PHYSICS OF FLUIDS 25, 125110 (2013)

Direct numerical simulation of turbulent heat transferin a fluid-porous domain

M. Chandesris,1,a) A. D’Hueppe,1 B. Mathieu,1 D. Jamet,1 and B. Goyeau2

1CEA, DEN, DM2S/STMF, 17 rue des Martyrs, F-38054 Grenoble, France2EM2C, UPR-CNRS 288, Ecole Centrale Paris, Grande Voie des Vignes, 92295Chatenay-Malabry, France

(Received 26 July 2013; accepted 5 December 2013; published online 30 December 2013)

Turbulent heat transfer in a channel partially filled by a porous medium is investigatedusing a direct numerical simulation of an incompressible flow. The porous mediumconsists of a three-dimensional Cartesian grid of cubes, which has a relatively highpermeability. The energy equation is not solved in the cubes. Three different heatingconfigurations are studied. The simulation is performed for a bulk Reynolds numberReb = 5500 and a Prandtl number Pr = 0.1. The turbulent flow quantities arecompared with the results of Breugem and Boersma [“Direct numerical simulationsof turbulent flow over a permeable wall using a direct and a continuum approach,”Phys. Fluids 17, 025103 (2005)] to validate the numerical approach and macroscopicturbulent quantities are analyzed. Regarding the temperature fields, original resultsare obtained. The temperature fields show an enhanced turbulent heat transfer justabove the porous region compared to the solid top wall, which can be related tothe large vortical structures that develop in this region. Furthermore, these largestructures induce pressure waves inside the porous domain which are responsible oflarge temperature fluctuations deep inside the porous region where the flow is laminar.Finally, macroscopic turbulent quantities are computed to get reference results for thedevelopment of macroscopic turbulent heat transfer models in fluid-porous domain.C© 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4851416]

I. INTRODUCTION

Heat transfer at the interface between a porous layer and an open channel is a problem encoun-tered in a wide range of technological applications such as electronic cooling, drying processes, andpacked-bed heat exchangers.

The flow resistance within the porous layer reduces the velocity and creates a sharp tran-sition between the two regions. When the flow in the open region is turbulent, large vorticalstructures, associated with Kelvin-Helmholtz type of instability, develop just above the porousregion.1–3 These structures are different from those encountered at a solid impermeable wall.They are very efficient in transporting streamwise momentum in the wall-normal direction, theyinfluence the penetration of the velocity into the porous layer and thus the overall transfers inthis interfacial region. As a consequence, the heat transfer is increased as well as the pressuredrop.

In order to quantify the effects of the porous region on the heat transfer and on the pres-sure drop and to optimize heat transfer processes using highly conductive porous inserts, alarge number of numerical studies have been performed.4–9 In these studies, the free fluid andthe porous region are usually considered as two different homogeneous domains; for turbulentflows, k − ε models are used in both regions and boundary conditions are imposed at the fluid-porous interface (usually continuity conditions). However, the ability of the macroscopic boundary

a)Electronic mail: marion.chandesris@cea.fr

1070-6631/2013/25(12)/125110/21/$30.00 C©2013 AIP Publishing LLC25, 125110-1

125110-2 Chandesris et al. Phys. Fluids 25, 125110 (2013)

conditions to capture the physical phenomena taking place at the microscopic scale is notstraightforward,10–12 and the validity of the turbulence k − ε model in this transition region is alsoquestionable.

To validate and develop better turbulent heat and mass transfer models in the fluid-poroustransition region, a thorough understanding of the turbulent transport is essential. Direct Numeri-cal Simulation (DNS) has been largely used for the analysis of turbulent heat transfer in channelswith impermeable walls. Since the pioneering work of Kim and Moin,13 who performed DNSof turbulent channel flows for Reynolds number Reτ = 180 and Prandtl number Pr = 0.1, 0.71,and 2.0, using isothermal boundary conditions at the walls, numerous DNS have been providedwith higher Reynolds number,14, 15 various Prandtl number,16–18 and heating boundary conditionsat the walls.19, 20 DNS is also used to investigate configurations such as turbulent thermal bound-ary layers,21, 22 turbulent heat transfer in rotating flows,23 turbulent heat transfer over riblets,24 orturbulent heat transfer in Couette flow.25

In the field of atmospheric or aquatic flows, the so-called “canopy flows” have been exten-sively studied. In the absence of a canopy, the flow resembles a rough-wall boundary layer,26, 27

but in the presence of an extensive or dense canopy, the flow behaves as a perturbed mixinglayer28, 29 and is thus very similar to flows in fluid-porous domain. Regarding the effect of aporous layer in a channel, some DNS have been performed but, to our knowledge, only the tur-bulent momentum transfer has been studied and almost never the heat transfer. Jimenez et al.3

study turbulent shear flow over active and passive surfaces by imposing a boundary conditionat the porous wall that is supposed to mimic the behavior of a Darcy-type porous wall. Togain more insight on the turbulence structure inside the fluid-porous transition region, Breugemet al.1, 30 performed DNS in a channel partially filled by a porous medium using two different ap-proaches. In the “continuum approach,” the porous medium is modeled using the volume-averagedNavier-Stokes equations and it is characterized by its porosity and permeability. In the “directapproach,” the porous medium consists of a three-dimensional Cartesian grid of cubes. The com-plete flow field is resolved in the clear channel and within the pores of the porous region. Itrequires a large computational power, but it provides reference results not hampered by modelingassumptions.

