Mixed convection study on the influence of low Prandtl numbers and buoyancy in turbulent heat transfer using DNS

Research Collection

Journal Article

Mixed convection study on the influence of low Prandtl numbers and buoyancy in turbulent heat transfer using DNS

Author(s):

Guo, Wentao; Prasser, Horst-Michael Publication Date:

2021-08

Permanent Link:

https://doi.org/10.3929/ethz-b-000479183

Originally published in:

Annals of Nuclear Energy 158, http://doi.org/10.1016/j.anucene.2021.108258

Rights / License:

Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International

This page was generated automatically upon download from the ETH Zurich Research Collection. For more information please consult the Terms of use.

ETH Library

Mixed convection study on the influence of low Prandtl numbers and buoyancy in turbulent heat transfer using DNS

Wentao Guo ^⇑ , Horst-Michael Prasser

Department of Mechanical and Process Engineering, ETH Zurich, Switzerland

a r t i c l e i n f o

Article history:

Received 28 November 2020

Received in revised form 31 January 2021 Accepted 10 March 2021

Keywords:

DNS

Turbulent heat transfer Mixed convection Liquid metal

a b s t r a c t

Direct numerical simulation (DNS) is performed to study turbulent heat transfer in Poiseuille-Rayleigh- Bénard (PRB) flows with low Prandtl numbers in this article. The mesh is Cartesian and a highly accurate finite difference sixth-order compact scheme is chosen to discretize the incompressible Navier–Stokes equations to perform DNS. Liquids with a fixed Richardson number of 0.25 and four different Prandtl number (Pr = 0.025, 0.05, 0.1, 0.71) are simulated and compared with Poiseuille flow to investigate the influence of Prandtl number and buoyancy on –PRB flows. Constant fluid properties and Boussinesq approximation are assumed. The obtained results are discussed and analysed in an extensive way in this study. Specifically, buoyancy initiate large scale circulation and the scale shrinks with the increasing of Prandtl number. Velocity fluctuations become stronger with PRB flow which indicate that buoyancy can strongly enhance the turbulent intensity. Re s is increased in the cases of low Pr. Moreover, when Pr decreases, temperature distributions are found to be more homogeneous and mixing of the fluids is more sufficient in the middle of the channel. Additionally, the scale of the large-scale structures is enlarged in mixed convection compared with forced convection. This can be observed in the temperature field of low-Prandtl-number fluids. It is also observed that Reynolds analogy cannot be used to predict the thermal field under mixed convection or forced convection with low Prandtl number. The research results can be used for the R&D of Gen IV nuclear fast reactors.

Ó 2021 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

The heat and momentum transfer caused by the combined effect of buoyancy and shear within a fluid flow is named as mixed convection. This is a classical phenomenon related to heat transfer in nuclear reactors. In some fast nuclear reactors such as sodium- cooled fast reactors (SFR) and lead-cooled fast reactors (LFR), liquid metal is used as coolant. It circulates in the primary circuit under mixed convection conditions and moves the heat generated from the reactor core to the primary heat exchanger. To improve the safety features and efficiency of LFR and SFR, it is important to understand the mixed convection phenomenon of liquid metal, which usually has a low Prandtl number. Mixed convection is the combination of forced and natural convection. However, the result- ing flow field and the heat transfer intensity are not simply the combination of the two. Reynolds-averaged Navier–Stokes (RANS) type turbulent models are helpful to study mixed convection. It can be used for simulating large domains and realistic engineering

problems due to the low cost of computational resources. How- ever, the results from the RANS models need to be validated against experimental or direct numerical simulation (DNS) data before we can trust them. Due to the absence of experimental data on turbulent parameters for liquid metals, such as Reynolds stress and turbulent heat fluxes, RANS approaches can only be validated and compared with DNS data. So far as we know, the RANS models cannot be directly used to simulate the turbulent heat transfer of liquid metals. So new RANS models could be developed from DNS data. In the framework of the EU Horizon 2020 thermal hydraulics Simulations and Experiments for the Safety Assessment of MEtal-cooled reactors (SESAME) project, a series of high-fidelity DNS databases are created. The PRB flow of low-Prandtl-number fluids in a horizontal channel is one of them, which is also the main topic of this article.

In order to study the heat transfer behavior in a turbulent flow, a number of simulations has been performed. As for the forced con- vection, Poiseuille flow is a standard configuration and has been used extensively. It represents the pressure-driven turbulent flow flowing through a plane channel. The DNS of forced convection channel flow started from Kim et al. (1987). Further studies are carried out such as Jiménez and Moin (1991), who studies https://doi.org/10.1016/j.anucene.2021.108258

0306-4549/Ó 2021 The Author(s). Published by Elsevier Ltd.

This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

⇑ Corresponding author.

E-mail addresses: guow@student.ethz.ch (W. Guo), prasser@lke.mavt.ethz.ch (H.-M. Prasser).

Contents lists available at ScienceDirect

Annals of Nuclear Energy

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / a n u c e n e

turbulent flows in various narrow boxes without heat transfer.

Zonta et al. (2012) finds that viscosity which depends on tempera- ture can affect turbulence in Poiseuille flow with differentially heated walls. The temperature is regarded as passive scalar and the turbulent heat transfer is with respect to Prandtl number and Reynolds number. Besides, studies with conjugate heat transfer (Flageul et al., 2015; Tiselj and Cizelj, 2012; Tiselj et al., 2001) and imposed thermal boundary conditions (Kasagi et al., 1992;

Kawamura et al., 1999; Tiselj et al., 2001; Bricteux et al., 2012;

Kozuka et al., 2009; Kasagi and Iida, 1999; Pirozzoli et al., 2016) are performed.

