Date of Award
Fall 2025
Access Type
Dissertation - Open Access
Degree Name
Doctor of Philosophy in Mechanical Engineering
Department
Mechanical Engineering
Committee Chair
Leitao Chen
Committee Chair Email
leitao.chen@erau.edu
First Committee Member
Leitao Chen
First Committee Member Email
leitao.chen@erau.edu
Second Committee Member
Eduardo Divo
Second Committee Member Email
divoe@erau.edu
Third Committee Member
Sandra K.S. Boetcher
Third Committee Member Email
boetches@erau.edu
Fourth Committee Member
Paul Crittenden
Fourth Committee Member Email
crittenp@erau.edu
Fifth Committee Member
R.R. Mankbadi
Fifth Committee Member Email
mankbadr@erau.edu
College Dean
James W. Gregory
Abstract
This dissertation explores the combination of two sophisticated techniques for addressing computational fluid dynamics: the discrete velocity Boltzmann equation (DVBE) and the localized collocation meshless model with upwinding (U-LCMM). The DVBE is a high-level model that describes the foundations of transport phenomena by addressing the microscale motions of particles themselves and the effect of their aggregate behaviors on continuum principles. This equation integrates multiple scales of phenomena; while it can be used for fluid flow at Navier-Stokes scales, it can also resolve fine features that can only be described at the molecular level. This type of model is necessary for a wide range of applications, e.g., rarefied gases, multiphase flows, and chemically reactive transport. Additionally, this equation has recently experienced a large increase in focus because it is a highly parallelizable transport equation. This characteristic avoids the iterative difficulties typically associated with the Navier-Stokes equations. Most of this focus has been on using a Lagrangian approach to the Boltzmann equation—the lattice Boltzmann method—which is primarily solved on a grid. The lattice Boltzmann method requires significant assumptions and restrictions that can impact the efficiency or accuracy of the approximation of the Boltzmann equation.
Consequently, the second technique, meshless modeling, offers a solution to these limitations and becomes prominent. Meshless modeling is a class of techniques that removes the requirements of mesh generation and mesh dependencies in domain discretization and geometric representation. The U-LCMM can be used at any scale, allowing for regional variations in point densities without any inherent dependencies on underlying assumptions about the domain. As a modeling method, it can be used on any partial differential equation, but on the Boltzmann equation, it becomes a powerful tool. Both the Boltzmann equation and the U-LCMM are highly effective at addressing multiscale features. Together, they create a technique that can resolve transport through the simplest of channels or the tortuosities of a packed bed.
The focus of this dissertation is the comprehensive description of the numerical theory and methods used to combine these tools to build a complex fluid flow solver. The dissertation begins with a review of each topic and the individual specifications unique to each. Here, the DVBE is described in detail, which converts the continuous phase space Boltzmann equation into a discrete system of equations that can be computationally modeled. While much research has been done on the Boltzmann equation on structured grids, significantly less attention has focused on treating unstructured grids or meshless techniques, leaving gaps to be addressed. The most difficult of these concerns relates to boundary treatment, addressing macroscale properties like velocity and pressure at the particle level. This work builds the framework of applying the U-LCMM to the DVBE, from geometric representation using point generation, to the application of the DVBE for fluid transport.
Next, this method demonstrates its rigor in two manuscripts. In the first manuscript, the stability of the solver in terms of the Boltzmann stability criterion is addressed, including its strengths and limitations. The solver's abilities are then demonstrated through a series of abecedarian exercises, including the Taylor-Green vortex flow, lid-driven square cavity flow, and channel flow over a circular cylinder. These illustrate the solver's ability to address multiple boundary types, including periodic, Dirichlet, and zero-gradient boundaries. In the second manuscript, the solver is applied to address porous media. The complex multiscale nature of porous media makes it a challenging topic for traditional CFD solvers, as geometric and transport complexities require greatly varying length scales — once more, precisely a challenge fit for both the Boltzmann equation and the meshless modeling. In this manuscript, the numerical model demonstrates its ability to use microscale transport to recover the Darcy equation, a macroscale equation to describe flow across porous media, and explores the distinctions in macroscopic behavior between ordered and disordered porous structures. Additionally, the order of convergence and the timing complexity of the numerical method are addressed. These studies conclude that applying U-LCMM to the DVBE shows great potential to address complex transport phenomena.
Scholarly Commons Citation
Maidenberg, Amandine, "Meshless Discrete Velocity Boltzmann Model for Porous Media Flow" (2025). Doctoral Dissertations and Master's Theses. 943.
https://commons.erau.edu/edt/943
Included in
Complex Fluids Commons, Fluid Dynamics Commons, Membrane Science Commons, Numerical Analysis and Scientific Computing Commons, Partial Differential Equations Commons, Transport Phenomena Commons
