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Fast, Adaptive Algorithms for Flow Problems

McLaughlin, Michael (2020) Fast, Adaptive Algorithms for Flow Problems. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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Time-accurate simulations of physical phenomena (e.g., ocean dynamics, weather, and combustion) are essential to economic development and the well-being of humanity. For example, the economic toll hurricanes wrought on the United States in 2017 exceeded $\$200$ billon dollars. To mitigate the damage, the accurate and timely forecasting of hurricane paths are essential. Ensemble simulations, used to calculate mean paths via multiple realizations, are an invaluable tool in estimating uncertainty, understanding rare events, and improving forecasting. The main challenge in the simulation of fluid flow is the complexity (runtime, memory requirements, and efficiency) of each realization. This work confronts each of these challenges with several novel ensemble algorithms that allow for the fast, efficient computation of flow problems, all while reducing memory requirements. The schemes in question exploit the saddle-point structure of the incompressible Navier-Stokes (NSE) and Boussinesq equations by relaxing incompressibility appropriately via artificial compressibility (AC), yielding algorithms that require far fewer resources to solve while retaining time-accuracy. Paired with an implicit-explicit (IMEX) ensemble method that employs a shared coefficient matrix, we develop, analyze, and validate novel schemes that reduce runtime and memory requirements. Using these methods as building blocks, we then consider schemes that are time-adaptive, i.e., schemes that utilize varying timestep sizes.

The consideration of time-adaptive artficial compressibility methods, used in the algorithms mentioned above, also leads to the study of a new slightly-compressible fluid flow continuum model. This work demonstrates stability and weak convergence of the model to the incompressible NSE, and examines two associated time-adaptive AC methods. We show that these methods are unconditionally, nonlinearly, long-time stable and demonstrate numerically their accuracy and efficiency.

The methods described above are designed for laminar flow; turbulent flow is addressed with the introduction of a novel one-equation unsteady Reynolds-averaged Navier-Stokes (URANS) model with multiple improvements over the original model of Prandtl. This work demonstrates analytically and numerically the advantages of the model over the original.


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Item Type: University of Pittsburgh ETD
Status: Unpublished
CreatorsEmailPitt UsernameORCID
McLaughlin, Michaelmem266@pitt.edumem2660000-0003-2694-3376
ETD Committee:
TitleMemberEmail AddressPitt UsernameORCID
Committee ChairLayton,
Committee MemberGivi,
Committee MemberNeilan,
Committee MemberTrenchea,
Date: 8 June 2020
Date Type: Publication
Defense Date: 30 March 2020
Approval Date: 8 June 2020
Submission Date: 30 March 2020
Access Restriction: No restriction; Release the ETD for access worldwide immediately.
Number of Pages: 163
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: artificial compressibility, ensemble simulations, one-equation models, timestepping
Date Deposited: 08 Jun 2020 16:44
Last Modified: 08 Jun 2020 16:44

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