Current Research
The projects currently underway span a broad spectrum of topics ranging from development of 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 (particularly heat transfer and combustion). Some specific areas include simulation of firewhirls and forest fire spread, solidification processes, flight of insects, and laser heating of materials on microscale/femtosecond scales, development of new numerical procedures (especially PDE solvers) 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 phe- nomenon 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 transition to and from turbulence and 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 generalizations of Kolmogorov scaling theories and discrete dynamical systems (so-called chaotic maps possessing a “strange attractor”) derived directly from the N.–S. equations and termed the "poor man's Navier–Stokes (PMNS) equations," 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 LES flow code, initially under GE Aircraft Engine and AFOSR sponsorship, and more recently with support from NASA. This code is based on delta-form quasilinearization implemented in the context of a time-split trapezoidal-midpoint/centered-difference finite-volume dis- cretization of a projection method solution of the Navi- er–Stokes equations in a multi-block, structured-grid form- ulation. 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 new SGS turbulence models described above and thus can predict transition and interactions of turbulence with other physical phenomena.
The figure displays small-scale streamlines colored with small-scale vorticity magnitude at one instant during decay of isotropic turbulence. Computations were performed on a very coarse 163 grid; of particular interest are the various small vortices of sizes smaller than the grid resolution. These have been produced by effects of the SGS model and provide an indication of the detail that can be produced by the new SGS modeling procedure.
Fire Whirl Simulation
The new LES code has begun to be used in various simulations. An example is shown in the figure at the right which displays the gridding used to model a fire whirl experiment, including the entire laboratory. The simulation was performed in a 3m cubed box with lid (which has been removed in the accompanying figure to permit an inside view) in which is an apparatus that induces swirling flow as combustion of a fuel causes rise of warm gases due to buoyancy. In the calculations reported here, the combustion process was replaced with heat input along the centerline of the apparatus of a magnitude to reproduce experimentally-observed temperatures.

This study was one of the first to examine details of the workings of the SGS model. In the figures shown below, we present horizontal slices of data at a fixed instant in time of computed Kolmogorov exponents (left) and the Reynolds number-related bifurcation parameter of the PMNS equations.
We remark that these are separate calculations, e.g., neither depends on the other; but both show the same thing: the SGS model successfully detects regions where turbulence should be present, viz., inside the experimental apparatus and, to a much lesser extent, along the walls of the laboratory where boundary layers are forming. The very realistic general symmetries of the flow, along with detailed asymmetries arising from turbulence, are apparent. The rest of the solution domain is filled with only very slow moving (and, hence, not turbulent) air. In this region the associated Kolmogorov exponent values lead to essentially zero amplitudes for turbulent velocity (and temperature) fluctuations, and at the same time the bifurcation parameters of the PMNS equations are such as to lead to a steady zero time series.
Simulation of Wildland Fire Spread
The LES code has also been used to sim- ulate larger-scale phenomena. The figure to the right displays a portion of the gridding employed to simulate forest-fire spread in a region 1.5 km wide, 7 km in the ambient wind direction and 2 km in altitude. The forest is modeled as a very porous medium with a heat source replacing actual combustion of forest fuels. In the adjacent figure it can be seen that uniform gridding with 1 m vertical resolution was used from the ground to the top of the forest canopy. Beyond this height the grid was gradually stretched to almost 50 m in the vertical direction. The horizontal grid spacing was slightly less than 50m x 50m. Despite this rather coarse gridding, the LES SGS model was able to extract sufficient in-
formation from resolved-scale calculations to produce realistic vortical structures that might be associated with fire whirls in actual forest fires. Two of these are seen in the figure. In the lower left side we see fluctuating velocity stream lines (shaded with small-scale vorticity magnitude) corresponding to a horizontal fire whirl. These are responsible for the characteristic streaks of alternately burned and unburned forest regions. Near the lower right-hand corner, one can see a vertical fire whirl appearing as a tornado-like structure. It is clear that it is smaller than the grid spacing, and thus has been generated entirely by the SGS model.