The aim of this study is to use DNS to get new information on the turbulent heat transfer ina channel partially filled with a porous medium. Using the geometry and the flow parameters ofBreugem and Boersma,30 we perform a similar DNS at Reb = 5500 and study the turbulent thermalfield with a Prandtl number Pr = 0.1. In order to investigate the effects of the fluid-solid boundaryconditions on the turbulent transfers, three different heating conditions are considered. To analyzethe obtained results, the instantaneous fields are time-averaged, to study the turbulence propertiesand they are also volume-averaged to obtain mean quantities, representative of the flow in the porousregion.31

II. PROBLEM DESCRIPTION AND NUMERICAL METHOD

A. Geometry and governing equations

Let us consider a fully developed incompressible turbulent flow with a passive temperature fieldin a channel partially filled with a model porous medium. The flow is driven by a mean pressuregradient in the x-direction and is thus tangential to the fluid-porous interface. The geometry andthe Reynolds number are identical to those studied by Breugem and Boersma.30 The geometry isillustrated in Fig. 1. The domain dimension is 3H × 2H × 2H, respectively, in the streamwise(x-direction), spanwise (z-direction), and wall-normal (y-direction) directions. The porous mediumis composed of a structured arrangement of 30 × 20 × 9 = 5400 cubes. The cube size is dp = H/20and the distance between cubes df is equal to dp. The domain is limited by solid walls at y = H andy = −H. The distance between the bottom wall and the first cube is equal to dp. Consequently, theporosity in the homogeneous porous region equals φp = 0.875. Periodicity conditions are used inthe streamwise and spanwise directions.

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FIG. 1. Sketch of the geometry. Averaging volume and corresponding extend of the transition region for the cellular filter(red) and the quadratic shaped filter (blue).

The Navier-Stokes and energy equations governing the flow are

∂ui

∂xi= 0, (1)

∂ui

∂t+ ∂ui u j

∂x j= − 1

ρ

∂p

∂xi+ ν

∂2ui

∂x2j

, (2)

∂T

∂t+ ∂u j T

∂x j= ∂

∂x j

(α

∂T

∂x j

). (3)

The temperature is assumed to be a passive scalar: the buoyancy effects and the temperature depen-dence of the fluid properties are neglected, especially viscosity and heat conductivity. Therefore, theobtained results are valid only for systems where the temperature differences are not too large.

The mean pressure gradient in the x-direction is adjusted at each computational time step tokeep a constant mean velocity in this direction. The corresponding Reynolds number ReT = UT 2H/νis equal to 5962, where UT is the superficial averaged streamwise velocity. This Reynolds numberwill correspond, once the flow is distributed between the porous and free regions, to a bulk Reynoldsnumber Reb = UbH/ν close to 5500, where Ub is the bulk velocity in the free channel, the freechannel being defined by Breugem and Boersma30 as y > 0, allowing comparison with their velocityfields. No-slip boundary conditions are used for velocities on the walls (top and bottom walls, andcubes).

The energy equation is not solved within the cubes. Furthermore, since the cubes are notconnected, the effective conductivity of the porous region is going to be low compared to the effectiveconductivity of the porous materials that are usually used to study heat transfer enhancement byporous inserts. However, we choose to keep this geometry to be able to compare our velocity resultsto available one and thus to validate our numerical approach. Meanwhile, three different heatingconfigurations are studied:

• Case 1: The cubes surfaces are adiabatic and a constant temperature is imposed at the top andbottom walls: T(y = H) > T(y = −H).

• Case 2: The cubes surfaces are adiabatic and a constant incoming heat flux is imposed at thetop and bottom walls.

• Case 3: The top and bottom walls are adiabatic and an incoming heat flux is imposed at thecubes.

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Z

X

YFlow

Heated wall: T=1

Heated wall: T=0

Case 1

y/H=1

y/H=−1

y/H=0

Case 3

Flow

Adiabatic wall

Adiabatic wall

Heated cubes

Case 2

Flow

Heated wall: Qw=−1

Heated wall: Qw=1

Adiabatic cubesAdiabatic cubes

FIG. 2. Heating configurations and schematic description of the geometry.

The top- and bottom-wall boundary conditions in cases 1 and 2 are classical when studyingturbulent heat transfer in channel flow and will allow comparison with DNS computations in freechannels. Case 3’s configuration gets closer to heat transfers existing in some applications, for whichthe solid may generate heat. The heating configurations are summarized in Fig. 2. In case 1, classicalboundary conditions of periodicity for the temperature are used at the domain inlet and outlet. Incases 2 and 3, heat is injected in the domain. To represent an infinite channel with established heattransfer, the instantaneous temperature T is divided into two parts: a fluctuating periodic componentθ and a linear gradient.14 T and θ satisfy identical boundary conditions for the heat flux at theupper and lower walls. However, θ satisfies a modified energy equation with a sink term, boundaryconditions of periodicity at the domain inlet and outlet, and modified boundary conditions at cubeswalls in the flow direction.

The choice of the Prandtl number has been dictated partially by the convergence time of thecomputation. Indeed, the time scale of the thermal diffusion in the y-direction in the porous regionis large compared to the time scale of the turbulent momentum diffusion. The time scale of thethermal diffusion in the porous region is of the order of H2/α since the effective thermal diffusivityof the porous medium, is very close to the fluid thermal diffusivity for this geometry. The timescale of the turbulent momentum diffusion in the free channel can be estimated by H/u p

τ . Thus, theratio between the two time scales is of the order of Rep

τ Pr . By increasing the thermal diffusivityα and thus decreasing the Prandtl number, the ratio between the two time scales can be reduced.The compromise has been to choose Pr = 0.1, since reference results in channels with impermeablewalls are available for this Prandtl number.13 The ratio between the two time scales is thus of theorder of 60. This choice for the Prandtl number gives a Peclet number Pe = RePr around 1 in theporous region and around 550 in the free region.

B. Numerical method

The computations are performed using the TRIO_U code developed at CEA-Grenoble.32 Thegoverning equations are discretized on a staggered and uniform Cartesian mesh by means of a finite-volume method. The interior of the cubes is not discretized. The discrete form of the incompressibleNavier-Stokes equations is solved using a projection method. A second-order centered advectionscheme is used for the momentum equation and for the temperature advection term, a Quick 3rd orderscheme is used. Time advancement is done using a 3rd order Runge-Kutta explicit time integrationscheme.

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FIG. 3. Averaging volume for an ordered porous media. Left: Averaging volume for the quadratic-shaped weighting function.Right: top-hat, triangle shaped, and quadratic shaped weighting functions.

