Direct numerical simulation of turbulent heat transfer in a wall-normal rotating channel flow

International Journal of Heat and Fluid Flow 80 (2019) 108480

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Direct numerical simulation of turbulent heat transfer in a wall-normal rotating channel flow Cale Bergmanna,b, Bing-Chen Wang a b

T

⁎,b

Defence R&D Canada, Suffield Research Centre, P. O. Box 4000, Medicine Hat, AB T1A 8K6, Canada Department of Mechanical Engineering, University of Manitoba, Winnipeg, MB R3T 5V6, Canada

ARTICLE INFO

ABSTRACT

Keywords: Rotating flow Turbulence Heat transfer Channel flow Direct numerical simulation

Investigations into the characteristics of turbulent heat transfer and coherent flow structures in a plane-channel subjected to wall-normal system rotation are conducted using direct numerical simulation (DNS). In order to investigate the influence of system rotation on the temperature field, a wide range of rotation numbers are tested, with the flow pattern transitioning from being fully turbulent to being quasilaminar, and eventually, fully laminar. In response to the Coriolis force, secondary flows appear as large vortical structures, which interact intensely with the wall shear layers and have a significant impact on the distribution of turbulence kinetic energy (TKE), turbulence scalar energy (TSE), temperature statistics, and turbulent heat fluxes. The characteristic length scales of turbulence structures responsible for the transport of TSE are the largest at the quasilaminar state, which demands a very large computational domain in order to capture the two-dimensional spectra of temperature fluctuations. The effects of the Coriolis force on the turbulent transport processes of the temperature variance and turbulent heat fluxes are thoroughly examined in terms of their respective budget balances.

1. Introduction In literature, pressure-driven turbulent plane-channel flows subjected to spanwise and streamwise system rotation have been studied extensively using both experimental and numerical methods. However, the number of detailed studies of wall-normal rotating flows is still limited in the current literature, and there are even fewer studies involving turbulent heat transfer. These flows exist in a wide variety of industrial and geophysical applications, including gas-turbine engines, centrifugal pumps, and Earth’s planetary atmospheric boundary layer. A deeper understanding of the physical mechanisms underlying turbulent heat transfer in a wall-normal rotating plane-channel flow requires a systematic analysis of the heat and fluid flow in both physical and spectral spaces. In rotating plane-channel flows, two Coriolis force components act on the fluid, both of which act in directions perpendicular to the rotation axis. For a spanwise-rotating flow, one component acts in the streamwise direction and one component in the wall-normal direction; for a streamwise-rotating flow, one component acts in the spanwise direction and one component acts in the wall-normal direction; and for a wall-normal rotating flow, one component acts in the spanwise direction and one component acts in the streamwise direction. It is because of these differences in the directions of the Coriolis force ⁎

components that the physics underlying streamwise-rotating, spanwiserotating, and wall-normal rotating flows are considerably different from each other. These differences in the flow patterns further lead to different mechanisms in turbulent heat transfer in these three types of rotating channel flows. In comparison with the spanwise- and streamwise-rotating planechannel flows, studies of wall-normal rotating plane-channel flows rely exclusively on numerical simulations. This is because it is rather difficult to use an experimental approach to create a wall-normal rotating channel flow driven by a streamwise pressure gradient while having two (streamwise and spanwise) unrestricted mean velocity components (Mehdizadeh and Oberlack, 2010). Compared with spanwise- and streamwise-rotating flows, wall-normal rotating flows are very sensitive to the system rotation speed. Wu and Kasagi (2004a) performed DNS to investigate turbulent plane-channel flow under combined wall-normal and streamwise rotations, as well as combined wall-normal and spanwise rotations. Mehdizadeh and Oberlack (2010) conducted DNS to study both turbulent and laminar wall-normal rotating channel flows for a wide range of rotation numbers of Ro2 = 0 1.82 . The rotation def

number is defined in a general manner as Roi = 2 i / u 0, where Ωi is the rotating speed about the xi-axis (i = 1, 2, or 3), δ is the half-channel height, and uτ0 is the wall friction velocity of a non-rotating channel flow. They recognized three distinct flow regimes: turbulent

Corresponding author. E-mail address: [email protected] (B.-C. Wang).

https://doi.org/10.1016/j.ijheatfluidflow.2019.108480 Received 1 June 2019; Received in revised form 13 September 2019; Accepted 18 September 2019 0142-727X/ Crown Copyright © 2019 Published by Elsevier Inc. All rights reserved.

International Journal of Heat and Fluid Flow 80 (2019) 108480

C. Bergmann and B.-C. Wang

Nomenclature

δ δij Δ Δt Δxi ϵijk ε εi εu

English symbols cp Ci Co Dt Dα Di p Dit Di Di Eii ETT h i, j, k ki K Li Ni Ntotal Nu p P PiT

Piu Pr qw Reb Reτ Roi t T Tτ Tτ0 ⟨T′2⟩ ub ui uτ,eff uτi uτ0 ui T ui uj xi

specific heat capacity Coriolis production term for turbulent heat flux Courant number: Δt · |ui|/Δxi turbulent diffusion term for temperature variance molecular diffusion term for temperature variance pressure diffusion term for turbulent heat flux turbulent diffusion term for turbulent heat flux molecular thermal diffusion term for turbulent heat flux viscous diffusion term for turbulent heat flux 2D energy spectrum of ui velocity fluctuations 2D energy spectrum of temperature fluctuations convective heat transfer coefficient tensor indices wavenumber in the xi-direction thermal conductivity computational domain size in the xi-direction number of grid points in the xi-direction total number of grid points: N1N2N3 Nusselt number: 2δh/K instantaneous pressure turbulent production term for temperature variance turbulent production term for turbulent heat flux due to the mean temperature gradients turbulent production term for turbulent heat flux due to the mean velocity gradients Prandtl number: ν/α wall heat flux: K (d T / dx2) w Reynolds number based on the bulk mean velocity: 2ubδ/ν Reynolds number based on the wall friction velocity of a non-rotating channel flow: uτ0δ/ν rotation number about the xi-axis: 2Ωiδ/uτ0 time instantaneous temperature, non-dimensionalized by the temperature difference between the top and bottom walls wall friction temperature: |qw |/ cp u 0 wall friction temperature for the non-rotating case temperature variance bulk mean velocity instantaneous velocity components in the xi-direction effective wall friction velocity: u 21 + u 23 wall friction velocity in the xi-direction: | wi |/ wall friction velocity for the non-rotating channel flow turbulent heat fluxes Reynolds stresses coordinate of the streamwise (i = 1), wall-normal (i = 2 ), and spanwise (i = 3 ) directions

u i ui xj xj

η λi ν Πi ρ Σ

tot 3

τwi τw0 ϕii ϕTT Ωi

Kolmogorov length: (ν3/εu)1/4 wavelength in the xi-direction kinematic viscosity pressure-temperature gradient correlation term for turbulent heat flux density sum of budget terms u 2 u3 ) total spanwise shear stress: ( d u3 /dx2 wall shear stress in the xi-direction: ρν(d⟨ui⟩/dx2)w wall shear stress in streamwise direction for non-rotating channel flow 2D premultiplied energy spectrum of ui velocity fluctuations 2D premultiplied energy spectrum of temperature fluctuations rotating speed about the xi-axis

Subscripts, Superscripts, and Symbols ( · )1, ( · )2, ( · )3 streamwise, wall-normal, and spanwise components, respectively ( · )0 value of the non-rotating channel flow ( · )b bulk mean quantity ( · )w value at the wall ⟨·⟩ quantity averaged over time and over the homogeneous (streamwise-spanwise) directions ^ (·) Fourier coefficient ( · )′ fluctuating component ( · )* complex conjugate ( · )+ quantity expressed in wall coordinates, through non-dimensionalization based on δ, ν, uτ0, and Tτ0 Abbreviations 2D DNS LES LETOT MPI TGL TKE TSE

Greek symbols α

half-channel height Kronecker delta mean grid spacing: (Δx1Δx2Δx3)1/3 time step grid resolution in the xi-direction Levi-Civita symbol molecular dissipation rate for temperature variance molecular dissipation term for turbulent heat flux molecular dissipation rate for turbulence kinetic energy:

thermal diffusivity

(Ro2 ≤ 0.091), quasilaminar (Ro2 = 0.145 0.273), and laminar (Ro2 ≥ 0.546). They also presented an analytical solution of laminar wall-normal rotating flow, and showed that because of the existence of the spanwise velocity, Ekman-like spirals develop in the flow field, where the flow direction rotates with an increasing distance from the wall towards the centre of the channel. They termed this type of flow “Poiseuille-Ekman flow”, because at high rotation numbers (Ro2 > 0.273), the flow exhibits a typical Ekman-like boundary layer behavior. Additionally, Mehdizadeh and Oberlack (2010) illustrated

two-dimensional direct numerical simulation large-eddy simulation large-eddy turnover time: δ/uτ0 Message passing interface Taylor-Görtler-like turbulence kinetic energy: ui ui /2 turbulence scalar energy: ⟨T′2⟩

elongated coherent structures found in the quasilaminar region of a wall-normal rotating flow. Relevant to this study, there are several DNS and large-eddy simulation (LES) studies of turbulent heat transfer of spanwise-rotating duct flows in the literature. Pallares and Davidson (2000) and Pallares et al. (2005) conducted LES to study turbulent heat transfer in square ducts subjected to spanwise system rotations. They investigated the effects of rotation number and Grashof number on the secondary flow pattern and heat transfer performance. Qin and Pletcher (2006) 2