The size of physical fire whirls observed in wildland fires varies greatly, but they are typically no more that a few meters to at most a few tens of meters in diameter; however, they have been seen to extend to very high altitudes. In the present simulation, our computed fire whirls were typically of O(10) meters in diameter, and usually no more than about 50 m in height. The following figures display small-scale fluctuating temperature contours and small-scale velocity stream lines (the black lines) at three instants in time shortly after a fire was started on a 100 m2 patch of forest. The time slices are approximately a minute apart in physical time. Deep orange color represents high-intensity temperature fluctuations with lighter orange and yellow corresponding to lower amplitude fluctuations. The green background is the forest, and light blue is the sky. There is a mild breeze (2 m/sec) blowing from left to right. It is evident that the turbulence model is producing quite erratic fluctuations, and the area affected by the fire is growing in time. Indeed, we have checked the computed spread rates against known results, and the computations are quite reasonable. Finally, in all three figures one can see numerous vortical structures, most of them away from the burning region, but usually twisting into the forest canopy. In the last figure, however, one can see a fire whirl situated nearly in the center of the region of most intense thermal fluctuations.
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 Yang & McDonough (2003) and consist of composition of a discrete mollifier with the discrete solution operator after each time step of a numerical simulation. This is a key part of our current LES procedure.
PMNS Equation

As noted above, the Advanced CFD Group is investigating use of discrete dynamical systems (DDSs) to inexpensively model detailed time-dependent fluctuations in turbulent flow fields. It has recently been shown by McDonough & Huang (2000, 2004) that a particular DDS, the "poor man's Navier–Stokes 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.

Recent work with the PMNS equations has included carrying out more detailed comparisons with experimental results. To facilitate this it has been useful to derive direct correspondences between the bifurcation parameters of the DDS and those of the complete system of equations governing the flow field physics. McDonough (2005) has produced such correspondences for the N.–S. equations and natural convection; in particular, direct relationships have been derived connecting Prandtl number Pr, Ray- leigh number Ra and Nusselt number Nu to the bifurcation parameters of the PMNS equations. This permits com- parisons of the sort shown in the figure to the right in which the plotted points are results from the PMNS plus thermal energy equations for a fixed low Pr (=0.025) and a range of very high Ra. The correlations were developed by Cioni et al. JFM 335, 111 (1997) based on their measured data at the same Pr and range of Ra.
The general success experienced with the incompressible PMNS equations as the fluctuating factor in SGS models has motivated using the same approach for constructing SGS models for compressible LES formulations. We note that our overall approach that avoids filtering the equations of motion makes treating the compressible N.–S. equations somewhat easier than would be the case with the more usual treatments. Our initial work, as was the case for incompressible flows, has focused on developing a fairly detailed understanding of the behavior of the DDS corresponding to compressible flow. The figure shown here demonstrates that the same kinds of solution behaviors seen for the incompressible PMNS equations are also found for the compressible case. In this figure, we present a regime map constructed by plotting a component of strain rate against the SGS Reynolds number defined in terms of normal strain.
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.
Short-Time Laser Pulse Heating of Thin Films
The Advanced CFD Group has recently performed heat transfer simulations in physical regimes in which the classical heat conduction equation does not yield results in agreement with laboratory observations. In particular, when space and or time scales are very small, such as in femtosecond laser pulse heating of films of thickness on the order of nanometers this equation must be replaced by other alternatives that allow finite-rate thermal front propagation. We have recently employed the "dual-phase lag" (DPL) equation for this purpose; and while it has recognized deficiencies, it nevertheless produces qualitatively correct results. The figure at the right presents a comparison among DPL, hyperbolic heat conduction equation (HHCE) and the classical parabolic heat equation for two different laser pulses, parts (a) and (b) of the figure. It can be seen that neither the classical of HHCE results indicate any significant depth of penetration of the thermal front; instead, it spread very rapidly in lateral directions. The DPL equation results, on the other hand, show far less lateral spreading and much deeper penetration of heat for the same laser pulse as used in the other two models. This is in better correspondence with physical observations.

Simulation of Insect Flight
One of our most recent new projects involves simulation of insect flight. It is by now recognized that insects are capable of generating far greater lift with their flapping wings than is possible for a fixed-position airfoil, but not all details of this are well understood. For example, it is still widely believed that insect flight occurs mainly in the laminar flow regime, while inspection of essentially any published data from insect flight clearly shows the hallmarks of turbulence. Our preliminary studies have employed a laminar analysis to enable comparison with previous computed results; an example consisting of a simple wing model with aspect ratio similar to that of a bumblebee is shown in the figure. This was computed with the commercial flow code Fluent and corresponds to instantaneous streamlines colored with pressure, looking from beneath the leading edge of the wing as it is in the upstoke part of its flapping cycle. The high-lift-producing leading-edge vortex is readily apparent, but no attempt was made to simulate turbulence.
We will soon begin simulations with the new LES code discussed above to development of an understanding of specific effects of interaction of wing-flapping generated turbulence with that of the ambient air and the overall effects of this on lift and drag. Such understanding is considered invaluable in the design of micro air vehicles (MAVs), but it is also important for answering many questions associated with insect flight in realistic conditions. Producing such a simulation capability will permit detailed studies of such topics as evolution of insect flight (e.g., in Earth's rather different early atmosphere), mechanisms for prey capture and evasion, and mating.
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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