For Prandtl number Pr > 1, the spatial resolution is imposed by the temperature and mustverify the Batchelor length scale η0, which is estimated as η0 = ηPr−1/2 where η is the Kol-mogorov length scale. For heat transfer with Prandtl numbers Pr < 1, the spatial resolution isimposed by the velocity, which is the present case. Thus, we use the same spatial resolution as inBreugem and Boersma.30 The mesh is made of 600 × 400 × 400 = 96 × 106 cells, whose sizeis H/200.

III. AVERAGING METHOD

A. Averaging operators

To analyze the obtained results, the instantaneous fields are both time-averaged and volume-averaged. The time (or Reynolds) average operator is used to study the turbulent properties of theflow field. For any quantity ψ , the Reynolds decomposition is noted ψ = ψ + ψ ′ where the overbardenotes the Reynolds-averaged value and the prime denotes the temporal fluctuation.

In the porous region, the volume-average operator of the volume averaging method is usedto obtain mean quantities representative of the flow. The volume averaging method is a techniqueused to rigorously derive continuous macroscopic equations from the description of the problem ata microscopic scale for multiphase systems. This technique also allows to determine the “effectivemacroscopic properties” of the porous medium.31

Applying the volume-average operator to the microscopic or DNS results produces the“reference” fields that are the expected output of the continuous macroscopic model with ap-propriate effective macroscopic properties. Hence, these properties can be obtained by matchingwith the volume-averaged DNS results.33–35

Two types of volume averages are commonly introduced: the volume average and the intrinsicaverage. A formal definition of the volume-average operator, at position x, for any quantity ψ of thefluid phase can be defined using a convolution product:36, 37

〈ψ〉 (x)= (m � (χ f ψ)

)(x) =

∫R3

m(r − x) χ f (r) ψ(r) dr, (4)

where m is a weighting function that is nonzero on the averaging volume V , x is the centroid ofthe averaging volume V , and χ f is the fluid indicator function (see Fig. 3). The intrinsic average isdefined by

〈ψ〉 f (x)=m � (χ f ψ)

m � (χ f )(x) = 〈ψ〉 (x)

φ(x), (5)

125110-6 Chandesris et al. Phys. Fluids 25, 125110 (2013)

where φ is the porosity:

φ(x)=∫R3

m(r − x) χ f (r) dr = V f

V. (6)

Here, V f is the volume of the fluid phase contained within the averaging volume V .

B. Filtering functions

The choice of the weighting function m has been largely studied.37–40 According to the theoryof distributions,41 choosing the weighting function among test functions (C∞ with compact support)will provide volume-averaged quantities with continuous derivatives of all orders.37, 38 However,from a physical view point, the weighting function should also match the topology of the porousmedium, such that the porosity, defined by φ = m�χ f, is constant or contains no small variations.For ordered porous medium, the support of weighting functions that are both infinitely differen-tiable and that match the topology of the porous medium are very large compared to the pore’ssize. Thus, weighting functions with less regularity can behave as a sharper filter and are oftenpreferred.

When applying volume-average operators to quantities defined on ordered, i.e., periodic, porousmedia, we have the following properties. For quantities that are the sum of a constant (low fre-quency component) and a zero-mean periodic function whose period matches the topology ofthe porous medium, the top-hat filter mV recovers the constant and filters entirely the periodiccomponent.40

If the low frequency component also contains a linear part, Quintard and Whitaker40 showthat to filter entirely the periodic component, it is necessary to use the cellular filter, mc, whichcorresponds to the top-hat filter convoluted twice, mc = mV � mV . It is a “triangle shaped” filter (seeFig. 3).

In the present study, the low frequency component of the velocity field contains a non-negligibleparabolic part. To filter out the periodic part and recover a low frequency component and itsderivatives not hampered by fluctuations on scales smaller than the filter size, we have observedthat it is necessary to increase the regularity of the weighting function. The quadratic shaped filter,which corresponds to the top-hat filter convoluted three times, m = mV � mV � mV , filters entirelythe high frequency component of functions that are the sum of a parabolic function and a zero-meanperiodic function. This filter proves to be sufficient for this present study (see in particular Fig. 8(c))and will thus be used in the y-direction. Its support is presented in Fig. 3. As a consequence, the sizeof the transition region is larger (δi = 5dp) than when using the cellular filter (δi = 3dp) as shownin Fig. 1.

C. Double-averaged equations

The macroscopic, or double-averaged, governing equations are obtained by applying the time-and the volume-average operators to the microscopic governing equations (1)–(3). To derive thevolume-averaged equations, it is necessary to relate the volumetric averages of derivatives and thederivatives of volumetric averages. These relations are known as the spatial averaging theorems42, 43

and can be written as follows:

〈∇ψ〉 = ∇ 〈ψ〉 +∫

Afs

m(r − x) ψ(r)n(r) dr, (7)

〈∇ · ψ〉 = ∇ · 〈ψ〉 +∫

Afs

m(r − x) ψ(r) · n(r) dr, (8)

where n is the unit normal vector oriented outward from the fluid into the solid phase andAfs is the fluid-solid surface contained in the averaging volume V . Using these theorems andthe velocity no-slip boundary conditions at solid walls, the double-averaged governing equations

125110-7 Chandesris et al. Phys. Fluids 25, 125110 (2013)

are

∂ 〈ui 〉∂xi

= 0, (9)

∂ 〈ui 〉∂t

+ ∂

∂x j

(〈ui 〉

⟨u j

⟩φ

)+ ∂τ

i ju

∂x j= −φ

ρ

∂ 〈p〉 f

∂xi+ ∂

∂x j

(ν∂ 〈ui 〉∂x j

)− ∂

∂x j

⟨u′

i u′j

⟩+ f l

i , (10)

φ∂

⟨T

⟩ f

∂t+ ∂

∂xi

(φ 〈ui 〉 f

⟨T

⟩ f)