International Journal of Heat and Fluid Flow 80 (2019) 108480

C. Bergmann and B.-C. Wang

also conducted LES of turbulent heat transfer in a spanwise-rotating square duct. They studied the effects of Coriolis and buoyant forces on the statistics of the velocity and temperature fields and secondary flow patterns. Recently, Fang and Wang (2018) performed a DNS study of turbulent heat transfer in a square duct at very high rotation numbers while the Reynolds number was fixed at Re = 150 (based on half duct width and the overall wall friction velocity averaged over four duct walls). El-Samni and Kasagi (2000) studied streamwise- and spanwise-rotating channel flow up to Ro1 = 15 and Ro3 = 15 in conjunction with a def

fixed Reynolds number of Re = 150 (Re = u

0

enriching the current literature by conducting a systematic DNS study of turbulent heat transfer in a wall-normal rotating channel flow for a wide range of rotation numbers. The rotation number varies significantly from Ro2 = 0 to 1.82, and as a result, turbulent heat transfer occurs under fully-turbulent, quasilaminar, and fully-laminar flow conditions. Ever since the pioneering studies of Jiménez and Moin (1991) and Lozano-Durán and Jiménez (2014) of stationary turbulent planechannel flow, it has become well understood that the dimensions of a computational domain must be properly selected to ensure that all relevant turbulent motions are captured by DNS. Very recently, Yang and Wang (2018) studied streamwise-rotating plane-channel flow for a wide range of rotation numbers varying from Ro1 = 0 to 150 with the Reynolds number fixed at Re = 180 . They observed the appearance of TGL vortices, which are elongated in the streamwise direction and increase in length with the rotation number. At their highest rotation number tested Ro1 = 150, an extremely long streamwise domain of L1 = 512 must to be used in order to fully capture the TGL vortices. Yang and Wang (2018) indicated that the minimal computational domain depends strongly on the specific physical quantity under investigation, and suggested that the selection of the computational domain size should be solidly based on evidence from both physical and spectral spaces. Specific to DNS of wall-normal turbulent channel flows, previous authors used a number of practical methods to determine their computational domain sizes. Mehdizadeh and Oberlack (2010) varied the size of the domain in the streamwise and spanwise directions until the two-point correlations of velocity fluctuations along the channel centre decayed rapidly to zero within half the domain. The computational domain used in the DNS of Mehdizadeh and Oberlack (2010) was L1 × L 2 × L 3 = 4 × 2 × 8 at rotation number Ro2 = 0.054 and Reynolds number Re = 180 . As the second objective of this research, we conduct DNS of wall-normal rotating plane-channel flow at the same Reynolds and rotation numbers of Mehdizadeh and Oberlack (2010), with a difference that we now extend the research from pure fluid flow to turbulent heat transfer. Furthermore, we will show evidence that although the domain size suggested by Mehdizadeh and Oberlack (2010) can be used successfully for the prediction of the velocity field for a couple of fully-turbulent flow test cases, it is undersized for several other fully-turbulent and quasilaminar flow test cases, and is in general, insufficient for capturing the energetic structures associated with the turbulent temperature field. In fact, in order to properly conduct DNS of turbulent heat transfer, the largest domain used in the current research must be significantly increased to L1 × L2 × L3 = 12 × 2 × 40 at Ro2 = 0.273, which is 60 times larger than that ( L1 × L 2 × L 3 = 4 × 2 × 2 ) used in Mehdizadeh and Oberlack (2010). Correspondingly, the total number of

/ where ν is the kidef

nematic viscosity) and a Prandtl number of Pr = 0.71 (Pr = / where α is the thermal diffusivity). They found that the heat transfer rate is enhanced for an increasing rotation number. Specifically, Nusselt number Nu increases with spanwise rotation up to a maximum at Ro3 = 7.5, while for streamwise rotation, Nu increases monotonically with Ro1. Liu and Lu (2007) performed DNS on spanwise-rotating turbulent heat and fluid flows in a plane channel at Re = 194 and Pr = 0.71. They varied the rotation number Ro3 between 0 and 7.5. They examined basic turbulent statistics and turbulent heat flux budget terms, and revealed how coherent structures, specifically Taylor-Görtler-like (TGL) vortices, significantly alter the turbulent temperature statistics. They also showed that Nu monotonically decreases with an increasing rotation number Ro3 in a spanwise-rotating channel flow, a result that contradicts the findings of El-Samni and Kasagi (2000). In the literature, only a couple of DNS studies can be found that are related to turbulent heat transfer in wall-normal rotating plane-channel flows. Wu and Kasagi (2004b) studied turbulent heat transfer in a channel flow subjected to arbitrary system rotation. They studied combined spanwise- and streamwise-, combined streamwise- and wallnormal, and combined wall-normal and spanwise-rotating flows, all at a def

Reynolds number of Reb = 2ub / = 4560 (based on the bulk mean velocity ub) and a Prandtl number of Pr = 0.71. They found that spanwise rotation dominates heat transfer in combined spanwise- and streamwise-rotating flows, while for combined wall-normal and spanwise-rotating flows, the wall-normal rotation reduces the effects of spanwise rotation on turbulent heat transfer. Li et al. (2006) studied heat transfer in wall-normal rotating flows for Ro2 = 0 0.1 at Re = 194 with a Prandtl number of Pr = 1. They observed that as Ro2 increases from 0 to 0.06, the magnitude of the spanwise turbulent heat flux u3 T is enhanced, coinciding with increased spanwise velocity fluctuations. Meanwhile, streamwise heat flux u1 T and wall-normal heat flux u2 T are reduced in magnitude, coinciding with the reduction in streamwise and wall-normal velocity fluctuations. As Ro2 increases further, all turbulent heat fluxes decrease, coinciding with the trend of laminarization. As an objective of this research, we aim at

Fig. 1. Schematic of the computational domain, coordinate system, orientation of the rotating axis, Coriolis forces ( 2 2 u3 and 2Ω2u1), direction of the driving u 20 / ), and temperature boundary conditions. The temperature has been normalized using the temperature difference between the pressure gradient ( p / x1 = , respectively. top and bottom walls located at x2 = and x2 = 3

International Journal of Heat and Fluid Flow 80 (2019) 108480

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ui =0 xi

Table 1 Summary of 10 test cases simulated using DNS. The comparative study is conducted at 10 rotation numbers in conjunction with a fixed Reynolds number of Re = 180 as in Mehdizadeh and Oberlack (2010). The DNS study of Mehdizadeh and Oberlack (2010) focused exclusively on pure fluid flows (instead of turbulent heat transfer), and as such, the computational domain size in their research varied from 4πδ × 2δ × 2πδ (for most test cases) to 4πδ × 2δ × 8πδ (for the case of Ro2 = 0.054 ). Test Case

Ro2

Flow Condition

L1 × L2 × L3

N1 × N2 × N 3

0 0018 0054 0091 0145 0182 0273 0546 0728 182

0 0.018 0.054 0.091 0.145 0.182 0.273 0.546 0.728 1.82

Turbulent Turbulent Turbulent Turbulent Quasilaminar Quasilaminar Quasilaminar Laminar Laminar Laminar

20πδ × 2δ × 6πδ 8πδ × 2δ × 6πδ 4πδ × 2δ × 8πδ 8πδ × 2δ × 20πδ 5πδ × 2δ × 50πδ 7πδ × 2δ × 50πδ 12πδ × 2δ × 40πδ 1.5πδ × 2δ × 15πδ 1.5πδ × 2δ × 15πδ 1.5πδ × 2δ × 15πδ

960 × 128 × 576 384 × 128 × 576 192 × 128 × 768 384 × 128 × 1920 240 × 128 × 4800 336 × 128 × 4800 576 × 128 × 2880 72 × 128 × 1440 72 × 128 × 1440 72 × 128 × 1440

u 20

.

1 p + xi

T T + uj = t xj

2T

x 2j

2u

i

2

x j2

,

ijk

j uk

,

(3) (4)

1 2

2 2 2 (x 1

+ x 32)

.

(5)

Following the usual convention for treatment of rotating channel flows, the centrifugal force component is absorbed into the pressure because of its conservative nature. In the above equations, mean variables are indicated using an operator ⟨ · ⟩, calculated by averaging over time and over the homogeneous x1- and x3-directions. The last term ( 2 ijk j uk ) of Eq. (3) represents the Coriolis force, which appears as a result of the chosen non-inertial rotating reference frame. Specific to this study, the reference frame rotates about the wall-normal axis, and there are two Coriolis force components: one component acts in the streamwise direction ( 2 2 u3), and the other acts in the spanwise direction (2Ω2u1). In order to determine the proper computational domain size for conducting DNS of turbulent heat transfer in a wall-normal rotating channel flow, a wide range of streamwise and spanwise domain sizes were tested under different rotation numbers. Table 1 summarizes the testing parameters under different conditions in our comparative study. The various Ro2 values are identical to those in the DNS study of the turbulent flow field conducted by Mehdizadeh and Oberlack (2010). At each rotation number, Mehdizadeh and Oberlack (2010) varied the size of their domain in the streamwise and spanwise directions until the two-point correlations of velocity fluctuations along the channel centre decayed rapidly to zero within half the domain. Their largest domain was L1 × L2 × L3 = 4 × 2 × 8 at Ro2 = 0.054 and Re = 180, which is substantially smaller than the domains used in the present study. In total, 10 test cases are considered in this study, with the rotation number varying considerably from Ro2 = 0 to 1.82. As the rotation number increases, the flow pattern transitions from being fully turbulent, to being quasilaminar, and eventually, fully laminar. In our comparative study of the computational domains, 7 streamwise domain sizes (ranging from L1 = 1.5 to 20πδ), and 6 spanwise domain sizes (ranging from L3 = 6 to 50πδ) are considered. An in-house pseudospectral code was developed using FORTRAN 90 to solve the governing equations, which is of high temporal and spatial accuracies. Message passing interface (MPI) libraries were used to parallelize the code. The velocity, temperature, and pressure fields were expanded into Fourier series in the streamwise and spanwise directions, and expanded in Chebyshev polynomials in the wall-normal direction. A third order time-splitting scheme was used to discretize the governing equations by following Karniadakis et al. (1991). Aliasing in the streamwise and spanwise directions was removed via the 2/3-rule. The algorithm alternates between the divergence form and the convective form when explicitly solving the non-linear convective term. This is done to increase the accuracy of the algorithm while reducing the computation time, as recommended by Zang (1991). The pressure and linear diffusion terms were solved implicitly. The number of grid points varied in the streamwise and spanwise directions, such that the grid resolution was maintained uniformly at x1+ = 11.8 and x 3+ = 5.89 ( x1 = 0.06545 and x3 = 0.03273 ) with the exception of case 0273, where the spanwise grid resolution was

