Large eddy simulation of turbulent heat transfer in a non-isothermal channel: Effects of temperature-dependent viscosity and thermal conductivity

International Journal of Thermal Sciences 146 (2019) 106094

Contents lists available at ScienceDirect

International Journal of Thermal Sciences journal homepage: http://www.elsevier.com/locate/ijts

Large eddy simulation of turbulent heat transfer in a non-isothermal channel: Effects of temperature-dependent viscosity and thermal conductivity Lei Wang *, Jian Liu, Safeer Hussain, Bengt Sund�en Division of Heat Transfer, Department of Energy Sciences, Lund University, 22100, Lund, Sweden

A R T I C L E I N F O

A B S T R A C T

Keywords: Turbulent forced convection Variable physical properties Constant gas density Temperature gradient Large eddy simulation

In this work, we perform large eddy simulations (LES) to study the influence of variable viscosity and thermal conductivity on forced convection in a non-isothermal channel flow. To prevent thermal dilatational effect, the gas density is assumed to be constant. The temperature ratio T2/T1 is varied from 1.01 to 2.2, where T2 and T1 are the temperatures of hot and cold walls, respectively. The mean turbulent Reynolds number is kept the same at 395. The results indicated that the mean flow fields are significantly affected by the temperature-dependent fluid properties. Despite the modified velocity and temperature profiles, it is interesting to note that the molecular momentum and heat transport across the channel remain unchanged. Meanwhile, pronounced differences are exhibited for various turbulence statistics such as root-mean-square velocity and temperature fluctuations, Reynolds shear stress, and correlation between streamwise velocity and temperature. Compared with the isothermal flows, it is also found that the presence of the temperature gradient tends to diminish heat transfer. With increasing the temperature ratio, the Nusselt numbers for both sides are reduced. Moreover, the hot side has a higher Nusselt number than the one at the cold side.

1. Introduction Accurate prediction of turbulent heat transfer with variable ther­ mophysical properties is of great importance in engineering and thus has received considerable attention. Typical examples include cooling sys­ tems in the gas turbine, rocket propulsion, supercritical water-cooled reactor, concentrating solar receivers, etc. When heating or cooling rates are high, the fluid properties are temperature-dependent, which, in turn, alters the flow and temperature fields and causes a departure of heat transfer coefficient from the data predicted by the constantproperty correlation, e.g., Dittus-Boelter equation. To account for this departure, Petukhov [1] presented a theoretical analysis of heat transfer to fluids with variable properties. In this work, Petukhov recommended a Sieder-Tate type correlation to calculate Nusselt number for liquids and gases, respectively. Since then, further experimental investigations into the influence of variable properties on heat transfer have been implemented and various equations have been proposed [2,3]. How­ ever, almost all these works were carried out under moderate heating conditions where wall-to-bulk temperature differences were large but not significant. Thus it is difficult to extend the existing knowledge to

circumstances where pronounced variation of fluid properties are encountered. More importantly, the earlier studies focused more on the determination of an empirical factor in the heat transfer correlation than the characterization of the underlying flow and thermal physics. For gas flows involving intensive heating or high temperature drops, significant variations of physical properties are expected. In this case, temperature cannot be considered as a passive scalar anymore because the velocity and thermal fields are strongly coupled. For intensively heated internal gas flows with uniform wall heat flux, turbulent flow may revert to a laminar-like state, causing considerable impairment of heat transfer and significant rise in wall temperature. For gas-cooled nuclear reactors, for example, this would cause a serious safety risk. In light of this, experimental and numerical studies have been extensively performed to determine the onset of criteria for its occurrence and throw insight on the mechanisms by which extreme property variations can lead to laminarization [4–6]. More recently, motivated by the devel­ opment of concentrating solar receivers which are characterized by the strong asymmetric heating manner, turbulent flow in non-isothermal channels has been numerically investigated. In these flows, the trans­ verse temperature gradient was established by having two plates with

* Corresponding author. E-mail address: [email protected] (L. Wang). https://doi.org/10.1016/j.ijthermalsci.2019.106094 Received 15 August 2018; Received in revised form 2 September 2019; Accepted 4 September 2019 Available online 12 September 2019 1290-0729/© 2019 Elsevier Masson SAS. All rights reserved.

