Large eddy simulations of wall jets with coflow for the study of turbulent Prandtl number variations and data-driven modeling

There is a continuing effort by turbulence researchers to provide improved turbulent heat flux predictions for Reynolds-averaged Navier-Stokes (RANS) calculations of heat transfer applications. In this paper, data-driven models are developed for the turbulent heat flux prediction in wall jets with coflow using a gene expression programming (GEP)–based machine-learning technique. The training data used as input to the optimization algorithm are obtained by performing highly resolved large eddy simulations (LES) of nine cases covering various flow and geometry conditions. The study examines whether predictive RANS-based heat transfer closures can be trained that are robust to these physically very different nine LES cases. The GEP heat flux closures were developed by adopting the gradient-diffusion hypothesis with the optimization target being a nondimensional parameter representing the inverse of a nonconstant turbulent Prandtl number ($P{r}_t$), with a functional dependence on the velocity and temperature gradients. First, examination of the turbulent Prandtl number calculated from time-averaged LES data showed a significant deviation from the commonly assumed constant value of 0.9, with a more significant dependence on the lip wall thickness than the blowing ratio. Second, a posteriori testing of the developed closures by solving the RANS-based scalar transport equation using as input the LES time-averaged velocity and turbulent viscosity showed a significant improvement in the prediction of adiabatic wall effectiveness not only for the cases they were trained on, but also for the entire matrix of LES cases. Finally, our best-performing model (trained on the thickest lip wall case) was also evaluated in a full RANS context and a significant improvement for the prediction of the adiabatic wall effectiveness was achieved, in particular for the medium and the thin lip thickness cases. The lack of improvement when testing the thickest lip wall case in a full RANS context indicates that for cases with strong vortex shedding the effect of organized unsteadiness on the turbulent flow field is important. In such cases, only modifying the heat flux model without improving the RANS velocity field is not sufficient and other methodologies like deriving a model for the Reynolds stress are necessary. Collectively, the current study demonstrates the ability of the presented model-development framework in creating bespoke models that can provide accurate predictions for a wide range of operating conditions.

Figure 1

Computational domain (not to scale) and LES blocks.

Figure 2

The GEP heat flux model development flowchart.

Figure 3

Instantaneous temperature fields obtained from LES for various lip wall thicknesses ($t/s$) and blowing ratios (BR). Note that for each plot the streamwise and the wall-normal extents are respectively $x/s=-4:30$ and $y/s=0:10$ (not to scale).

Figure 4

Instantaneous $Q$-criterion field (nondimensional isolevel 2) colored with the temperature field obtained from LES cases A1 (a), B2 (b), and C3 (c).

Figure 5

Mean coefficient of friction along the lower wall.

Figure 6

Normalized mean streamwise velocity (i) and temperature (ii) in a number of locations downstream of the jet exit.

Figure 7

Reynolds stresses [normal stress $R_{xx}$ (i) and shear stress $R_{xy}$ (ii)] in a number of locations downstream of the jet exit.

Figure 8

Turbulent heat fluxes [streamwise component $\overline{u^{\prime }{T}^{\prime }}$ (i) and wall-normal component $\overline{v^{\prime }{T}^{\prime }}$ (ii)] in a number of locations downstream of the jet exit.

Figure 9

Probability density function (PDF) of the turbulent Prandtl number ${\text{Pr}}_t$ calculated from time-averaged LES data using Eq. (16).

Figure 10

The mean turbulent Prandtl numbers for the LES cases as a function of lip thickness ($t/s$) for different blowing ratios (BR).

Figure 11

Comparison of ${\eta }_{wall}$ obtained from LES, and the solution of the scalar transport equation [Eq. (9)] as input the time-averaged LES velocity field with the standard eddy diffusivity model with ${\text{Pr}}_t=0.9$ for cases A1 (a), B2 (b), and C3 (c).

Figure 12

Training regions for the turbulent heat flux model development.

Figure 13

Comparison of the adiabatic wall effectiveness obtained from different flux models for case A1.

Figure 14

Turbulent thermal diffusivity ${\alpha }_t$ calculated from time-averaged LES (${\alpha }_t^{LES,opt}$), standard eddy diffusivity model with ${\text{Pr}}_t=0.9$ (${\alpha }_t^{EDM}$), and the developed GEP closures [i.e., ${\alpha }_t^{mod,1}$, Eq. (18); ${\alpha }_t^{mod,2}$, Eq. (19); and ${\alpha }_t^{mod,3}$, Eq. (20)] for cases A1 (i), B2 (ii), and C3 (iii) at $x/s=30$ (a) and $x/s=45$ (b).

Figure 15

The PDF of the turbulent Prandtl number along with the contribution from only the constant term in each developed data-driven model [i.e., ${\alpha }_t^{mod,1}$, Eq. (18); ${\alpha }_t^{mod,2}$, Eq. (19); and ${\alpha }_t^{mod,3}$, Eq. (20)] illustrated with arrows.

Figure 16

Comparison of the adiabatic wall effectiveness obtained from different flux models.

Figure 17

PDF of basis functions (a) $I_1$, (b) $I_2$, (c) $J_1$, and (d) $J_2$ calculated from time-averaged LES data.

Figure 18

Comparison of the normalized temperature field obtained from different flux models.

Figure 19

Comparison of the adiabatic wall effectiveness (${\eta }_{wall}$) obtained from time-averaged LES and RANS for cases A1 (a), B2 (b), and C3 (c).