+ ∂τ iT

∂xi= ∂

∂xi

(αφ

∂⟨T

⟩ f

∂xi

)− ∂

∂xi

⟨u′

i T′⟩+ Tor + P, (11)

where τ u corresponds to the momentum dispersion tensor:

τ i ju = ⟨

ui u j⟩ − 〈ui 〉〈u j 〉

φ, (12)

⟨−u′

i u′j

⟩is the volume-average Reynolds stress tensor, fl is the friction force that results from the

interaction between the solid and fluid phases:

f li (x) =

∫Afs

m

(ν

∂ui

∂x j− p − 〈p〉 f (x)

ρδi j

)· n j dr, (13)

τ T is the thermal dispersion vector defined by

τ iT = ⟨

ui T⟩ − φ 〈ui 〉 f

⟨T

⟩ f, (14)⟨

−u′i T

′⟩

is the xi-component of the volume-average turbulent heat flux, P corresponds to the wall

heat flux:

P =∫

Afs

m α∂T

∂xini dr, (15)

and Tor is the tortuosity that measures how the thermal diffusivity is modified by the solid matrix:

Tor (x) = ∂

∂xi

(∫Afs

m α(

T − ⟨T

⟩ f(x)

)ni dr

). (16)

In the above equation, the intrinsic averaged temperature⟨T

⟩ fis evaluated at x, the centroid of the

averaging volume. Thus, it does not depend on the surface Afs. This writing, classically used byOchoa-Tapia and co-workers,44, 45 allows the derivation of Eq. (11) without imposing any length-scale constraint.

In the present DNS, the turbulent flow is fully developed in the x- and z-directions. Thus,the double-averaged variables do not depend on x and z. From the double-averaged continuityequation (9) and the fact that the velocity vanishes at the upper wall, it comes that the y-componentof the double-averaged velocity is zero, 〈v〉 = 0. Thus, the system (9)–(11) reduces to

∂τxyu

∂y= −φ

ρ

∂⟨p�

⟩ f

∂x+ ∂

∂y

(ν∂ 〈u〉∂y

)− ∂

∂y

⟨u′v′⟩ + f l

x , (17)

〈u〉∂

⟨T

�⟩ f

∂x+ ∂τ

yT

∂y= ∂

∂y

(αφ

∂⟨T

⟩ f

∂y

)− ∂

∂y

⟨v′T ′⟩ + Tor + P, (18)

where∂〈p�〉 f

∂x and∂⟨T

�⟩ f

∂x are the uniform pressure and temperature gradients used to compensate forthe frictions and thermal exchanges at the walls.

125110-8 Chandesris et al. Phys. Fluids 25, 125110 (2013)

D. Double-averaging operators for the DNS results

In order to study the turbulent transfer characteristic of the free-porous domain, the DNS resultsare filtered twice, as the governing equations. The time-averaged quantities are computed for eachmesh cell of the whole domain. Furthermore, since the geometry is a periodic arrangement of30 × 20 identical patterns in the x- and z-directions, a faster temporal convergence is obtained byaveraging the values of ψ found in each pattern as

ψnew(x, y, z) = 1

600

30∑i=0

20∑k=0

ψ(x + i2dp, y, z + k2dp). (19)

The geometry is thus “reduced” to a column of 9 cubes in the y-direction. The volume averagingprocess is then performed applying a spatial filter in the three directions. A top-hat filter is used inthe x- and z-directions, since there is no mean gradient in these directions and a quadratic-shapedfilter (top-hat convoluted three times) is used in the y-direction.

IV. RESULTS AND DISCUSSIONS

In this section, the DNS results are presented. The velocity statistics are compared with thoseof Breugem and Boersma30 to validate our numerical method and then the temperature results arepresented and analyzed.

The velocity field is initialized with a parabolic mean velocity profile in the free fluid channel,perturbed with a random incompressible fluctuation in all directions. For each case, the temperaturefield is initialized using results obtained on a reduced similar configuration and a coarse mesh. Thesimulation is run for about 3500 H/Ub, until the thermal field is fully developed. The convergenceof both the thermal fields and the second order statistics has been checked. The mean thermal fieldsshow no variations and the variations on the rms temperature are negligible. No difference betweenthe three cases has been noticed. This time is rather large and corresponds approximately to 60times the simulation time needed to obtain a fully developed velocity field. This is coherent withthe time scale analysis presented in Sec. II A. The statistical sampling is then started. The numberof instantaneous data fields used for the statistics is equal to 3000, spanning a total time interval of2000 H/Ub. Fig. 4 shows examples of the obtained instantaneous velocity and temperature fields.

A. Statistics of the velocity field

The different Reynolds numbers that characterize the flow in a fluid-porous domain are presentedin Table I. Ret

τ = utτ H/ν is the friction Reynolds number at the top wall, where ut

τ = √−ν∂u/∂yy=H

is the friction velocity at the top wall. Repτ is the friction Reynolds number at the “permeable” wall.

The “permeable wall” is a fictitious wall whose location has been defined by Breugem and Boersma30

as the location above the porous region for which the porosity is equal to one. However, this choiceremains arbitrary, since any other location inside the transition region could have been chosen.33

Furthermore, since we are using a different weighting function with a larger support when post-processing the results, the location above the porous region for which the porosity is equal to one isnot y = 0 in the present post-processing. However, to be able to perform comparisons, we keep this

definition. Repτ is equal to u p

τ H/ν, where u pτ =

√− ⟨

u′v′⟩ + ν∂ 〈u〉 /∂yy=0

is the friction velocity at

the “permeable wall.”The total Reynolds numbers, ReT, are identical, since we are imposing Breugem’s value.

Regarding the bulk Reynolds number and the top and permeable friction Reynolds numbers, theirdifference is within 1%. The Darcy number given by Dac = Kc/H2, where Kc is the permeability ofthe grid of cubes, is also presented Table I.

