Conference Agenda

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Session Overview
Session
MS 18a: Advances in high order methods for fluid dynamic
Time:
Tuesday, 13/July/2021:
12:00pm - 2:00pm

Session Chair: Henar Herrero
Session Chair: Jezabel Curbelo
Virtual location: Zoom 2


Session Abstract

On approaching turbulence and geophysical domains, different scales appear in

the problems (Aurnou et al, 2019). Small scales and large domains are difficult to

reach with spectral methods because they are ill conditioned so that it is not

possible to increase suitably the size of the expansions. A strategy to increase

the size of the mesh avoiding the increase of the expansion are domain

decomposition (DD) techniques (Quarteroni, 1999, 2002). Among them, the

alternating Schwarz methods respect the original computation (Blayo, 2016; Xu,

2005). Herrero et al. applied this methodology to a stationary natural convection

problem (Herrero, 2018). The method is expensive in time and unstable for some

values of the parameters. A way of stabilization and optimization consists of using

two levels of mesh in the alternating Schwarz method (Axelsson, 2019).

Geophysical problems are studied using direct numerical simulations (GL Kooij

et al. 2019, Curbelo et al. 2019). The study of instabilities requires the tracking of

bifurcations varying parameters. These methods are based on the observation

that varieties calculated by means of proper orthogonal decomposition (POD)

from solutions at some values of the parameters also contain solutions to different

values of the parameters. If the equations that govern the evolution of a system

are dissipative, the behavior of the system for long times is contained in a finite-

dimensional inertial variety, which is often of a low dimension (Foias, 1988).

Herrero et al. applied this type of methods for a Rayleigh-Bénard problem in a

rectangular domain with reduced basis (Herrero, 2013) and taking as snapshots

transient states of a temporal evolution towards a stationary solution or transient

states of Newton's iteration for the treatment of non-linearity for a fixed parameter

value, managing to reproduce the diagram of bifurcations with very low

computational cost (Pla, 2015). Gutierrez-Castillo et al. analyzed the use of POD

for temporal flow patterns in non-newtonian fluids.


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Presentations
12:00pm - 12:30pm

HORSES3D: A HIGH ORDER DISCONTINUOUS GALERKIN SOLVER FOR FLOW SIMULATIONS AND MULTIPHYSICS APPLICATIONS

Esteban Ferrer1, Juan Manzanero1, Andres Rueda2, Wojciech Laskowski1, Gerasimos Ntoukas1, Oscar Mariño1, Gonzalo Rubio1, David A Kopriva3, Eusebio Valero1

1ETSIAE-UPM (School of Aeronautics), Spain; 2University of Cologne - Mathematical Institute; 3San Diego State University

We present the latest developments of our high order discontinuous Galerkin solvers: HORSES3D, capable of solving a range of flow applications including compressible, incompressible, turbulent (LES) and multiphase flows.

We provide an overview of the different models implemented for turbulent flows (iLES, Smagorinsky, SVV-Smagorinsky, WALE) [1, 7] and multiphase flows [2-4] (Cahn-Hilliard). Additionally, we detail the capabilities to perform local p-adaption [5-6], implicit strategies and multigrid to advance the solution in time, efficiently.

REERENCES

[ 1 ] - J Manzanero, E Ferrer, G Rubio, E Valero, "Design of a Smagorinsky Spectral Vanishing Viscosity turbulence model for discontinuous Galerkin methods", Computers & Fluids, Vol 200, 2020

[ 2 ] - J Manzanero, G Rubio, DA Kopriva, E Ferrer, E Valero, "An entropy–stable discontinuous Galerkin approximation for the incompressible Navier–Stokes equations with variable density and artificial compressibility", Journal of Computational Physics, Vol 408, 2020

[ 3 ] - J Manzanero, G Rubio, DA Kopriva, E Ferrer, E Valero, "A free-energy stable nodal discontinuous Galerkin approximation with summation-by-parts property for the Cahn-Hilliard equation", Journal of Computational Physics, Vol 403, 2020

[ 4 ] - J Manzanero, G Rubio, DA Kopriva, E Ferrer, E Valero, "Entropy-stable DG approximation with SBP property for the incompressible Navier-Stokes/Cahn-Hilliard system", Accepted at Journal of Computational Physics 2020

[ 5 ] - AM Rueda-Ramirez, J Manzanero, E Ferrer, G Rubio, E Valero, "A p-Multigrid Strategy with Anisotropic p-Adaptation Based on Truncation Errors for High-Order Discontinuous Galerkin Methods", Journal of Computational Physics, Vol 378, p 209-233, 2019