Fig. 1 shows a schematic of the computational domain, coordinate system, orientation of the rotation axis, Coriolis forces, direction of the driving pressure gradient, and temperature boundary conditions. The temperature field is treated as a passive scalar with Pr = 0.71. The temperature has been normalized using the temperature difference between the top and bottom walls (located at x2 = and x2 = , respectively), which is held constant at T = 0.0 and 1.0, respectively. In this DNS study, we focus our attention on the effect of wall-normal system rotation on turbulent heat transfer. To this purpose, a wide range of rotation numbers are considered, varying from Ro2 = 0 to 1.82, in conjunction with a fixed Reynolds number Re = 180 . The standard no-slip boundary conditions were imposed at either wall (i.e., ui (x1, ± , x3) = 0 ), and periodic boundary conditions are applied to the streamwise and spanwise directions. The flow is driven by a constant streamwise pressure gradient, proportional to the wall shear stress of a non-rotating channel flow τw0, i.e.

=

ui u + uj i = t xj

p= p +p

2. Test cases and numerical method

w0

(2)

where ρ is the fluid density, Ωj is the speed of system rotation about the xj-axis, and ϵijk is the Levi-Civita symbol. The instantaneous velocities ui and non-dimensionalized temperature T can be decomposed into a mean and a fluctuating component, i.e. ui = ui + ui and T = T + T , while the instantaneous pressure p can be decomposed into a mean component ⟨p⟩, a fluctuating component p′, and a cen1 trifugal force component 2 22 (x12 + x32) as follows:

grid points for the current DNS is Ntotal = 576 × 128 × 2880, which is 100 times that used in Mehdizadeh and Oberlack (2010) at the same rotation number. The remainder of this paper is organized as follows: the numerical method and test cases are described in Section 2. The results of this DNS study are presented and analyzed in Section 3, which includes a detailed discussion of coherent structures, energy spectra of temperature fluctuations, characteristic length scales of the temperature field and proper computational domain sizes required for capturing them, the mean velocity profiles, Reynolds stresses, the mean temperature profiles, Nusselt number, temperature variance, turbulent heat fluxes, and underlying mechanisms of the transport processes of the temperature variance and turbulent heat fluxes. Finally, major findings of this study are summarized in Section 4.

p = x1

,

(1)

Here, uτ0 is the wall friction velocity of the non-rotating channel flow. It should be noted that although the direction of the bulk fluid velocity changes with the speed of wall-normal system rotation, for convenience, the x1- and x3-directions are still referred to as the “streamwise” and “spanwise” directions, respectively (such that the terminologies are consistent with those used in the test case of a non-rotating channel flow). The equations that govern the turbulent heat and fluid flow in a plane channel subjected to a system rotation take the following form in the context of an incompressible fluid: 4

International Journal of Heat and Fluid Flow 80 (2019) 108480

C. Bergmann and B.-C. Wang

x 3+ = 7.85 ( x3 = 0.04363 ). The grid points are defined in the wallnormal direction using Chebyshev Gauss-Lobatto points, where x2 (j) = cos( j/N2) (j = 0, …, N2 ) . The number of grid points in the wallnormal direction was 128 with a resolution that varied from x 2+ = 0.055 ( x2 = 0.0003056 ) at the first node off the wall to x 2+ = 4.45 ( x2 = 0.02472 ) at the channel centre. The total number of grid points Ntotal varied between 13.3 M (for a small domain of 1.5πδ × 2δ × 15πδ for the three laminar flow cases) and 212.3 M (for a large domain of 12πδ × 2δ × 40πδ for the quasilaminar case 0273). The grid resolution, quantified by the mean grid spacing

initialized using one of three methods: laminar velocity and scalar fields with random perturbations added to the velocity field, a previous run with an identical Ro2 or at a lower Ro2. The time step Δt was defined such that the maximum Courant number was less than 0.3, i.e. max(Co) = max( t ·|ui |/ x i ) 0.3. Samples for turbulent statistics were collected after the velocity and temperature fields reached statistically stationary states. For each run, at least 220 samples were taken over approximately 43 large-eddy turnover times (LETOTs), where one LETOT is δ/uτ0. All simulations were conducted on the supercomputing and storage facilities of Western Canada Research Grid (WestGrid). The CPU time and memory required to perform a simulation depended on the total number of grid points, varying from 1,300 core-hours to 60,000 corehours, and from 20 GB to 800 GB for cases with the smallest and largest numbers of grid points (i.e., 13.3 M and 212.3 M grid points, respectively).

def

= ( x1 x2 x3)1/3, was sufficiently fine to ensure that the smallest scale of turbulence, i.e. Kolmogorov length η, was captured as demanded by DNS. The DNS was conducted in a strict manner by ensuring that the computational grid resolved scales down to = following the recommendation of Grötzbach (1983) and Pope (2000). The velocity and temperature fields of each simulation were

Fig. 2. Instantaneous turbulent structures visualized using isosurfaces of constant temperature (T = 0.7 ) at Ro2 = 0.091, 0.182, and 0.273, colored according to the wall-normal coordinate x2/δ. To facilitate a direct comparison, length scales are identical in panels (a), (b), and (c). The length of the half-channel height δ is identified in each magnified region. Only the bottom half of the channel is shown for each case, such that contours extending through to the top half of the channel are cut off, appearing as empty (or white) spaces. 5

International Journal of Heat and Fluid Flow 80 (2019) 108480

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Prior to the present simulations of 10 test cases, the computer code had been validated by reproducing three benchmark cases: the turbulent channel flow of Moser et al. (1999), the wall-normal rotating channel flow of Mehdizadeh and Oberlack (2010), and the heated channel flow of Kasagi et al. (1992). In presenting the results, all variables denoted with a superscript “+” are non-dimensionalized using δ, ν, uτ0, and Tτ0. The friction temperature Tτ is defined as follows: def

T =

|qw | = cp u 0 u

0

d T dx2

. w

different rotation numbers, all temperature values, unless otherwise noted, are non-dimensionalized using the friction temperature of the non-rotating channel-flow case, Tτ0. 3. Result analysis 3.1. Turbulence scales, energy spectra, and computational domain sizes Large energy-containing eddy motions dominate the turbulent transport of momentum and thermal energy. Thus, in DNS of turbulent heat convection, the accuracy of the numerical algorithm itself is insufficient to warrant correct predictions of the velocity and temperature fields. In order to simulate the physics correctly, large energy-containing scales must be captured by a properly sized computational domain; otherwise, the turbulence energy level will be miscalculated

(6)

where qw = K (d T / dx2) w is the wall heat flux, K = cp is the thermal conductivity, and cp is the specific heat capacity at constant pressure of the fluid. The value of Tτ is sensitive to the rotation number Ro2. In order to facilitate a fair comparison between test cases of

Fig. 3. 2D premultiplied energy spectra of temperature fluctuations ϕTT of the turbulent flow cases (0, 0018, 0054, and 0091) at the x 2+ -location corresponding to the peak value max (ϕTT) (marked using a cross symbol “+” in the figure panel). Isopleths: 0.625max (ϕTT) (innermost), 0.25max (ϕTT) (middle), and 0.125max (ϕTT) (outermost). The black box demarcates the boundaries of the current computational domain listed in Table 1 for conducting DNS of turbulent heat transfer, while the dashed green lines demarcate the boundaries of the computational domain used in Mehdizadeh and Oberlack (2010) for conducting DNS of turbulent fluid flows. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) 6