Figure 5 shows the Reynolds- and volume-averaged velocity profiles and the volume-averagedrms velocity. We recall that the volume averaging operator does not give meaningful values close tothe top and bottom walls on a distance equal to half the filter size. For a quadratic filter, the zonesimpacted by the wall effect correspond to −1 < y/H < −1 + 3dp/H and 1 − 3dp/H < y/H < 1. These

125110-9 Chandesris et al. Phys. Fluids 25, 125110 (2013)

(a) (b)

(c) (d)

(e)

FIG. 4. Instantaneous velocity and temperature fields. Cross section along the streamwise direction of the velocity field,normalized by the bulk velocity Ub: (a) streamwise and (b) wall-normal velocity. 3D view of the temperature fields. Crosssections along the 3 directions and isocontours of the temperature field: (c) case 1, (d) case 2, and (e) case 3.

zones are delimited by dotted lines in all the following figures. Breugem and Boersma30 choose toreduce the wall-normal extent of the averaging volume close to the top and bottom walls. This canexplain the difference between Breugem’s and our profiles in these zones.

The difference between the porous medium and the free region is recovered. In the free re-gion, the velocity profile is skewed with a maximum located above the center of the channel aty/H = 0.69. This skewed mean velocity profile is a direct consequence of the larger skin friction atthe permeable wall, Rep

τ = 664 than at the top wall, Retτ = 390.

TABLE I. Reynolds numbers characteristic of Breugem’s DNS and of the present work.

DNS φ dp/H Reb ReT Retτ Rep

τ Dac

Breugem 0.875 0.05 5460 5963 394 669 3.4 × 10−4

Present work 0.875 0.05 5423 5962 390 664 3.4 × 10−4

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FIG. 5. Volume-averaged mean velocity and rms velocity profiles. (a) Streamwise averaged velocity, (b) variance of thestreamwise velocity, (c) variance of the wall-normal velocity, and (d) variance of the spanwise velocity.

The rms values are low in the porous region, increase to reach a peak at the free-porousinterface around y = 0, then decrease in the free region to reach a smaller peak near the solid walland decrease again at the top wall. From this profile description, one can identify two zones ofproduction and three zones of dissipation of fluctuations (see Fig. 6). The production of fluctuationsis characterized by two peaks, one close to the fluid-porous interface and one close to the solid topwall. The difference between the width of the two peaks is coherent with the fact that the fluctuationsare created by two different mechanisms: a Kelvin-Helmholtz type of instability above the porousregion, which creates large vortices, as also observed in canopy flows,28, 29 and typical elongatedstreaks at the solid top wall. This difference can be seen in Fig. 7 that presents a snapshot of the axialvelocity field on two borders of the domain and the isocontour of the q criterion. The iso-surface ofthe q criterion helps to locate the turbulent structures. As can be seen in this figure, the turbulentstructures are more numerous and larger above the porous region than at the solid top wall andthey slightly penetrate inside the porous region. This can also be observed on the isocontours ofthe instantaneous temperature fields presented in Fig. 4. The zones of dissipation of the fluctuationsare located in the free region near the peaks of production where viscous dissipation is large andin the porous region. In the free channel, two dissipation zones can be distinguished: one abovethe permeable wall and one below the solid top wall. The border between these zones correspondsto the maximum of the mean velocity. In the porous region, the dissipation of the fluctuations islarger than in the free channel. This enhancement is due to the drag created by the solid matrix.Breugem, Boersma, and Uittenbogaard1 performed a detailed analysis of the velocity, pressure, andvorticity fluctuations. In particular, they study the spectra of the velocity and pressure fluctuationsand show that the wave components of the fluctuations decrease exponentially when penetratinginside the porous region, with the larger-scale fluctuations decreasing more slowly than the small-scale fluctuations. This result confirms the observation of large vortices entering in the upper part

125110-11 Chandesris et al. Phys. Fluids 25, 125110 (2013)

y/H=0.69

y/H=0

y/H=−1

y/H=1

Boundary layer

porous wall

Boundary layer top wall

Production zone

Production zone

Dissipation zone

Dissipation zone

Dissipation zone

FIG. 6. Production and dissipation zones.

of the porous medium. This effect is particularly important because of the large permeability valueof the porous structure considered. Studying the pressure fluctuations, they also show that insidethe porous region the velocity fluctuations are induced by pressure fluctuations. These velocityfluctuations are thus inactive in the sense that they do not contribute to the Reynolds shear-stress orto the vorticity fluctuations. For a detailed analysis of this point, the interested reader is referred toBreugem, Boersma, and Uittenbogaard.1 Thus, large scale velocity and pressure fluctuations existinside the porous region, but they do not contribute to the Reynolds shear stress and thus to themixing, as we will see when studying the temperature statistics.

The profiles of the turbulent quantities 〈k〉 and 〈ε〉 are presented in Figs. 8(a) and 8(b). For thevolume-averaged turbulent kinetic energy, 〈k〉, the profiles obtained by Trio U and Breugem are invery good agreement. This quantity illustrates the intensity of the temporal velocity fluctuations inthe domain. The values are null in the porous medium, increase to reach a peak around y = 0, thendecrease in the free region to a minimum around y = 0.7, which corresponds to the maximum of

FIG. 7. Cross section of the streamwise velocity and isocontour of the q criterion Q = 7 × 108 s−2.

125110-12 Chandesris et al. Phys. Fluids 25, 125110 (2013)

FIG. 8. Turbulent quantities: (a) Volume averaged turbulent kinetic energy, (b) volume averaged dissipation rate, and (c)turbulent viscosity.

the velocity field. The peak width at y = 0 corresponds to large vortical structures that are rapidlydamped in the porous medium (−0.5 < y < 0) due to the friction force created by the solid matrix.In the free region, the vortical structures elongate and eventually disappear. The volume-averageddissipation rate 〈ε〉 is compared to the one obtained by Moser, Kim, and Mansour46 for Reτ = 360in a DNS with solid walls (Breugem et al.1 did not publish this profile). The observed behavior isconsistent with the physics of turbulent transfer in a free-porous domain. For −1 < y/H < −0.5,the dissipation rate is very weak since there is no turbulence; for −0.5 < y/H < 0, the dissipationrate increases to reach a peak at y/H = −0.1; it characterizes the dissipation of the turbulence bythe drag due to the presence of the solid matrix; 0 < y/H < 0.7 corresponds to the dissipation ofthe Kelvin-Helmholtz vortices created above the porous region; 0.7 < y/H < 1 corresponds to thedissipation of the turbulent structures created at the solid top wall.

