Current Research
The projects currently underway span a broad spectrum of topics ranging from new computational techniques to solution of practical engineering problems; but most involve developing models for turbulence prediction and especially interaction of turbulence with other physical phenomena. Other specific areas include simulation of solidification processes, development of new numerical procedures and use of 3-D immersive visualization tools.
Turbulence Modeling

The problem of turbulence is often called “the last unsolved problem of classical mathematical physics.” Turbulence is a flow phenomenon already recognized by da Vinci more than 500 years ago, as evidenced by his numerous sketches, an example of which is shown here. Not only did da Vinci recognize the phenomenon, but he also characterized and named it: “turbolenza.”

Despite a tremendous expenditure of effort during the past century this problem remains unsolved today in the sense that we are not able to reliably predict the behavior of hardly any important, nontrivial fluid flows.

The equations that govern turbulent flows, the Navier–Stokes equations, have been known since 1840, but it was not recognized until the late 20th Century that they could actually exhibit turbulent solutions. In part because of this, and more so because the early views of turbulence mistakenly categorized it as a random phenomenon, attempts to model turbulence beginning in the early 20th Century and extending even to the present have involved statistical properties of the flow field, rather than the instant-to-instant detailed prediction of physical variables that is often needed when analyzing such effects as flame extinction or re-ignition in combustion processes.

The Advanced CFD Group has begun investigations into use of modern mathematical theories on the nature of turbulence to develop models that are both inexpensive to evaluate, and at the same time physically realistic. The approach provides an alternative to usual large-eddy simulation (LES) and focuses on development and use of high-fidelity subgrid-scale (SGS) models constructed from Kolmogorov scaling theories and discrete dynamical systems (so-called chaotic maps possessing a “strange attractor”) in conjunction with filtering solutions rather than equations.

3-D General-Purpose CFD Code Development
The Advanced CFD Group is currently developing a new 3-D incompressible flow code under AFOSR sponsorship. This code is based on delta-form quasilinearization implemented in the context of a time-split trapezoidal-midpoint/centered-difference finite-volume discretization of a projection method solution of the Navier–Stokes equations in a multi-block, structured-grid formulation. The code will include all forms of heat transfer, chemical reactions, porous media and rotating coordinate systems and will be applied to a wide range of practical problems—from analysis of cooling turbine blades, to spread of forest fires and formation of ice in turbulent streams and rivers. The code utilizes the new turbulence models based on generalized Kolmogorov scalings and the PMNS equations and thus can predict transition and interactions of turbulence with other physical phenomena.
Solution Filtering
Filtering the equations of motion leads to many difficulties, as has long been recognized, while filtering the solutions is a relatively simple “signal-processing” problem. The accompanying figure shows that results from the two different approaches are very similar; but the latter leads to far easier implementations and more flexibility in SGS model construction, thus opening the way to many new approaches in turbulence modeling. The mathematical foundations for such an approach are very solid, as described in McDonough& Yang (2003) and consist of composition of a discrete mollifier with the discrete solution operator.
PMNS Equation

The Advanced CFD Group is investigating use of discrete dynamical systems (DDSs) for the purpose of inexpensively modeling detailed time-dependent fluctuations in turbulent flow fields. It has recently been shown by McDonough & Huang (2000, 2003) that a particular DDS, the "poor man's Navier–Stokes equation," (PMNS equation) derived from the basic equations of fluid motion is capable of exhibiting all of the known temporal behaviors of the full system of partial differential equations. Thus, intensive study of this system is now underway. The figures below provide "regime maps" (two- and three-dimensional bifurcation diagrams) of such systems.

Other Discrete Dynamical Systems
Our studies of DDSs as models for small-scale turbulent fluctuations emphasize utilization of mathematical analyses leading to the poor man’s Navier–Stokes equation and analogous DDSs arising from other transport equations. The goal of these investigations is to establish mappings from physical flow variables to bifurcation parameters of the DDSs to permit their automatic and reliable use as turbulence models. The first step in this process is detailed analysis of the DDSs, themselves, and in particular, learning their behaviors as bifurcation parameters are varied. Such analyses can be summarized in regime maps as displayed here for a DDS associated with the thermal energy equation.

McDonough & Joyce (2002), McDonough & Zhang (2002), Xu et al. (2002) and McDonough et al. (2003) have conducted studies of DDSs involving interactions of the PMNS equations with similar equations representing heat transfer and chemical reactions. Results shown below include an instantaneous snapshot of a modeled pool fire and comparison of modeled and measured time series of temperature and species concentrations in a hydrogen–air jet non-premixed flame.

Fitting Experimental Data to Chaotic Maps

McDonough, et al. (1998) presented a technique whereby turbulent experimental data could be fit to specified chaotic maps. Yang et al. (2003) have recently applied this procedure to laboratory experiments to obtain two-dimensional velocity fields fit to a discrete dynamical system derived by McDonough & Huang (2000), the “poor man’s Navier–Stokes equation.” The figure to the left indicates the accuracy that can be obtained with such a fit.

The portion of the time series prior to t =0.4 is from modeled results, while that following t=0.4 is experimental data. It is evident that the basic turbulent features of the data have been preserved in the fitted model, and it is important to recognize that the ability to capture such detail is essential to constructing the physically-based models required in our overall LES formalism. It is also important to recognize that the fitting procedure does not attempt a pointwise "exact" fit, which would be completely inappropriate for turbulence models. Rather, the fitting process is formulated so that the model results preserve the statistical properties of the experimental data, and at the same time exhibit a similar qualitative appearance.
Freezing of Super-Cooled Liquids
Freezing of liquids and melting of solids are important phenomena in many industrial processes, bio-medical applications and even in household activities. The Advanced CFD Group is developing very general techniques for predicting these phenomena based on extension of ideas associated with the phase-field method. Innovations being introduced at UK include a new numerical procedure known as discrete operator interpolation (DOI) for more accurately tracking the melt front, and inclusion of effects of thermodynamic fluctuations. The adjacent figure depicts a dendrite, a complicated crystalline structure that commonly forms as supercooled liquids freeze—a well-known example of which is a snowflake.
Three-Dimensional Immersive Visualization
The Advanced CFD Group in collaboration with faculty and students from the UK Computer Science Department, and with support from the National Science Foundation (NSF), is developing a state-of-the-art visualization system, called the Metaverse, that permits investigators to literally walk inside their 3-D data sets. This is expected to facilitate better understanding of complicated phenomena from many different disciplines, but at the same time it demands a different way of thinking about how data should be represented, rendered and displayed. The Advanced CFD Group is providing data sets to be examined with these ideas in mind in support of this visualization research.

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Lecture Notes