L. Wang et al.

International Journal of Thermal Sciences 146 (2019) 106094

incompressible air flow in a plane channel where x-, y-, and z-axis correspond to streamwise, wall-normal, and spanwise direction, respectively. The channel dimensions considered are 2πH � 2H � πH, where H is the half-channel height (H ¼ 1). To solve a fully developed flow, we use a periodic boundary condition in the streamwise and spanwise directions. As a result, a constant pressure gradient along the streamwise direction is required to drive the flow. The constanttemperature thermal boundary condition is applied for the top and bottom walls, where the no-slip condition is used for the fluid velocity. The temperature ratio is T2/T1 ¼ 1.01, 1.3, 1.7, and 2.2. In the following, the subscript 1 refers to the bottom wall (cold) and the subscript 2 represents the top wall (hot). In LES, the momentum and energy equations are averaged by a spatial filtering procedure. The filtering operation is denoted by (~). After filtering and introducing models for subgrid-scale (SGS) stress tensor and scalar flux vector, the following set of time-dependent vari­ able-property equations is obtained:

Fig. 1. Schematic of channel flow configuration.

high temperature at the top wall and low temperature at the bottom wall. The temperature difference between the hot and cold plate is high enough so that the gas physical properties undergo dramatic variation inside the flow passage. Due to the inherent coupling between mo­ mentum and energy equations, these flows deviate significantly from those under the isothermal condition, even in the forced convection regime. Wang and Pletcher [7] performed large eddy simulation (LES) of a planar channel flow using temperature ratios of 1.02 and 3. The results indicated that the higher temperature ratio has a more impact on the velocity and temperature fluctuations as well as the velocity-temperature correlation. The direct numerical simulation (DNS) results of Nicoud [8] and Toutant and Bataille [9] indicated that the mean and turbulent profiles are asymmetric even a semi-local scaling was used. The LES study of Serra et al. [10] showed that the heat transfer coefficient increases at the cold side and decreases at the hot side as the temperature ratio increases. Serra et al. [11] and Yahya et al. [12] demonstrated that turbulence transfer is strongly affected by the high temperature gradient because they found that the 5/3 Kol­ mogorov scaling law is not valid. The recent studies [13,14] showed that the high temperature gradient can be considered as a strong external agency in modifying the turbulence production and transfer. It should be pointed out that, in the above-stated studies, the gas density is greatly dependent on the temperature. Thus the pronounced dilatational effect owing to the variable density is inevitably mingled with the effect of variable viscosity and thermal conductivity. To assess the isolated in­ fluence produced by the variable viscosity and thermal conductivity on the forced convection, it is necessary to carry out simulations in which the dynamic effect caused by the gas dilatation is assumed to be vanishing. The main objective of this study is to explore the effects of variable viscosity and thermal conductivity on the turbulent heat transport in a non-isothermal channel with different temperature ratios. To provide detailed information of flow dynamics and thermal structures, the approach of LES is adopted. In the simulation, the viscosity and thermal conductivity are dependent on the temperature variation, while the gas density is assumed to be constant. We remark here that the variableproperty algorithm employed in this work is different from the recent DNS studies of Lee et al. [15,16], and Zonta et al. [17], in which liquids were selected as the working medium, thus only dynamic viscosity varies considerably with temperature; all the other physical properties depend on temperature rather weakly. In the present study, the tem­ perature ratio between the hot wall and cold wall is varied from 1.01 to 2.2, which is large enough to influence the gas physical properties. The mean turbulent Reynolds number is kept at 395. The Prandtl number is Pr ¼ 0.71. Sutherland law is employed to obtain the temperature-dependent viscosity. The governing equation, numerical method, and validation are described in Section 2. Results are presented and discussed in Section 3. Finally, the conclusions are given in Section 4.

∂~ui ¼0 ∂xi

(2.1) �

~ ∂~ui ∂ ~ui ~uj ∂P ∂ ¼ þ þ ∂t ∂xj ∂xi ∂xj ~ ui Þ ∂T~ ∂ðT~ ∂ ¼ þ ∂xi ∂t ∂xi

�

�

α þ αsgs

v þ νsgs

� � ∂~ ui ∂~ uj þ ∂xj ∂xi

� �� � ∂T~ ∂xi

uk 2 ∂~ δij 3 ∂xk

�� þ Fδi1

(2.2) (2.3)

where ui is the ith component of the velocity vector, T is the tempera­ ture, P is the fluctuating kinematic pressure, F is the driving pressure gradient, ν and α are kinematic viscosity and thermal diffusivity, respectively. To account for the temperature-dependent properties, we use Sutherland law to calculate viscosity which is linked to the thermal diffusivity via Prandtl number

νðTÞ ¼ 1:461⋅10 α¼

6

T 1:5 T þ 111

ν

(2.4) (2.5)

Pr

where Pr ¼ 0.71. Sutherland’s law is typically used for air and gives fairly accurate results with an error less than a few percent over a wide range of temperatures [18]. In the simulation, we assume that the fluid density and heat capacity are constant (ρ ¼ 1 kg m 3, cp ¼ 1005 J kg 1 K 1). The SGS viscosity νsgs is calculated by means of Smagorinsky model [19]: qffiffiffiffiffiffiffiffiffiffiffi (2.6) νsgs ¼ ðCs ΔÞ2 2S~ij ~Sij where Cs ¼ 0:167, Δ is the grid size Δ ¼ ðΔx Δy Δz Þ1=3 , where Δx, Δy, and � ~ij ¼ 1 ∂u~i þ Δz are the mesh sizes along the corresponding axes, and S 2 ∂ xj � ∂u~j ∂xi . However, the standard Smagorinsky model results in the non-zero value of νsgs at the solid boundary. This defect can be avoided by introducing a van-Driest wall damping function [20] into the length scale Cs Δ, f ¼1

expð

yþ =Aþ Þ

(2.7)

where Aþ is taken as 26. The turbulent heat flux is modeled with a gradient transport model [21], with αsgs ¼ νsgs =Prt , where the turbulent Prandtl number is set equal to Prt ¼ 0.85. We carried out LES with an open source OpenFOAM solver. Herein we limit ourselves to a summary of the numerical method. The gov­ erning equations are discretized on a collocated grid system. To avoid spurious pressure-velocity coupling, the Rhie-Chow interpolation is