125110-13 Chandesris et al. Phys. Fluids 25, 125110 (2013)

In the DNS, no model is used to compute the Reynolds stress. All the turbulence scales aresolved. However, to get reference results for macroscopic models based on a “macroscopic turbulentviscosity” hypothesis, we compute the following quantity:

νtφ = − ⟨u′v′⟩

∂ 〈u〉 /∂y. (20)

The result is presented in Fig. 8(c). The profile νtφ (y) can be decomposed into three regions. Inthe porous region where y/H ≤ −0.5, the macroscopic turbulent viscosity hypothesis is not verifiedsince the signs of the numerator and the denominator of Eq. (20) are not always equal. However,both the volume-averaged Reynolds-shear stress and the velocity gradient are negligible in thisregion. In the region where −0.5 ≤ y/H ≤ 0, the macroscopic turbulent viscosity decreases almostlinearly. In this region, a difference between the present results and Breugem’s data is observed.This is due to the use of different volume-averaging filters. In Breugem’s profile, the observedfluctuation follows the geometry, which is a sign that the order of the filter was not high enough.In the free fluid region, where y/H ≥ 0, the two results are similar. Around y/H = 0.7, where thevelocity gradient vanishes, the macroscopic turbulent viscosity hypothesis is not verified. Indeed,the volume-averaged Reynolds-shear stress and the velocity gradient do not vanish exactly at thesame position and the ratio tends to infinity. However, this discrepancy lies in a very small regionwhere both the volume-averaged Reynolds-shear stress and the velocity gradient vanish.

B. Statistics of the temperature field

This section presents the low-order turbulence statistics and characteristic turbulent structuresof the heat transfer for the three studied heating configurations. For case 1, the computed quantityis the temperature T. For cases 2 and 3, where heat is injected in the system, only the periodiccomponent θ of the temperature is computed. It will be noted θ0 for case 2 and θ1 for case 3. Theobtained results are made non-dimensional using the friction temperature at the top or permeablewall or the temperature difference, when the temperature gradient at the wall is zero. The top wallfriction temperature is defined by

T tτ =

α ∂T∂y (y = H )

ρcputτ

. (21)

The friction temperature at the permeable wall is defined as

T pτ =

(− ⟨

v′T ′⟩ + α∂〈T〉∂y

)(y = 0)

ρcpu pτ

. (22)

The temperature difference is defined as

�T = [T ]H−H ,�θ = [θ ]H

−H . (23)

1. Mean temperature profiles

The volume-averaged mean temperature profiles are presented in Fig. 9. In order to show theeffect of the turbulence in the porous medium, the obtained temperature profiles are comparedwith temperature profiles obtained with a laminar model based on the effective thermal diffusivityof the porous region. For that, we compute using the DNS results the tortuosity term Tor(y) (seeEq. (16)). The tortuosity term is nonzero only in the porous and transition regions. It can be modeledintroducing a tortuosity coefficient,47 noted here αor:

Tor (y) = ∂

∂y

(φαor (y)

∂⟨T

⟩ f

∂y

). (24)

In the homogeneous porous region, it has been proven that the tortuosity coefficient, αor, dependsonly on the porous structure and on the fluid properties.47 Thus, the tortuosity coefficient must be

125110-14 Chandesris et al. Phys. Fluids 25, 125110 (2013)

FIG. 9. Volume averaged mean temperature (ψ∗ = 〈ψ〉 f −(ψ(H )+ψ(−H ))/2�ψ

, ψ being T , θ0, and θ1) and tortuosity coefficient.

constant and independent of the temperature configuration, result that we recover as can be seenin Fig. 9. In the transition region, this result cannot be proven because length scale constraints,necessary for the proof, are no longer valid. A nice result is the fact that the tortuosity coefficientremains independent of the temperature configuration also in the transition region. The value of thetortuosity coefficient is rather weak compared to the fluid thermal diffusivity, which means that theeffective diffusivity of the homogeneous porous region is almost equal to the fluid diffusivity.

Regarding the y-component of the thermal dispersion vector, τyT , it should be null in the

homogeneous porous region, given the geometry and the fact that the y-component of the velocity iszero. This result can be observed in Fig. 10. Thus, the laminar model presented in Fig. 9 correspondsto the solution of Eq. (18), the dispersion being null, the tortuosity and the wall heat flux beingknown and neglecting the turbulent heat flux to highlight the effect of the turbulence in the transitionregion.

For cases 1 and 2, with adiabatic cubes, the mean temperature profile is linear in the homogeneousporous region; for case 3, with heating cubes, the mean temperature profile is parabolic. In the free

fluid region, the mean temperature T is very close to the volume-averaged mean temperature⟨T

⟩ f

except close to the top wall for 1 − 3dp/H < y/H < 1, where the filter support spans beyond thewall. Thus, we add the mean temperature T for y/H > 0.5 in Fig. 9 to show the evolution of the

mean temperature close to the top wall. T and⟨T

⟩ fare indeed equal, except for a very thin region

close to the top wall.Comparing the DNS result and the solution obtained with the laminar model, we see that

the effect of the turbulence is visible deeply inside the porous region, on a region larger than thezone of variation of the porosity, until y/H = −0.5 which corresponds to approximately 4 rows ofcubes. Once again, this effect is particularly important because of the large permeability value ofthe porous structure considered. For case 1 with fixed temperature at wall and adiabatic cubes, the

125110-15 Chandesris et al. Phys. Fluids 25, 125110 (2013)

FIG. 10. Profiles of the heat flux budget and volume averaged rms temperature.

mean temperature profile is strongly skewed in the channel region with the inflexion point locatedat y/H = 0.12. For case 2, with fixed flux at walls and adiabatic cubes, the position of the minimumtemperature is slightly skewed at y/H = 0.44. As shown in Fig. 10, these points correspond to theminimum location of the rms temperature fluctuations.

