Carolina C. Manica

Numerical analysis of Leray-Tikhonov deconvolution models of fluid motion

This report develops and studies a new family of Navier-Stokes equation regularizations: Leray-Tikhonov regularizations with time relaxation models. This new family of turbulence models is based on a modification (consistent with the large scales) of Tikhonov-Lavrentiev regularization. With this approach, we obtain an approximation of the unfiltered solution by one filtering step. We introduce the modified Tikhonov deconvolution operator and study its mathematical properties. We also perform rigorous numerical analysis of a computationally attractive algorithm for this family of models and present numerical experiments using it.

On Crank-Nicolson Adams-Bashforth timestepping for approximate deconvolution models in two dimensions

We consider a Crank-Nicolson-Adams-Bashforth temporal discretization, together with a finite element spatial discretization, for efficiently computing solutions to approximate deconvolution models of incompressible flow in two dimensions. We prove a restriction on the timestep that will guarantee stability, and provide several numerical experiments that show the proposed method is very effective at finding accurate coarse mesh approximations for benchmark flow problems.

Métodos Temporais e Modelo de Deconvolução de Leray para as Equações de Navier-Stokes em Fluidos Incompressíveis via Elementos Finitos

O estudo das equacoes de Navier-Stokes desperta interesse dos estudiosos da area da analise numerica, visto que a partir destas pode-se determinar os campos de velocidade e pressao de um escoamento. Com estas equacoes tambem pode-se aproximar coecientes aerodinâmicos, fato de grande interesse nas industria automobilistica e aeronautica. Propoe-se estudar a aproximacao das equacoes de Navier Stokes via o metodo de elementos nitos. Estudam-se duas propostas de metodos de discretizacao temporal para as equacoes dadas. Introduz-se um modelo de regularizacao e atraves do calculo dos coecientes de arrasto e sustentacao comprova-se a sua efetividade.

On the regularized modeling of density currents

In this work we studied four regularization models with deconvolution for density currents, namely, Boussinesq-α, Boussinesq-ω, Boussinesq-Leray and Modified-BoussinesqLeray. A Crank-Nicolson in time and Finite Element in space algorithm is proposed and proved to be unconditionally stable and optimally convergent, which is also verified through convergence rates in computational simulations. Finally, the models are compared through the Marsigli’s flow problem. We found that all regularization models produced accurate solutions for low Reynolds number. Moreover, as expected, we observe that increasing deconvolution order improves solution. On the other hand, for high Reynolds number BoussinesqLeray and Boussinesq-α with deconvolution produced the most accurate solutions. However, from the computational viewpoint, the Boussinesq-Leray model presented advantage due to its decoupling between momentum and filter equations which permits to increase the deconvolution order with no significant increase in the computational cost.

On a Computational Advantageous Voigt Regularization for Geophysical Flows

In this paper we study a linearized Crank-Nicolson in time and Finite Element in space algorithm for the BV-Voigt regularization model of geophysical flows, which presents interesting advantages from the computational point of view. We prove the algorithm conserves energy and is unconditionally stable and optimally convergent. Lastly, we show that the BV-Voigt model provides accurate solutions and compares favorably with a related regularization model in a coarse mesh, a case in which the BV model solution degenerates.

Convergence Analysis of a Fully Discrete Family of Iterated Deconvolution Methods for Turbulence Modeling with Time Relaxation

We present a general theory for regularization models of the Navier-Stokes equations based on the Leray deconvolution model with a general deconvolution operator designed to fit a few important key properties. We provide examples of this type of operator, such as the (modified) Tikhonov-Lavrentiev and (modified) Iterated Tikhonov-Lavrentiev operators, and study their mathematical properties. An existence theory is derived for the family of models and a rigorous convergence theory is derived for the resulting algorithms. Our theoretical results are supported by numerical testing with the Taylor-Green vortex problem, presented for the special operator cases mentioned above.

Architecture of Approximate Deconvolution Models of Turbulence

This report presents the mathematical foundation of approximate deconvolution LES models together with the model phenomenology downstream of the theory. This mathematical foundation now begins to be complete for the incompressible Navier–Stokes equations. It is built upon averaging, deconvolving and addressing closure so as to obtain the physically correct energy and helicity balances in the LES model. We show how this is determined and how correct energy balance implies correct prediction of turbulent statistics. Interestingly, the approach is simple and thus gives a road map to develop models for more complex turbulent flows. We illustrate this herein for the case of MHD turbulence.