A Correlation for the Discontinuity of the Temperature Variance Dissipation Rate at the Fluid-Solid Interface in Turbulent Channel Flows

Abstract

Discontinuity of the dissipation rate associated with the temperature variance at the fluid-solid interface is analyzed in a turbulent channel flow at a Reynolds number, based on the friction velocity of 395 and a Prandtl number of 0.71. The analysis is performed with a wall-resolved Large Eddy Simulation and the results are used to derive a regression for the dissipation rate discontinuity, which depends only on the fluid-solid thermal diffusivity and conductivity ratios. Wall-resolved Large Eddy Simulations at a higher Reynolds number and a higher Prandtl number are used to investigate the validity of two correlations derived from the regression for the selected thermal properties ratios. The present results are obtained with the open-source Computational Fluid Dynamics solver Code_Saturne, and use the fully conservative fluid-solid thermal coupling capability introduced by the authors in version 5.0.

Appendices

Appendix

A Coupling Strategy

Since version 5.0, Code_Saturne can be used stand-alone to perform conjugate heat transfer with monolithically solving the fluid and solid temperatures in a fully conservative way, both in space and time, as described in this appendix. The coupling strategy developed contains two main entities. The geometric entity contains information related to the coupling interface and allows cells on both sides of it to exchange information. The physical entity contains information related to the coupled fields and their physical properties and allows to solve the coupled fields in a fully conservative way.

We start with a global mesh containing both the fluid and the solid. Using a given criterion, we split the volume into a solid part and a (complementary) fluid one: each computational cell is either inside the fluid or inside the solid. The internal faces separating fluid and solid cells are duplicated, alongside with the corresponding vertices and edges. Then, one of the duplicates is associated with the fluid cell while the other is associated with the solid cell. Both faces are then converted into boundary faces. This first step is performed by the preprocessor and changes the connectivity of the mesh. At the end of this step, the fluid-solid interface appears as an infinitely thin boundary which no variable can cross.

Then the solver starts. The fields specified by the user as coupled are flagged as such. Then, each time the convection-diffusion equation of a coupled field is solved, or when its gradient is computed, the explicit and implicit parts of the corresponding linear system are modified on the fly to take into account the coupling contributions. As the velocity vanishes on the fluid-solid interface, the coupling will be fully conservative if the diffusive term is correctly estimated. To further discuss this point, we must define a number of quantities specific to the finite volume discretization, as shown in Fig. 12.

Fig. 12

Quantities used in the finite volume discretization. Left: cell Ω_i, cell Ω_j and the internal face S_ij connecting them. Right: boundary cell Ω_i and associated boundary face S_b. Code_Saturne 5.0.0 theory guide [28]

For an internal face — i.e. S_ij —, the thermal flux from cell Ω_i to cell Ω_j per unit surface, written D_ij, is

$$ D_{ij} = \lambda_{ij} \frac{T_{J^{\prime}}-T_{I^{\prime}}}{I^{\prime}J^{\prime}} $$

(33)

where λ_ij is the thermal conductivity on the internal face S_ij. For a boundary face — i.e. S_b —, the thermal flux out of cell Ω_i per unit surface, written D_ib, is

$$ D_{ib} = \lambda_{i} \frac{T_{F}-T_{I^{\prime}}}{I^{\prime}F} $$

(34)

Thus, the thermal flux from a boundary face to a neighbouring cell Ω_j per unit surface, written D_bj, is

$$ D_{bj} = \lambda_{j} \frac{T_{J^{\prime}}-T_{F}}{FJ^{\prime}} $$

(35)

If the field is coupled, the fluxes per unit surface D_ib and D_bj are equal. This flux will be written D_ibj. The faces temperature T_F in Eqs. 34 and 35 are also equal. This leads to

$$ \left( \frac{F J^{\prime}}{\lambda_{j}} + \frac{I^{\prime} F}{\lambda_{i}} \right) D_{ibj} = \left( T_{J^{\prime}}-T_{I^{\prime}} \right) $$

