Ingram, Ross Nicholas (2011) *Approximating fast, viscous fluid flow in complicated domains.* Doctoral Dissertation, University of Pittsburgh.

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## Abstract

Typical industrial and biological flows often occur in complicated domains that are either infeasible or impossible to resolve. Alternatives to solving the Navier-Stokes equations (NSE) for the fluid velocity in the pores of these problems must be considered. We propose and analyze a finite element discretization of the Brinkman equation for modeling non-Darcian fluid flow by allowing the Brinkman viscosity tends to infinity and permeability K tends to 0 in solid obstacles, and K tends to infinity in fluid domain. In this context, the Brinkman parameters are generally highly discontinuous. Furthermore, we consider inhomogeneous Dirichlet boundary conditions and non-solenoidal velocity (to model sources/sinks). Coupling between these two conditions makes even existence of solutions subtle. We establish conditions for the well-posedness of the continuous and discrete problem. We also establish convergence as Brinkman viscosity tends to infinity and K tends to 0 in solid obstacles, as K tends to infinity in fluid region, and as the mesh width vanishes. We prove similar results for time-dependent Brinkman equations for backward-Euler (BE) time-stepping. We provide numerical examples confirming theory including convergence of velocity, pressure, and drag/lift.We also investigate the stability and convergence of the fully-implicit, linearly extrapolated Crank-Nicolson (CNLE) time-stepping for finite element spatial discretization of the Navier-Stokes equations. Although presented in 1976 by Baker and applied and analyzed in various contexts since then, all known convergence estimates of CNLE require a time-step restriction. We show herein that no such restriction is required. Moreover, we propose a new linear extrapolation of the convecting velocity for CNLE so that the approximating velocities converge without without time-step restriction in l^{infty}(H^1) along with the discrete time derivative of the velocity in l^2(L^2). The new extrapolation ensures energetic stability of CNLE in the case of inhomogeneous boundary data. Such a result is unknown for conventional CNLE (usual techniques fail!). Numerical illustrations are provided showing that our new extrapolation clearly improves upon stability and accuracy from conventional CNLE.

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## Details | ||||||||||||||||

Item Type: | University of Pittsburgh ETD | |||||||||||||||
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ETD Committee: |
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Title: | Approximating fast, viscous fluid flow in complicated domains | |||||||||||||||

Status: | Unpublished | |||||||||||||||

Abstract: | Typical industrial and biological flows often occur in complicated domains that are either infeasible or impossible to resolve. Alternatives to solving the Navier-Stokes equations (NSE) for the fluid velocity in the pores of these problems must be considered. We propose and analyze a finite element discretization of the Brinkman equation for modeling non-Darcian fluid flow by allowing the Brinkman viscosity tends to infinity and permeability K tends to 0 in solid obstacles, and K tends to infinity in fluid domain. In this context, the Brinkman parameters are generally highly discontinuous. Furthermore, we consider inhomogeneous Dirichlet boundary conditions and non-solenoidal velocity (to model sources/sinks). Coupling between these two conditions makes even existence of solutions subtle. We establish conditions for the well-posedness of the continuous and discrete problem. We also establish convergence as Brinkman viscosity tends to infinity and K tends to 0 in solid obstacles, as K tends to infinity in fluid region, and as the mesh width vanishes. We prove similar results for time-dependent Brinkman equations for backward-Euler (BE) time-stepping. We provide numerical examples confirming theory including convergence of velocity, pressure, and drag/lift.We also investigate the stability and convergence of the fully-implicit, linearly extrapolated Crank-Nicolson (CNLE) time-stepping for finite element spatial discretization of the Navier-Stokes equations. Although presented in 1976 by Baker and applied and analyzed in various contexts since then, all known convergence estimates of CNLE require a time-step restriction. We show herein that no such restriction is required. Moreover, we propose a new linear extrapolation of the convecting velocity for CNLE so that the approximating velocities converge without without time-step restriction in l^{infty}(H^1) along with the discrete time derivative of the velocity in l^2(L^2). The new extrapolation ensures energetic stability of CNLE in the case of inhomogeneous boundary data. Such a result is unknown for conventional CNLE (usual techniques fail!). Numerical illustrations are provided showing that our new extrapolation clearly improves upon stability and accuracy from conventional CNLE. | |||||||||||||||

Date: | 28 September 2011 | |||||||||||||||

Date Type: | Completion | |||||||||||||||

Defense Date: | 25 April 2011 | |||||||||||||||

Approval Date: | 28 September 2011 | |||||||||||||||

Submission Date: | 05 April 2011 | |||||||||||||||

Access Restriction: | No restriction; Release the ETD for access worldwide immediately. | |||||||||||||||

Patent pending: | No | |||||||||||||||

Institution: | University of Pittsburgh | |||||||||||||||

Thesis Type: | Doctoral Dissertation | |||||||||||||||

Refereed: | Yes | |||||||||||||||

Degree: | PhD - Doctor of Philosophy | |||||||||||||||

URN: | etd-04052011-153616 | |||||||||||||||

Uncontrolled Keywords: | extrapolation; analysis; backward; Brinkman; convergence; Crank-Nicolson; Darcy; error; Euler; finite element; implicit; inhomogeneous; linearization; non-Darcy; non-solenoidal; porous media; stability; Stokes; very porous media; volume penalization; existence; Navier | |||||||||||||||

Schools and Programs: | Dietrich School of Arts and Sciences > Mathematics | |||||||||||||||

Date Deposited: | 10 Nov 2011 14:34 | |||||||||||||||

Last Modified: | 19 Jan 2012 16:29 | |||||||||||||||

Other ID: | http://etd.library.pitt.edu/ETD/available/etd-04052011-153616/, etd-04052011-153616 |

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