Award Date


Degree Type


Degree Name

Doctor of Philosophy (PhD)


Mathematical Sciences

First Committee Member

Monika Neda

Second Committee Member

Jichun Li

Third Committee Member

Hongtao Yang

Fourth Committee Member

Pengtao Sun

Fifth Committee Member

Pushkin Kachroo

Number of Pages



This dissertation consists of two parts. The first part consists of research on accurate and efficient turbulent fluid flow modeling via a family of regularizations of the Navier-Stokes equation which are known as Time Relaxation models. In the second part, we look into the modeling application for the filtration/backwash process at the River Mountains Water Treatment Facility in Henderson, NV.

In the first two chapters, we introduce the Time Relaxation models and their associated differential filter equations. In addition, we develop the regularization method which employs the Nth van Cittert deconvolution operator, which gives rise to the family of models. We also justify theoretically and computationally the use of an effective averaging length scale δ in the time relaxation model when using the van Cittert operator for higher orders of deconvolution N, by presenting experimental results from our use of this model in the Shear Layer Roll Up benchmarking problem. In addition, we will perform a sensitivity analysis with respect to the time relaxation coefficient χ which appears as a scaling factor for the regularization term in the model, and show how sensitivities with respect to χ are improved when utilizing the effective averaging length scale δ.

In the third chapter, we develop the time relaxation model with the newly proposed energy-momentum-angular momentum conservation (EMAC) discretization of the non-linear term. We will present energy, momentum and angular momentum balances for the continuous formulation of the TRM with EMAC as well as the full discretized scheme using TRM with EMAC, and we will show that the fully discrete balances for TRM with EMAC reduce to the fully discrete analogues of the conservation of energy, momentum and angular momentum for the continuous Navier-Stokes equations under the assumption of no viscosity, no regularization, and no body force. In addition, we will present the stability and error estimate of the TRM with EMAC, and we will compare these results with the stability and error estimate for the TRM with the well known skew symmetric formulation for the non-linear term. We show that the error estimate for the EMAC scheme under high Reynolds number is much improved over the skew-symmetric scheme. In particular, we will show that the error for the EMAC scheme is O(eν−1), while under the same conditions, the skew-symmetric scheme error is O(eν−3), which is a significant improvement for high Reynolds number, i.e. low values of kinematic viscosity ν. We will then present numerical experiments on the Taylor Green vortex problem to verify the convergence rates for our error estimates, and experiments on the 3D Ethier-Steinman problem and the 2D Lattice Vortex problem to show that the numerical errors produced by EMAC are much smaller than the skew-symmetric scheme.

In the fourth chapter, We begin the work of part two by introducing the general run-to-run control model, which is used in a wide array of applications in addition to water treatment. Then we will introduce the specific run-to-run control model which is formulated specifically for a general filtration/backwash system, and we will modify it to fit the parameters, specifications, and measured data that is available from the River Mountains facility. In particular, we will discuss the implementation of a least squares problem to fit parameters for a filtration cycle ODE model. We will also discuss the backwashing component of the proposed model, and the difficulties in implementing such a model with real time plant data. From there, we formulate a cost of power objective function for the whole filtration/backwashing cycle in terms of the setpoints of filtration/backwashing operation, namely the length of both filtration and backwashing cycles, and the setpoints for the fluid flux during each of these cycles. We show the process of minimizing this objective function with respect to the setpoints of plant operation, including our various implementations of this model using standard function optimization with constraints, genetic algorithms, and MCMC methods.

In the final chapter, we will draw some conclusions and summarize the findings of all the work contained in the dissertation.


Fluid Modeling; Navier-Stokes; Run-to-Run Control; Time Relaxation; Wastewater Treatment


Mathematics | Other Physics | Physics | Water Resource Management

File Format


File Size

2500 KB

Degree Grantor

University of Nevada, Las Vegas




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