2. Governing equations and numerical method With reference to the schematics of Fig. 1, we consider an 2

L. Wang et al.

International Journal of Thermal Sciences 146 (2019) 106094

Table 1 Turbulent Reynolds numbers, friction velocity, and viscosity. Case 1 2 3 4

T2/T1 1.01 1.3 1.7 2.2

Reτ1 396 428 460 488

Reτ2 396 366 330 294

Reτm

uτ1

396 397 395 391

uτ2

0.00791 0.00786 0.00772 0.00759

adopted [22]. Time integration is implemented by a second-order im­ plicit scheme. The convective terms for the momentum and energy equations are discretized with a second-order Gauss limited linear scheme [23]. Poisson equation is solved for the pressure following both the predictor and corrector steps to satisfy the continuity equation at each step. Once the velocity field is known, the temperature field is computed as a solution of the energy equation and finally viscosity is updated using Sutherland law. The computation domain contains 102 � 129 � 128 grid points. The mesh is non-uniform in the wall-normal direction and uniform in the streamwise and spanwise directions. The grid points in the y direction are determined by a hyperbolic tangent transformation [24]. �� � � �� � � 1 2k 1 yk ¼ H 1 þ tanh tanh 1 ðaÞ ; k 2 1; Ny (2.8) 1þ a Ny

0.00792 0.00804 0.00815 0.00825

ν1

ν2 5

2.0 � 10 1.837 � 10 1.681 � 10 1.554 � 10

5 5 5

2.0 � 10 5 2.195 � 10 2.470 � 10 2.806 � 10

5 5 5

where qw is the wall heat flux. The profiles on the cold side are normalized using uτ1 and Tτ1, and the profiles on the hot side are normalized using uτ2 and Tτ2, respectively, which are symbolized by a superscript ‘þ’. Moreover, yþ is defined as yþ ¼

uτ y

νw

(2.11)

The turbulent Reynolds number Reτ is based on the uτ and the vis­ cosity at the wall: Reτ ¼

uτ H

νw

(2.12)

Reτm is the average value between the hot side and cold side (Reτm ¼ Table 1 shows the turbulent Reynolds number, friction veloc­ ity, and viscosity for different temperature ratios. At the hot side, the increased viscosity decreases the turbulent Reynolds number Reτ2 and tends to laminarize the flow if its value is not high enough [12,25]. In the present study, the flow remains turbulent for all the considered cases. The mean turbulent Reynolds number is kept nearly the same (Reτm � 395). For the sake of convenient comparison, the case of T2/T1 ¼ 1.01 is set as benchmark in which the physical properties are constant, while variable properties are used at higher temperature ratios. In the fully developed flow, as shown in Fig. 1, the heat transferred out of the hot wall is removed from the channel through the cold wall, thus there is no change in the bulk temperature. To distinguish the heat transfer capacity between the two sides, the hot and cold bulk temper­ atures are calculated separately, RH R 2H U⋅Tdy U⋅Tdy Tb1 ¼ R0 H (2.13) ; Tb2 ¼ RH 2H Udy Udy 0 H Reτ1 þReτ2 ). 2

where Ny ¼ 129 is the number of grid points on the y axis, and a is an adjustable parameter depending on the mesh dilatation. In our case, þ þ a ¼ 0.96135. The dimensionless mesh sizes are Δþ x ¼ 25, Δz ¼ 10, Δy ¼ 0:5 at the wall to 12.5 at the center. Here, the superscript þ indicates the quantities normalized by the wall variables, which are described in the following. For the characterization of our simulations, the classic wall scaling is used. In particular, the friction velocity uτ is solved with the near-wall mean velocity gradient, ffiffiffiffiffiffiffiffiffiffi� ffiffiffiffiffi rffiffiffiffi sffiffiffiffiffi� τw ∂U uτ ¼ (2.9) ¼ νw ρ ∂y w

where the subscript w refers to wall quantities: w → 1 on the bottom side and w → 2 on the top side, τw is the wall shear stress. The friction temperature Tτ is defined as � � qw νw ∂T 1 Tτ ¼ ¼ (2.10) ρCP uτ Pr ∂y w uτ

The Nusselt number for the hot and cold sides is thus calculated based on the respective bulk temperature and the half-height of the channel,

