Siddiqua, Farjana
(2024)
Spurious Numerical Dissipation and Time Accuracy.
Doctoral Dissertation, University of Pittsburgh.
(Unpublished)
Abstract
In this thesis, we do the numerical analysis of an advection-diffusion-reaction problem in bioseparation and a corrected Smagorinsky model for turbulence. Numerical dissipation due to time discretization schemes often contributes to or causes overdissipation. The goal of this dissertation is to control spurious numerical dissipation and to acquire long-time high-order accuracy.
In the first project, we analyze an advection-diffusion-reaction problem with the non-homogeneous boundary conditions (useful for practical settings) that model the chromatography process, a vital stage in bioseparation. We prove stability and error estimates using finite elements for spatial discretization and the midpoint method for time discretization. These yield a second-order convergence rate and better total mass conservation. The numerical tests validate the theoretical results.
In the second project, we develop a turbulence model named the Corrected Smagorinsky Model (CSM) and analyze it. When the ratio of dissipation of turbulent kinetic energy (TKE) and the production of TKE is equal to $1$, we call it statistical equilibrium. We extend a classical model for turbulence at statistical equilibrium to non-equilibrium turbulence and propose and analyze algorithms for the solution of the extended model. The classical Smagorinsky model's solution is an approximation to a (resolved) mean velocity. Since it is an eddy viscosity model, it cannot represent a flow of energy from unresolved fluctuations to the (resolved) mean velocity. This classical Smagorinsky model was corrected to incorporate this flow and still be well-posed. The computational experiments verify the properties of the algorithms and show that the model captures the non-equilibrium effects.
In the third project, we analyze the one-leg, two-step variable time step methods of Dahlquist, Liniger, and Nevanlinna (DLN) for the time discretization in the Corrected Smagorinsky Model. Turbulent flows strain computational resources in terms of memory usage and CPU (central processing unit) speed. The adaptive DLN methods are second-order accurate and allow large timesteps, hence requiring less memory and fewer FLOPS (floating point operations per second). We demonstrated the method's second-order accuracy, quantified its numerical dissipation, demonstrated error estimates in addition to proving the kinetic energy is bounded for various time steps and illustrated theoretical results by numerical tests.
Share
| Citation/Export: |
|
| Social Networking: |
|
Details
| Item Type: |
University of Pittsburgh ETD
|
| Status: |
Unpublished |
| Creators/Authors: |
|
| ETD Committee: |
|
| Date: |
27 August 2024 |
| Date Type: |
Publication |
| Defense Date: |
8 June 2024 |
| Approval Date: |
27 August 2024 |
| Submission Date: |
20 June 2024 |
| Access Restriction: |
No restriction; Release the ETD for access worldwide immediately. |
| Number of Pages: |
193 |
| 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: |
Advection, Diffusion, Reaction, Chromatography, Adsorption, Bioseparation, Eddy Viscosity, Corrected Smagorinsky, Complex turbulence, Backscatter, Complex turbulence, Backscatter, the DLN method, G-stability, variable time-stepping. |
| Date Deposited: |
27 Aug 2024 13:46 |
| Last Modified: |
27 Aug 2024 13:46 |
| URI: |
http://d-scholarship.pitt.edu/id/eprint/46585 |
Metrics
Monthly Views for the past 3 years
Plum Analytics
Actions (login required)
 |
View Item |
![[feed]](/images/feed-icon-32x32.png){kind=icon}