Helicity and Physical Fidelity in Turbulence Modeling

This thesis is a study of physical fidelity in turbulence modeling. We first consider conservation laws in several popular turbulence models and find that of the Leray, Leray-deconvolution, Bardina and Stolz-Adams approximate deconvolution model (ADM), all but the Bardina model conserve a model energy. Only the ADM conserves a model helicity. Since the ADM conserves a model energy and helicity, we then investigate a joint helicity-energy spectrum in the ADM. We find that up to a filter-dependent length scale, the ADM cascades energy and helicity jointly in the same manner as the Navier-Stokes equations.We also investigate helicity treatment in discretizations of turbulence models. For inviscid, periodic flow, we implement energy conserving discretizations of the ADM, Leray, and Leray-deconvolution models as well as the Navier-Stokes equations (NSE) and observe helicity treatments. We find that of none of the models conserve helicity (or model helicity) in the discretizations. Since the Leray-deconvolution model of turbulence is newly developed, our implementation is new and thus we analyze the trapezoidal Galerkin scheme that we implement and compare it to the usual Leray model.Lastly, we develop an energy and helicity conserving trapezoidal Galerkin scheme for the Navier-Stokes equations. We prove conservation properties for the scheme, stability, and show the scheme does not lose asymptotic accuracy compared to the usual trapezoidal Galerkin scheme. We also present numerical experiments that compare the energy and helicity conserving scheme to more typical schemes

Numerical analysis and phenomenology of homogeneous, isotropic turbulence generated by higher order models of turbulence

Turbulence appears in many processes in the nature and it is connected with many engineering, biophysical and climate applications. Therefore, the accurate, efficient and reliable simulation of turbulent flows is an essential difficulty in many current applications. Fundamental and universal (i.e. mathematical) insights into fluid structures will enable such simulations. To that end, we apply the phenomenology of homogeneous, isotropic turbulence to a family of Large Eddy Simulation (LES) models, the so-calledfamily of Approximate Deconvolution Models (ADM). We establish that the models themselves have an energy cascade with two asymptotically different inertial ranges. Delineation of these gives insight into the resolution requirements of using ADM. A correct prediction of a 3D turbulent flow means getting the energy balance and rotational structures correct, i.e., it means (in the large) matching the energy and helicity statistics. Thus, we consider the prediction of energy and helicity statistics of the family of Approximate Deconvolution Models of turbulence. We show that the family of ADM has a helicity cascade that it is linked to its energy cascade and predicted correctly over the large/resolved scales. Turbulent flows are very rich in scales and to be able to capture all of them, we need to use a very fine mesh. Unfortunately, even with the amazing development of the computer power, we are not able to perform such simulations. Thus, many numerical regularization (aiming to truncate the small scales) have been explored in computational fluid dynamics. We investigated one of such regularization, called the Time Relaxation Model (TRM). We apply the phenomenology of homogeneous, isotropic turbulence to understand how the time relaxation term, by itself, acts to truncate solution scales and to use this understanding to give insight into coefficient selection. We also study the stability and convergence analysis of a finite element discretization of TRM. Next we complement this with an experimentalstudy of the convergence rates and of the effect the time relaxation term has on the large scales of a flow near a transitional point

Numerical Methods in Turbulence

Fluid motion and its richness of detail are described by theNavier-Stokes equations. Most of the numerical analysis existent todate is applicable for strong solutions (typically small body forceand initial data). We prove that statistics of weak solutions areoptimally computable in the simple but important case of small bodyforce and large initial data. These estimates are used to predictdrag and lift statistics, quantities of great interest inengineering. In the case of arbitrarily large body force and initialdata, for shear flows, statistics of the computed solution are shownto behave according to the Kolmogorov theory.Many times, in turbulent fluid flow, a direct numerical simulationbecomes expensive. One alternative is Large Eddy Simulation (LES).It exploits the decoupling of scales, achieved via introduction of afilter, thus reducing the number of degrees of freedom in asimulation. A relatively new family of LES models is the ApproximateDeconvolution Models (ADM). They have remarkable mathematicalproperties and perform well in computations. However, some reportsclaim that they are unstable for simulations with walls and requirethe addition of explicit stabilization.We show that, given the right formulation, variationaldiscretizations of the Zeroth Order Model, a member of the ADMfamily, are indeed stable. We present evidence that stability of oneformulation is sensitive to the exact way in which filtering isperformed and show some numerical results. An alternativeformulation, which does not depend on the way filtering isperformed, is also presented. In both cases we perform convergencestudies. This is a first step in determining stable and robustdiscretizations for the whole family of ADM, as well as guidance fordealing with arbitrary geometries/domains that arise in practicalapplications.Getting a prediction of a turbulent flow right also means gettingthe energy balance and the rotational structures correct, whichmeans (in the large) matching the energy and helicity statistics. Weapply similarity theory to the ADM and show that the model has ahelicity cascade, linked to its energy cascade, which predicts thecorrect helicity statistics up to the cut-off frequency

Convergence of approximate deconvolution models to the mean Magnetohydrodynamics Equations: Analysis of two models

We consider two Large Eddy Simulation (LES) models for the approximation of large scales of the equations of Magnetohydrodynamics (MHD in the sequel). We study two $\alpha$-models, which are obtained adapting to the MHD the approach by Stolz and Adams with van Cittert approximate deconvolution operators. First, we prove existence and uniqueness of a regular weak solution for a system with filtering and deconvolution in both equations. Then we study the behavior of solutions as the deconvolution parameter goes to infinity. The main result of this paper is the convergence to a solution of the filtered MHD equations. In the final section we study also the problem with filtering acting only on the velocity equation

Mathematical Architecture for Models of Fluid Flow Phenomena

This thesis is a study of several high accuracy numerical methods for fluid flow problems and turbulence modeling.First we consider a stabilized finite element method for the Navier-Stokes equations which has second order temporal accuracy. The method requires only the solution of one linear system (arising from an Oseen problem) per time step. We proceed by introducing a family of defect correction methods for the time dependent Navier-Stokes equations, aiming at higher Reynolds' number. The method presented is unconditionally stable, computationally cheap and gives an accurate approximation to the quantities sought. Next, we present a defect correction method with increased time accuracy. The method is applied to the evolutionary transport problem, it is proven to be unconditionally stable, and the desired time accuracy is attained with no extra computational cost. We then turn to the turbulence modeling in coupled Navier-Stokes systems - namely, MagnetoHydroDynamics. Magnetically conducting fluids arise in important applications including plasma physics, geophysics and astronomy. In many of these, turbulent MHD (magnetohydrodynamic) flows are typical. The difficulties of accurately modeling and simulating turbulent flows are magnified many times over in the MHD case. We consider the mathematical properties of a model for the simulation of the large eddies in turbulent viscous, incompressible, electrically conducting flows. We prove existence, uniqueness and convergence of solutions for the simplest closed MHD model. Furthermore, we show that the model preserves the properties of the 3D MHD equations. Lastly, we consider the family of approximate deconvolution models (ADM) for turbulent MHD flows. We prove existence, uniqueness and convergence of solutions, and derive a bound on the modeling error. We verify the physical properties of the models and provide the results of the computational tests