International Journal of Heat and Fluid Flow 80 (2019) 108480

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(although the simulation can still be highly accurate in terms of the algorithm). The imposed wall-normal system rotation has a significant impact on turbulent flow structures and heat transfer. In comparison with the a non-rotating heated channel flow, mean and turbulent secondary flow structures are induced by the Coriolis forces. As the first step of DNS, a proper computational domain size must be determined based on the analysis of the mean and turbulent secondary flow structures associated with turbulent heat convection. We will demonstrate the turbulent coherent structures based on instantaneous velocity and temperature fields through vivid contour visualizations, and then the turbulent coherent structures will be rigorously analyzed by using two-dimensional (2D) premultiplied energy spectra of temperature fluctuations. Fig. 2 compares the instantaneous turbulence structures, visualized using the isosurfaces of a constant non-dimensional temperature value (T = 0.7 ), for a turbulent (Ro2 = 0.091) and two quasilaminar cases (Ro2 = 0.182 and 0.273). From Fig. 2, it is seen that for all three cases, the high temperature fluid near the bottom wall is forced upwards by large flow structures. The turbulence structures are oriented in the streamwise-spanwise directions, and become increasingly elongated as the rotation number increases from Ro2 = 0.091 to 0.182. Furthermore, it is observed that these coherent structures cluster together at Ro2 = 0.091, and become more separated with increasing rotation numbers of Ro2 = 0.182 and 0.273. These large coherent flow structures act to transport turbulent thermal energy within the channel. The turbulence structures shown in Fig. 2 are qualitative. In order to conduct a rigorous analysis, a further study of turbulence structures in the spectral space is necessary. The premultiplied energy spectrum of temperature fluctuations (ϕTT) is a useful tool for determining the level of TSE or temperature variance, defined as ⟨T′2⟩. The 2D premultiplied energy spectrum of temperature fluctuations, ϕTT, over a homogeneous (x1-x3) plane is defined as follows: TT (k1, x2 ,

k3) = k1 k3 ETT (k1, x2 , k3)

conducted by Yang and Wang (2018), if the computational domain size used in conducting DNS is too small to even capture a complete innermost isopleth of 0.625max (ϕTT), the numerical simulation is considered rather inaccurate with respect to the prediction of large-scale flow structures. Ideally speaking, it should be further required that all scales of eddy motions within the outermost isopleth (corresponding to 0.125max (ϕTT)) be fully captured by using a properly sized computational domain, thus ensuring the DNS to be reliable with respect to simulation of both large and moderate scale eddy motions associated with the turbulent transport process of the thermal energy. However, in our simulations, it is observed that for this criterion (of capturing the majority of the turbulence energy down to 1/8-th of the peak value, i.e. capturing the full outermost isopleth of 0.125max (ϕTT)), a much larger domain is required in order to capture the energy spectrum associated with the temperature fluctuations than that associated with the velocity fluctuations as in Mehdizadeh and Oberlack (2010). In view of this challenge in DNS of turbulent heat transfer, we require that the majority of the TSE of the temperature fluctuations be captured down to at least 2/8-th of its peak value in a simulation. This implies that in the present DNS, it is required that, at a minimum, the isopleth of 0.25max (ϕTT) be fully captured, if it is not entirely possible to fully capture the isopleth of 0.125max (ϕTT). As shown in Fig. 3, for the turbulent cases, all large scales of the turbulent temperature field (where the TSE level varies from the peak value max (ϕTT) to 0.125max (ϕTT)) are readily captured by the domains used in the present study (demarcated using solid black lines). To achieve this, it is necessary to significantly increases the computational domain size in cases 0, 0018, and 0091 in comparison with those used in Mehdizadeh and Oberlack (2010). In fact, as is evident in Fig. 3a and 3d, if the original computational domain of Mehdizadeh and Oberlack (2010) (demarcated using dashed green lines) were used, the DNS would not be able to fully capture the large TSE-containing scales of the turbulent temperature field. In Fig. 3a, both the outermost and middle isopleths (corresponding to 0.125max (ϕTT) and 0.25max (ϕTT), respectively) cannot be fully contained by the domain demarcated using dashed green lines. In Fig. 3d, the situation worsens, because even the mode of ϕTT is outside of the domain demarcated using dashed green lines, indicating that if the smaller computational domain of Mehdizadeh and Oberlack (2010) (originally used for conducting DNS of fluid flows) were used here for conducting DNS of turbulent heat transfer, DNS would be very inaccurate such that even the most energetic turbulence scales associated with temperature fluctuations would be missed in the prediction. As shown in Fig. 3b, the original domain of Mehdizadeh and Oberlack (2010) seems to be large enough to capture the outermost isopleth of 0.125max (ϕTT) at x 2+ = 16 (where the peak value of ϕTT is located), this original domain is insufficient to fully capture the outermost isopleth at the wall-normal location of x 2+ = 178 (effectively, the channel centre. Not shown). Hence, we extended the domain in the streamwise direction from L1 = 4 (original) to 8πδ (current), demarcated using a solid black line. There is a special case: for case 0054 (shown in Fig. 3c), it is interesting to observe that the original domain of Mehdizadeh and Oberlack (2010) is actually large enough such that the large TSE-containing scales of the turbulent temperature field (as indicated by all three isopleths of different TSE levels) are properly captured throughout the entire domain (at all x2locations). From Fig. 3a–3d, it is very interesting to observe a general trend that all three isopleths (innermost, middle, and outermost) become more concentrated as the rotation number increases. Furthermore, there is another general trend in the streamwise and spanwise characteristic wavelengths (i.e., λ1/δ and λ3/δ) corresponding to the mode of the 2D premultiplied energy spectrum, which vary significantly with the rotation number. Specifically, the turbulent temperature field is dominated by relatively small scales at low rotation numbers, but as the rotation number increases, ϕTT isopleths cover a greater range of λ1 and, to a much greater extent, λ3. Later, we will refine the study of the rotation number effect on the characteristic

(7)

,

where k1 and k3 are the streamwise and spanwise wavenumbers, respectively, and ETT is the energy spectrum of temperature fluctuations, defined as:

^ ^* ETT (k1, x2 , k3) = 2 T (k1, x2 , k3) T (k1, x2 , k3)

,

(8)

^* ^ where T denotes a Fourier coefficient of T′, and T represents the ^ complex conjugate of T at different wavenumbers. In Eqs. (7) and (8), both the premultiplied energy spectrum and energy spectrum are expressed as functions of wave numbers k1 and k3. Alternatively, they can also be expressed as functions of streamwise and spanwise wavelengths, defined as i = 2 /ki (for i = 1 and 3, respectively). Fig. 3 shows the isopleth pattern of ϕTT of the temperature fluctuations for the turbulent cases at four rotation numbers (Ro2 = 0, 0.018, 0.054, and 0.091). The figure is plotted in the x1-x3 plane at the wall-normal position where the peak value of ϕTT occurs. The peak location (or, mode) is marked using a cross symbol “+” in all figure panels. Following the general approaches of Hoyas and Jiménez (2006), Avsarkisov et al. (2014), and recently, Yang and Wang (2018), three isopleths are plotted for each spectrum here, corresponding to 5/8-th, 2/8-th, and 1/8-th of the peak value (or, 0.625max (ϕTT), 0.25max (ϕTT), and 0.125max (ϕTT), respectively). The most energetic turbulent motions associated with the turbulent transport process of the thermal energy are found within the innermost isopleth where the 2D premultiplied energy spectrum decays to 5/8-th (or, 62.5%) of its peak value. The outermost isopleth identifies the less energetic (but still important) eddy scales, where the 2D premultiplied energy spectrum decays to 1/8-th (or, 12.5%) of its peak value. In the DNS studies of non-rotating channel flows conducted by Hoyas and Jiménez (2006) and Avsarkisov et al. (2014), and streamwise-rotating channel flows 7

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wavelengths of the dominant TSE-containing motions (as indicated by the mode of the 2D premultiplied energy spectrum of temperature fluctuations) by use of Fig. 5. Fig. 4 shows the isopleths of ϕTT of temperature fluctuations for all three quasilaminar cases (0145, 0182, and 0273) and a laminar case (0546) at the wall-normal location where the peak value max (ϕTT) occurs. From Fig. 4 it is clear that the mode of ϕTT is outside of the dashed green boxes, indicating that the original domain sizes used in Mehdizadeh and Oberlack (2010) for DNS of pure fluid flows are too small for conducting current DNS of turbulent heat transfer (because even the peak value of ϕTT would be missed in the predictions). In contrast, the computational domain sizes used in the current study (shown using solid black lines) have been significantly extended to fully capture large scales of turbulent temperature fluctuations at the TSE level of at least 0.25max (ϕTT) in Fig. 4a and 4c, and of at least 0.125max (ϕTT) in Fig. 4b and 4d. Compared to the turbulent cases (shown in Fig. 3), the quasilaminar cases require computational domains that are much larger in the spanwise direction and slightly smaller in the streamwise direction. This implies that the turbulent temperature field is dominated by fluctuations that are correlated over large spanwise distances. Large coherent structures elongated in the spanwise direction act to propagate turbulent thermal energy in the form of temperature variance over large distances, i.e. wavelengths. The maximum value of ϕTT occurs at a much greater spanwise wavelength in the three quasilaminar cases (0145, 0182, and 0273), meaning that the turbulence structures of the highest TSE span a large portion of the domain in the spanwise dimension. Compared to the turbulent cases shown previously in Fig. 3, it is clear that turbulent structures are dominated by larger scales at higher rotation numbers (in the quasilaminar cases). As a result, the computational domains of quasilaminar cases are much larger than those for the four turbulent cases in Table 1. Although case 0546 is classified artificially as laminar, there still appears to be an isolated region of temperature fluctuations at high wavelengths, even though the Reynolds stresses at Ro2 = 0.546 are trivial (the maximum value of the non-dimensionalized TKE at Ro2 = 0.546 is on the order of 10 26 ). The length scales of turbulence that contribute the greatest amount of TSE to the temperature variance can be ascertained by identifying the characteristic wavelengths corresponding to max (ϕTT). Fig. 5 compares the streamwise and spanwise characteristic wavelengths (λ1 and λ3, respectively) corresponding to the mode of ϕTT for Ro2 ≤ 0.728. From Fig. 5, it is apparent that the largest characteristic length scales with respect to both λ1 and λ3 occur in the quasilaminar states (for 0.145 ≤ Ro2 ≤ 0.273). Furthermore, the spanwise characteristic wavelength λ3 is always greater than the streamwise characteristic wavelength λ1 for Ro2 > 0.054. This coincides with the observation of coherent structures being elongated in the spanwise direction in a wallnormal rotating flow, and most notably in a quasilaminar state. This also further clearly indicates that there is a necessity to use a domain that is large enough (most importantly, in the spanwise direction) in DNS of heated turbulent wall-normal rotating channel flows in order to fully capture the largest scales of turbulent temperature fluctuations. Thus far, we have thoroughly analyzed the ranges of dominant turbulence scales associated with instantaneous temperature fluctuations. In order to develop a deeper understanding of the spectral characteristics of the temperature field, a spectral analysis of the turbulent velocity field is also needed. The premultiplied energy spectrum of velocity fluctuations (ϕii), where i represents the streamwise (i = 1), wall-normal (i = 2 ), or spanwise (i = 3 ) direction, is a useful tool for determining the level of TKE, defined as ui ui /2 . The 2D premultiplied energy spectrum of velocity fluctuations, ϕii, over a homogeneous (x1x3) plane is defined as follows: ii (k1, x2 ,

k3) = k1 k3 Eii (k1, x2 , k3)