2. Total heat flux profiles

Fig. 10 presents the different contributions to the total heat flux in the wall-normal direction.The exact total heat flux, obtained from the integration of the averaged energy equations, is alsopresented. There is a slight difference between the two total heat fluxes, which may be causedby very low frequency components of the turbulent fluctuations. This difference can be regardedas a measure of the computation uncertainty. For further studies, we would advise to increase the

125110-16 Chandesris et al. Phys. Fluids 25, 125110 (2013)

FIG. 11. Volume-averaged turbulent heat flux.

sampling time interval, 2000 H/Ub, given the observed convergence time scale of the thermal fields(around 3500 H/Ub).

For the three heating configurations, the dispersion is null. The tortuosity is nonzero in theporous and transition regions. As expected given the low Prandtl number and the low Reynoldsnumber in the porous region, the molecular diffusion dominates the heat transfer in the porousregion and is not negligible compared to the turbulent diffusion in the transition zone. Even in thefree region, the molecular diffusion is not negligible. The turbulent diffusion dominates the heattransfer in the free region and is still important in the upper part of the porous region. It is negligiblefor y/H < −0.5. The Peclet numbers, Pep = Pr Rep = 1 in the porous medium and Peb = Pr Reb

= 550 in the free region, are not large enough to have a negligible diffusive flux compared to theturbulence flux. For cases 1 and 2, the turbulent contribution to the total heat flux is higher at thepermeable wall (y = 0) than at the solid top wall. This result can also be seen in Fig. 11 where thewall-normal turbulent heat flux is presented. For case 3, we cannot observe a similar behavior sincethe total heat flux is null at the top wall.

3. RMS temperature profiles

The volume-averaged root mean square temperature fluctuations defined by ψrms =√⟨

ψ ′2⟩ f

are presented in Fig. 10. As expected for cases 1 and 2, the rms profile presents a local maximumclose to the top wall. This maximum cannot be observed for case 3, since the top wall is adiabatic.For the three heating configurations, the rms profile presents a local maximum inside the porousmedium, whose value is larger than the top wall maximum value. This maximum in the temperaturefluctuation is located deep inside the porous medium, around y/H = −0.4, in a region where the flow

125110-17 Chandesris et al. Phys. Fluids 25, 125110 (2013)

is laminar. This result is surprising at first glance, because the turbulent quantities (turbulent kineticenergy k, rms velocities, etc.) are low at this location.

In fact, this maximum in the rms temperature should not be associated with a high level ofturbulence. As highlighted by Breugem and Boersma30 and recalled in Sec. IV A, the cubes act as afilter for the small-scale fluctuations of velocity. At the same time, large scale fluctuations penetratedeep inside the porous medium, but these fluctuations are inactive in the sense that they are inducedby pressure waves. Thus, they do not contribute to the Reynolds shear stress or to the temperaturemixing, keeping the temperature gradient relatively large. As a consequence, the presence of theselarge scale waves in a relatively high temperature gradient induces large temperature fluctuationsthat are not related to turbulent motion, but to these pressure waves. The amplitude of the pressurewaves decreases inside the porous medium, but at the same time, for the three cases, the temperaturegradient is the largest for −1 ≤ y/H ≤ −0.5, where there is no turbulence mixing. This combinationcan explain why the maximum in the rms temperature is located around y/H = −0.5.

For the rms velocities, such a behavior cannot be observed. Indeed, there is no large velocitygradient inside the porous region because of the drag induced by the solid matrix. The maximum inrms velocities is thus located just above the porous region, around y/H = 0.

4. Turbulent heat flux profiles

The streamwise volume averaged turbulent heat fluxes are presented in Fig. 11. In the porousmedium, the behaviors are identical for the three heating configurations. The values are null atthe bottom wall and decrease to a minimum around y/H = −0.4, which corresponds to the maxi-mum of the rms temperature. The values increase to reach a peak close to the porous interface aty/H = −0.05 and in this zone, the profile’s superposition is impressive. In the free region, the behav-iors are different depending on the applied boundary conditions. For cases 1 and 2, the streamwiseturbulent heat flux is larger and the peak is wider at the permeable wall compared to the solid topwall which is coherent with the existence of large vortices at the permeable wall and elongatedstreaks at the solid top wall. The noteworthy point around y/H = 0.7, where the three profiles arecrossing each other, corresponds to the maximum of the velocity field.

The wall-normal averaged turbulent heat flux is also presented in Fig. 11. As for the streamwisecomponent, the profiles are identical in the porous region for the three heating configurations. Thevalues are null at the wall and then increase to a maximum around −0.05 < y/H < 0. In thefree region, the behaviors depend on the applied boundary conditions. For case 2, the wall-normalaveraged turbulent heat flux vanishes at y/H = 0.5, but the profile is not symmetric. The wall-normalturbulent heat flux is larger at the permeable wall (y = 0) than at the solid top wall.