As for the natural convection regime, the Rayleigh-Bénard con- vection (RBC) is a prototype case and has been widely studied, both numerically (Grötzbach, 1982; Silano et al., 2010; Scheel and Schumacher, 2016) and experimentally (Otic´ and Grötzbach, 2005; Akashi et al., 2019). The fluid is heated up from bottom and cooled down from top so it is driven by buoyancy. The two controlling parameters for RBC flow are Prandtl and Rayleigh number.

A prototype case for mixed convection study is Poisueille-Ray leigh-Bénard (PRB) flow. In PRB flows, a complex flow structure is produced by the combination of pressure driven (Poiseuille) and buoyancy driven (Rayleigh-Bénard) effects. The structures depend on the relative intensity of Grashof number and Reynolds number. Mixed convection has been investigated both numerically and experimentally, but publications are less frequent as forced and natural convection due to its challenging nature. Iida and Kasagi (1997) build a DNS database to study PRB flow of air with Pr = 0.71. Davidson et al. (2003) performed DNS of fluid flowing in a vertical channel for a unitary Prandtl number. They point

out that the shear stress is strongly affected by buoyancy in mixed convection. Kath and Wagner (2014) studied PRB flows with the smallest acceptable computational box size for a Prandtl number equals to 0.71 using LES and DNS. Zonta and Soldati (2014) used DNS to investigate PRB and Poiseuille flow with Boussinesq and non-Boussinesq assumption with a Prandtl number equals to 3.

Pirozzoli et al. (2016) performed DNS for PRB flows at a unitary Prandtl number in a wide range of Re and Ra numbers. Besides simulations, some experimental studies on unstable stratification effects and PRB flow can be found in Komori et al. (1982), Keisuke and Masamoto (1985) and Keisuke et al. (1991).

As mentioned above, for PRB flows, most studies focus on Pr close to or larger than 1. The experiments for low-Prandtl- number fluids are difficult due to: 1. For Pr < 0 : 2, liquid metals such as mercury, gallium and liquid sodium are opaque. So optical access to the flow by laser-imaging techniques such as laser Dop- pler velocity measurement (LDV) or particle image velocimetry (PIV) are inapplicable; 2. the thermal conductivity of the boundary materials, which are usually copper plates, is not significantly higher than that of the liquid metals. The temperature of the plates varies during the experiment and will change the magnitude of the heat transfer because of the Robin boundary condition (Scheel and Schumacher, 2016). So fundamental experimental studies at isothermal boundary conditions are practically impossible. DNS of PRB flows of liquid metals is also challenging due to the appear- ance of large-scale structures. A larger domain compared to cases with a unitary Prandtl number is necessary for capturing them (Guo et al., 2020; De Santis et al., 2018). This increases the compu- tational cost significantly. This challenge has to be faced in studies of the influence of low Prandtl number and buoyancy on heat Nomenclature

Roman symbols

C p specific heat at constant pressure d r averaged relative difference g gravitational acceleration Nu Nusselt number

Pr Prandtl number

Pr

turbulent Prandtl number

p pressure

q w wall heat flux Ra Rayleigh number Re Reynolds number Re s friction Reynolds number Ri Richardson number T temperature

T _h temperature on the hot wall T c temperature on the cold wall

t time

U b bulk velocity u _s friction velocity u ; v ; w velocity component x streamwise direction y wall-normal direction z spanwise direction Greek symbols

a thermal diffusivity

b coefficient of thermal expansion d channel half height

h dissipation rate of temperature variance h non-dimensional temperature

h _s friction temperature k thermal conductivity

t kinematic viscosity

q density

g statistical deviation rate Subscripts

()

root-mean-square ()

wall value Superscripts

()

normalized by u s or h _s () 0 fluctuation component ðÞ statistically averaged d averaged over Acronyms

CFD computational fluid dynamics CFL Courant-Friedrichs-Lewy DNS direct numerical simulation FTT flow-through times FVM finite volume method

LDV laser Doppler velocity measurement LES large eddy simulation

LFR lead-cooled fast reactor PIV particle image velocimetry PRB Poiseuille-Rayleigh-Bénard

RANS Reynolds-averaged Navier–Stokes equations

transfer for turbulent PRB flows which are needed to build DNS databases for RANS validation.

Most simulation studies mentioned above used spectral method or second-order finite volume method (FVM) to discretize the gov- erning equations. The advantage of the spectral method is that it has a high-order derivative approximation, but it requires the geometry to be simple, and on sharp boundaries it will lose accu- racy. Moreover, implementing physical models in a spectral solver is not easy. The second-order FVM is widely used in commercial computational fluid dynamics (CFD) software. However, when per- forming a DNS study, second-order accuracy schemes are not ideal for equation discretization in space. For simulating small turbulent scales down to the Kolmogorov length scale, the mesh density required is huge. It requires excessive computational resources.

The code used in our DNS study is Incompact3d ( https://www.in- compact3d.com/). It is an open source code developed at the University of Poitiers and Imperial College London by Laizet and Lamballais (2009) and Laizet and Li (2011) It has a superior versa- tility than spectral codes and a higher accuracy than industrial codes. The code uses a six-order-accurate finite difference compact scheme (Laizet and Lamballais, 2009) to discretize governing equa- tions and a hybrid second-order Crank-Nicolson and Adam- Bashforth scheme for the temporal discretization. Crank-Nicolson is an implicit scheme which is only used for treating the second- order derivative in the wall-normal direction.

In this study, the result of several DNS of the mixed convection channel flow at moderate Reynolds number and low Prandtl num- bers are presented. The influence of Pr on PRB flow in a horizontal channel is studied and the results are shown in Section 3. A DNS database of PRB flows for low-Prandtl-number fluids is generated and can be used to analyze the mechanism of heat transfer within liquid metals and to improve and validate RANS models. The data- base follows the creative commons attribution license (CC-By) and it is accessible from the following website: https://doi.org/10.

3929/ethz-b-000468723.

