Ensemble time-stepping algorithms for natural convection

Fiordilino, Joseph (2018) Ensemble time-stepping algorithms for natural convection. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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Abstract

Predictability of fluid flow via natural convection is a fundamental issue with implications for, e.g., weather predictions including global climate change assessment and nuclear reactor cooling. In this work, we study numerical methods for natural convection and utilize them to study predictability. Eight new algorithms are devised which are far more efficient than existing ones for ensemble calculations. They allow for either increased ensemble sizes or denser meshes on current computing systems.

The artificial compressibility ensemble (ACE) family produce accurate velocity and temperature approximations and are fastest. The speed of second-order ACE degrades as $\epsilon \rightarrow 0$ or $\Delta t \rightarrow 0$ due to the iterative solver. However, first-order ACE has a uniform solve time since $\gamma = \mathcal{O}(1)$. The ensemble backward differentiation formula (eBDF) family are most accurate and reliable. The penalty ensemble algorithm (PEA) family are strongly affected by the timestep and are least accurate. In particular, $\gamma = \mathcal{O}(1/ \Delta t^2)$ for second-order PEA leads to solver breakdown. We also propose an ACE turbulence (ACE-T) family of methods for turbulence modeling which are both fast and accurate.

A complete numerical analysis is performed which establishes full-reliability. The analysis involves techniques that are novel and results that subsume, elucidate, and expand previous results in closely related fields, e.g., iso-thermal fluid flow. Numerical tests show predicted accuracy is consistent with theory.

Predictability is a highly complex and problem-dependent phenomenon. Predictability studies are performed utilizing the new second-order ACE algorithm. We perform a numerical test where the flow reaches a steady state. It is found that increasing the size of the domain increases predictability. Also, spatial averages increase predictability with increasing filter radius. We also study a problem with a manufactured solution. Sufficiently large rotations increase the predictability of a flow. Further, spatial averages decrease predictability with increasing filter radius.

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Details

Item Type: University of Pittsburgh ETD
Status: Unpublished
Creators/Authors:
Fiordilino, Josephjaf150@pitt.edujaf1500000-0003-3239-1010
ETD Committee:
Committee ChairLayton, Williamwjl@pitt.eduwjl0000-0002-2539-4478
Committee MemberNeilan, Michaelneilan@pitt.eduneilan0000-0002-1564-1041
Committee MemberTrenchea, Catalintrenchea@pitt.edutrenchea
Date: 26 September 2018
Date Type: Publication
Defense Date: 26 June 2018
Approval Date: 26 September 2018
Submission Date: 27 June 2018
Access Restriction: No restriction; Release the ETD for access worldwide immediately.
Number of Pages: 157
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: Finite element method, numerical analysis, fluid flow, uncertainty quantification
Date Deposited: 26 Sep 2018 23:28