5. Cross-correlation profiles

The cross-correlation coefficients are used to compare the different turbulence mechanisms. Fora free-porous domain, we define the cross-correlation coefficients as

Ruψ =⟨u′ψ ′⟩

urmsψrms, Rvψ =

⟨v′ψ ′⟩

vrmsψrms, Rvu =

⟨v′u′⟩

vrmsurms,

where ψ corresponds to the temperatures T, θ0, and θ1, respectively. The profiles of the cross-correlation coefficients are presented in Fig. 12. For the heating configuration with imposed tem-peratures at walls, case 1, the profiles of the cross-correlation RuT, RvT , and Ruv are presentedin the first subfigure of Fig. 12. The comparison between the profiles of RvT and Ruv shows animportant difference. It reveals a lack of correlation between the averaged turbulent heat flux

⟨v′T ′⟩

and the averaged Reynolds stress⟨v′u′⟩. On the contrary, the profiles Ruv and RuT have a closer main

behavior.Thus, the Reynolds shear stress and the streamwise turbulent heat flux

⟨u′T ′⟩ seem to be

generated by similar turbulence mechanisms. In order to analyze the similarity, the two quantities⟨u′v′⟩ and

⟨u′T ′⟩ are compared in Fig. 12. The Reynolds stress is scaled such that the maxima

125110-18 Chandesris et al. Phys. Fluids 25, 125110 (2013)

FIG. 12. Profiles of the cross-correlations and comparison of the Reynolds stress and turbulent heat flux.

of the two quantities are equal at the permeable wall. It turns out that the same scaling factor isused for the three cases. The comparison shows an impressive identical decrease between the twostresses just below the permeable wall, for −0.25 ≤ y/H ≤ −0.05. Deeper inside the porous region,the two fluxes no longer scale. The streamwise turbulent heat flux becomes negative whereas theReynolds stress goes to zero which could be explained by the different behavior of the temperatureand normal velocity fluctuations as discussed in Sec. IV B 3: the velocity fluctuations decreaserapidly inside the porous region because of the drag induced by the solid matrix, whereas nosimilar mechanism exists for the thermal fluctuations. To study the coupling between T′ and v′,it could be interesting to study in further works the budgets of the temperature variance and ofthe stream-wise and wall-normal heat fluxes. For the two other heating configurations, cases 2and 3, the profiles are presented in Figs. 12(c)–12(f). It leads to the same observations as forcase 1.

125110-19 Chandesris et al. Phys. Fluids 25, 125110 (2013)

FIG. 13. Profile of the volume averaged thermal turbulent diffusivity.

6. Macroscopic turbulent diffusivity profile

Macroscopic turbulent diffusivities are often modeled using a gradient diffusion hypothesis thatrelates the turbulent heat flux − ⟨

v′T ′⟩ to the gradient of the mean temperature via a macroscopicturbulent diffusivity αtφ as

αtφ = − ⟨v′T ′⟩ f

∂⟨T

⟩ f/∂y

. (25)

Using the DNS results, we compute this quantity for the three heating configurations. In the porousregion and in the transition zone, the same value is obtained as shown in Fig. 13. For the constantwall-heat flux condition, the value of the macroscopic turbulent diffusivity diverges around y/H= 0.44. The volume-averaged turbulent heat flux and the gradient of the volume-averaged meantemperature do not vanish exactly at the same position and the ratio tends to infinity. However, as forthe macroscopic turbulent viscosity, it happens in a small region where both the volume-averagedturbulent heat flux and the gradient of the volume-averaged mean temperature vanish. Except forthat, an identical main behavior is observed for the three cases in the free fluid region, which makesthe use of macroscopic turbulent models based on this hypothesis attractive.

It can be noted that the macroscopic turbulent diffusivity decreases rapidly when entering theporous medium and is rather weak around y/H = −0.4, at the location where the rms temperatureprofiles present a maximum (see Fig. 10). Thus, as already discussed in Sec. IV B 3, even though,large scale temperature fluctuations penetrate inside the porous medium, as is illustrated by therms temperature profiles, these fluctuations do not contribute to the temperature mixing and themacroscopic turbulent diffusivity remains low in this region.

V. CONCLUSION

In this study, a direct numerical simulation of passive turbulent heat transfer in a channel partiallyfilled by a porous medium is performed. The porous medium is made of a Cartesian grid of cubesthat exhibits a relatively high permeability. The energy equation is not solved in the cubes. Sincethe cubes are not connected, the effective thermal diffusivity of the homogeneous porous region isalmost equal to the fluid thermal diffusivity. The simulation is performed for a bulk Reynolds numberReb = 5500 and a Prandtl number Pr = 0.1. The results coming from the DNS are time-averagedquantities which are then volume-averaged on the whole domain using an appropriate weightingfunction, to obtain mean quantities representative of the flow in the porous region.

For the flow, the double-averaged quantities (volume-averaged mean velocity and rms velocity)are compared with the results of Breugem and Boersma.30 The profile comparison shows a goodagreement validating the DNS computation of the turbulent flow. We recover the observations thatlarge vortical structures are created at the permeable wall. They are responsible for momentum

125110-20 Chandesris et al. Phys. Fluids 25, 125110 (2013)

exchange between the free region and the porous region and they induce a strong increase in theReynolds shear stress. The turbulence quantities used in macroscopic k − ε models are computedto get reference results. In particular, the macroscopic turbulent viscosity hypothesis has beenverified. We have also shown that a quadratic shaped weighting function must be used to get rid ofnon-physical subfilter fluctuations.

For the temperature, the large vortical structures created above the porous region are alsoresponsible for an increase of the turbulent heat flux above the porous region compared to the caseof a solid wall. The effect of the large scale structures is visible deeply inside the porous regionuntil y/H = −0.5 which corresponds to approximately 4 rows of cubes, which is a region largerthan the zone of variation of the porosity. The rms temperature profiles present a strong peak deepinside the porous region. However, it is shown that this peak is not related to turbulent mixing, but tolarge scale pressure waves that penetrate deeply inside the porous region, inducing large temperaturefluctuations where the temperature gradient is large, whereas a similar behavior cannot be observedfor velocity fluctuations because of the drag.

Finally, the macroscopic turbulent diffusivity hypothesis has been verified. The same diffusivityhas been obtained for the three heating configurations, in the porous region, in the transition region,and in the free fluid region which is very encouraging for the development of accurate macroscopicheat transfer model in fluid-porous domain.

ACKNOWLEDGMENTS

The computations were performed on the cluster SGI Altix ICE 8200, JADE computer, of theCentre Informatique National de l’Enseignement Superieur (CINES) in Montpellier, France. Theauthors wish to thank M. Phillipe and O. Cioni, who contributed to launch this study, as well asW.-P. Breugem who provided his results.

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