Session A13: CFD: General I

Accurate Higher-order Embedded Boundary Methods without Small Cell Issues

We present a cut-cell finite volume approach for discretizing fluid dynamics conservation laws in complex geometries. We use an embedded boundary method, where the geometry is defined by the contour of an implicit function on a Cartesian grid, which greatly simplifies meshing for constructive solid geometry or data-driven interfaces. However, it does introduce the problem of arbitrarily "small cells": if different flux reconstruction functions are used on a given small volume, careful cancellations must be maintained to leading order. Naive approaches, such as one-sided differences, redistribution, or grid-line interpolations, can deteriorate accuracy and stability. We introduce a higher-order least squares reconstruction that maintains high accuracy and creates stencils with good stability and conditioning properties. We demonstrate this with classical fluids test problems on simple domains, where we can demonstrate stability and convergence. We will also show results for very complex geometries that still maintain accuracy and stability. With careful refinement of the geometric representation, we can also show very high convergence rates, with at least 4th order in space and time for a conservative, arbitrarily small-cell mesh.   

Numerical reflections in hp-nonuniform Discontinuous Galerkin methods

The h and p adaptivity of Discontinuous Galerkin (DG) methods can reduce the cost of a simulation, but numerical reflections may occur at interfaces between regions of different resolution. We analyze eigensolutions of the DG method for linear hyperbolic systems and couple the solutions on either side of a change in resolution to find the amplitude of the reflections. We find that numerical reflections can be significant. Especially for high p, the reflections can be large at resolutions where the wave's propagation is resolved on the fine grid. We extend prior work to changes in h and p, and find that the reflections can be decreased by certain combinations of changes in h and p, but not eliminated entirely. In the 1D case, the reflections can still be eliminated by use of an exact characteristics-splitting flux.   

Application of Multi-Fidelity Methods to Uncertainty Quantification of Gravity Currents

Density variations due to sources such as heat or solute concentration are often observed in physical systems, including deposition of saline solutions into fresh water and smoke generation from wildfires. These belong to a class of flows known as gravity currents, where differences in density may induce flow due to buoyancy effects. In this study, we consider a simplified model of a lock-exchange problem, where a heavy fluid is placed in a lighter ambient with an impermeable barrier that is then removed. A nonlinearly varying ambient density profile is assumed, leading to a three-dimensional parameter space for the strength, shape, and height of the ambient profile. A system of shallow water equations (SWEs) has been derived to model the front location, area, and concentration of the current. Comparisons with two- and three-dimensional direct numerical simulations (DNSs) shows the SWEs are able to capture trends with variations in ambient stratification within the slumping phase. Finally, we discuss the applicability of using a bi-fidelity method to correct these quantities with a limited amount of DNS, which will enable for efficient exploration of the parameter space.   

Multi-fidelity sensitivity analysis using locally calibrated RANS model

Forward propagation of uncertainty can be estimated through the sensitivity of a quantity of interest to uncertain variables. Multi-fidelity sensitivity analysis combines the high accuracy of high-fidelity models (DNS/LES) with the low computational cost and non-chaotic nature of low-fidelity models (RANS). In this study, we increase the model-complexity of the RANS closure model by allowing its coefficients to vary spatially. This model, with spatially varying coefficients, is calibrated using high-fidelity data at a base condition. The calibration is performed using a gradient-based optimization process. An adjoint solver is used to expedite the calibration process. The calibrated model is then used for sensitivity estimation.   

The impact of Omnidirectional Integration on the pressure solver in subdomain data assimilation for incompressible flows.

The divergence-free projection in incompressible Navier-Stokes solvers is essential for the fractional-step method in incompressible flow simulations. When reconstructing velocity and pressure fields in a subdomain through data assimilation, challenges arise particularly with unknown boundary conditions, where the projection step changes accordingly. Assuming an orthogonal projection would lead to zero normal pressure gradient at boundaries, which is compensated with a large domain, often requiring extra costly computation and grid storage even when the region of interest is small. Additionally, using Neumann boundary conditions can degrade performance with noisy velocity measurements. Liu and Moreto (2020) demonstrated that Omnidirectional Integration (ODI) can achieve Dirichlet boundary conditions and as a result significantly outperforms Neumann boundary conditions in ensuring pressure reconstruction accuracy. This study integrates ODI into a 2D incompressible Navier-Stokes solver by taking advantage of the pressure Dirichlet boundary conditions enabled by ODI. This method demonstrates that the improved projection provides results for both velocity and pressure closest to the true divergence-free flow.   

