Date of Award


Document Type

Open Access Dissertation

Degree Name

Doctor of Philosophy in Mathematical Sciences (PhD)

Administrative Home Department

Department of Mathematical Sciences

Advisor 1

Alexander Labovsky

Committee Member 1

Jiguang Sun

Committee Member 2

Zhengfu Xu

Committee Member 3

Aleksey Smirnov


In the first chapter of this dissertation, we give some background on the Navier-Stokes equations and turbulence modeling. The next two chapters in this dissertation focus on two important numerical difficulties arising in fluid flow modeling: poor mass-conservation and nonphysical oscillations. We investigate two different formulations of the Crank-Nicolson method for the Navier-Stokes equations. The most attractive implementation, second order accurate for both velocity and pressure, is shown to introduce non-physical oscillations. We then propose two options which are shown to avoid the poor behavior. Next, we show that grad-div stabilization, previously assumed to have no effect on the target quantities of the test problem used, can significantly alter the results even on standard benchmark problems. We also propose a work-around and verify numerically that it has promise.

The next two chapters of this dissertation focus on developing the theory and computational evidence for an exciting new class of turbulence models. These models rely on a defect correction loop of predictor-corrector type to produce high order turbulence models. Owing to their defect correction formulation, these models are inherently parallelizable. Four models are investigated on an extensive suite of classic benchmark problems, and we show that all models significantly out-compete their natural competitors. In chapter 4 one of these models is given a full analytical workup where it is shown to be unconditionally stable and optimal order accurate.

Finally, in the penultimate chapter we derive an unconditionally stable and optimal order accurate turbulence model for the fluid-fluid interaction system with a rigid lid. Using the same defect-deferred correction procedure we improve the new model to produce a high order accuracy model in chapter 7. Both models are verified on a series of computational tests.

Creative Commons License

Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License
This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License.