Leray-deconvolution regularization for Navier-Stokes flow of the step problem
Vorticity at Re=3900 for cylinder problem
Velocity of the Eithier-Steinman problem
Sensitivity of velocity for the Eithier-Steinman problem
- Computational Fluid Dynamics
- Numerical Solution of Partial Differential Equations
- Numerical Analysis of Continuous and Discontinous Finite Element Methods
- Large Eddy Simulations
- Sensitivity Problems
- Statistical Analysis
- Scientific Computing
- STEM education
My thesis area:
Applied Computational Fluid Dynamics.
Thesis advisor:
Professsor William J. Layton.
I am involved in the following interconnected projects:
- Theoretical and computational studies for continuous and discontinuous finite element analysis of fluid flow models. These studies involve the stability bounds for the approximate solutions and error estimates between the true solution and the approximated finite element solution. The numerical studies are verified in 2D and 3D computational experiments.
- The physical fidelity of turbulence models has been explored by examining the energy cascade and also the joint energy-helicity cascade that models possess. These studies include the derivation of models' micro-scale too. The physical fidelity of fluid flow models has been also explored by examining the computations of drag/lift coefficients and pressure drop on the body. My work investigated the sensitivity of these quantities, and how can these quantities be corrected using sensitivity results.
- There are several reasons (as conservation properties) so that one may wish to compute with rotational form of the nonlinearity of Navier-Stokes equations. We showed that not accounting for Bernoulli pressure (i.e. approximating it like usual pressure in numerical schemes) can lead to bad results, but this effect can be eliminated with the right choice of elements or grad-div stabilization.
- I have been working on a sensitivity model for the steady-state neutron diffusion equation, with heat transfer and fixed fluid flow. Also, on sensitivity computations for fluid flow regularizations of Navier-Stokes equations, that helps identifying the reliable interval values for parameter with respect to which the sensitivity is investigated.
- I have been working on statistical analysis of significance of data for solar efficiency and impact of soiling on the production of solar energy
- I have been working on enhancing STEM education at UNLV through the NSF grant "Improving STEM Student Fundamental Math Skills with Tailored Activity-Based Instruction".
MY STUDENTS
Graduate Students
- Jorge Reyes
PhD thesis in Computational Mathematics.
Expected graduation Spring 2023
Research: Jorge has been working on an improved generalization of the Smagorinsky model based on Kolmogorov theory. He is also working on projection methods for Navier-Stokes equations with a specific discretization of nonlinearity.
I am advising his PhD thesis.
- Jeffrey Belding
PhD thesis in Computational Mathematics.
Graduated Fall 2021
Research: Jeffrey has been working on regularized fluid flow modeling based on partial differential equations of motions, i.e. the so-called Time Relaxation equations. He is developing finite element schemes for this system of equations under proper scaling of filtering parameters, deriving error estimates for them, and implementing the numerical schemes for benchmark simulations. He has been running simulations for convergence verification and full step benchmark problem. We are also investigating sensitivity studies of these models with respect to their filtering parameters.
I advised his PhD thesis "Numerical Studies of regularized Navier-Stokes equations and an application of a run-to-run control model for membrane filtration".
- Tahj Hill
Mater thesis in Computational Mathematics.
Graduated Spring 2019
Research: Studies of energy and enstrophy for a class of fluid flow models that represent regularizations of Navier-Stokes equations. He derived the energy and enstrophy balances for a fluid flow model based on filtering, and ran a 3D benchmark problem investigating the energy and enstrophy of the fluid flow.
I advised his Master Thesis "Energy and enstrophy investigations in regularized Navier-Stokes equations".
- Sean Breckling
PhD thesis in Computational Mathematics.
Graduated Spring 2017
Research: Sean has been working on fluid flow modeling based on partial differential equations of motions, i.e. Navier-Stokes equations. He is developing finite element schemes for regularized Navier-Stokes equations, deriving error estimates for them, and implementing the numerical schemes for benchmark simulations. He has been running simulations for convergence verification, full step benchmark problem, and flow over an obstacle to draw conclusions about the important characteristic, such as drag, lift, pressure drop etc. We are also investigating sensitivity studies of these models with respect to their parameters.
I advised his PhD thesis "Numerical and Sensitivity Analyses of Navier Stokes-alpha Models".
- Shaurya Agarwal
Undergraduate: Indian Institute of Technology, Guwahati
PhD in Electrical & Computer Engineering from University of Nevada Las Vegas (2015)
I co-advised his Master Thesis in Mathematics: "Inverse Problem for Non-viscous Mean Field Control: Example from traffic"
- Puneet Lakhanpal
Undergraduate: Indian Institute of Technology, Guwahati
Double Master of Science (Electrical & Computer Engineering, and Mathematical Sciences) from University of Nevada Las Vegas (2014)
Research: Numerical investigation of traffic flow problems using Godunov method and finite element method. Traffic flow has been considered to be a continuum flow of compressible liquid having a certain density profile and an associated velocity. Lighthill Witham and Richards (LWR)model combined with the Greenshield's model was applied in simultion.