Fig. 2. Validation with the DNS data of Moser et al. [26]. (a) Streamwise mean velocity. (b) Reynolds shear stress. 3

L. Wang et al.

International Journal of Thermal Sciences 146 (2019) 106094

profile plotted in semi-logarithmic coordinates. The LES results of Serra et al. [10] are presented here for the sake of cross comparison. Our simulations slightly overestimate the mean velocities in the log region. This is probably due to the insufficient spatial resolution in the spanwise direction. As shown in Ref. [28], Δþ z ¼ 10 is the minimum requirement to perform a proper simulation with LES approach. In Fig. 2b, the pro­ files of Reynolds shear stress are plotted. To investigate the effect of grid size on the turbulent quantities, the LES results of Mukha and Liefven­ dahl [29] are also presented. In Ref. [29], the cell numbers along the axes were 240 � 200 � 180, which are very close to the resolution (256 � 193 � 192) used in the DNS study [26]. It is noted that all the LES results under-predict the values of the shear stress in the near-wall re­ gion. Finer meshes would reduce the disagreement, but they cannot completely removed it. One can speculate that such under-predictions of shear stress are caused by the SGS model which does not take into ac­ count the energy transfer from small scales to the large ones, i.e., the back scatter. In Ref. [29] and the present study, the SGS models are absolutely dissipative, thus only the energy transfer from the large to the small scales (forward scatter) can be accounted for. Piomelli et al. [30] showed that, in the wall-bounded turbulent flow, the region with high Reynolds shear stress correlates fairly well with the strong backward and forward scatter events. Therefore, the accurate modelling of back scatter is desirable in turbulent flows. This issue can be investigated in the future studies. Fig. 3 shows the profiles of mean and fluctuation tem­ perature in a non-isothermal channel. Compared against with Seki’s DNS data, the essential feature of scalar quantities is captured. The mean temperature profiles match well close to the wall. However, a slight overestimation is seen in the central part of the channel. Due to the presence of temperature gradient, the root-mean-square temperature fluctuation Trms exhibits a local maximum at the channel center. Our simulations overestimate the inner peak close to the wall and underes­ timate the outer peak. Nevertheless, the discrepancies are about 10%. Finally, we calculated the Nusselt number to give an indication of the integral parameter. Based on the bulk velocity (not shown here) and the whole channel height, we obtain Nu ¼ 31.4, which is 7% lower than the one predicted by the Petukhov correlation [1]. Overall, a fair agreement is obtained between our results and the reference database.

Fig. 3. Validation with the DNS data of Seki et al. [27].

� � k1 Nu1 ¼

∂T ∂y

� � H

1

kb1 ðTb1

k2

∂T ∂y

; Nu2 ¼ kb2 ðT2 T1 Þ

H 2

Tb2 Þ

(2.14)

where kb1 and kb2 thermal conductivities which are evaluated by the bulk temperature Tb1 and Tb2, respectively. Prior to collecting data, each case has been run for approximately 120 diffusion time units (H/uτ) to reach a statistically steady state which is identified by the convergence of the root-mean-square velocity and temperature fluctuations. Subsequently, data are collected over roughly 40 diffusion time units for velocity and temperature fields. Finally, the mean scalar quantities and various statistical moments are computed by integrating in time and in the homogenous x and z directions (i.e., the Reynolds average). 2.1. Validation We performed an extensive validation against DNS database avail­ able for the case of isothermal channel flow provided by Moser et al. [26] and non-isothermal channel flow with constant physical properties by Seki et al. [27] at Reτ ¼ 395. Fig. 2a shows the streamwise velocity

Fig. 4. The mean profiles in global coordinates. (a) Velocity. (b) Temperature. 4

L. Wang et al.

International Journal of Thermal Sciences 146 (2019) 106094

Fig. 5. Mean velocity profiles in wall units. (a) Cold side. (b) Hot side.

3. Results and discussions

side. The dissymmetrical velocity distributions were also reported in the previous studies [7–10]. However, in their studies, the impact of the thermal dilatation was important. Zonta et al. [17] run a DNS to study the behavior of incompressible Newtonian fluids with temperature-dependent viscosity in forced convection. They also found the dissymmetrical velocity profiles. This suggests that the mean flow variables can be modified by the variations of fluid viscosity, even though its density is kept constant. In Fig. 4b, the mean temperature normalized by the temperature difference (T2 – T1) is plotted. As apparent from Fig. 4b, the mean temperature profiles become dissym­ metrical and shift upward as the temperature ratio rises. Due to the stratification of thermal conductivity, the temperature gradient at the cold side is higher than the one at the hot side. This is also in agreement with previous observations [7,11,12]. In Fig. 5, the mean velocity profiles normalized by the wall units are presented. Since the friction velocities are not equal at the cold and hot walls, as shown in Table 1, the profiles at the cold and hot side of the