[ 6 ] - AM Rueda-Ramirez, G Rubio, E Ferrer, E Valero, "Truncation Error Estimation in the p-Anisotropic Discontinuous Galerkin Spectral Element Method", Journal of Scientific Computing, Vol 78(1), p 433-466, 2019

[ 7 ] - E Ferrer, "An interior penalty stabilised incompressible Discontinuous Galerkin - Fourier solver for implicit Large Eddy Simulations", Journal of Computational Physics, Vol 348, p 754-775, 2017



12:30pm - 1:00pm

Implicit shock tracking for unsteady flows by the method of lines

Per-Olof Persson1,2, Andrew Shi1,2, Matthew Zahr3

1Department of Mathematics, University of California, Berkeley, USA; 2Mathematics Group, Lawrence Berkeley National Laboratory, USA; 3Department of Aerospace and Mechanical Engineering, University of Notre Dame, USA

We present the time-dependent extension of our recently developed high-order implicit shock tracking (HOIST) framework for resolving discontinuous solutions of inviscid conservation laws. Central to the framework is an optimization problem which simultaneously computes a discontinuity-aligned mesh and the corresponding high-order approximation to the flow, which provides nonlinear stabilization and a high-order approximation to the solution. The time discretization is based on method of lines and diagonally implicit Runge-Kutta (DIRK) methods, and we formulate and solve an optimization problem that produces a feature-aligned mesh and solution at each Runge-Kutta stage of each timestep. A Rankine-Hugoniot based prediction of the shock location together with a high-order untangling mesh smoothing procedure provides a high-quality initial guess for the optimization problem at each time, which results in Newton-like convergence of the sequential quadratic programming (SQP) optimization solver. This method is shown to deliver highly accurate solutions on coarse, high-order discretizations without nonlinear stabilization and recovers the design accuracy of the Runge-Kutta scheme. We demonstrate our framework using several inviscid unsteady conservation laws, and we verify that our method is able to recover the design order of accuracy of our time integrator in the presence of strong discontinuities.



1:00pm - 1:30pm

Impact of wall modeling on kinetic energy stability for the compressible Navier-Stokes equations

Vikram Singh1,2, Steven Frankel2, Jan Nordström3,4

1Ocean in the Earth System, Max-Planck Institute for Meteorolgy, Hamburg, Germany; 2Faculty of Mechanical Engineering, Technion - Israel Institute of Technology, Haifa Israel; 3Department of Mathematics, Computational Mathematics, Linköping University, SE-581 83 Linköping, Sweden; 4Department of Mathematics and Applied Mathematics, University of Johannesburg, PO Box 524, Auckland Park 2006, South Africa

Affordable, high order simulations of turbulent flows on unstructured grids for very high Reynolds' number flows require wall models for efficiency. However, different wall models have different accuracy and stability properties. Here, we develop a kinetic energy stability estimate to investigate stability of wall model boundary conditions. Using this norm, two wall models are studied, a popular equilibrium stress wall model, which is found to be unstable and the dynamic slip wall model which is found to be stable. These results are extended to the discrete case using the Summation-by-parts (SBP) property of the discontinuous Galerkin method. Numerical tests show that while the equilibrium stress wall model is accurate but unstable, the dynamic slip wall model is inaccurate but stable.



1:30pm - 2:00pm

POD analysis of temporal flow patterns in different regimes

Paloma Gutierrez-Castillo1, Manuel Garrido-Martin1, Becca Thomases2

1University of Malaga, Spain; 2University of California, Davis

Proper Orthogonal Decomposition (POD) has been used broadly in analyzing turbulent flows at high Reynolds numbers, such as flow in a pipe. However, there exists a lack of knowledge in analyzing some other regimes which contain interesting temporal behaviors. We present two study cases with completely different flow regimes showing the advantages of analyzing them using POD. First, we describe an application in creeping flow (very low Reynolds number) in Non-Newtonian fluids. POD helps characterize the different bifurcations of the flow directly related to the movement of stagnation points of the problem. We have also proved the efficiency of this method to store data recovering 90% of the temporal evolution with only a few geometric modes (time-independent) and some temporal modes, which are a single value for each time. Second, we analyze experimental data of a wing tip vortex at moderate Reynolds numbers. The possible attenuation of this kind of vortices is a key criterion for any airport design. By using POD, we were able to describe the vortex and isolate a mode representing the global attenuation of the vortex.



 
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