Exploring the Potential and the Practical Usability of a Machine Learning Approach for Improving Wall Friction Predictions of RANS Wall Functions in Non-equilibrium Turbulent Flows

Abstract

A data-driven wall function estimation approach is proposed, aimed at accounting for non-equilibrium effects in turbulent boundary layers in RANS simulations of wall bounded flows. While keeping key simplifying hypothesis of standard wall functions and their general structure, the law-of-the-wall is replaced by a fully connected feed-forward neural network. The latter is trained to infer wall friction from the local flow state at the first of-wall nodes, described by an extended set of flow variables and gradients. For this purpose, the neural network is trained on high-fidelity wall resolved simulation data. It is then applied to formulate two different wall functions trained on high-fidelity data: a backward-facing step and a round jet impacting a flat wall. After integration into an industrial CFD code, they are applied to perform RANS simulations of the flow configurations they were trained for, and are shown to yield a largely improved prediction of wall friction as compared to standard wall functions. Finally, key issues related to the practical usability in RANS applications of the proposed data-driven approach are critically discussed.

Appendices

Appendix: Formulation of the LWWF

The LWWF is based on the Launder and Spalding wall model (Launder and Spalding 1974). The balance between production and dissipation rates of turbulent kinetic energy in the wall bounbary layer leads to the following estimation of the friction velocity:

$$\begin{aligned} u_{k} = C_{\mu }^{1/4}k_1^{1/2} \end{aligned}$$

(A1)

with $C_{\mu }=0.09$ and $k_1$ the value of the turbulent kinetic energy at the centroid of the first off-wall cell p. This friction velocity $u_{k}$ is used to compute the dimensionless wall distance $h^{+}$:

$$\begin{aligned} h^{+}=\frac{h u_{k}}{\nu } \end{aligned}$$

(A2)

with h the wall normal distance to the centroid of the first off-wall cell. Combining the logarithmic law-of-the-wall with Eq. A2 yields a second estimation of the friction velocity $u_{\tau }$:

$$\begin{aligned} u_{\tau }=\left\{ \begin{array}{ll} \frac{\left\Vert \varvec{U^p_t} \right\Vert }{\frac{1}{\kappa }ln(E h^{+})} &{} \text{ if } h^{+}>11.2 \\ \\ \frac{\left\Vert \varvec{U^p_t} \right\Vert }{h^{+}} &{} \text{ if } h^{+} \le 11.2 \end{array} \right. \end{aligned}$$

(A3)

with $E=e^{\kappa B}$, $\kappa = 0.4187$ and $B=5.5$. The wall shear stress is then computed as follows:

$$\begin{aligned} \varvec{\tau _w}^{LWWF} = - \frac{\varvec{U^p_t}}{\left\Vert \varvec{U^p_t} \right\Vert } \rho u_{k} u_{\tau }. \end{aligned}$$

(A4)

Regarding the wall boundary conditions applied to the turbulence quantities, a zero-gradient Neumann condition is used for k and a wall law for the specific dissipation rate $\omega$:

$$\begin{aligned} \omega = \frac{u_{\tau }}{C_{\mu }^{1/2} \kappa h } \end{aligned}$$

(A5)

Appendix B: Filtering High-Fidelity Data to Coarse RANS Meshes

Fig. 14

Schematic principle of a pre-filtering on coarse RANS meshes along the wall normal distance of HiFi data available on a highly resolved mesh

A two-dimensional filtering along the local normal distance to the wall is applied to HiFi data in order to map the mean flow variables from the highly resolved mesh having served to generate it, to the coarser mesh resolution typical of practical RANS simulations.

As illustrated in Fig. 14, the unfiltered value $I^{hf}(y_i)$ of any mean flow variable I in the initial HiFi data is replaced by an area-averaged value $I^{filt}(y_i)$ over all cells of the HiFi mesh intersecting with the coarse RANS cell of area $S_{filt} = y_i ^2$, as:

$$\begin{aligned} I^{filt}(y_i) = \frac{1}{S_{filt}} \sum _{j=1}^{N^{hf}} I^{hf}_j (S^{hf}_j \cap S_{filt}), \end{aligned}$$

(B6)

where $N^{hf}$ is the number of cells in the HiFi mesh.