Projection based Variational Multiscale Methods for Incompressible Navier-Stokes Equations to Model Turbulent Flows in Time-dependent Domains

Abstract

Numerical solution of differential equations having multitude of scales in the solution field is one of the most challenging research areas, but highly demanded in scientific and industrial applications. One of the natural approaches for handling such problems is to separate the scales and approximate the solution of the segregated scales with appropriate numerical method. Variational multiscale method (VMS) is a predominant method in the paradigm of scale separation schemes. In our work we have used the VMS technique to develop a numerical scheme for computations of turbulent flows in time-dependent domains. VMS allows separation of the entire range of scales in the flow field into two or three groups, thereby enabling a different numerical treatment for the different groups. In the context of computational fluid dynamics(CFD), VMS is a significant new improvement over the classical large eddy simulation (LES). VMS does away with the commutation errors arising due to filtering in LES. Further, in a three-scale VMS approach the model for the subgrid scale can be contained to only a part of the resolved scales instead of effecting the entire range of resolved scales. The projection based VMS scheme that we have developed gives a robust and efficient method for solving problems of turbulent fluid flows in deforming domains, governed by incompressible Navier {Stokes equations. In addition to the existing challenges due to turbulence, the computational complexity of the problem increases further when the considered domain is time-dependent. In this work, we have used an arbitrary Lagrangian-Eulerian (ALE) based VMS scheme to account for the domain deformation. In the proposed scheme, the large scales are represented by an additional tensor valued space. The resolved large and small scales are computed in a single unified equation, and the effect of unresolved scales is confined only to the resolved small scales, by using a projection operator. The popular Smagorinsky eddy viscosity model is used to approximate the effects of unresolved scales. The used ALE approach consists of an elastic mesh update technique. Moreover, a computationally efficient scheme is obtained by the choice of orthogonal finite element basis function for the resolved large scales, which allows to reformulate the ALE-VMS system matrix into the standard form of the NSE system matrix. Thus, any existing Navier{Stokes solver can be utilized for this scheme, with modifications. Further, the stability and error estimates of the scheme using a linear model of the NSE are also derived. Finally, the proposed scheme has been validated by a number of numerical examples over a wide range of problems.