In this section, we illustrate the influence of the variable viscosity and thermal conductivity on the mean flow variables, statistical prop­ erties, and heat transfer characteristics in the non-isothermal channel flow under different temperature ratios. 3.1. Mean flow variables The mean dimensionless velocity and temperature profiles in global coordinates are presented in Fig. 4. In Fig. 4a, the streamwise velocity is normalized using the maximum velocity and plotted against y/H. For the case of T2/T1 ¼ 1.01, the velocity distribution is symmetrical. With increasing the temperature ratios, the symmetry is gradually lost. On the cold side where the viscosity decreases with temperature, the velocity profiles are seen to overshoot the benchmark. The largest deviation from the reference is the case of T2/T1 ¼ 2.2. The effect is reversed at the hot

Fig. 6. Molecular transport. (a) Viscous shear stress. (b) Conductive heat flux. 5

L. Wang et al.

International Journal of Thermal Sciences 146 (2019) 106094

Fig. 7. Reynolds shear stress. (a) Cold side. (b) Hot side.

channel are plotted respectively. It is found that the velocity profiles collapse well in the inner-wall region (yþ < 10) for both sides. However, further away from the wall, there is a noticeable divergence. At the cold side, the velocity profiles shift upward from the benchmark case with increasing the temperature ratio. At the hot side, the velocity profiles are found to collapse better, albeit a visible difference still exists. This suggests that the validity of the classic wall scaling seems to be in question when the fluid properties undergo the large variations. Similar results have been reported in previous studies on various flow configu­ rations. In the compressible flow with strong heat transfer, Wang and Pletcher [7] employed the van Driest effective velocity transformation to collapse the data. Whereas the re-scaled velocity profiles become closer, the fundamental disagreement cannot be completely removed. In studying the wall heating effect on turbulent boundary layer, Lee et al. [15] modified the inner length scale, lν ¼ νðyÞ=uτ , based on the local mean viscosity which is temperature-dependent and the wall friction velocity. It was found that the velocity profiles were shifted upward from the constant state with the same inclination angle. These findings imply the variable fluid properties not only change the mean flow

variables, but also the turbulent flow characteristics. This will be visited later. Since the viscosity and thermal conductivity are temperaturedependent, it is of interest to investigate whether the molecular mo­ mentum and heat transport are also modified by the presence of tem­ perature gradient within the flow. Fig. 6 presents the local distribution of viscous shear stress (τvis ¼ νðyÞ dU dy ) and conductive heat flux (qcon ¼

kðyÞ dT dy ) across the channel, which are scaled by the wall quantities τw

and qw, respectively. Most strikingly, it is noted that various profiles collapse in a perfect manner, indicating that the molecular transport for momentum and heat is independent of the variations of fluid properties. Moreover, the profiles are symmetric with respect to the geometric channel centerline, suggesting that the manner of wall heating or cooling has little influence on the molecular transport. This is incon­ sistent with the results of Wang and Pletcher [7]. They found that the molecular heat conduction in the cold wall region was suppressed at high temperature ratio, which was attributed to the increase of mean density due to fluid cooling. Further inspection of Fig. 6 shows that the molecular transport loses its importance in the central part of the

Fig. 8. Wall-normal turbulent heat flux. (a) Cold side. (b) Hot side. 6

L. Wang et al.

International Journal of Thermal Sciences 146 (2019) 106094

Fig. 9. Streamwise turbulent heat flux. (a) Cold side. (b) Hot side.

Fig. 10. Cross-correlation coefficients. (a) Ruv. (b) RvT.

channel. The viscous shear stresses vanish or nearly so, while the conductive heat fluxes still exist but have a small value (about 0.05).

good manner, indicating that the turbulence-driven heat transport from the hot side to the cold side is almost unaffected by the variable physical �> � remains almost properties. In the central part of the channel, < vT constant at about 0.95. In this region, the turbulent heat transport is dominant over the molecular one (qcond � 0.05). Note that the sum of molecular and turbulent heat fluxes should be equal to 1 (or nearly so). This is required by the fully developed condition because the wall heat fluxes between the two sides are equal. Fig. 9 plots the profiles of � >. � It is seen that the profiles collapse streamwise turbulent heat flux < uT well in the wall region (yþ < 30). Further away from the wall, the pro­ files shift upward from the benchmark case at the cold side, while they shift downward at the hot side. It is noted that a similar trend is also found in the Reynolds shear stress, as shown in Fig. 7. In the following, the cross-correlation coefficient, which is defined as �> � =φrms ϕrms , where φ� and ϕ� are fluctuation quantities at a Rφϕ ¼ < φϕ