,

defined as:

^ ^* Eii (k1, x2 , k3) = 2 ui (k1, x2 , k3) ui (k1, x2 , k3)

,

(10)

^* ^ where ui denotes a Fourier coefficient of ui , and ui represents the ^ complex conjugate of ui at different wavenumbers. Fig. 6 shows the isopleths of both ϕ11 and ϕ33 of velocity fluctuations for all four turbulent flow cases (0, 0018, 0054, and 0091) at the wall-normal locations where the peak values, max (ϕ11) and max (ϕ33), occur. Similarly, Fig. 7 shows the isopleths of both ϕ11 and ϕ33 of velocity fluctuations for all three quasilaminar flow cases (0145, 0182, and 0273) and a laminar flow case (0546). From Fig. 6, it is clear that the computational domains used in the DNS study of wall-normal rotating channel flows by Mehdizadeh and Oberlack (2010) fail to capture the energetic scales associated with turbulent velocity fluctuations of the non-rotating turbulent channel flow case (Ro2 = 0 ), but can be used successfully for the other three fully-turbulent flow cases (Ro2 = 0.018, 0.054 and 0.091). In contrast, as shown in Fig. 7, their domain is insufficient for all three quasilaminar flow cases (0145, 0182, and 0273) and a laminar flow case (0546), especially with respect to the analysis based on the 2D premultiplied energy spectra of the spanwise velocity fluctuations, ϕ33. The above conclusions on the adequacy of the computational domain sizes used in the current research and that of Mehdizadeh and Oberlack (2010) obtained based on the 2D premultiplied energy spectra of temperature and velocity fluctuations are consistent in general. However, a careful comparison of Figs. 6 and 7 with Figs. 3 and 4 shows that the ranges of turbulence structure scales as indicated by the isopleths of the 2D premultiplied energy spectra of temperature and velocity fluctuations are apparently different. This difference can be understood from the turbulent transport processes of TKE and TSE, which are different in terms of their production, dissipation and turbulent diffusion mechanisms, etc. The Coriolis force term has a significant impact on both the velocity and temperature fields, which directly appears in the momentum equation only. Furthermore, it should be indicated that there is ambiguity in comparing turbulence structures inferred from the velocity spectra and those inferred from the temperature spectra. Turbulence structures related to the velocity field are characterized using two components of the velocity spectra here, ϕ11 and ϕ33; whereas turbulence structures relating to the temperature field are characterized using only the temperature spectrum ϕTT. In view of these factors, the scales of turbulence structures, when characterized by the velocity spectra and the temperature spectra, can be different. This indicates that the minimum computational domain required for accurately conducting DNS of a turbulent fluid flow and that for accurately conducting DNS of a heat convection process associated with the flow are not necessarily identical. 3.2. Statistics of the velocity field The focus of this paper is on the study of turbulent heat transfer. However, before we start the discussion of the temperature field and the transport processes of turbulent heat fluxes, it is important to address the basic characteristics of the velocity field. The non-dimensionalized mean streamwise velocity u1 + and mean spanwise velocity u3 + profiles of all test cases listed in Table 1 are plotted and compared in Fig. 8. The effects of the Coriolis force on the mean flow profiles are apparent by comparing rotating and non-rotating flows. As is well known, in a non-rotating channel flow, u3 + 0 holds strictly. However, as is seen in Fig. 8, the mean streamwise velocity u1 + monotonically decreases with an increasing value of Ro2; while there is a mean spanwise secondary flow, with u3 + > 0 . Furthermore, the positively-valued mean spanwise velocity u3 + increases monotonically with the rotation number up to Ro2 = 0.054, and then continuously decreases due to the increasingly stronger laminarization trend of the flow. This is a clear

(9)

where k1 and k3 are the streamwise and spanwise wavenumbers, respectively, and Eii is the energy spectrum of velocity fluctuations, 8

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Fig. 4. 2D premultiplied energy spectra of temperature fluctuations ϕTT of the quasilaminar cases (0145, 0182, and 0273), and laminar flow case (0546) at the x 2+ -location corresponding to the peak value max (ϕTT) (marked using a cross symbol “+” in the figure panel). Isopleths: 0.625max (ϕTT) (inner), 0.25max (ϕTT) (middle), and 0.125max (ϕTT) (outer). The black box demarcates the boundaries of the current computational domain listed in Table 1 for conducting DNS of turbulent heat transfer, while the dashed green lines demarcate the boundaries of the computational domain used in Mehdizadeh and Oberlack (2010) for conducting DNS of turbulent fluid flows. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

reflection of the Coriolis effect on the mean velocity field in the sense that the mean streamwise Coriolis force component 2 2 u3 is negatively valued, and as the rotation number increases, it tends to suppress the streamwise momentum of the flow. Meanwhile, the other Coriolis force component 2Ω2u1 induces the spanwise secondary flow motion (shown in Fig. 8b), a pattern that is distinctively different from that of a non-rotating channel flow. The profiles of the non-dimensionalized Reynolds stresses are plotted in Fig. 9. In contrast to the non-rotating plane channel flow (Ro2 = 0 ), all six Reynolds stresses are non-zero if Ro2 > 0 because of the effects of not only the streamwise pressure gradient but also the two Coriolis forces. For Ro2 ≥ 0.546, all Reynolds stresses are negligible, hence, the flow is classified as laminar by Mehdizadeh and Oberlack (2010). In view of this, the profiles of Reynolds stresses are plotted for 0 ≤ Ro2 ≤ 0.273 in Fig. 9. As shown in the figure, both

u1 u1 + and u2 u2 + monotonically decrease as Ro2 decreases in value. Similarly, the magnitude of the shear stress u1 u2 + is the highest at Ro2 = 0, and then steadily diminishes with an increasing rotation number Ro2. The remaining Reynolds stresses, u3 u3 + , u1 u3 + , and u2 u3 + , all increase in value from the non-rotating case Ro2 = 0 to a maximum at Ro2 = 0.054, then decrease with Ro2 as the flow gradually transitions from a turbulent to a quasilaminar state. From Fig. 9e, it is clear that the profile of shear stress u1 u3 + peaks at Ro2 = 0.145. Furthermore, the value of u1 u3 + switches sign and becomes negatively valued around the channel centre once the flow transitions from turbulent to quasilaminar, but remains positively valued in the near-wall region. The characteristics of first- and second-order flow statistics observed here are consistent with those reported in Mehdizadeh and Oberlack (2010). 9

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evident that the absolute value of the wall-normal gradient of the mean temperature, |d T + /dx2 |, decreases monotonically with an increasing Ro2 value in the near-wall region only (for 0.8 < |x2/δ| ≤ 1.0). Outside this region (near the channel centre, for |x2/δ| ≤ 0.8), the slope of the mean temperature profile varies in a more complex manner with Ro2: the value of |d T + /dx2 | initially decreases with Ro2 until the rotation number reaches Ro2 = 0.145 (at which the flow has transitioned to a quasilaminar state); then, it increases with Ro2 until the flow becomes laminarized at Ro2 = 0.546 . Given the central symmetry of the mean temperature profile (about x2 / = 0 ), we further study its distribution by rescaling it with respect to the wall coordinate only in the top half of the channel (for 0 ≤ x2/δ ≤ 1.0) in Fig. 10b. As shown in Fig. 10b, in the viscous sublayer, the profiles of ⟨T⟩/Tτ at all rotation numbers strictly obey the linear law-of-the-wall, i.e. T / T = Pr·x 2+ . Furthermore, as the rotation number exceeds Ro2 = 0.546, the flow becomes laminarized, and as a consequence, the temperature profiles (for Ro2 ≥ 0.546) collapse to the linear law-of-the-wall over the entire channel width. To further quantify the effect of wall-normal system rotation on the mean temperature field as well as the heat transfer rate, the profile of the Nusselt number Nu is plotted in Fig. 11, which is defined as

Fig. 5. Variation of the streamwise and spanwise characteristic wavelengths (λ1 and λ3, respectively) corresponding to max (ϕTT) with the rotation number.