2. Numerical methodology 2.1. Flow configuration

The flow configuration is a horizontal channel which is shown in Fig. 1. An iso-thermal Dirichlet-type boundary condition is applied for the temperature at the wall boundaries. The bottom wall is the hot one and kept at T

, while the top wall is cold and kept at T

. The letter d represents the channel half height. This is a typical PRB flow, where the buoyancy forces induced by the tem- perature difference D ^T

¼ ð T

T

Þ cause an unstable stratifica- tion (the acceleration of gravity acts downward along y - direction). The letters x ; y and z represent the streamwise, the wall-normal, and the spanwise coordinates respectively. In the x and in z-direction, periodic boundary conditions are applied and in the y-direction, a no-slip boundary condition for velocity is used.

The fluid moves at an imposed constant mass flow rate.

2.2. CFD solver

In our DNS with the code Incompact3d, (1) the Oberbeck- Boussinesq approximation is applied, which means the fluid has uniform thermal-physical properties and (2) the working fluid is assumed to be incompressible. The continuity equation, the momentum equation and the energy balance equation are the gov- erning equations for the simulation and they are defined in a non- dimensional way as:

@u i

@x i ¼ 0; ð1Þ

@u i

@t þ @

@x j

u j u i

¼ @p

@x i þ 1 Re b

@ ² u i

@x j @x j þ fd i1 þ Ri b d i2 h; ð2Þ

@h

@t þ @

@x j

u j h

¼ 1 Re b Pr

@ ² h

@x j @x j : ð3Þ

Reference velocity and length are non-dimensionalized by the bulk velocity, U

and the channel height, 2d respectively. The pres- sure is non-dimensionalized by q U

. In Eq. (2), fd

is added as the momentum source term to keep a constant mass flow rate. f is a time-dependent value. It is constant in space and it is varied in a certain way to keep the averaged bulk velocity at unity. d

is the symbol of the Kronecker delta. Ri

d

hrepresents the buoyancy and his the non-dimensional temperature which is defined as h ¼ ð T T

Þ= D ^T

. The expression of bulk Reynolds number is Re

¼ 2U

d = t . The Richardson number and the Grashof number are defined as Ri ¼ Gr = ð Re

Þ

and Gr ¼ gb D T

ð Þ 2d

= t

respectively.

The spatial discretization schemes are six-order compact finite difference schemes based on Cartesian grids. The continuity, momentum and energy equations of the working fluid with con- stant properties are solved using a projection method (Flageul et al., 2015):

u

u

Dt ¼ 3

2 1 2

@ u

u

@x

1

2 u

@ u

@x

þ 1 Re

@

u

@x

þ 1 Re

@

u

@z

0

@

1 A

1 2 1

2

@ u

u

@x

1

2 u

@u

@x

þ 1 Re

@

u

@x

þ 1 Re

@

u

@z

0

@

1 A

þ 1 Re

@²u

@y²i

þ

2h0

1 Re

h00 h0

3 2

@u

@y 1 2

@u

@y

þ F

ð p

; h

Þ;

ð4Þ

u ^nþ1 _i u _i D t ¼ @

@x i

p ^{n þ 1} þ @

@x i

p ⁿ ð5Þ

Eq. (4) can be reformed in a condensed way:

1 D ^t 2Re b h0 ² @ yy

u _i ¼ G i p ⁿ ; u ⁿ _j ; u ⁿ _j ¹

D ^t ð6Þ

The same strategy can be used for the energy equation:

1 D t 2Re b Prh0 ² @ yy

T ^nþ1 ¼ G T T ⁿ ; T ⁿ¹ ; u ⁿ _j ; u ⁿ¹ _j

D ^t ð7Þ

In Eq. (4), u

represents the tentative velocity. F

ð p

; h

Þ contains pressure, buoyancy force and forcing acceleration terms. In the momentum equation, the convective term is computed using the skew-symmetric form.

Owing to the stability limitation related to the diffusion term in the energy equation for low-Prandtl-number fluid, a semi-implicit scheme is applied. The viscous and convective terms are integrated through a second order Adams–Bashforth scheme except for the wall-normal viscous term, which is integrated through a Crank- Fig. 1. Sketch of the computational domain with the coordinate system and the

flow (Guo et al., 2020).

Nicolson time advancing scheme. Dairay et al. (2014) and Flageul et al. (2017).

A validation DNS of PRB flow is performed to evaluate the precision of Incompact3d when running DNS. The DNS data from Iida and Kasagi (1997) is used as the benchmark data for this purpose. The simulation results from Incompact3d are in line with the benchmark DNS data, which validates the capabilities of Incompact3d to perform high fidelity DNS, especially for PRB flow.

2.3. Test cases

To figure out the influence of low Prandtl number and buoy- ancy, four test cases (case 2–5) with different Pr numbers are con- sidered under a constant Re

which is 4667. A forced convection case (case 1) with Pr = 0.025 is used for comparison. Pressure drop and Re _s change with Pr numbers at the constant mass flow rate.

The bulk velocity U

is 0.667. The size of the domain is 10 p d 2d 4 p d. A similar meshing strategy, which follows the experience gained from the validation case in A, is used for all five test cases. Therefore, the mesh for the simulation has the size of 512 256 512. For each case, time-steps are different and CFL numbers are kept below 0.1. Due to the high resolution of the mesh, the consumption of the computational resource is relatively large. The simulations are run on the supercomputer Cray XC40 with 2048 cores for parallel computing and 5.25 million core hours are consumed. Table 1 reports the parameters used in the simula- tions for the five cases.