3.2. Statistical properties In this section, the effect of variable viscosity and thermal conduc­ tivity on the turbulent statistical properties is illustrated. Fig. 7 shows � at the cold and hot sides, the profiles of Reynolds shear stress < u�v> which are normalized by u2τ . Unlike the molecular shear stress, the turbulent momentum transport is significantly modified. At the cold side, profiles collapse well close to the wall (yþ < 50). Further away from the wall, the profiles with higher temperature ratios shift upward from the case of constant properties. The situation is different at the hot side, especially in the central region (yþ > 50) where the profiles shift downward as the temperature ratios increases. Fig. 8 presents the wall�> � at the cold and hot sides, normal turbulent heat fluxes < vT respectively, which are normalized by uτ Tτ . The profiles agree in a fairly

single point, is calculated. In Fig. 10a, Ruv is plotted to give an indication of the correlation between the streamwise and wall-normal velocity

7

L. Wang et al.

International Journal of Thermal Sciences 146 (2019) 106094

Fig. 11. Root-mean-square velocity fluctuations. (a) Cold side. (b) Hot side.

fluctuations. With increasing the temperature ratio, it is clear that the profiles become asymmetrical with respect to the centerline. Overall, the correlation coefficient of the hot side is higher than the one of the cold side. In Fig. 10b, RvT is plotted. Different from Ruv, the profiles of RvT are

more symmetrical. The profiles collapse fairly well in the central region, however, divergence is found close to the wall. To inspect the impact of fluid properties on the turbulence in­ tensities, the root-mean-square streamwise, wall-normal, and spanwise 8

L. Wang et al.

International Journal of Thermal Sciences 146 (2019) 106094

294 at the hot side. Abe et al. [31] studied a fully developed turbulent channel flow with Reτ ¼ 180, 395, and 640. It was observed that the turbulence intensities, Reynolds stresses, and their budget terms, normalized by the wall units, are strongly affected by the Reynolds number. In the next, attention is given to the temperature fluctuations. Fig. 13 plots the profiles of Trms. For both sides, the profiles are nearly unchanged in the wall region (yþ < 50). In the central part, however, the levels of temperature fluctuations at the cold side are subdued, while they are enhanced at the hot side. This is expected because the sup­ pressed levels of velocity fluctuations at the hot side tend to reduce the turbulent mixing, thus the temperature is less homogeneous and the magnitudes of Trms are higher. Finally, to shed a more light on the temperature fluctuations, we calculate the production term for the �> � ∂∂Ty , which is normalized by temperature variance, that is, Pθ ¼ < vT u2τ T 2τ ν 1 and plotted in Fig. 14. With increasing the temperature ratio, it is found that the Pθ increases at the hot side and reduces at the cold side. This is consistent with the temperature fluctuations observed in Fig. 13. 3.3. Heat transfer characteristics

Fig. 12. Turbulent kinetic energy at cold and hot sides for T2/T1 ¼ 2.2.

Another important concern is the effect of temperature ratios on heat transfer. Based on the simulated velocity and temperature fields, the Nusselt numbers for both sides are calculated, as shown in Table 2. Also included in this table are the wall heat fluxes. As stated above, the heat input from the hot wall should be always balanced by the heat removal from the cold one in the fully developed flow under consideration. Thus it is important to check the heat balance before we present the heat transfer coefficients. For all the tested cases, it is found that the dis­ crepancies of wall heat flux between the cold and hot sides are less than 2%, indicating that the heat balance has been achieved. Compared with the benchmark case in the isothermal flow where Nu0 ¼ 15.7, which is also based on the half-height of the channel, Table 2 illustrated that the presence of the transverse temperature gradient tends to reduce the heat transfer on both sides. A fitting equation of the Nusselt number for the cold and hot side, respectively, is given by

velocity fluctuations is presented in Fig. 11. We again observe that the profiles become dissymmetrical when the temperature ratio rises. On the cold side, the levels of velocity fluctuations are enhanced. On the hot side, on the contrary, their levels are suppressed. In particular, vrms and wrms are more sensitive to the wall heating than cooling. Meanwhile, it is found that the peak location of the hot side is closer to the wall than the one of the cold side. Qualitatively, these findings are in agreement with previous observations [9,10]. In particular, Toutant and Bataille [9] used the semi-local scaling to collapse the data so that the influence of the variable density and viscosity can be taken into account. However, the fluctuation profiles of the hot and cold sides do not collapse. This suggests that the fundamental nature of turbulence in forced convection is altered by the variable physical properties. Fig. 12 plots the turbulent kinetic energy (TKE), Ek ¼ < u�i u�i > =2, normalized by u2τ , for the cold and hot sides. For the sake of clarity, only the case of T2/T1 ¼ 2.2 is selected. It is found that the turbulence intensities at the hot side is significantly suppressed by the temperature gradient. Two physical mechanisms may be responsible for this behavior. First, the increased viscosity at the hot side tends to dissipate TKE; second, the Reynolds number effect becomes more pronounced. Note that the turbulent Reynolds number at the cold side is 488 which is much higher than

Nu1 =Nu0 ¼ ðT2 =T1 Þ

0:282

(3.1)

Nu2 =Nu0 ¼ ðT2 =T1 Þ

0:156

(3.2)

Further inspection of Table 2 indicated that the hot side has a higher Nusselt number than the cold side for all the tested cases. This finding is contrary to the observation of Serra et al. [10]. In their studies, they