I co-advised his Master Thesis in Mathematics: "Numerical Simulation of Traffic Flow Models"
- Sergio Contreras
BS Electrical Engineering, University of Nevada Las Vegas, 2010 (Magna cum laude), Outstanding Graduating Senior (May 2010)
Dual Master of Science (Electrical & Computer Engineering, Mathematical Sciences) from University of Nevada Las Vegas (2010-2014)
Research: Observability studies for Eulerian and Lagrangian framework for traffic flow problems. The kinematic wave model was applied.
I co-advised his Master Thesis in Mathematics: "Observability in Traffic Modeling: Eulerian and Lagrangian Coordinates"
- Elena Nikonova
BS Mathematics (cum laude), University at Buffalo, 2009
Research: Theoretical and computational investigation of regularized Navier-Stokes equations based on finite element method and its sensitivity computations
Poster:"Fluid Flow Regularization of Navier-Stokes Equations", presented at the 2011 Graduate College and Professional Association Annual Research Forum (received an Honoroble Mention) and at the Student Poster Session at the Spring 2011 Southern California-Nevava MAA Section Meeting
Talk:"Finite element analysis and computations of regularized Navier-Stokes equations by time relaxation", AMS Spring Western Sectional Meeting, University of Nevada Las Vegas, April 2011
Undergraduate Students
- Delon Roberts
Major: Mathematics
Research: Investigating systems of ordinary differential equations of Lotka-Volterra (predator-prey) type with and without harvesting. Numerical solutions are obtained using numerical methods for solving differential equations, such as Runge-Kutta methods.
Poster: "A Computational Analysis of the Lotka-Volterra Equations", presented at the OUR-UNLV Undergraduate Research Showcase, UNLV, November, 2016.
- Jonathan Kim
Major: Mathematics
Research: Applications of the fluid flow problems in different biological/medical settings and numerical study of the governing equations of the fluid flow, Navier-Stokes equations, based on the finite element method.
Poster: "Applications and Computations of Fluid Flow Problems", presented at the OUR-UNLV Undergraduate Research Showcase, UNLV, October, 2015.
- Tahj Hill
Major: Mathematics
Research: Studies of energy and enstrophy for a class of fluid flow models that represent regularizations of Navier-Stokes equations.
Poster: "Energy and Enstrophy Investigations in Regularized Navier-Stokes Equations", presented at the UNLV Undergraduate Research and Creative Activities Symposium, Festival of Communities, UNLV, April 2013.
- Louisa Owuor
Major: PreProfessional
Research: Applications of ordinary differential equations (such as Verhulst (logistic) equation and Gompertz equation) in area of cancer modeling. These equations have been shown to provide a good fit for the growth data of numerous tumors. Also, application of the method of least squares for the prediction of the overall cancer incidences for male and female in Nevada and Clark County.
Poster: "Modeling and Prediction of Cancer Growth", presented at the UNLV Undergraduate Research and Creative Activities Symposium, Festival of Communities, UNLV, April 2013.
- Eva Arnold
Major: Mathematics
Research: Applications of logistic differential equation in areas, such as, biology, medicine, psychology and economics.
Poster: "Mathematical Analysis and Applications of Logistic Differential Equation", presented at the UNLV Undergraduate Research and Creative Activities Symposium, Festival of Communities, UNLV, April 2011.
- Sabrina Beckman and Stefan Cline
Major of Sabrina Beckman: Mathematics with minor in Hospitality
Major of Stefan Cline: Mathematics with minor in German language
Research: Predictions of undergraduate student population at the UNLV, College of Science, and Department of Mathematical Sciences using least squares method.
Poster: "UNLV Enrollment Forecasting", presented at the UNLV Undergraduate Research and Creative Activities Symposium, Festival of Communities, UNLV, April 2011.
- Shipra De
Majors: Mathematics, Computer Science and Economics.
Research: Computations of drag and lift in incompressible cylinder flow problem based on regularization of Navier-Stokes equations, report in progress.
Recipient of National Science Foundations Experimental Program for the Stimulation of Competitive Research at the University of Nevada Las Vegas, Summer 2007 (accepted) and 2008 (declined due to accepting other summer internship).
Goldwater Awardee for academic year 2010-2011.
UNLV 2011 Outstanding Graduate.
- David Hannasch
Majors: Mathematics and Computer Science
Research: Simulations of fluid models for different incompressible flow benchmark problems
Recipient of National Science Foundations Experimental Program for the Stimulation of Competitive Research at the University of Nevada Las Vegas, Summer 2009.
Recipient of "2010 MAA Undergraduate Poster Session Prize Winner" award at the Joint Mathematics Meetings, San Francisco, January 2010.
Recipient of National Physical Science Consortium Fellowship.
Paper: D. Hannasch and M. Neda, On the accuracy of the viscous form in simulations of incompressible flow problems, Numerical Methods for Partial Differential Equations, accepted for publication.
Honors College Thesis: "Efficient Flow Simulations", May 2010, (advised by MN).