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Approximating fast, viscous fluid flow in complicated domains

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 (Unpublished)

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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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Item Type: University of Pittsburgh ETD
Status: Unpublished
CreatorsEmailPitt UsernameORCID
Ingram, Ross Nicholasrni1@pitt.eduRNI1
ETD Committee:
TitleMemberEmail AddressPitt UsernameORCID
Committee ChairLayton, Williamwjl@pitt.eduWJL
Committee MemberTrenchea, Catalintrenchea@pitt.eduTRENCHEA
Committee MemberGaldi, Giovannigaldi@pitt.eduGALDI
Committee MemberYotov, Ivanyotov@math.pitt.eduYOTOV
Date: 28 September 2011
Date Type: Completion
Defense Date: 25 April 2011
Approval Date: 28 September 2011
Submission Date: 5 April 2011
Access Restriction: No restriction; Release the ETD for access worldwide immediately.
Institution: University of Pittsburgh
Schools and Programs: Dietrich School of Arts and Sciences > Mathematics
Degree: PhD - Doctor of Philosophy
Thesis Type: Doctoral Dissertation
Refereed: Yes
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
Other ID:, etd-04052011-153616
Date Deposited: 10 Nov 2011 19:34
Last Modified: 15 Nov 2016 13:38


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