The friction Reynolds number is Re s ¼ u s d

m ^ð8Þ

where u s is the friction velocity:

u s ¼ U b

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Re b

du dy s

ð9Þ Y

is the non-dimensional wall distance between the wall and the first node next to it. In the wall normal direction (y- direction), a stretched mesh is used. In the other two directions, the mesh has an equal spacing. D ^x

; D ^y

^and D ^z

are the non- dimensional cell spacings in all three directions, which are perpen- dicular to each other. Due to the fact that the mesh is stretched in y-direction, here D ^y

^{represents the maximum cell size.}

Y

; D ^x

; D ^y

^and D ^z

are non-dimensionalized in the following way (all three coordinates are treated the same way):

x ^þ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi Re b

du dy s

x: ð10Þ

du = dy in Eq. (9) and Eq. (10) is calculated at the wall boundary.

The Nusselt number is defined below:

Nu ¼ 2q w d

k ð h i T T w Þ ð11Þ

where is the bulk-averaged value over d. FTT represents flow- through times, which means the number of passes of the fluid through the domain during the averaging time of the statistics:

FTT ¼ U b t

L x ð12Þ

2.4. Computational domain

A low-Prandtl-number fluid has higher thermal diffusivity or conductivity compared with the fluids with Pr P 1. Accordingly, in the thermal field of low-Prandtl-number flows, the scales of the structures are larger compared to those in the velocity field (Shams, 2017). The particularity has to be considered when choos- ing a computational domain to simulate low-Pr fluids. As indicated by Tiselj (2014), the computational domain selected for investigat- ing the Poiseuille flow of low-Pr fluid needs to be larger than for unitary Prandtl number fluids. He performed simulations of forced convection with Pr equals to 0.01 and presumed that a computa- tional domain with L

L

L

¼ 12 p d 2d 4 p d should be ade- quate to hold large-scale structures in the thermal field

compared to the conventional domain

(L

L

L

¼ 4 p d 2d 2 p d) selected for the unitary Prandtl number Poiseuille flow simulation (Shams and Komen, 2018). In A, the DNS of Iida and Kasagi (1997) is discussed, where the domain is L

L

L

¼ 5 p d 2d 2 p d, which is large enough for Pr equals to 0.71. Nevertheless, if the same domain is used for Pr = 0.025, non-physical results might appear. So in this study, L

L

L

¼ 10 p d 2d 4 p d is used as the computational domain. In order to verify if the domain is large enough for the con- sidered flow configuration, a two-point correlation is calculated for Ri equals to 0 and 1 with Pr = 0.025 and the results are presented in our previous publication. The details of the two-point correlation study can be found in Guo et al. (2020). It is adequate to support the claim that the domain is sufficient for DNS of PRB flow with a low Prandtl number.

2.5. Convergence

The data is averaged in two temporal windows with the same period of time. They are compared in order to prove the results are statistically converged. The definition of the statistical devia- tion rate g is shown below:

g ¼ 1 n

X ⁿ

i ¼ 1

f 2 ð Þ y i f 1 ð Þ y i

f ₁ ð Þ y _i

; ð13Þ

where f

ð Þ y

and f

ð Þ y

are the averaged data from the plane at posi- tion y

in the temporal windows 1 and 2 respectively. n represents the index of the node in y-direction. The simulations have run an adequate time (FTT > 5) before testing the convergence. The statis- tics from both time windows have been averaged in space and time and they are compared to calculate g . The statistical deviation rates of the data shown in this paper are less than 1%. This proves the data presented are converged statistically. In Section 3, the tempo- ral window applied is the FTT given in Table 1.

Table 1

Summary of the parameters used in the cases

Ri Pr Gr Re s h _s Nu Y

Dx

(const.) Dy

(max.) Dz

(const.) FTT Dt

Case 1 0 0.025 0 151 0.145 4.25 0.42 9.27 3.32 3.71 25 4:1 10

Case 2 0.25 0.025 5:3 10

171 0.361 7.90 0.47 10.49 3.76 4.20 25 3:1 10

Case 3 0.25 0.05 5:3 10

170 0.241 9.74 0.47 10.43 3.74 4.17 21 3:1 10

Case 4 0.25 0.1 5:3 10

167 0.154 11.82 0.46 10.27 3.67 4.11 21 3:0 10

Case 5 0.25 0.71 5:3 10

163 0.039 19.70 0.45 9.98 3.58 3.99 21 3:0 10

3. Results and discussions 3.1. Flow field analysis

Figs. 2–4 demonstrate the velocity distributions for the five cases shown in Table 1 respectively. The velocity distributions on a constant z and x plane are shown on the left and right side. The distribution of the velocity shows that the flow is turbulent and small back-flows (u < 0) can be observed due to turbulence.

The streamwise velocity component for Poiseuille flow (case 1) and PRB flows (case 2) with Pr = 0.025 has similar flow topology.

Mixed convection enhances the level of turbulence. When Pr increases while Ri is kept constant at 0.25 (case 2–5), the level of turbulence is slightly decreased in the near wall region and the velocity distribution tends to be more homogeneous. The forced convection case (case 1) has the lowest turbulence level, and the topology is close to mixed convection with relatively high Pr num- ber (case 5). This is consistent with the Re _s shown in Table 1. In Fig. 3, the v -velocity is close to zero for Poiseuille flow (case 1).

Mixed convection will strengthen the large scale circulation for low-Prandtl-number fluid. The v -component of the velocity field for mixed convection with Ri = 0.25 (case 2–5) shows that fluid rises from the lower wall and shows periods of descending streams everywhere in the domain. The initial velocity of the plume close to the bottom wall is low, but it is accelerated by buoyancy. However, this phenomenon becomes less obvious for higher Pr fluids. This is due to the larger momentum diffusivity and velocity tends to be more homogeneous for higher Prandtl number fluids. The span- wise velocity field also reveals that the scales of circulations and the turbulence levels are enlarged for PRB flow.

Fig. 5 displays the mean velocity of the 5 cases. The normalized velocity is defined below:

u ^þ ¼ u

u s ð14Þ

It demonstrates that the velocity profile in the bulk region for PRB flows is flatter compared to forced convection due to higher turbulence levels, which reveals that the fluid is mixed better.