Fig. 13. Root-mean-square temperature fluctuations. (a) Cold side. (b) Hot side. 9

L. Wang et al.

International Journal of Thermal Sciences 146 (2019) 106094

transfer on both sides. In particular, the hot side has a higher Nusselt number than the one at the cold side. Acknowledgement The authors would like to acknowledge the support of Swedish En­ ergy Agency. They are also thankful to the Swedish National Infra­ structure for Computing (SNIC) at HPC2N and LUNARC for providing computation resources. Nomenclature

Δ

turbulent kinetic energy [m2 s 2] half-height of channel, [m] thermal conductivity [W m 1 K 1] Nusselt number molecular Prandtl number turbulent Prandtl number production term for temperature variance [K2 s 1] wall heat flux [W m 2] turbulent Reynolds number mean turbulent Reynolds number between the hot and cold sides correlation coefficient between streamwise and wall-normal velocity fluctuations correlation coefficient between wall-normal velocity and temperature fluctuations mean temperature [K] temperature fluctuation [K] friction temperature [K] friction velocity [m s 1] velocity fluctuation in x-y-, z-direction, respectively [m s 1] mean streamwise velocity [m s 1] streamwise direction [m] wall-normal direction [m] spanwise direction [m] viscosity [m2 s 1] thermal diffusivity [m2 s 1] wall shear stress [m2 s 2] mesh size [m]

Subscript 1 2 b cond rms sgs vis w

cold wall hot wall bulk conductive root mean square subgrid-scale viscous wall

Ek H k Nu Pr Prt Pθ qw Reτ Reτm

Fig. 14. Production of temperature variance.

Ruv

Table 2 Nusselt number at hot and cold sides, and wall heat fluxes. Case

Nu1

Nu2

qw1

qw2

T2/T1 ¼ 1.3 T2/T1 ¼ 1.7 T2/T1 ¼ 2.2

14.69 13.79 12.66

14.92 14.53 13.74

9.05 22.47 39.68

8.89 22.36 40.17

RvT T T0 Tτ uτ u0 , v0 , w0 U x y z

stated that the heat transfer coefficient at the cold side (25 WK 1m 2) exceeds the hot side (16 WK 1m 2) under the condition of T2/T1 ¼ 2. We suppose that such a discrepancy can be attributed to the pronounced thermal dilatational effect in Ref. [10] which probably modifies the nature of forced convection. Anyway, this is an interesting issue and will be investigated in the future studies.

ν α τw

4. Conclusions We performed a series of LES simulations to investigate the turbulent heat transfer in a non-isothermal flow where the fluid viscosity and thermal conductivity undergo large variations. To prevent the complex involvement of dilatational effect caused by the temperature-dependent gas density, we carried out simulations based on the incompressible Navier-Stokes equations. In this study, the turbulent Reynolds number is kept constant at 395, and the temperature ratio is varied from 1.01 to 2.2. The molecular Prandtl number Pr ¼ 0.71. Sutherland law is employed to calculate the temperature-dependent viscosity. Despite the absence of the dilatational effect, the simulations show that the forced convection behavior is significantly modified by the variable viscosity and thermal conductivity. With increasing the tem­ perature ratio, the mean velocity and temperature profiles from the both sides lose symmetry. Even with the aid of the wall unit scaling, the mean streamwise velocity profiles do not collapse. It is also noted that the molecular transport of momentum and heat across the channel remains unchanged. Meanwhile, various turbulence statistics exhibits pro­ nounced differences. The velocity fluctuations, Reynolds shear stress as well as the streamwise turbulent heat flux are enhanced at the cold side, while they are dampened at the hot side. With respect to the scalar statistics, on the contrary, the hot side is seen to intensify the levels of temperature fluctuations. This is mainly due to the dampened velocity fluctuations on the hot side which retard the turbulent mixing. Finally, we examine the heat transfer characteristics under different temperature ratios. Compared with the isothermal flow, the results reveal that the presence of the transverse temperature gradient tends to reduce heat

Subscript þ quantities normalized by wall variables References [1] B.S. Petukhov, Heat transfer and friction in turbulent pipe flow with variable physical properties, Adv. Heat Transf. 6 (1970) 503–564. [2] C.A. Sleicher, M.W. Rouse, A convenient correlation for heat transfer to constant and variable property fluids in turbulent pipe flow, Int. J. Heat Mass Transf. 18 (1975) 677–683. [3] O. Buyukalaka, J.D. Jackson, The correction to take account of variable property effects on turbulent forced convection to water in a pipe, Int. Heat Mass Transf. 41 (1998) 665–669. [4] A.M. Shehata, D.M. McEligot, Mean structure in the viscous layer of stronglyheated internal gas flows. measurements, Int. J. Heat Mass Transf. 41 (1998) 4297–4313. [5] D.P. Mikielewicz, A.M. Shehata, J.D. Jackson, D.M. McEligot, Temperature, velocity and mean turbulence structure in strongly heated internal gas flows.

