Bayesian inference of mean velocity fields and turbulence models from flow MRI

TL;DR

The paper develops a Bayesian inverse framework to simultaneously reconstruct the mean velocity field and learn unknown RANS closure parameters from mean-flow data, using adjoint-accelerated inference and a Laplace approximation for efficient uncertainty quantification. It formulates the RANS problem with an algebraic eddy-viscosity closure $\mu_e = \mu_\ell + \mu_t$, where $\mu_t = \ell_c^2\dot{\gamma}$ and $\dot{\gamma} = \sqrt{2\nabla^s\bm{u}:\nabla^s\bm{u}}$, and estimates a six-parameter closure together with geometry and boundary inputs via a Gaussian prior and a MAP solution. The methodology is tested on flow MRI data of a confined turbulent jet in an FDA nozzle, demonstrating successful mean-flow reconstruction and parameter learning with quantified uncertainties, and showing that the learned viscosity fields adapt to capture jet development and wall-region behavior. The work highlights the potential to calibrate turbulence closures from MRI measurements and points to extensions with more sophisticated closures (e.g., $k$-$\epsilon$) or inclusion of turbulent kinetic energy data to improve extrapolation across flow conditions.

Abstract

We solve a Bayesian inverse Reynolds-averaged Navier-Stokes (RANS) problem that assimilates mean flow data by jointly reconstructing the mean flow field and learning its unknown RANS parameters. We devise an algorithm that learns the most likely parameters of an algebraic effective viscosity model, and estimates their uncertainties, from mean flow data of a turbulent flow. We conduct a flow MRI experiment to obtain mean flow data of a confined turbulent jet in an idealized medical device known as the FDA (Food and Drug Administration) nozzle. The algorithm successfully reconstructs the mean flow field and learns the most likely turbulence model parameters without overfitting. The methodology accepts any turbulence model, be it algebraic (explicit) or multi-equation (implicit), as long as the model is differentiable, and naturally extends to unsteady turbulent flows.

Figures (7)

• Figure 1: (a) A 2D slice of 3D three-component velocity data from flow MRI through a 3D-printed aorta. This data has typical signal-to-noise-ratio (SNR) and is colored by the velocity into the page, $u_y$. (b) A diagram of our prior knowledge that this velocity data comes from a fluid that obeys the Navier--Stokes equations within a domain $\Omega$ bounded by a no-slip surface $\Gamma$, a Dirichlet velocity condition on $\Gamma_i$, and a stress-free boundary on $\Gamma_o$. This prior knowledge is expressed through a finite element CFD solver in which the free parameters define the shape of $\Gamma$, the velocity on $\Gamma_i$, and the viscosity $\mu$. The optimal parameter values and their uncertainties are found with adjoint-accelerated Bayesian inference giving (c) the inferred velocity field and boundary position, which can be compared with (d) high SNR data gathered over several hours, which was not seen by the assimilation algorithm. The reconstructed field in panel (c) is a higher resolution twin of the high SNR data in panel (d) despite assimilating only the low SNR data in panel (a) and the prior knowledge in panel (b). This figure is adapted from kontogiannis2024.
• Figure 2: Bayesian inversion of RANS problems: we assimilate mean flow data, $\bm{u}^\star$, in order to jointly reconstruct the mean flow field, $\bm{u}$, and learn the unknown RANS parameters, $\bm{x}$. The most likely RANS parameters, $\bm{x}^\circ$, are found by solving problem \ref{['eq:map_opt_problem']}, and the inferred mean velocity field is given by $\bm{u}^\circ=\mathcal{Q}\bm{x}^\circ$kontogiannis2024.
• Figure 3: Phantom design schematic (length units in centimeters).
• Figure 4: Nozzle phantom assembled on the benchtop. (a) Assembled phantom embedded into a solid ballistics gel. (b) Bench setup schematic.
• Figure 5: Optimization log of the data-model discrepancy of each velocity component.