Besides, this phenomenon is stronger at lower Prandtl numbers.

As indicated by the variation of Re s in Table 1, the main effect of a decreasing Prandtl number is the enhancement of the turbulent level near the wall as well as of the wall-normal momentum trans-

port. The boundary layer becomes thinner due to the increase of skin friction in PRB flow. The viscous sublayer, the buffer layer and the log-law region are distinguishable in case of forced convection.

The root-mean-square (RMS) of the velocity components fluctu- ation and the Reynolds shear stress are shown in Fig. 6. The plots show that the values of u

; v

; w

and u

v

in PRB flows are higher than those in Poiseuille flow. The difference becomes more obvious when Pr is decreased. In other words, the velocity fluctu- ation and the turbulent intensity is enhanced with decreasing Pr.

In the Poiseuille flow regime, u

is reduced in the bulk region.

However, in the mixed convection regime, the profile of u

is flat.

The profile of v

is different in forced and mixed convection. v

increases monotonously along the whole half-channel height in the mixed convection regime because turbulence is enhanced by buoyancy. The increase of the RMS of the velocity fluctuations in three directions of mixed convection is because of the ejection of plumes near the wall. The plumes that rise up from the bottom plate initiate cross-stream eddies. The increase of wall shear stress and stronger perturbation effects due to the cross-stream eddies enhance the fluctuation of the velocity. The Reynolds shear stresses reach the maximum at y

30 in the Poiseuille flow regime. In the PRB flow regime, the peak moves towards the wall and increases when the Prandtl number decreases. These results indicate that the fluxes in wall-normal direction of the streamwise turbulent momentum is intensified by the synergistic effect of thermal diffu- sion and buoyancy so that momentum is spread vertically.

3.2. Temperature field analysis

The temperature field is shown in Fig. 7. The forced convection and mixed convection display different temperature distribution pattern. In forced convection flow (case 1), temperature propaga- tion in wall-normal direction is dominated by thermal conduction.

So the temperature drops along the wall-normal direction of the channel almost linearly (see Fig. 8). In the mixed convection regime (case 2–5), both conduction and convection contribute to the energy transport and thermal plumes are observed. The tem- perature gradient in the PRB flow is larger than that of forced con- vection and large eddies are generated from the unstable thermal boundaries. The hot fluid has a lower density and rises from the bottom of the wall due to buoyancy, while cold fluid moves down to the bottom. The conduction from the hot wall heats the cold

Fig. 2. Instantaneous distribution of the streamwise (u) velocity component at z ¼ 3:14 (left) and x ¼ 7:85 (right).

fluid and generates the next eruptions. The hot fluid rising from the bottom of the domain is called thermals, or thermal plumes (Xi et al., 2004), which is a remarkable feature of mixed convection (Ahlers et al., 2009). The thermals penetrate through the entire domain and are deflected near the upper wall. The deflection of thermals causes a strong divergence of the velocity and produces the observed large scale convection. Coherently, small-scale circu- lations occur within thermals and between adjacent thermals. This typical phenomenon of mixed convection is highlighted in Parodi et al. (2004). The characteristic dimension of the circulation is affected by the Prandtl number. When Pr decreases, the scale of the circulation becomes larger and the characteristic dimension becomes limited by the channel height.

In Fig. 8, the temperature profiles are reported in bulk and fric- tion units. The temperature in Fig. 8(b) is normalized by the fric- tion temperature:

h ^þ ¼ h

h s ð15Þ

Fig. 3. Instantaneous distribution of the wall-normal ( v ) velocity component at z ¼ 3:14 (left) and x ¼ 7:85 (right).

Fig. 4. Instantaneous distribution of the spanwise (w) velocity component at z ¼ 3:14 (left) and x ¼ 7:85 (right).

Fig. 5. Mean velocity profile.

h _s is the friction temperature:

h s ¼ q w

q C p u s ð16Þ

From Fig. 8(a) we can see that the major influence of increasing Pr consists in increasing the gradient near the heated plate and decreasing it in the bulk region, which reduces the thermal bound- ary layer thickness. Consequently, higher heat transfer rates are

observed. In the bulk region in case of mixed convection, the tem- perature has a flat profile but in case of forced convection, the tem- perature decreases along the wall-normal direction of the channel.

This is because PRB flow has an enhanced turbulent mixing because of buoyancy and the thermal diffusion of forced convec- tion mainly depends on conductivity alone. The mean temperature profile for PRB flow consists of two zones. The bulk region is well mixed, which spreads over a sizable part of the channel, and Fig. 6. (a) RMS of u

(b) RMS of v

^{(c) RMS of w}

(d) Reynolds shear stresses.

Fig. 7. Instantaneous temperature fields at z ¼ 3:14 (left) and x ¼ 7:85 (right).

separate boundary layers near the two plates. The friction temper- ature decreases with increasing Pr. Moreover, when increasing Pr, the wall heat flux increases, which means turbulent heat transfer is enhanced with higher Pr number.

The temperature fluctuation profile is plotted in Fig. 9. The RMS of temperature fluctuation is more significant for mixed convec- tion. It rises in the viscous and the buffer regions and descends in the bulk flow region. A better turbulent mixing due to buoyancy and a higher thermal diffusivity can reduce the intensity of thermal fluctuation globally. In the center of the domain, temperature fluc- tuations are distributed more homogeneously for the same reason.

The drag force near the wall generates more turbulence and because a high Pr fluid has less thermal diffusivity compared to the momentum diffusivity, the temperature fluctuation increases in the bulk region.

The streamwise and wall-normal turbulent heat fluxes are plot- ted in Fig. 10. The peak of the streamwise component for mixed convection appears in the buffer region and it is higher compared to the Poisueille flow regime. For a low-Pr fluid, the ratio of ther- mal energy transported through conduction to convection is higher. Consequently, the turbulent heat flux is reduced with a decreasing Prandtl number. The transport of energy by convection in streamwise direction is stronger for PRB flow than Poisueille flow. The buoyancy force generates thermal plumes near the wall boundary, so more energy is transported in wall normal direction by convection and the wall-normal turbulent heat flux increases.