10

L. Wang et al.

[6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]

International Journal of Thermal Sciences 146 (2019) 106094 [20] E.R. van Driest, On the turbulent flow near a wall, J. Aeronaut. Sci. 23 (1956) 1007–1011. [21] P. Moin, K. Squires, W. Cabot, S. Lee, A dynamic subgrid-scale closure method for compressible turbulence and scalar transport, Phys. Fluids 3 (1991) 2746–2757. [22] C.M. Rhie, W.L. Chow, A numerical study of the turbulent flow past an airfoil with trailing edge separation, AIAA J. 21 (1983) 1525–1532. [23] P.K. Sweby, High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM J. Numer. Anal. 21 (1984) 995–1011. [24] P. Moin, J. Kim, Numerical investigation of turbulent channel flow, J. Fluid Mech. 118 (1992) 341–377. [25] B. Lessani, M. Papalexanderis, Time-accurate calculation of variable density flows with strong temperature gradients and combustion, J. Comput. Phys. 212 (2006) 218–246. [26] R.D. Moser, J. Kim, N.N. Mansour, Direct numerical simulation of turbulent channel flow up to Reτ ¼ 590, Phys. Fluids 11 (1999) 943–945. [27] Y. Seki, H. Abe, H. Kawamura, in: DNS of Turbulent Heat Transfer in a Channel Flow with Different Thermal Boundary Conditions, the 6th ASME-JSME Thermal Engineering Joint Conference, March 16-20, 2003. TED-AJ03-226. [28] F. Kremer, C. Bogey, C. Bailly, Investigation of turbulent channel flow using large eddy simulation, in: Proc. Acoustics Conference, 23-27 April, 2012 (Nantes, France). [29] U. Piomelli, W.H. Cabot, P. Moin, S. Lee, Subgrid-scale backscatter in turbulent and transitional flows, Phys. Fluids A 3 (1991) 1766–1771. [30] T. Mukha, M. Liefvendahl, Large-eddy Simulation of Turbulent Channel Flow, Technical report 2015-014, Uppsala University, 2015. [31] H. Abe, H. Kawamura, Y. Matsuo, Direct numerical simulation of a fully developed turbulent channel flow with respect to the Reynolds number dependence, ASME J. Fluids Eng. 123 (2001) 382–393.

Comparison of numerical predictions with data, Int. J. Heat Mass Transf. 45 (2002) 4333–4352. J.H. Bae, J.Y. Yoo, H. Choi, D.M. McEligot, Effects of large density variation on strongly heated internal air flow, Phys. Fluids 18 (2006), 075102. W.P. Wang, R.H. Pletcher, On the large eddy simulation of a turbulent channel flow with significant heat transfer, Phys. Fluids 8 (1996) 3354–3366. F.C. Nicoud, Numerical Study of a Channel Flow with Variable Properties, Center for Turbulence Research, Annual Research Briefs, 1998, pp. 289–310. A. Toutant, F. Bataille, Turbulence statistics in a fully developed channel flow submitted to a high temperature gradient, Int. J. Therm. Sci. 74 (2013) 104–118. S. Serra, A. Toutant, F. Bataille, Thermal large eddy simulation in a very simplified geometry of a solar receiver, Heat Transf. Eng. 33 (2012) 505–524. S. Serra, A. Toutant, F. Bataille, Y. Zhou, High-temperature gradient effect on a turbulent channel flow using thermal large-eddy simulation in physical and spectral spaces, J. Turbul. 13 (2012) 1–25. S.M. Yahya, S.F. Anwer, S. Sanghi, Turbulent forced convective flow in an anisothermal channel, Int. J. Therm. Sci. 88 (2015) 84–95. F. Aulery, A. Toutant, F. Bataille, Y. Zhou, Energy transfer process of anisothermal wall-bounded flows, Phys. Lett. A 379 (2015) 1520–1526. F. Aulery, D. Dupuy, A. Toutant, F. Bataille, Y. Zhou, Spectral analysis of turbulence in anisothermal channel flows, Comput. Fluids 151 (2017) 115–131. J. Lee, S.Y. Jung, J.J. Sund, T.A. Zaki, Effect of wall heating on turbulent boundary layers with temperature-dependent viscosity, J. Fluid Mech. 726 (2013) 196–225. J. Lee, S.Y. Jung, J.J. Sund, T.A. Zaki, Turbulent thermal boundary layers with temperature-dependent viscosity, Int. J. Heat Fluid Flow 49 (2014) 43–52. F. Zonta, C. Marchioli, A. Soldati, Modulation of turbulence in forced convection by temperature-dependent viscosity, J. Fluid Mech. 697 (2012) 150–174. F.M. White, Viscous Fluid Flow, 3d ed., McGraw-Hill, New York, 2005. J. Smargorinsky, General circulation experiments with the primitive equations, Mon. Weather Rev. 91 (1963) 99–164.

11