Parameterizing Particle Dispersion in a Granular Bed Mobilized by Oscillatory Flow

Oscillatory bottom flows induced by surface waves produce a time-dependent transport of bottom sediments that consists of two main phases. As sediments are mobilized by the accelerating flow, grain-grain interactions result in an upward dispersion of the surface sediments. When fluid velocity slows, collisions become more infrequent and the particles fall back to the bed as a result of settling. At any given time, the net vertical flux of sediment is a combination of the collision-driven vertical flux and the settling flux of the particles. While the process of settling is well studied and modeled by empirical relationships, there has been significantly less attention to modeling the dispersion process. Here we use discrete element method (DEM) simulations of unimodal sediments in oscillating sawtooth waves to develop an empirical model of the dispersive flux. We posit that the dispersive flux can be modeled using Fick’s Law with a diffusivity that will scale with the energy of the particles and by extension the free stream velocity. The time-dependent diffusivity is determined by regressing the flux against the concentration gradient in the DEM simulations and expressed as a function of the freestream velocity. The Fick’s law diffusivities are then compared to particle dispersion rates using analysis of Lagrangian particle motion through all phases of the oscillating waves which are used to both validate the diffusion term model and to characterize the horizontal and vertical spread of particles under different forcing conditions.  This vertical diffusive flux model is then used to reproduce the DEM simulations, showing that the parameterization of the diffusion accurately predicts the changes in concentration through the wave cycle. The diffusive rates are compared with known empirical sediment transport predictions.   

Variational Physics-Informed Neural Networks for Unsteady Incompressible Flows

Recent advances in Physics-Informed Neural Networks (PINNs) applied to fluid mechanics have largely relied on the Newtonian framework, utilizing Navier–Stokes equations or their derivatives to train the neural network. Here, we propose an alternative approach based on variational methods; specifically employing the principle of minimum pressure gradient to turn the fluid mechanics problem into a minimization problem whose solution can be used to predict the flow field in unsteady incompressible viscous flows.

We apply this novel approach to study the unsteady flow field in a lid-driven cavity at Reynolds numbers ranging from 100 to 5000. The computational performance of the proposed method is compared to conventional PINNs showcasing its accuracy and efficiency. Computational results indicate that the variational approach to PINNs offers a robust and efficient way to solve incompressible fluid mechanics problems. Moreover, it exhibits the potential for extension to turbulent and non-Newtonian fluids, paving the way for broader implementations in fluid mechanics.   

Learned Corrections for Discretization in Partial Differential Equations

Direct numerical solutions to partial differential equations are often computationally impractical due to the need for very fine grid sizes. As a result, one must rely upon effective equations on coarser grids that can approximate the small-scale dynamics. Traditional approaches to derive these dynamics posed significant challenges, but recent advances in differentiable physics and machine learning have enabled data-driven approaches to automatically learn them. In this work, we use convolutional neural networks to learn corrections for errors introduced by finite difference approximations of spatial derivatives on coarser grids. The correction network is trained against high-resolution solutions of the underlying equation, allowing the low-resolution solver to accurately simulate dynamics on small spatiotemporal scales. This entire process is implemented within a fully differentiable physics framework, integrating the correction network into the time integration steps and the training process. We demonstrate improved accuracy in various advection-diffusion and fluid systems and investigate the generalizability of these trained correction networks.   

Influence of biorthogonal wavelets on adaptive simulations of turbulent flows

We present a wavelet-based adaptive approach for computing flows in complex geometries, in particular the aerodynamics of flapping insect flight. This approach is implemented in our open-source code WABBIT (Engels et al. Commun. Comput. Phys., doi:10.4208/cicp.OA-2020-0246). Dynamically evolving grids using regular Cartesian blocks allow significant reduction of memory and CPU time requirements while monitoring the precision of the computation. Distributing the blocks among MPI processes permits an efficient parallelization on large scale supercomputers. Our adaptation strategy is based on lifted interpolating biorthogonal wavelets, and in this talk we discuss how the choice of wavelet influences on the spatio-temporal dynamics of the flow. We demonstrate that an insufficient vanishing moments of the reconstructing wavelet results in a reduced convergence order. Applying lifting to the wavelets is therefore important for nonlinear problems. The classical three-vortex problem and the Taylor-Green vortex serve as test cases in 2D and 3D, respectively. Finally, we perform 3D simulations of a model bumblebee and discuss the impact of the wavelet on the vortical structures. We conclude with guidelines on the choice of wavelets for adaptive CFD simulations.