(36)

To conclude, the flux D_ibj of the coupled field should be equal to the flux D_ij we would have obtained on an internal face. Thus, using Eq. 33 and writing $\eta _{ij} = \frac {I^{\prime } F}{I^{\prime } J^{\prime }}$, one gets

$$ \frac{1}{\lambda_{ij}} = \left( \frac{1-\eta_{ij}}{\lambda_{j}} + \frac{\eta_{ij}}{\lambda_{i}} \right) $$

(37)

This demonstrates that an harmonic interpolation for the face thermal conductivity gives a fully conservative coupling strategy. Second order accuracy in space on non-orthogonal meshes is reached thanks to an iterative process.

The main advantage of the present coupling strategy is that the solver remains fully conservative in case of conjugate heat transfer. In addition, second-order accuracy in time is preserved. The main disadvantage is that the velocity and the pressure are also defined in the solid domains. As both remain exactly zero inside the solid, this is not an issue, but it does lead to higher memory usage compared with a dedicated solver for the solid thermal diffusion. This coupling strategy will provide a valuable framework for future RANS models able to tackle conjugate heat transfer.

B Impact of the SGS Model and P r _t

Prior to the present study, several LES of a turbulent channel flow at Re_τ = 1020 were performed to select a SGS model for the momentum equation. The mesh used corresponds exactly to the one described in the Section 2.2. SGS models tested already implemented in Code_Saturne were the Standard Smagorinsky (Std.Smag), the WALE model (Wale), and 2 variants of the Dynamic Smagorinsky (Dyn.Smag.1 and Dyn.Smag.2). The two variants of the Dynamic Smagorinsky SGS model are using different clippings. The authors also tested some variants of the Dynamic Smagorinsky model with spatial averaging over homogeneous directions. As they did not bring significant improvements, they are simply omitted here.

Overall, the SGS models tested only provide a qualitative agreement for the Reynolds stresses. This is illustrated in Fig. 13 with the variance of the streamwise velocity ($\overline {{u^{\prime }_{x}}^{2}}$). Regarding the dissipation rate (ε) associated with the turbulent kinetic energy, Fig. 14 shows that the Wale SGS model provides the best estimation near the wall. The Wale SGS model was selected on this criterion.

Fig. 13

LES of a turbulent channel flow at Re_τ = 1020. Impact of the SGS model on the variance of the streamwise velocity

Fig. 14

LES of a turbulent channel flow at Re_τ = 1020. Impact of the SGS model on the dissipation rate associated with the turbulent kinetic energy

The authors have also looked at the impact of the turbulent Prandtl number (Pr_t). Two simulations were performed at Re_τ = 395 and Pr = 1 using conjugate heat transfer with unit ratios of fluid-solid thermal properties (K = G = G₂ = 1). The mesh used corresponds exactly to the one described in the Section 2.2. One simulation were performed with Pr_t = 0.5 and the other with Pr_t = 1. As visible on Figs. 15 and 16, the turbulent Prandtl number has no significant impact on the statistics in the near-wall layer (y⁺ < 5).

Fig. 15

LES of a turbulent channel flow at Re_τ = 395 and Pr = 1. Conjugate heat transfer with K = G = G₂ = 1. Impact of the turbulent Prandtl number (Pr_t) on the variance of the temperature

Fig. 16

LES of a turbulent channel flow at Re_τ = 395 and Pr = 1. Conjugate heat transfer with K = G = G₂ = 1. Impact of the turbulent Prandtl number (Pr_t) on ε_θ

When performing a LES, the choice of the SGS models used in the momentum equation and in the energy equation is of paramount importance. However, development and testing of refined SGS models is out of the scope of the present study. Thus, the authors have decided to combine a fine mesh with relatively simple SGS models already implemented in Code_Saturne, and to carefully assess the obtained results using DNS.