Direct Numerical Simulation in CFD via Lagrangian Formulations and Multi-Scale Homogenization

NIGRO, NORBERTO M.; RYZHAKOV, PAVEL; IDELSOHN, SERGIO R.; GIMENEZ, JUAN M.; LARRETEGUY, AXEL

Boulder

Workshop; Advances in Numerical Methods for Simulation, Optimization, and Uncertainty Quantification of Coupled Physics Problems; 2018

Although the Navier-Stokes equations are equivalently applicable to both laminar and turbulent fluid flows, the current computing power precludes employing fine meshes that would allow to simulate turbulent flows without introducing empirical approximations. The prediction power of the models is therefore restricted to problems within the margins of the selected empirical approximation. Solving the problem without such approximations on a mesh sufficiently fine so as to represent the whole expected range of eddy sizes is known as ?Direct Numerical Simulation? (DNS). Taking into account that a wide range of fluid flow problems of industrial interest are indeed turbulent, it is worthwhile to continue improving the models so that they fit more and more with the physics of the problem. Simulating a CFD problem in a given domain with a ?fine enough? DNS mesh introduces an unmanageable number of unknowns for current computers. The project we are working on involves modelling turbulent flow in a DNS fashion, i.e. without any additional turbulence model. However, we strive to develop an approach considerably more computationally in a more efficient way than a classical DNS. The basic idea is as follows. The global (macro) domain, can be divided into many equally-shaped small (micro) domains, the so-called RVEs for ?Representative Volume Elements? (micro problems). These RVEs, in principle, can be solved individually and independently from each other. The RVEs may even be previously solved off-line for different time-dependant loads. The micro and macro problems require to be coupled. This will be performed via the theory known as ?homogenization?. Another important aspect of the solution process in our approach at the macro level is the use of a Lagrangian formulation for the fluid [1]. The Lagrangian particles will take care not only of the transport of the macro-velocity, but also of the micro-eddies that may appear in some RVEs, thus convecting the turbulent energy. This is a very important feature in turbulent flows where turbulence is produced in some high gradient regions and is then convected and diffused to other regions with different gradients. Finally, concerning the off-line solution of the RVEs for different time-depending loads, a Reduction Order Model (ROM) [2], a Machine Learning technique or a pseudo-spectral (FFT + Chebychev) will be tested in order to obtain a sensible speed-up. This presentation is a work in progress, aimed more at opening a discussion on the topic, rather than at presenting elaborate results.