On a well-posed turbulence model

Abstract

Abstract. This report considers mathematical properties, important for practical computations, of a model for the simulation of the motion of large eddies in a turbulent flow. In this model, closure is accomplished in the very simple way:uu ∼ūū, yielding the modelIn particular, we prove existence and uniqueness of strong solutions, develop the regularity of solutions of the model and give a rigorous bound on the modelling error, ||ū − w||. Finally, we consider the question of non-physical vortices (false eddies), proving that the model correctly predicts that only a small amount of vorticity results when the total turning forces on the flow are small. 1.Introduction. The great challenge in simulation of turbulent flows from applications ranging from geophysics to biomedical device design is that equations for the pointwise flow quantities are well-known but intractable to computational solution and sensitive to uncertainties and perturbation in problem data. On the other hand, closed equations for the averages of flow quantities cannot be obtained directly from the physics of fluid motion. Thus, modeling in large eddy simulation (meaning the approximation of local, spacial averages in a turbulent flow) is typically based on guesswork (phenomenology), calibration (data fitting model parameters) and (at best) approximation.

Regularization modeling for large-eddy simulation of homogeneous isotropic decaying turbulence

“…The modified Leray regularization was proposed in [24] and the simplified Bardina model was put forward in [21][22][23]. Each of these modifications of the convective fluxes induces a particular sub-filter model to be used in the large-eddy template.…”

“…For each of these models, the rigorous existence, uniqueness and regularity of the solution to the modeled equations have been established [11,[21][22][23][24]. Whenever the unfiltered solution u appears in one of the regularization model tensors, we imply that an approximate inversion of the Helmholtz filter is used.…”

“…We include the Leray principle [11,12,14], the NS-α formulation [15][16][17][18][19][20], the simplified Bardina model [21][22][23] and a recent proposal based on a modified Leray principle [24]. These inviscid, i.e., non-hyper-viscous, regularization principles are appealing as the corresponding system of nonlinear partial differential equations is known to possess a unique solution with known regularity [11,14,17,18,[21][22][23][24]. These models were shown to predict large scales of ensemble averages of turbulent flows in channels and pipes at very large Reynolds numbers in close agreement with experimental data [15,16,25] (see also [26] for the Clark-α model).…”

Numerical study of a regularized barotropic vorticity model of geophysical flow

“…The Bardina model emerged in 1980 as a particular closure model to approximate the Reynolds stress tensor introduced by Bardina et al [17]. It was later studied analytically in a simplified form by Layton and Lewandowski [18] and Cao et al [19]. Following [19], the simplified Bardina model can be written as…”

Global regularity and convergence of a Birkhoff‐Rott‐α approximation of the dynamics of vortex sheets of the two‐dimensional Euler equations

“…In [13,14,15,31,32] the corresponding Navier-Stokes-˛(NS-˛) (also known as the viscous Camassa-Holm equations or the Lagrangian-averaged Navier-Stokes-( LANS-˛)) model, was obtained by introducing the appropriate viscous term into the Euler-˛equations. The extensive research of the˛-models (see, e.g., [2,7,10,11,13,14,15,16,17,18,20,31,32,34,35,36,40,42,43,48,49,51,52,63,77]) stems, on the one hand, from the successful comparison of their steady state solutions to empirical data, for a large range of huge Reynolds numbers, for turbulent flows in infinite channels and pipes [13,14,15]. On the other hand, thę -models can also be viewed as numerical regularizations of the original, Euler, or Navier-Stokes systems [7,11,44,52].…”