The dissipation rate of the temperature variance and the turbu- lent Prandtl number is shown in Fig. 11. The dissipation rate of temperature variance is defined below:

e h ¼ a ^{@h0 þ}

@x ^þ

2

þ @h0 ^þ

@y ^þ

2

þ @h0 ^þ

@z ^þ

2 !

ð17Þ It increases in the viscous and the buffer regions with increasing Pr. The buoyancy forces and the enhanced momentum diffusivity lead to a higher temperature fluctuation gradient in the viscous and the buffer regions.

The definition of turbulent Prandtl number is the ratio between the momentum eddy diffusivity and temperature eddy diffusivity:

Pr t ¼ u0 v ⁰

v ^{0h0 dh=dy du=dy ð18Þ}

Pr

is assumed to be a constant number around 0.9 according to the Reynolds analogy, which is applied in a lot of turbulent models (Kasagi and Ohtsubo, 1993; Kays, 1994; Shams et al., 2014). How- ever, this simplification becomes invalid when simulating mixed convection flows and low-Prandtl-number fluid (Grötzbach, 2007). Because the temperature profile for Pr = 0.1 and Ri

= 0.25 is inversed in the bulk region, the sign of d h = dy is changed along the height of the channel. Consequently, the value of Pr

becomes negative in the bulk region, which is also observed by other researchers (Sid et al., 2015). Fig. 11(b) shows that Pr

in the mixed convection regime varies along the channel and it devi- ates from 0.9 significantly in the buffer and the bulk regions. The variation of Pr

along the channel is qualitatively comparable in the mixed convection regime and the upper bound is around 1 appearing in viscous sublayer. In bulk region, the wall-normal tem- perature gradient approximates to zero because buoyancy force enhances turbulent mixing. There, Pr

drops to zero. It means that the momentum eddy diffusivity is dominant in the near wall region, whereas in the bulk region, the heat transfer by eddy diffu- sivity is dominant, because of an enhanced turbulent mixing by buoyancy. In the forced convection regime of a low-Pr fluid, Pr

also varies along the channel. It increases starting from the wall to reach a maximum value of 3.03 and decreases afterwards mono- tonically towards the middle of the channel. Compared with mixed convection regime, its profile displays much higher amplitudes (Kasagi and Ohtsubo, 1993). This is mainly due to the higher tem- perature gradient in the bulk region in forced convection flow than mixed convection flow.

Fig. 8. (a) Mean temperature profile h (b) Mean temperature profile in friction units.

Fig. 9. RMS of temperature fluctuations.

4. Conclusions

Mixed convection is a common and important phenomenon in the nuclear industry. However, the mechanisms behind it are still not clear and the number of studies on this topic is limited. The present study investigates the PRB convection inside a channel with systematically varied Pr numbers and compares its influence on the turbulence statistics and the turbulent heat transfer. We focus our study on low-Prandtl-number fluids because they are characteristic for relevant coolant options for fast reactors. Turbu- lent statistics are calculated from velocity and temperature fields generated by DNS and analysed. These are the main findings:

1. In the mixed convection regimes, the velocity field is affected by turbulence and thermal diffusion. The velocity profile is flat- tened under the influence of buoyancy. The RMS of velocity fluctuations are larger in bulk region than Poiseuille flow regimes and the difference is accentuated with decreasing the Prandtl number.

2. The increase of Re _s indicates that buoyancy contributes to tur- bulence near the wall. This effect is attenuated with increasing the Prandtl number. A possible explanation of this phenomenon could be that the low-speed vortices are intensified due to buoyancy-driven instabilities and they are confined by the large scale circulation generated by thermal plumes.

3. Buoyancy can contribute to thermal convection. The heat and momentum transfer in the wall-normal direction is enhanced.

Large-scale thermal structures are observed in the PRB flow regime for low-Pr fluids. The size of large-scale structures grows

with decreasing the Prandtl number. Compared to forced con- vection, the thermal plumes are also larger in the mixed con- vection regimes. For the DNS of mixed convection with low- Prandtl-number fluids, a larger simulation domain is thus necessary.

4. The thermal field is related to the combined effects of the Rey- nolds and Prandtl numbers. The temperature is distributed more homogeneously in the bulk region in case of a PRB flow.

When increasing the Prandtl numbers, the temperature gradi- ent grows in the viscous sublayer and falls off in the buffer and the bulk regions. The thermal boundary layer thickness attenuates with increasing Pr.

5. Temperature fluctuations in the mixed convection regime are accentuated compared with forced convection due to the gener- ation of thermal flumes rising from the hot plate. The tempera- ture fluctuations are intensified when Pr increases because the conduction filtering is less effective for fluids with a smaller thermal conductivity.

Fig. 10. (a) Streamwise turbulent heat flux (b) Wall-normal turbulent heat flux.

Fig. 11. (a) Dissipation rate of temperature variance (b) Turbulent Prandtl number.

Table A.2

Flow and domain condition for the validation simulation.

Boundary conditions Periodic (x-, z-direction), Non-slip (y-direction)

Number of cells 256 256 256

Computational domain 5 p d 2d 2 p d Friction Reynolds number Re _s ¼ 150

Reynolds number Re ¼ 4680

Prandtl number Pr ¼ 0:71

Grashof number Gr ¼ 1; 300;000

Richardson number Ri ¼ 0:06

6. The Reynolds analogy breaks down in simulating mixed convec- tion and low-Prandtl-number fluids. The turbulent Prandtl number profile for diverse Prandtl numbers could be helpful

for developing new turbulent Prandtl number models, for the use in RANS models. This may help to improve the accuracy of modeling turbulent heat transfer.

