Session J15: CFD: Shock Capturing, DG, Higher Order Schemes

Formulation of a new dynamic slip wall model in a Consistent Discontinuous Galerkin Framework

Slip wall modeling (Bose et al. 2014, Phy. Fluids 26(1), 015104; Bae et al. 2019, J. Fluid Mech. 859, 400-432) has been suggested as a promising approach to realize LES as a feasible computational tool for practically relevant high Reynolds number flows. In this work, building on the slip wall model of Pradhan and Duraisamy 2023 (J. Fluid Mech. 955, A6), we propose a new modeling procedure in which the dynamic part of the model is based on a modified form of Germano identity, and makes use of a model for the slip wall model coefficients obtained using an optimal finite-element projection framework. The dynamic slip wall model is tightly integrated with Discontinuous-Galerkin (DG) operators. A sharp modal cut-off filter is used as the test filter for the dynamic procedure. The performance of the new model is validated using a series of statistically stationary turbulent flows at high Reynolds numbers. Numerical experiments show that, similar to traditional wall-stress models, grid convergence studies are possible for the slip wall model provided the size of the element adjacent to the wall is fixed. The results show that the new dynamic model is able to consistently predict mean velocity profiles along with the Reynolds normal and shear stresses that match with the available DNS for various under-resolved coarse LES meshes.   

Coupled Discontinuous Galerkin - Direct Simulation Monte Carlo Simulation of Rocket Plume Impingement on Non-Flat Lunar Surfaces.

A coupled code using Discontinuous Galerkin (DG) and Direct Simulation Monte Carlo (DSMC) methods is employed to simulate axisymmetric impingement of Rocket Plume on Non-Flat Lunar Surfaces[1]. Better resolution of flow discontinuities within the rocket nozzle is attempted with the high order DG approach. A one-way coupling is employed by locating the interface in a high Mach Number region [2]. The coupling is affected by overlapping the continuum and DSMC solution domains at the interface and starting the DSMC particles from the DG Nodal locations at the interface. Geometries such as conoids and paraboloids are employed to model cratered and mounded lunar surfaces. The sensitivity of the shock structures and hence the flow fields to the surface topography is examined. Comparisons are made with results from plume interaction with the flat surface[3]. Among other things, the exhaust-gas mass-flux in the radial direction (perpendicular to the axis of the lander) for the conical surfaces was found to be much more sensitive to the surface slope than that for the parabolic surfaces, and flow reversal past the lander in terms of the mass flux was found to be much higher for the cratered surfaces and the tendency for viscous erosion was found to be much higher for the cratered surfaces.   

High-order discontinuous Galerkin methods with finite volume subcells for compressible flows on simplices

This work presents GPU-accelerated nodal discontinuous Galerkin methods for supersonic and hypersonic flows with finite volume subcell stabilization on simplices. Our approach merges the favorable attributes of the discontinuous Galerkin method in smooth flow regions with the ideal characteristics of a total variation diminishing finite volume methods for resolving shocks. A modal decay rate-based regularity estimator is used to detect high-wavenumber solution components near shocks and under-resolved regions. The methodology is implemented in the libParanumal library which includes a set of finite element flow solvers for heterogeneous (GPU/CPU) systems through OCCA, an open-source library that provides the portability layer to offload targeted kernels across different architectures and vendor platforms. The efficiency, scalability, and local high-order accuracy of the method are confirmed through distinct supersonic and hypersonic test cases. The kernel performance metrics will be demonstrated for various GPU architectures, including NVIDIA, Intel, and AMD GPUs, as well as for different programming models, CUDA, OpenCL, and SYCL.   

Warp-DG: A Differentiable Discontinuous Galerkin Solver for Compressible Flows

The rapid growth in machine learning and deep learning has opened numerous opportunities to advance computational fluid dynamics for compressible flows. The goal of this work is to reduce the gap between the state-of-the-art CFD techniques and latest advancement in the learning community by developing a differentiable PDE platform for complex geometry. Existing works that either rely on low-order finite volume or high-order finite-difference discretization face challenges when higher numerical accuracy over complex geometry is desired. Thus, we build a high-order discontinuous Galerkin (DG) solver under the framework of differentiable programming from Nvidia Warp.

In this presentation, we will show the convergence results of the developed solver over several classical PDE examples over both structured and unstructured grids. Finally, an end-to-end learning of the hyperparameters in the DG scheme will be demonstrated to show the effectiveness of the proposed framework on advancing CFD for compressible flows with machine learning techniques.    

An Extended Discontinuous Galerkin Method for High-order shock treatment

We present an implicit high-order shock fitting approach based on a cut-cell method for the direct simulation of compressible flows. We formulate a suitable Constraint Optimization Problem and develop a Sequential Quadratic Program solver aiming to fit the shock front represented by the zero iso-contour of a Level Set function. In the Extended Discontinuous Galerkin method the approximation space is enriched by basis functions which are discontinuous alongside this interface, therefore a sharp representation of the shock-induced jumps can be obtained. As a consequence, no shock capturing has to be used and high-order accuracy is reached.

