Amiri Margavi, Alireza and Babaee, Hessam
(2024)
Low-Rank Approximation With Time-Dependent Bases For Fluid Flows.
Doctoral Dissertation, University of Pittsburgh.
(Unpublished)
Abstract
Accurately and efficiently describing the response of transitional flows to external forcing has numerous applications in climate science, flow control, sensor/actuator placement, and flow stability analysis. However, solving such high-dimensional systems can be costly, and many current reduced-order modeling techniques fail to make faithful low-dimensional representations. Flow response to external forcing is typically high-dimensional when expressed by static modal reduction techniques; however, for transitional flows, this approach could be ineffective because many modes are required. This research investigates and explores the reduced-order description of the linear response for time-dependent flows subjected to high-dimensional external excitations.
In the second chapter, we demonstrate that recently developed reduced-order modeling techniques based on time-dependent bases (TDB-ROM) can effectively extract the instantaneous correlated structures. In particular, we use forced optimally time-dependent decomposition (f-OTD), which extracts the time-dependent correlated structures of the flow response to various excitations. The correlated structures are used to build a reduced-order model that can be used for diagnostic and predictive purposes. We show how f-OTD enables us to address several inquiries, such as (i) identifying the excitation that leads to maximum amplification, (ii) reconstructing the full-state flow, and (iii) using a sparse selection measurement strategy to estimate the flow structure. We also demonstrate that in the case of a steady-state mean flow subject to harmonic forcing, the f-OTD subspace converges to the dominant resolvent modes. The theoretical results are illustrated with four cases: a toy model, the Burgers equation, the 2D temporally evolving jet, and 2D decaying isotropic turbulence flow.
In the third chapter, we address some of the outstanding challenges of TDB-ROMs. While they hold significant promise in reducing the computational burden of massive matrix differential equations (MDEs), their practical application encounters notable challenges such as computational efficiency, ill-conditioning, intrusiveness, and rank adaptivity. Recent advancements inspired by interpolation and hyper-reduction techniques from the vector differential equations, by using the CUR factorizations of low-rank matrices, have successfully addressed these challenges. This innovative approach, termed TDB-CUR, effectively mitigates the aforementioned difficulties. However, when applied to the Navier-Stokes equations, TDB-CUR faces computational limitations. The inherent coupling of velocity and pressure fields within these equations, combined with the absence of an independent transport equation for pressure, presents a unique challenge. Enforcing the divergence-free condition necessitates solving a linear equation system that requires full access to the column's components, thus aligning its computational load similar to that of the full-order model(FOM). In addressing this challenge, rather than imposing the divergence-free condition on a full-rank matrix of dimensions $n \times s$, we strategically apply it to a more manageable orthonormal basis set with a reduced size of $n \times r$. This adaptation significantly diminishes the computational overhead, yielding the TDB-CUR method as practical and efficient for tackling linearized Navier-Stokes problems. Additionally, due to the minimally intrusive nature of this technique, it has been easily extended to solve nonlinear forced flow problems. To demonstrate the performance of the proposed technique, we apply it to a 2D decaying isotropic turbulence flow subjected to a high-dimensional external forcing.
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Details
| Item Type: |
University of Pittsburgh ETD
|
| Status: |
Unpublished |
| Creators/Authors: |
|
| ETD Committee: |
|
| Date: |
6 September 2024 |
| Date Type: |
Publication |
| Defense Date: |
25 June 2024 |
| Approval Date: |
6 September 2024 |
| Submission Date: |
7 August 2024 |
| Access Restriction: |
No restriction; Release the ETD for access worldwide immediately. |
| Number of Pages: |
85 |
| Institution: |
University of Pittsburgh |
| Schools and Programs: |
Swanson School of Engineering > Computational Modeling and Simulation |
| Degree: |
PhD - Doctor of Philosophy |
| Thesis Type: |
Doctoral Dissertation |
| Refereed: |
Yes |
| Uncontrolled Keywords: |
Unsteady base flows, low-rank approximation, time-dependent bases, forced optimally time-dependent decomposition |
| Date Deposited: |
06 Sep 2024 20:05 |
| Last Modified: |
06 Sep 2024 20:05 |
| URI: |
http://d-scholarship.pitt.edu/id/eprint/46857 |
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