3.3. Statistics of the temperature field The non-dimensional mean temperature profiles are displayed in Figs. 10a and 10b, across the entire channel with respect to the nondimensional coordinate x2/δ, and in the top half of the channel with respect to the wall coordinate x 2+, respectively. From Fig. 10a, it is

def

Nu =

2 ·h 2 = K Tb Tw

d T dx2

, w

(11)

where h is the convective heat transfer coefficient, Tw is the wall surface temperature, and Tb is the bulk mean temperature of the fluid. Clearly,

Fig. 6. 2D premultiplied energy spectra of the streamwise and spanwise velocity fluctuations ϕ11 and ϕ33 of the turbulent flow cases (case 0, 0018, 0054, and 0091) at the x 2+ -location corresponding to the peak values of max (ϕ11) and max (ϕ33), respectively (marked using a cross symbol “+” in the figure panel). Isopleths: 0.625max (ϕii) (innermost), 0.25max (ϕii) (middle), and 0.125max (ϕii) (outermost). The black box demarcates the boundaries of the current computational domain listed in Table 1 for conducting DNS of turbulent heat transfer, while the dashed green lines demarcate the boundaries of the computational domain used in Mehdizadeh and Oberlack (2010) for conducting DNS of turbulent fluid flows. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) 10

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Fig. 7. 2D premultiplied energy spectra the streamwise and spanwise velocity fluctuations ϕ11 and ϕ33 of the quasilaminar flow cases (0145, 0182, and 0273), and laminar flow case (0546) at the x 2+ -location corresponding to the peak values of max (ϕ11) and max (ϕ33), respectively (marked using a cross symbol “+” in the figure panel). Isopleths: 0.625max (ϕii) (innermost), 0.25max (ϕii) (middle), and 0.125max (ϕii) (outermost). The black box demarcates the boundaries of the current computational domain listed in Table 1 for conducting DNS of turbulent heat transfer, while the dashed green lines demarcate the boundaries of the computational domain used in Mehdizadeh and Oberlack (2010) for conducting DNS of turbulent fluid flows. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

the Nusselt number decreases monotonically with an increasing Ro2. The general trend indicates the effect of wall-normal system rotation is to reduce heat transfer. Eventually, as the flow reaches a laminar state at Ro2 = 0.546, the Nusselt number approaches an asymptotic value of Nu = 2, a characteristic of a heated non-rotating laminar channel flow. To gain further insight into the effect of wall-normal rotation on the

Fig. 8. Profiles of non-dimensionalized mean streamwise velocity u1

wall shear stress and wall-normal heat transfer rate, the wall friction velocities in the streamwise direction (uτ1) and spanwise direction (uτ3), def

effective wall friction velocity (u ,eff = u 21 + u 23 ), and the wall friction temperature (Tτ) are plotted in Fig. 12. Here, the wall friction def

def

(d ui / dx2) w is the velocity is defined as u i = | wi |/ where wi = wall shear stress in the xi-direction. It is clear that the wall friction

+

and mean spanwise velocity u3

11

+

profiles (non-dimensionalized by uτ0).

International Journal of Heat and Fluid Flow 80 (2019) 108480

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Fig. 9. Profiles of Reynolds stresses ui uj

temperature Tτ decreases monotonically with rotation number Ro2, as well, the profile matches the Nusselt number profile (given in Fig. 11). This can be understood directly from the definition equations of the wall friction temperature and Nusselt number, i.e. Eqs. (6) and (11), respectively. Both values of Tτ and Nu are proportional to the magnitude of the wall-normal mean temperature gradient, |d T /dx2 |w . The streamwise wall friction velocity uτ1 decreases monotonically with rotation number Ro2, coinciding with the trend of the mean streamwise velocity u1 + (see Fig. 8a). The spanwise wall friction velocity uτ3 is

+

(non-dimensionalized by u 20 ).

zero identically in a non-rotating flow. As soon as the system rotation is imposed, the value of uτ3 starts to increase and reaches its maximum at Ro2 = 0.054, and then decreases monotonically as the rotation number Ro2 continues to increase. These trends in the spanwise wall friction velocity uτ3 coincide with the trend of the mean spanwise velocity u3 + shown in Fig. 8b and that of Reynolds shear stress u2 u3 + shown in Fig. 9f. Figs. 8b and 9f show that the magnitudes of both d u3 + / dx2 and u2 u3 + peak in the near-wall region at Ro2 = 0.054 . This is not surprising, simply because the following linear equation holds for the 12

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C. Bergmann and B.-C. Wang

Fig. 10. Profiles of the non-dimensional mean temperature, obtained by non-dimensionalizing ⟨T⟩ using either the wall friction temperature of the non-rotating case Tτ0 in panel (10a), or by the wall friction temperature corresponding to the specific rotation number of each case Tτ in panel (10b).

spanwise shear stress balance: tot 3

=

d u3 dx2

u 2 u3 =

2

2

u1 x2 +

w3

,

(12)

where 3tot represents the total spanwise shear stress at an arbitrary walltot normal position x2. This equation indicates that u 23 = w3 3 . Therefore, as Ro2 increases, it is guaranteed that the magnitude of uτ3 u 2 u3 )). peaks together with that of 3tot (or, that of ( d u3 /dx2 Finally, from Fig. 12, it is seen that the effective wall friction velocity uτ,eff reaches a maximum value at Ro2 = 0.018 as a consequence of the combined trends of both streamwise and spanwise wall friction velocities. Fig. 13 compares the profiles of the non-dimensional temperature variance T 2 + at different rotation numbers. The temperature variance for the laminar flow cases (for Ro2 ≥ 0.546) is trivial, and hence, only the temperature variance for a single laminar flow case (Ro2 = 0.546 ) is shown for clarity. Fig. 13 shows that in the region off the wall (0.52 ≤ |x2/δ| ≤ 0.92), the temperature variance T 2 + increases monotonically as the rotation number increases from Ro2 = 0 to 0.145 (when the flow becomes quasilaminar), after which the trend reverses. Furthermore, it is observed that the value of T 2 + is the greatest in the channel centre at Ro2 = 0.273. Later in Section 3.4, we will refine the study by conducting a full budget balance analysis of the temperature

Fig. 11. Effect of wall-normal system rotation on the Nusselt number, Nu.

Fig. 12. Variation of wall friction velocities in the x1- and x3-directions, uτ1 and uτ3, respectively, effective wall friction velocity uτ,eff, and wall friction temperature Tτ with respect to rotation number Ro2. Wall friction temperature has been naturally normalized by the temperature difference between the top and bottom walls (see Fig. 1).

Fig. 13. Profiles of non-dimensional temperature variance ⟨T′2⟩ (non-dimensionalized by T 20 ) for the turbulent cases (0, 0018, and 0054), quasilaminar cases (0145, 0182, and 0273), and a laminar case (0546). 13

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Fig. 14. Turbulent heat fluxes, non-dimensionalized by uτ0Tτ0.

variance T 2 +. It will be explained that the greatest contributor to the temperature variance is the turbulent production term whose magnitude is dominated by both the mean temperature gradient d⟨T⟩/dxj and turbulent heat flux uj T . Fig. 14 shows profiles of the non-dimensionalized streamwise ( u1 T + ), wall-normal ( u2 T +), and spanwise ( u3 T +) turbulent heat fluxes, non-dimensionalized by uτ0Tτ0. Clearly, u1 T + peaks at Ro2 = 0, then decreases and eventually switches sign when the flow becomes quasilaminar (Ro2 = 0.145 0.273). In the near-wall region, the value of u1 T + decreases monotonically with an increasing rotation number Ro2. From Fig. 14b, it is clear that the profile of u2 T + remains almost flat in the channel centre at all rotation numbers, and furthermore, the value of u2 T + decreases monotonically as Ro2 increases. However, from Fig. 14c, it is interesting to see that u3 T + gradually increases with Ro2 until it reaches a maximum at the onset of the quasilaminar state (Ro2 = 0.145) . The magnitude of u3 T + then starts to decrease with an increasing Ro2 value (from Ro2 = 0.145 to 0.273), reflecting the trend of laminarization under the enhanced Coriolis effects. As the rotation number further increases beyond 0.273 (or, for Ro2 ≥ 0.546), the temperature variance and turbulent heat fluxes are negligible due to the laminarization of the flow, and therefore, the profiles of the turbulent heat fluxes at the very high rotation numbers are not plotted in the figure. On assuming the heat and fluid flows are both statistically steady and homogeneous in the x1-x3 plane, the following mean thermal energy equation can be obtained from the thermal energy equation (Eq. (4)), viz.

d u2 T dx2

=

d2 T dx 22

.

(13)

By integrating this equation with respect to the wall-normal coordinate x2, and considering the boundary condition (i.e., u 2 T = 0 at either wall), the following equation for heat flux can be derived:

q d T = w dx2 cp

u2 T

.

(14)

The above equation has a very clear physical meaning that the summation of the turbulent and molecular heat fluxes is equal to the wall heat flux identically. Considering further that d⟨T⟩/dx2 is always negatively valued, and the direction of the wall coordinate x2 measured from the upper and lower walls is opposite of each other with reference to the general coordinate system defined in Fig. 1, the following nondimensional heat-flux balance equation can be obtained by dividing both sides of Eq. (14) using uτ0 and Tτ0:

u2 T

+

±

1 d T + = T+ Pr dx 2+

.

(15)

The “+” and “−” signs between the turbulent and molecular heat flux terms on the left hand side of the above equation account for the wall coordinates measured from the upper and lower walls, respectively, and the wall friction temperature has been non-dimensionalized using the wall friction temperature of the non-rotating case, i.e. T + = T / T 0, on the right hand side of the above equation. Eq. (15) clearly indicates that the wall-normal turbulent heat flux u2 T + is linearly proportional to the wall-normal gradient of the mean temperature d T + / dx 2+ shown 14

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previously in Fig. 10. From Fig. 14b, it is seen that the magnitude of u2 T + is approximately 0.9 (close to unity) in the channel centre at low rotation numbers. This is because at very low rotation numbers, the flow is fully turbulent, and as such, heat transfer is dominated by the turbulent heat flux in the wall-normal direction in the channel centre. However, as the rotation number increases, the flow tends to become laminarized, associated with a decrease and an increase of the turbulent and molecular heat fluxes, respectively. As is evident in Fig. 10, after Ro2 exceeds 0.273, the flow becomes fully laminarized and all profiles 0 and the heat transfer proof T + collapse, consequently, u2 T + cess is dominated by the molecular heat flux term. Thus, from Eq. (15), the familiar linear law-of-the-wall (i.e., T / T = Pr·x 2+ ) can be readily Pr·x 2+ recovered with reference to the upper wall (and T / T = 1/ T with reference to the bottom wall).

diffusion, molecular diffusion, and molecular dissipation rate, respectively, defined as

= P + Dt + D + t

,

2

(17)

,

T2 x j2

(18)

,

T T xj xj

(19)

.

