Residual‐based turbulence models for moving boundary flows: hierarchical application of variational multiscale method and three‐level scale separation

Abstract

SUMMARYThis paper presents residual‐based turbulence models for problems with moving boundaries and interfaces. The method is developed via a hierarchical application of variational multiscale ideas and the models are cast in an arbitrary Lagrangian–Eulerian (ALE) frame to accommodate the deformation of domain boundaries. An overlapping additive decomposition of velocity and pressure fields into coarse and fine scale components leads to coarse and fine scale mixed‐field problems. The problem governing fine scales is subjected to a further decomposition of the fine scale velocity into overlapping components termed as fine scales level I and level II. In turn, in the bottom‐up integration of scales, the model for level II fine scales serves to stabilize the problem governing level I fine scales, and model for level I fields yields the turbulence models. From the computational perspective, the coarse scales are represented in terms of the standard Lagrange shape functions, whereas level I and level II scales are represented via quadratic and fourth order polynomial bubbles, respectively. Because of the bubble functions approach employed in the consistently derived fine scale models, the resulting method is free of any embedded or tunable parameters. The proposed turbulence models share a common feature with the LES models in that the largest scales in the flow are numerically resolved, whereas the subgrid scales are modeled. The method is applied to flow around a plunging airfoil at Re = 40,000, and results are compared with experimental and numerical data published in the literature. Also presented are the results for the plunging airfoil at Re = 60,000 to show the robustness and range of applicability of the method. Copyright © 2013 John Wiley & Sons, Ltd.

Residual-based turbulence models and arbitrary Lagrangian–Eulerian framework for free surface flows

“…To derive a formulation that successfully encompasses all these attributes we employ the VMS framework, which decomposes the solution fields into components of different scales. In the following section we present a synopsis of the method that was derived in Masud and Calderer 29 and we extend it to the treatment of the free surface condition that is considered here.…”

“…Equations (28)- (29) depend only on the coarse scales and the level-I fine scales. Using a second set of bubble functions that is linearly independent of the first set of bubbles employed to derive the model (27), we can solve (28)- (29) and derive a model for the level-I scales in terms of the coarse-scale fields:…”

“…This has been shown in a variety of turbulent flow problems over fixed 8,28 and moving domains. 29 In this paper we extend 29 to problems with free surfaces and show the applicability of the proposed method via application to two sets of turbulent free surface flow problems.…”

“…To accommodate free surface motions the formulation is cast in an ALE frame of reference. 7,23 Underlying idea of the residual-based turbulence models is the VMS method [4][5][6]14,19,20,29,34,38 wherein the solution fields are decomposed in components of overlapping scales, namely, the coarse and the fine scales. The fine scales are modeled in terms of the residuals of the Euler-Lagrange equations of the coarse scales, and are variationally embedded in the coarse-scale equations.…”

Projection‐based variational multiscale method for incompressible Navier–Stokes equations in time‐dependent domains

“…In particular, the Smagorinsky eddy viscosity model is used to model the effects of unresolved scales. Alternatively, the small-scale solution is approximated in a space enriched with bubble functions, and the solution is transferred to the large-scale equation in the form of a stability tensor in [16,17]. Further, the choice of orthogonal finite element basis function for the resolved large scale leads to a computationally efficient scheme.…”