Further, two additional robustness measures inspired by numerical experiments and literature are discussed. Cell Agglomeration and solution reinitialization are used to prevent the optimization algorithm from stagnating at an undesired local minimum. Lastly, the methods capabilities are demonstrated on a sequence of two-dimensional problems of increasing difficulty (Space-Time Advection, Space-Time Burgers, Steady 2D Euler Equations) and discussed by means of a convergence study.   

A Discontinuous Galerkin Method for Compressible Gas/Liquid Interfacial Flows with Consistent and Conservative Phase-Fields

Simulating compressible gas/liquid interfacial flows efficiently and with high accuracy is a challenging multi-physics problem due to large gradients, variable material properties, and disparate time and length scales. To address these challenges, we develop a discontinuous Galerkin method to solve the compressible Navier-Stokes equations using the five-equations multiphase model. The temporal scheme is explicit (Runge-Kutta) and the spatial scheme relies on a discontinuity sensor to identify regions where high-order limiting is applied, i.e., at interfaces and shock waves. Viscous effects and heat transfer are included, and adaptive mesh refinement via AMReX provides efficient resolution of sharp flow features. Further, we demonstrate a novel consistent and conservative phase-field method that controls the numerical diffusion of the material indicator function in a physically accurate manner. We illustrate the viability of our method through a variety of one- and multi-dimensional compressible gas/liquid interfacial problems, including high-speed impact of a liquid droplet onto a rigid wall.   

MUSCL and THINC hybrid scheme for strong and very weak shock waves in steady and unsteady flows

The monotonic upstream-centered schemes for conservation laws (MUSCL) are widely used to obtain second-order accuracy in finite volume methods [1]. Although the fundamental flow can be computed by MUSCL, it is known that MUSCL is too dissipative to solve a density discontinuity [2,3]. In addition, a recent study by the authors found that MUSCL cannot solve weak shock waves due to the excessive dissipation [4]. To overcome the problem, we focused on the tangent of hyperbola interface capturing (THINC) technique and found that the hybrid MUSCL-THINC scheme [5], which was developed for two-phase flow computations, can sharply capture weak shock waves [6]. This scheme was also valid for some unsteady compressible flows. However, the hybrid MUSCL-THINC scheme showed oscillations behind strong stationary shock waves, and the convergence was very poor in steady problems. In this study, we modified the weighting strategy of the cell boundary values computed by MUSCL and THINC to avoid the excessive usage of THINC around strong shocks. As a result, the modified scheme showed better convergence in the steady, two-dimensional, blunt-body problem with M = 3.0. We confirmed that the modification did not significantly affect high-resolution density discontinuity and weak shock capturing in unsteady problems. With this modification, high-resolution and robust computation was firstly achieved in the MUSCL and THINC hybrid scheme.   

Information geometric regularization of the barotropic Euler equation

A key numerical difficulty in compressible fluid dynamics is the formation of shock waves in supersonic flows. Shock waves feature jump discontinuities in the velocity and density of the fluid and thus preclude the existence of classical solutions to the compressible Euler equations. Weak solutions are commonly defined by viscous regularization, but even small amounts of viscosity can substantially change the long-term behavior of the solution through excessive dissipation, motivating the search for inviscid regularizations.

We revisit this classical problem by interpreting the Euler equations as flows on diffeomorphism manifolds based on the pioneering work of Vladimir Arnold. From this perspective, the formation of shock waves results from the lack of geodesic completeness of the diffeomorphism manifold. We restore the geodesic completeness of the diffeomorphism manifold by embedding it into a suitable information geometry. By deriving the Euler equations on this modified geometry, we obtain a regularization that avoids the formation of singularities without adding viscosity.   

A quantitative comparison of shock-capturing oscillations caused by high-order finite-difference schemes

Based on a quantitative evaluation process we have designed for assessing the robustness of shock/discontinuity-capturing finite-difference schemes, we compare the performance of various high-order schemes ranging from the classical weighted essentially non-oscillatory schemes to the novel targeted essentially non-oscillatory schemes. The results show that the schemes known for having better resolution may indeed produce stronger overshooting oscillations in various scenarios, where the CFL number, wave frequency, and distance between discontinuities vary. More importantly, the differences between the oscillations of different schemes are quantified systematically. Specifically, as the evaluation process includes a complete set of wave-interaction scenarios, we reveal that: (1) in certain ranges of wavenumber, a higher-order scheme may show smaller oscillations, and (2) in a certain range of CFL number, a smaller CFL number may lead to stronger oscillation. These two findings are difficult to observe while using the typical conventional evaluation process for robustness, which is in a case-by-case manner and without sufficient quantitative information. Potential usages of the present approach are also introduced.