(20)

Because both the velocity and temperature fields are statistically steady and homogeneous in the x1- and x3-directions, the transport equation of the temperature variance reduces to:

In order to develop a deeper understanding of turbulent heat transfer under the wall-normal rotating condition, the transport processes of temperature variance ⟨T′2⟩ and turbulent heat fluxes ui T can be further studied. The transport equation of temperature variance reads:

D T Dt

2

,

2

xj

D =

3.4. Transport processes of temperature variance and turbulent heat fluxes

2

uj T

Dt =

=

T xj

2 uj T

P=

0=

D T2 = Dt

2 u2 T 2

d T dx2

d u2 T dx2

2

+

d2 T 2 dx 22

T T T T T T + + x1 x1 x2 x2 x3 x3

.

(21)

Fig. 15 compares the temperature variance T 2 + budget balance for the non-rotating (Ro2 = 0 ), turbulent (Ro2 = 0.054 ), and quasilaminar (Ro2 = 0.182 ) cases. By comparing the budget terms of T 2 + at these

(16)

α

where P, D , D , and ε are the turbulent production rate, turbulent

Fig. 15. Budget balance of the transport equation of temperature variance ⟨T′2⟩ for the non-rotating (Ro2 = 0 ), turbulent (Ro2 = 0.054 ), and quasilaminar (Ro2 = 0.182 ) cases. The summation of all terms on the right hand side of Eq. (21) are represented by Σ. All terms are non-dimensionalized by u 20 T 20/ . 15

International Journal of Heat and Fluid Flow 80 (2019) 108480

C. Bergmann and B.-C. Wang

three rotation numbers, it is apparent that the budget balance of T 2 + is dominated by the turbulent production term P + and molecular dissipation term ε. It is clear that as the rotation number increases from Ro2 = 0 to 0.182, although the peak value of the turbulent production term P + changes very little, the peak location moves towards the channel centre, furthermore, the level of P + increases in the channel centre. Correspondingly, the magnitude of the negatively-valued molecular dissipation term + also increases to counterbalance the turbulent production term in the channel centre. The turbulent and molecular diffusion terms, Dt + and D +, respectively, also exhibit a similar trend, in the sense that their peak values increase slightly and their peak location moves towards the channel centre as the rotation number increases. From our analysis of the velocity and temperature fields, it is understood that as the rotation number increases from Ro2 = 0 to 0.182, the flow transitions from a turbulent to a quasilaminar state under the enhanced Coriolis effects, and as discussed in Section 3.1, turbulence structures become larger which in turn facilitate near-wall transport of TSE, or temperature variance T 2 +. The turbulent production term P + is the dominant source term for the temperature variance T 2 +. In order to refine our study, the profiles of the turbulent production term at seven rotation numbers are compared in Fig. 16. The effects of wall-normal system rotation on the turbulent production term, in terms of the pattern of the gradual shift of its peak location from the wall towards the channel centre and the decrease of its magnitude in the near-wall region, are evident. In the near-wall region, the production term P + monotonically decreases in amplitude with an increasing rotation number Ro2. This has a profound impact on the magnitude of temperature variance T 2 + in the same region. A perusal of Fig. 13 indicates that in the near-wall region for |x2/δ| < 0.92, the magnitude of T 2 + indeed slightly decreases as the rotation number increases. Furthermore, because the near-wall peak value of P + shown in Fig. 16 reduces sharply as Ro2 increases, the relative importance of the production term P + in the channel centre compared to its peak value in the near-wall region enhances. Eventually at Ro2 = 0.273, the near-wall peak of P + diminishes almost entirely. Consequently, the relative importance of P + to the generation T 2 + is the strongest. This explains the previous observation that T 2 + peaks at Ro2 = 0.273 in the channel center in Fig. 13. Finally, from Figs. 15 and 16, it is interesting to observe that the profile of the turbulent production term P + is quasi-flat in the central region of the channel at both low and high rotation numbers. This can be explained as follows: by definition, P + = 2 u2 T + · d T + / dx 2+. As are evident in Figs. 10b and 14b, the values of both u2 T + and d T + / dx 2+ are approximately constant in the channel centre. Therefore, the value of P + must remain approximately constant in the same region. The observation that the values of both u2 T + and d T + / dx 2+ are approximately constant in the channel center at a given rotation number is not a coincidence, which can be well explained through Eq. (15). From Eq. (15), it is clear that the relationship between u2 T + and d T + / dx 2+ is strictly linear. Thus, if the value of one of these two variables is constant, so will be the other. The transport equation of turbulent heat flux ui T is:

D ui T Dt

= Piu + PiT + Dit + Di p + Di + Di +

i

+ Ci +

i

,

PiT =

ij

Di =

xj

Di =

Ci =

i

=

pT xj

1

2

(25)

,

(26)

T

ui xj

,

ui

T xj

,

xj

1

(24)

,

xj

Di p =

=

,

ui uj T

Dit =

i

T xj

ui uj

p

T xi

ijk

j

( + )

(27)

(28)

,

uk T

(29) (30)

,

ui T xj x j

.

(31)

Given that the flow and heat transfer are statistically steady and homogeneous in the x1-x3 plane, the transport equations for the three turbulent heat fluxes ui T (for i = 1, 2, and 3) can be further simplified, which are given in Appendix A. The budget balances of the three turbulent heat fluxes in the wall-normal direction are shown in Figs. 17–19. In each figure, three types of flows are compared for the non-rotating (Ro2 = 0 ), turbulent (Ro2 = 0.054 ), and quasilaminar (Ro2 = 0.182 ) cases. For the spanwise turbulent heat flux u3 T + budget, the non-rotating case is not shown because all terms are negligible. From Fig. 17, it is clear that in the budget balance of the streamwise turbulent heat flux u1 T + , the profiles of all budget terms are symmetrical about the channel centre ( x2 / = 0 ), a pattern that is consistent with the profile of u1 T + shown in Fig. 14a. As is evident in Figs. 17a and 17b, in the turbulent flow cases (for Ro2 = 0 and 0.054), the dominant sources of u1 T + are the two turbulent production terms, P1u + and P1T + (due to the mean velocity and temperature gradients, respectively). These two turbulent production terms are primarily balanced by the molecular dissipation term 1+, pressure-temperature t+ gradient correlation term + 1 , and turbulent diffusion term D1 , especially in the near-wall region. However, as the rotation number increases to Ro2 = 0.182, the flow becomes quasilaminar under the influence of the Coriolis force. Consequently, the profiles of the budget

(22)

where Piu represents the turbulent production due to the mean velocity gradients, PiT represents the turbulent production due to the mean temperature gradients, Dit denotes the turbulent diffusion, Di p is the pressure diffusion, Di is the viscous diffusion, Di is the molecular thermal diffusion, Πi represents the pressure-temperature gradient correlation, Ci is the Coriolis production, and εi denotes the molecular dissipation, which are defined as follows:

Piu =

uj T

ui xj

,

Fig. 16. Effect of system rotation on the production term of the temperature variance. All terms are non-dimensionalized by u 20 T 20/ .

(23) 16

International Journal of Heat and Fluid Flow 80 (2019) 108480

C. Bergmann and B.-C. Wang

Fig. 17. Budget balance of the transport equation of the streamwise turbulent heat flux u1 T in the wall-normal direction, for the non-rotating (Ro2 = 0 ), turbulent (Ro2 = 0.054 ), and quasilaminar (Ro2 = 0.182 ) cases. All budget terms shown in the figure have been non-dimensionalized using u 30 T 0/ .

produced by P2T + is primarily balanced by the pressure-temperature gradient correlation term +2 . As the rotation number increases, the magnitude of the pressure diffusion term D2p + increases rapidly, acting to transport u2 T + in the wall-normal direction. It is interesting to observe that although the wall-normal system rotation has a significant influence on the budget terms of u2 T + and the profile of u2 T + itself (shown in Figs. 18 and 14b, respectively), the direct contribution from the Coriolis term C2+ is zero identically. This is because from Eq. (30), it 2 u3 T + , C2+ 0, and C3+ 2 u1 T +. is straightforward that C1+ Clearly, the roles of the Coriolis production terms C1+ and C3+ are to transfer TSE (mutually) between the spanwise and streamwise turbulent heat fluxes u3 T + and u1 T + . The budget balance of the spanwise turbulent heat flux u3 T + for the turbulent (Ro2 = 0.054 ) and quasilaminar case (Ro2 = 0.182 ) are compared in Fig. 19. Because of the homogeneity of the velocity and temperature fields in the x1- and x3-directions, u3 T + 0 holds for non-rotating channel flows (shown clearly in Fig. 14c). Therefore, different from Figs. 17 and 18, the budget balance of u3 T + is not shown at Ro2 = 0 . From Fig. 19, it is clear that the budget balance of u3 T + is dominated by the two turbulent production terms due to the mean velocity and temperature gradients, balanced primarily by the molecular dissipation term 3+. The influence of the Coriolis term on u3 T + becomes greater with an increasing Ro2 value. From previous analysis of Fig. 8, it is understood that driven by the Coriolis force, there is a

