heat transfer coefﬁcient, and λ ∗ isthe thermal conductivity. The asterisk means the dimensionalvariables. In addition, the dissimilarity is evaluated by j/ ffactor, as
1
j/ f = 1/3,
4C
Pr RebfNu(7)

where ϕx = k(x − ct) − 2π n(0 < ϕx < 2π , n ∈ Z) is the wave
coordinate along the streamwise direction and Nϕx = kLx /(2π )is the number of streamwise locations belonging to the samephase. The RSS (−u′ v′ ) and THF (−v′ T ′ ) are decomposedusing the three-component decomposition method as(6)where Reb is the bulk Reynolds number. These are normalized
by the uncontrolled values. Figure 2(a) shows the dependencyof c, and the other parameters are ﬁxed at hw = 0.5, ∆ = 0.1,
A = 500, and k = 2. We found that the downstream traveling2−u′ v′ = −ũṽ − u′′ v′′ ,(10)−v′ T ′ = −ṽT̃ − v′′ T ′′ .

(11)12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022(b)(a)

Figure 2. Dependency of (a) c (hw = 0.5, ∆ = 0.1, A = 500, and k = 2) and (b) hw (∆ = 0.1, A = 500, c = 11, and k = 2) on theskin-friction coefﬁcient, Stanton number, and j/ f factor. They are normalized to those of the uncontrolled values.

(a)

(b)

Figure 3. Instantaneous streamwise velocity in (a) the uncontrolled case and (b) the reference case on the x − y plane between z = 0to z = π /2. The threshold values of the vortical structures are Q+ = 0.03 and Q+ = 0.17 for uncontrolled case and reference case,respectively.

peaks at c = 10. In addition, depending on the control position,
the j/ f factor increases, and the maximum value is obtained at
hw = 0.6. The three-component decomposition indicates thatthe traveling wave generates rotation of the velocity. The contribution of the coherent component is different be- tween theRSS and THF: the coherent-RSS decreases, and the coherentTHF increases. It is responsible for the dissimilarity between
momentum and heat transfers.

The LHS is referred to as the “total” component, and the ﬁrst
and second terms on the RHS are referred to as the “coherent”
and “random” components.
The RSS and THF signiﬁcantly inﬂuenced on the skinfriction coefﬁcient and Nusselt number. The decomposed RSS
and THF in each case are shown in Fig. 4. For the controlled cases (Cases 1∼3), other control parameters are ﬁxed
at ∆ = 0.1, A = 500, c = 11, and k = 2. The total-RSS in
Case 1 is slightly larger than that of Case NC; the randomRSS almost agrees with the total-RSS, and the coherent-RSS
is very small. When hw increases, the coherent-RSS decreases,
and the random-RSS increases in Case 2; the peaks slightly
decrease and shift toward the channel center in Case 3. The
random-THF and coherent-THF are positive. The coherentTHF peaks in the region near the wall in Case 1, whereas it attenuates and moves toward the channel center as hw increases
in Cases 2 and 3. However, the random-THF had double peaks
in Case 1, but the near-wall peak grows, and the other peak decreases and disappears in Cases 2 and 3.

CONCLUSION
The direct numerical simulations of the turbulent channel
ﬂow controlled by traveling waves are performed. We employ
the traveling wave-like body force to mimic the self-excitedthin ﬁlm and investigate the effects on skin-friction drag andheat transfer. The downstream traveling wave obtains the dissimilar effect between momentum and heat transfers, and it3

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

(b)

(c)

(d)

(e)

(f)

Figure 4. Decomposed Reynolds share stress (left) and turbulent heat ﬂux (right): top, Case 1; center, Case 2; bottom, Case 3. Black,total-RSS; red, random-RSS; blue, coherent-RSS; broken line, Case NC.

REFERENCES

ﬂow. Physics of Fluids 26 (115104).
Murakami, T. & Hara, S. 2020 Inﬂuence of the installation of arubber sheet on pressure drop and heat transfer characteristics in turbulent channel ﬂow. In Proc. 57th annual meetingof the heat transfer society of Japan (B211).
Yamamoto, A., Hasegawa, Y. & Kasagi, N. 2013 Optimal control of dissimilar heat and momentum transfer in a fully developed turbulent channel ﬂow. Journal of Fluid Mechanics
733, 189–220.

Hasegawa, Y. & Kasagi, N. 2011 Dissimilar control of momentum and heat transfer in a fully developed turbulent
channel ﬂow. Journal of Fluid Mechanics 683, 57–93.
Kaithakkal, A. J., Kametani, Y. & Hasegawa, Y 2020 Dissimilarity between turbulent heat and momentum transfer induced by a streamwise travelling wave of wall blowing andsuction. Journal of Fluid Mechanics 886, 1045.
Mamori, H. & Fukagata, K. 2014 Drag reduction effect bya wave-like wall-normal body force in a turbulent channel4