A probabilistic, data-driven closure model for RANS simulations with aleatoric, model uncertainty

TL;DR

The paper addresses the closure problem in Reynolds-averaged Navier–Stokes (RANS) simulations by introducing a probabilistic, data-driven closure that blends a parametric RS model with a stochastic discrepancy term to capture aleatoric model uncertainty. The parametric RS closure uses an invariant tensor-basis neural network to map velocity-gradient invariants to the anisotropic RS, while a sparsity-promoting ARD prior controls region-specific model errors through a reduced vector of latent variables. A fully Bayesian framework with Stochastic Variational Inference (SVI) and an adjoint-enabled differentiable RANS solver enables learning from sparse, indirect data (mean velocities and pressures) and propagates predictive uncertainty to all quantities of interest. The approach is demonstrated on a backward-facing step BFS benchmark, where predictions, including reattachment lengths, are probabilistic and envelop LES references, illustrating robust performance in regions with model errors. The work advances model-consistent learning for turbulence closures by explicitly accounting for aleatoric uncertainty and enabling end-to-end differentiable training and prediction with limited high-fidelity data.

Abstract

We propose a data-driven, closure model for Reynolds-averaged Navier-Stokes (RANS) simulations that incorporates aleatoric, model uncertainty. The proposed closure consists of two parts. A parametric one, which utilizes previously proposed, neural-network-based tensor basis functions dependent on the rate of strain and rotation tensor invariants. This is complemented by latent, random variables which account for aleatoric model errors. A fully Bayesian formulation is proposed, combined with a sparsity-inducing prior in order to identify regions in the problem domain where the parametric closure is insufficient and where stochastic corrections to the Reynolds stress tensor are needed. Training is performed using sparse, indirect data, such as mean velocities and pressures, in contrast to the majority of alternatives that require direct Reynolds stress data. For inference and learning, a Stochastic Variational Inference scheme is employed, which is based on Monte Carlo estimates of the pertinent objective in conjunction with the reparametrization trick. This necessitates derivatives of the output of the RANS solver, for which we developed an adjoint-based formulation. In this manner, the parametric sensitivities from the differentiable solver can be combined with the built-in, automatic differentiation capability of the neural network library in order to enable an end-to-end differentiable framework. We demonstrate the capability of the proposed model to produce accurate, probabilistic, predictive estimates for all flow quantities, even in regions where model errors are present, on a separated flow in the backward-facing step benchmark problem.

Figures (13)

• Figure 1: Probabilistic graphical model of the proposed model including model parameters ($\bm \theta$, $\bm \Lambda$), latent variables ($\bm E_{\tau}$) and observables $\hat{\bm{z}}$ from $M$ flow scenaria. Deterministic nodes are indicated with circles with dashed line, stochastic with circles with solid line and known/observed are shaded.
• Figure 2: Schematic illustration of the training/inference (left block) and probabilistic prediction (right block) framework proposed.
• Figure 3: Backward facing step flow configuration with step height $h$ and the total channel height $H$. The origin of the $x-y$ plane is placed at the corner of the step. The axes are depicted at the bottom left to avoid clutter.
• Figure 4: (Random) grid points where LES velocities/pressure were used for training data. We note that the total number of LES grid points is $\mathcal{O}(10^5)$ whereas LES simulation data at approximately $1000$ grid points were employed.
• Figure 5: Instantaneous velocity magnitude $||\bm{U}||$ at different time-instants $t=\{30,70,100\}$ obtained from the LES simulation performed at $Re=1100$. We note that the flow eventually reaches a stationary state.