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

Abstract

Communicated by K. TakizawaWe present a residual-based turbulence model for problems with free surfaces. The method is derived based on variational multiscale ideas that assume a decomposition of the solution fields into overlapping scales that are termed as coarse and fine scales. The fine scales are further split hierarchically into fine-scales level-I and fine-scales level-II. The hierarchical variational problems that govern the two fine-scale components are modeled employing bubble functions approach. The model for level-II scales is variationally embedded in the mixed field level-I problem to yield a stable level-I formulation. Subsequently, the model for level-I scales that in fact constitutes the fine-scale turbulence model is then variationally injected in the coarse-scale variational form. A significant feature of the method is that it does not contain any embedded tunable parameters. To accommodate the moving boundaries we cast the formulation in an arbitrary Lagrangian-Eulerian frame of reference. The free surface boundary condition is imposed weakly which results in a formulation that conserves the volume of the fluid. A variety of benchmark problems show the accuracy and range of applicability of the proposed formulation and results are compared with published data. A wavy bed problem is investigated to show the interaction of turbulence generated at the bottom surface with the free surface thereby leading to irregular free surface elevations. Keywords: Irregular free surfaces; open channel; residual-based turbulence models; variational multiscale method; large eddy simulation. † Corresponding author 1 Math. Models Methods Appl. Sci. Downloaded from www.worldscientific.com by UNIVERSITY OF PITTSBURGH on 08/18/15. For personal use only. 2 R. Calderer et al.

Variationally derived closure models for large eddy simulation of incompressible turbulent flows

“…To ensure global conservation of mass for higher Reynolds number flows, a least-squares form of local mass conservation was also imposed. Subsequently, hierarchical application of the notion of scale separation resulted in the three-level VMS method [24][25][26] which facilitated a more flexible fine-scale subsystem to derive closed form expressions for the turbulence models. In this method the fine scale pressure field that emanates from the mixed form of fine-scale variational equations helps with the conservation of mass in highly turbulent flows.…”

“…In the earlier works by the senior author it was shown that variationally derived fine scale models that provide a mechanism to transfer energy between resolved and unresolved scales also satisfy the governing momentum balance equations. [23][24][25][26] The magnitude of the fine scales is dictated by the residual of the Euler-Lagrange equations of the resolved coarse scales and therefore the resulting numerical method is variationally consistent. From the modeling perspective the fine-scale variational equation provides a mathematical zoom to focus on the finer features in the flow physics, and an application of ideas from the residual-free bubbles (RFB) method directly to the fine-scale variational equation provides an opportunity that can be exploited to derive the residual-based turbulence models.…”

“…In the approach being proposed here, we apply a generalization of RFB directly at the level of the fine-scale variational equation. [23][24][25][26] In order to keep the computational cost manageable, we do not resort to the development of the ideal bubbles or the residual free bubbles. Rather we employ higher-order basis functions, typically polynomial bubble functions, wherein the space of fine scales is selected to be independent of the space of coarse scales to ensure the stability of the fine-scale formulation.…”

“…Furthermore, application of the bubble functions approach directly at the fine-scale level helps in localizing the fine scale problem to the sum of element interiors 29,30 that yields many options for the derivation of the closure models. [23][24][25][26] In this context the fine-scale field (or centered fluctuations) can be viewed as the perturbation part of the total solution while coarse-scale field represents the mean solution.…”

An immersogeometric formulation for free-surface flows with application to marine engineering problems

“…The free-surface topological changes can be automatically handled by solving an additional scalar partial differential equation. Interface capturing methods have been widely applied to a wide range of interfacial problems, including bubble dynamics [17][18][19], jet atomization [20], and free-surface flows [21,22].…”