Featured Research

Machine Learning for Fluid Dynamics: Turbulence Modeling

In the past few decades, an exponential increase in computational power, algorithmic advances and experimental data collection strategies have seen an explosion in modelling efforts which leverage information obtained from physical data. In CFDLab, we are developing physics-constrained machine learning (ML) tools to identify a nonlinear relationship between the filtered and unfiltered quantities to obtain reliable closure models for large eddy simulations.

Read more about our ML ideas for LES closure modeling at:

Adaptive Mesh Refinements: Multilevel Wavelet Framework

A peculiarity of conservation laws with nonlinearities is the possibility of discontinuous solutions. These pose considerable challenges to computational and mathematical frameworks due to the requirement of very high degrees of freedom and the corresponding increase in computational time and storage requirements. Discrete wavelet-based techniques are commonly used for mesh refinement purposes. The general body of wavelet transform algorithms with their inherent ability of scale localization are a natural tool for application to problems such as the detection of localized structures and active error control. In our lab, we are developing wavelet such enabled numerical methods and grid adaptation methodologies for solving conservation laws efficiently.

Read more about our neural networks based ROM ideas at:

Closure Modeling of Turbulent Flows

Large eddy simulation (LES) is a popular technique for simulating turbulent flows. As the name says, in LES only the large spatial structures are approximated, whereas the small scales are modeled. This allows for much coarser spatial meshes and thus a computational cost that is significantly lower than that of a direct numerical simulation (DNS).  In practice, DNS, which resolves every scale of the solution, is prohibitively expensive for nearly all systems with complex geometry or flow configurations. To achieve the same order of physical accuracy as DNS, however, LES needs to correctly treat the closure problem: the effect of the small scales on the large ones needs to be modeled. Most such closure models are derived based on heuristic assumptions relevant to the physics of 3D turbulent flows (e.g., vortex stretching mechanism and forward energy cascade). We are developing and applying purely mathematical closure models, with no additional phenomenological arguments being used. This is particularly appealing for LES of geophysical flows, which are approximately 2D with different energy transfer characteristics (e.g., inverse energy cascade), in which stratification and rotation suppress vertical motions in the thin layers of fluid.

Numerical Methods for Incompressible Flows

Incompressible flows are fluid flows in which the flow speed is small compared to the speed of sound. A major subset of computational methods for solving these flows are pressure-based projection methods that decouple the computations of the pressure and velocity fields. The computational cost per time step of these pressure-based methods is that of solving a parabolic-type advection-diffusion equation and an elliptic-type Poisson equation. The number of Poisson equations that must be solved at each time step varies with the method and the problem dimensions, but for all the pressure-based methods, solving the Poisson equation takes considerably more computational resources than solving the advection-diffusion time dependent equations, especially for large scale problems and high Reynolds number flows. We have developed a coarse-grid projection (CGP) method for accelerating these incompressible flow computations, which is applicable to methods involving Poisson equations as incompressibility constraints. The CGP methodology is a modular approach that solves the pressure part of the problem on a coarse grid, and interpolates the results back to a fine grid to compute the velocity field. The method has been shown to drastically reduce the computational cost, with retaining a level of accuracy close to that of the fine resolution field, which is significantly better than the accuracy obtained for a similar computation performed solely using a coarse grid.

Numerical Methods for Compressible Flows

Computational studies of compressible flow problems are important in basic scientific research, and for a multitude of engineering applications. In the decades since the first compressible flow computations were performed, many successful shock-capturing algorithms have been proposed for computing these flows. The holy grail is to obtain a sharp discontinuity effectively, extending over as few grid points as possible. Terms like essentially non-oscillatory, monotone upwind, dispersion relation preserving, or total variation diminishing are applied to various methods for capturing shocks via some sort of numerical dissipation which is a consequence of physics that produce an upwind bias in these algorithms. This numerical dissipation can provide stability and robustness, without significantly deteriorating accuracy. A numerical scheme must possess sufficient dissipation to capture strong shocks without developing overshoots and oscillations in the vicinity of the discontinuity. Such numerical schemes used for the integration of compressible flow simulations should provide accurate solutions for the long time integrations these flows require. To this end, we have investigated performance of extensions of the state-of-the-art high-resolution shock capturing schemes by solving hyperbolic conservation laws in gas dynamics. 