Fig. A.12. Profiles of (a) mean velocity (b) mean temperature.

Fig. A.13. (a) RMS of Reynolds normal stresses (b) RMS of temperature variance (c) Streamwise turbulent heat flux (d) Wall-normal turbulent heat flux.

Table A.3

Averaged relative difference compared with DNS benchmark data from Iida and Kasagi (1997).

u

h

u0

h0

h0

v ⁰

^h0

^u0

v ⁰

w0

d

0.0222 0.0188 0.0601 0.0181 0.0298 0.0124 0.0350 0.0185

Based on this study, a DNS database for PRB flow at diverse Prandtl numbers has been established. This database could also be extended by studying other parameter variations and geome- tries, which could be beneficial for developing and validating RANS models.

CRediT authorship contribution statement

Wentao Guo: Conceptualization, Data curation, Formal analy- sis, Investigation, Methodology, Project administration, Software, Validation, Visualization, Writing - original draft. Horst-Michael Prasser: Funding acquisition, Resources, Supervision, Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing finan- cial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This research is sponsored by the EU Horizon 2020 SESAME pro- ject, PSI and ETHZ. The simulations carried out in this study are performed on Swiss National Supercomputing Centre (CSCS). The authors would like to thank Dr. Bojan Niceno for his kind support.

Appendix A. Validation of Incompact3d for PRB flow

For the purpose of proving the reliability of the DNS study in this research, a validation simulation is performed and compared with the benchmark data on mixed convection from Iida and Kasagi (1997). They study PRB flow using a similar geometry as it is shown in Fig. 1. The boundary conditions and flow configura- tion are the same as given in Section 2. A spectral method is used.

The Fourier modes are 128 128 and 96th order Chebyshev poly- nomials are used in the wavenumber space.

The parameters, which are set in Incompact3d are listed in Table A.2. They are consistent with the settings of Iida and Kasagi (1997). The CFL number is kept below 0.5. The size of the mesh used in the validation simulation is N

N

N

¼ 256 256 256 and D x

; D y

and D z

are 8.9, 1.1 and 3.6 respectively. A stretched mesh is used in the wall-normal direction and y

¼ 0 : 40. The semi-implicit temporal scheme, which is mentioned in Section 2.2, is applied.

The turbulent statistics are compared with the results of Iida and Kasagi (1997) in Fig. A.12 and Fig. A.13.

From the comparison shown in Fig. A.12 and Fig. A.13, we can tell that the results from Incompact3d have a very good consis- tency with the benchmark data. The averaged relative difference (d

) between the benchmark DNS data and Incompact3d using semi-implicit scheme is listed in Table A.3. The definition of d

is analogous to g which is defined in Eq. (13). n represents the node number in the wall-normal direction. f

ð Þ y

is the result from the benchmark data which represents the time and space averaged data in the x z plane at y

; f

ð Þ y

is the result from Incompact3d.

The results in general match the benchmark data quite well and it proves that the code is suitable for the DNS study discussed in this paper.

References

Ahlers, G., Grossmann, S., Lohse, D., 2009. Heat transfer and large scale dynamics in turbulent rayleigh-bénard convection. Rev. Mod. Phys. 81, 503.

Akashi, M., Yanagisawa, T., Tasaka, Y., Vogt, T., Murai, Y., Eckert, S., 2019. Transition from convection rolls to large-scale cellular structures in turbulent rayleigh- bénard convection in a liquid metal layer. Phys. Rev. Fluids 4, 033501.

Bricteux, L., Duponcheel, M., Winckelmans, G., Tiselj, I., Bartosiewicz, Y., 2012.

Direct and large eddy simulation of turbulent heat transfer at very low prandtl number: Application to lead–bismuth flows. Nucl. Eng. Des. 246, 91–

97.

Dairay, T., Fortuné, V., Lamballais, E., Brizzi, L., 2014. Les of a turbulent jet impinging on a heated wall using high-order numerical schemes. Int. J. Heat Fluid Flow 50, 177–187.

Davidson, L., Cuturic, D., Peng, S.-H., 2003. Dns in a plane vertical channel with and without buoyancy. Turbul. Heat Mass Transfer 4, 401–408.

De Santis, D., De Santis, A., Shams, A., Kwiatkowski, T., 2018. The influence of low prandtl numbers on the turbulent mixed convection in an horizontal channel flow: Dns and assessment of rans turbulence models. Int. J. Heat Mass Transfer 127, 345–358.

Flageul, C., Benhamadouche, S., Lamballais, É., Laurence, D., 2015. Dns of turbulent channel flow with conjugate heat transfer: Effect of thermal boundary conditions on the second moments and budgets. Int. J. Heat Fluid Flow 55, 34–44.

Flageul, C., Benhamadouche, S., Lamballais, E., Laurence, D., 2017. On the discontinuity of the dissipation rate associated with the temperature variance at the fluid-solid interface for cases with conjugate heat transfer. Int. J. Heat Mass Transfer 111, 321–328.

Grötzbach, G., 1982. Direct numerical simulation of laminar and turbulent bénard convection. J. Fluid Mech. 119, 27–53.

Grötzbach, G., 2007. Anisotropy and Buoyancy in Nuclear Turbulent Heat Transfer:

Critical Assessment and Needs for Modelling. Forschungszentrum, Karlsruhe.

Guo, W., Shams, A., Sato, Y., Niceno, B., 2020. Influence of buoyancy in a mixed convection liquid metal flow for a horizontal channel configuration. Int. J. Heat Fluid Flow 85, 108630.

Iida, O., Kasagi, N. 1997 Direct numerical simulation of unstably stratified turbulent channel flow (1997)..

Jiménez, J., Moin, P., 1991. The minimal flow unit in near-wall turbulence. J. Fluid Mech. 225, 213–240.

Kasagi, N., Iida, O. Progress in direct numerical simulation of turbulent heat transfer, in: Proceedings of the 5th ASME/JSME Joint Thermal Engineering Conference, American Society of Mechanical Engineers San Diego, pp. 15–19..

Kasagi, N., Ohtsubo, Y. 1993 Direct numerical simulation of low prandtl number thermal field in a turbulent channel flow, in: Turbulent Shear Flows 8, Springer, pp. 97–119..

Kasagi, N., Tomita, Y., Kuroda, A. 1992. Direct numerical simulation of passive scalar field in a turbulent channel flow..

Kath, C., Wagner, C., 2014. Dns and les of turbulent mixed convection in the minimal flow unit. In: New Results in Numerical and Experimental Fluid Mechanics IX. Springer, pp. 123–131.

Kawamura, H., Abe, H., Matsuo, Y., 1999. Dns of turbulent heat transfer in channel flow with respect to reynolds and prandtl number effects. Int. J. Heat Fluid Flow 20, 196–207.

Kays, W.M., 1994. Turbulent prandtl number. where are we? ATJHT 116, 284–295.

Keisuke, F., Masamoto, N., 1985. Unstable stratification effects on turbulent shear flow in the wall region. Int. J. Heat Mass Transfer 28, 2343–2352.

Keisuke, F., Masamoto, N., Hiromasa, U., 1991. Coherent structure of turbulent longitudinal vortices in unstably-stratified turbulent flow. Int. J. Heat Mass Transfer 34, 2373–2385.

Kim, J., Moin, P., Moser, R., 1987. Turbulence statistics in fully developed channel flow at low reynolds number. J. Fluid Mech. 177, 133–166.

Komori, S., Ueda, H., Ogino, F., Mizushina, T., 1982. Turbulence structure in unstably-stratified open-channel flow. Phys. Fluids 25, 1539–1546.

Kozuka, M., Seki, Y., Kawamura, H., 2009. Dns of turbulent heat transfer in a channel flow with a high spatial resolution. Int. J. Heat Fluid Flow 30, 514–524.

Laizet, S., Lamballais, E., 2009. High-order compact schemes for incompressible flows: A simple and efficient method with quasi-spectral accuracy. J. Comput.

Phys. 228, 5989–6015.

Laizet, S., Li, N., 2011. Incompact3d: A powerful tool to tackle turbulence problems with up to o (105) computational cores. Int. J. Numer. Methods Fluids 67, 1735–

1757.

Otic´, I., Grötzbach, G. 2005 Statistical analysis of turbulent natural convection in low prandtl number fluids. In: Progress in Turbulence, Springer, pp. 203–206..

Parodi, A., von Hardenberg, J., Passoni, G., Provenzale, A., Spiegel, E.A., 2004.

Clustering of plumes in turbulent convection. Phys. Rev. Lett. 92, 194503.

Pirozzoli, S., Bernardini, M., Orlandi, P., 2016. Passive scalars in turbulent channel flow at high reynolds number. J. Fluid Mech. 788, 614–639.

Pirozzoli, S., Bernardini, M., Verzicco, R., Orlandi, P. 2016 Mixed convection in turbulent channels with unstable stratification, arXiv preprint arXiv:1609.

02860..

Scheel, J.D., Schumacher, J., 2016. Global and local statistics in turbulent convection at low prandtl numbers. J. Fluid Mech. 802, 147–173.

Shams, A. 2017 Rans modelling of turbulent heat transport in liquid metal flow. sl, vki lecture series on thermohydraulics and chemistry of liquid metal cooled reactors, the von karman..

Shams, A., Komen, E., 2018. Towards a direct numerical simulation of a simplified pressurized thermal shock. Flow Turbul. Combust. 101, 627–651.

Shams, A., Roelofs, F., Baglietto, E., Lardeau, S., Kenjeres, S., 2014. Assessment and calibration of an algebraic turbulent heat flux model for low-prandtl fluids. Int.

J. Heat Mass Transfer 79, 589–601.

Sid, S., Dubief, Y., Terrapon, V., 2015. Direct numerical simulation of mixed convection in turbulent channel flow: on the reynolds number dependency of momentum and heat transfer under unstable stratification. In: Proceedings of the 8th International Conference on Computational Heat and Mass Transfer ICCHMT.

Silano, G., Sreenivasan, K., Verzicco, R., 2010. Numerical simulations of rayleigh–

bénard convection for prandtl numbers between 10–1 and 10 4 and rayleigh numbers between 10 5 and 10 9. J. Fluid Mech. 662, 409–446.

Tiselj, I., 2014. Tracking of large-scale structures in turbulent channel with direct numerical simulation of low prandtl number passive scalar. Phys. Fluids 26, 125111.

Tiselj, I., Cizelj, L., 2012. Dns of turbulent channel flow with conjugate heat transfer at prandtl number 0.01. Nucl. Eng. Des. 253, 153–160.

Tiselj, I., Bergant, R., Mavko, B., Bajsic, I., Hetsroni, G., 2001. Dns of turbulent heat transfer in channel flow with heat conduction in the solid wall. J. Heat Transfer 123, 849–857.

Tiselj, I., Pogrebnyak, E., Li, C., Mosyak, A., Hetsroni, G., 2001. Effect of wall boundary condition on scalar transfer in a fully developed turbulent flume. Phys. Fluids 13, 1028–1039.

Xi, H.-D., Lam, S., Xia, K.-Q., 2004. From laminar plumes to organized flows: the onset of large-scale circulation in turbulent thermal convection. J. Fluid Mech.

503, 47–56.

Zonta, F., Soldati, A., 2014. Effect of temperature dependent fluid properties on heat transfer in turbulent mixed convection. J. Heat Transfer 136.

Zonta, F., Marchioli, C., Soldati, A., 2012. Modulation of turbulence in forced

convection by temperature-dependent viscosity. J. Fluid Mech. 697, 150.