terms are significantly different from those of turbulent flow cases at lower rotation numbers (at Ro2 = 0 and 0.054). Specifically, in comparison with Fig. 17a and 17b, it is clear in Fig. 17c that the magnitudes of the pressure-temperature gradient correlation term + 1 , turbulent diffusion term D1t +, viscous diffusion term D1 +, and Coriolis production term C1+ have increased significantly, to the extent that they are either larger than or on par with those of the two production terms, P1u + and P1T +. In particular, the pressure-temperature gradient correlation term + 1 acts as the dominant sink to balance the two turbulent production terms. Furthermore, it is very interesting to observe the profile trends of t+ these dominant terms (P1u +, P1T +, + 1 , and D1 ) at Ro 2 = 0.182 shown in Fig. 17c are opposite to those at a lower rotation number of Ro2 = 0.054 shown in Fig. 17b. This explains the physical phenomenon observed previously in Fig. 14a. Owing to the trend reversal of these budget terms, the streamwise turbulent heat flux u1 T + (shown in Fig. 14a) decreases in magnitude and eventually switches sign when the flow becomes quasilaminar as the rotation number increases. Fig. 18 compares the budget balances of the wall-normal turbulent heat flux u2 T + for the non-rotating (Ro2 = 0 ), turbulent (Ro2 = 0.054 ), and quasilaminar (Ro2 = 0.182 ) cases. From the figure, it is evident that the turbulent production term P2T + due to mean temperature gradient is the dominant source term, especially at lower rotation numbers. Because there is no mean wall-normal velocity gradient, the production due to mean velocity gradients is zero identically, i.e. P2u + 0 . The TSE

17

International Journal of Heat and Fluid Flow 80 (2019) 108480

C. Bergmann and B.-C. Wang

Fig. 18. Budget balance of the transport equation of the wall-normal turbulent heat flux u2 T in the wall-normal direction, for the non-rotating (Ro2 = 0 ), turbulent (Ro2 = 0.054 ), and quasilaminar (Ro2 = 0.182 ) cases. All budget terms shown in the figure have been non-dimensionalized using u 30 T 0/ .

Fig. 19. Budget balance of the transport equation of spanwise turbulent heat flux u3 T in the wall-normal direction, for the turbulent (Ro2 = 0.054 ) and quasilaminar (Ro2 = 0.182 ) cases. All budget terms shown in the figure have been non-dimensionalized using u 30 T 0/ .

mean spanwise velocity, whose profile shape is similar to that of the mean streamwise velocity. Further from the analysis of Fig. 14, it is apparent there is a common feature between the profiles of u1 T + and u3 T + in the sense that they are both symmetrical about the domain

centre ( x2 / = 0 ). By comparing Fig. 19 with Fig. 17, it is clear that the budget balance of u3 T + has many similar features to that of u1 T + , and therefore, we shall skip the detailed discussion of the rotation number effect on u3 T + to keep the analysis concise. 18

International Journal of Heat and Fluid Flow 80 (2019) 108480

C. Bergmann and B.-C. Wang

4. Conclusion

dimensional mean temperature T + decreases monotonically with an increasing Ro2 value in the near-wall region. At low rotation numbers, the flow is turbulent and the profile of ⟨T⟩/Tτ at all rotation numbers strictly obeys the linear law-of-the-wall only in the vicinity of the wall. However, as the rotation number exceeds Ro2 = 0.546, the flow becomes laminarized, and consequently, the temperature profiles over the entire channel width collapse to the linear law-of-the-wall. The Nusselt number decreases monotonically with an increasing rotation number, implying the effect of wall-normal system rotation is to reduce heat transfer. Eventually, as the flow reaches a laminar state at Ro2 = 0.546, the Nusselt number approaches an asymptotic value of Nu = 2 . The effects of the Coriolis force on the turbulent transport processes of the temperature variance and turbulent heat fluxes have been thoroughly investigated in terms of their budget balances. The budget balance of temperature variance is dominated by the turbulent production P + and molecular dissipation + . It is interesting to observe that the profile of the turbulent production term P + is quasi-flat in the central region of the channel at both low and high rotation numbers. This physical phenomenon is a consequence of the linear relationship between u2 T + and d T + / dx 2+ . For the streamwise and spanwise turbulent heat fluxes, turbulent production due to the mean temperature and velocity gradients (i.e., PiT + and Piu +, for i = 1 and 3) are the most important mechanisms for generating turbulent heat fluxes over all rotation numbers. However, for the wall-normal turbulent heat flux, the turbulent production due to mean velocity gradients is zero identically (i.e., P2u + 0 ), and turbulent production is solely due to the mean temperature gradients (i.e., the P2T + term). For the streamwise turbulent heat flux, as the rotation number increases, the magnitudes of the pressure-temperature gradient correlation term + 1 , turbulent diffusion term D1t +, viscous diffusion term D1 +, and Coriolis production term C1+ increase significantly, to the extent that they are either larger than or on par with those of the two production terms, P1u + and P1T +. For the wallnormal turbulent heat flux, the direct contribution from the Coriolis term is zero identically (i.e, C2+ 0 ). The role of the streamwise and spanwise Coriolis production terms C1+ and C3+ are to transfer TSE mutually between the spanwise and streamwise turbulent heat fluxes u3 T + and u1 T + .

The effect of wall-normal system rotation on turbulent heat transfer in a plane channel flow has been studied systematically using DNS for a wide range of rotation numbers, varying from Ro2 = 0 to 1.82 in conjunction with a fixed Reynolds number of Re = 180 . A pseudospectral method code has been used to perform the DNS, which offers high temporal and spatial accuracies in the calculation of the velocity and temperature fields. As the rotation number increases, heat transfer occurs under fully-turbulent, quasilaminar, and fully-laminar flow conditions. Our study shows that although the domain sizes used in the previous studies on DNS of wall-normal rotating channel flows conducted by Mehdizadeh and Oberlack (2010) can be used successfully for the prediction of the velocity field for a couple of fully-turbulent flow test cases, they are undersized for several other fully-turbulent and quasilaminar flow test cases, and are in general, insufficient for capturing the energetic structures associated with turbulent temperature fluctuations. The instantaneous turbulence structures associated with the temperature field have been studied qualitatively based on direct visualization of the temperature field and quantitatively based on 2D premultiplied spectra of temperature fluctuations. It is clear that the high temperature fluid near the bottom wall is forced upwards by large flow structures, which facilitate the transport of turbulent thermal energy across the channel. Based on a thorough spectral analysis, it is observed that the turbulence structures are oriented in the streamwise-spanwise directions, and their characteristic length scales vary as the rotation number increases. The largest characteristic length scales (with respect to both streamwise and spanwise wavelengths λ1 and λ3) occur in the quasilaminar state (for 0.145 ≤ Ro2 ≤ 0.273). Furthermore, the spanwise characteristic wavelength λ3 is always greater than the streamwise characteristic wavelength λ1 for Ro2 > 0.054. This also clearly indicates that there is a necessity to use a domain that is large enough (most importantly, in the spanwise direction) in DNS of heated turbulent wall-normal rotating channel flows in order to fully capture the largest length scales of turbulent temperature fluctuations. In a wall-normal rotating flow, the mean streamwise velocity u1 + monotonically decreases with an increasing value of Ro2 under the influence of a negatively-valued mean streamwise Coriolis force component; meanwhile, driven by the other spanwise Coriolis force component, there is a mean spanwise secondary flow with u3 + > 0, a pattern that is distinctively different from that of a non-rotating channel flow (featuring u3 + 0 ). Furthermore, in contrast to the non-rotating plane channel flow (Ro2 = 0 ), all six Reynolds stresses are non-zero if Ro2 > 0 because of the combined effects of the streamwise pressure gradient and the two Coriolis force components. Both u1 u1 + and u2 u2 + monotonically decrease as Ro2 increases in value. Once the rotation number increases beyond Ro2 = 0.546, all Reynolds stresses become negligible due to the laminarization of the flow. It is observed that the absolute value of the gradient of the non-

Declaration of Competing Interest The authors do not claim any conflict of interest with any people or party. Acknowledgments The financial support from Natural Sciences and Engineering Research Council (NSERC) of Canada to B.-C.Wang is gratefully acknowledged. Additionally, the authors would like to thank Western Canada Research Grid (WestGrid) for access to supercomputing and storage facilities.

Appendix A. Turbulent heat flux transport equations Under the condition that the flow and heat transfer are statistically steady and homogeneous in the x1-x3 plane, the transport equations for the three turbulent heat fluxes ( u1 T , u2 T , and u3 T ) can be further simplified from Eq. (22) to:

0=

D u1 T = Dt

u2 T +

d u1 dx2

d T u1 dx2 x2 ( + )

u1 u2 +

1

dT dx2 p

T x1

d u1 u2 T dx2 2

2

u d T 1 dx2 x2

+

u3 T

u1 T u1 T u1 T + + x1 x1 x2 x2 x3 x3

,

(32)

19

International Journal of Heat and Fluid Flow 80 (2019) 108480

C. Bergmann and B.-C. Wang

0=

D u2 T = Dt

u2 u2 +

d T dx2

d T u2 dx2 x2 ( + )

0=

D u3 T = Dt

u2 T +

d u2 u2 T dx2 1

p

+

u d T 2 x2 dx2

T x2

u2 T u2 T u2 T + + x1 x1 x2 x2 x3 x3

,

d u 2 u3 T dx2

+

d u3 dx2

d T u3 dx2 x2 ( + )

+

1d pT dx2

u 2 u3 +

1

d T dx2 p

T x3

+2

2

(33)

u d T 3 dx2 x2

u1 T

u3 T u3 T u3 T + + x1 x1 x2 x2 x3 x3

.

(34)

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