Please click on the following links to learn more about our studies listed here:

Machine Learning for Fluid Dynamics: Model Reduction

Many reduced-order models are neither robust with respect to parameter changes nor cost-effective enough for handling the nonlinear dependence of complex dynamical systems. In our lab, we are developing robust machine learning frameworks for both intrusive and non-intrusive reduced order modeling paradigms of such nonlinear and nonstationary systems. 

Read more about our neural networks based ROM ideas at the following links:

Reduced Order Modeling

Our primary goal in developing model reduction strategies is to replace large dynamical systems with lower dimensional systems having similar range of validity and input/output characteristics. Proper orthogonal decomposition (POD) is one of the most successful reduced order modeling techniques in dynamical systems. POD has been used to generate a representative reduced-order model (ROM) for the control, optimization, and analysis of a large number problems involving fluids. Because POD typically extracts the most energetic modes of a given system, projecting these systems and their solutions onto these low-dimensional bases (i.e., POD modes) often produces ROMs that capture the dominant characteristics of these systems. The resulting systems are low dimensional but dense and provide an efficient framework for applications where either small size or fast simulations are required. 

Two-dimensional Turbulence

Two-dimensional turbulence, to the first approximation, is a reduced dimensional version of 3D turbulence, where the flow is constrained to two dimensions. In reality, 2D turbulence is never realized in nature or in the laboratory, both of which have some degree of three-dimensionality. Nevertheless, many aspects of idealized 2D turbulence appear to be relevant for physical systems in geophysics, astronomy and plasma physics. The physics of 2D turbulence have been elucidated substantially during the past decades by theoretical models, intensive numerical investigations, and dedicated soap film experiments. 

One of the most important reasons for studying 2D turbulence is to improve our understanding of geophysical flows in the atmosphere and ocean. We may also find two-dimensional flows in a wide variety of situations such as flows in rapidly rotating systems and flows in a fluid film on top of the surface of another fluid or a rigid object. From a theoretical perspective, 2D turbulence behaves in a profoundly different way from 3D turbulence due to different energy cascade behavior, and follows the Kraichnan-Batchelor-Leith (KBL) theory. In 3D turbulence, energy is transferred forward, from large scales to smaller scales, via the vortex stretching and tilting mechanism. In two dimensions that mechanism is absent, and it turns out that under most forcing and dissipation conditions energy will be transferred from smaller scales to larger scales. Despite the apparent simplicity of dealing with two rather than three spatial dimensions, 2D turbulence is possibly richer in its dynamics due to its conservation properties, such as its inverse energy and forward enstrophy cascading mechanisms, which 3D turbulence does not possess.

Further details can be found at the following links:

Physiological Fluid Flows

We have arrived at an era where the biology of the macroscopic systems in the human body is of great interest in many fields, including engineering, medicine, and physics. Our research intends to develop a physiological-systems based computational approach to human biology. It involves studying the coupled behavior of the respiratory and circulatory systems in the human body developing geometric multiscale models for these systems. While fully three-dimensional simulations such as these are invaluable to our understanding of human physiology, and to treating common circulatory system disorders such as arterial stenosis and heart valve malfunction, most hospitals and clinics do not have ready access to supercomputing clusters, and an approach that is still very accurate, but less computationally intensive is desired in the clinical or hospital setting. 

With this in mind, we have developed a mathematical model that incorporate fluid-structure interactions and branching effects aimed at providing accurate and fast-to-compute global models for human physiological systems represented as networks of quasi one-dimensional fluid flows. It was shown that the proposed model can be used as an efficient tool for investigating the dynamics of flows in the circulatory and respiratory systems and would, in particular, be a good candidate for the one-dimensional, system-level component of geometric multiscale models of physiological systems.

Please click on the following links to read further details:

Eulerian Modeling for Laser Shock Wave Assisted Patterning

In laser-material interactions, the laser is initially absorbed by surface atoms leading to a temperature gradient in its penetration depth, a distance that heat can be transferred to during the laser pulse. It is well known that lasers can produce high temperature and high pressure when absorbed by a material. In most of the practical applications, it is critical to determine the heat transfer mechanism for accuracy of laser processing. Here, we are developing numerical tools to better understand the underlying physics and laser irradiation processes inside the medium. 

Related works can be found at the following links: