"Shear-thinning" may sound like something done to overly woolly sheep, but in the world of fluid dynamics it is the behavior of a certain class of fluids, like ketchup and, more importantly, blood, that become runnier and less viscous as they flow in response to an applied force.

"Understanding the physics of such fluids is important in many fields," says Dr. William L. Barth, TACC Research Associate. The class includes a lot of industrial lubricants, many fluids that are pumped into oil wells to improve oil recovery, and some fluids like blood that are important in biology. Barth is a scientist at TACC whose computations are shedding new light on the behavior of shear-thinning fluids while also exercising and extending the capacities of high-performance computers

The pictures here come from several of Barth's recent calculations, most of which were done this year for his Ph.D. dissertation at UT Austin under the direction of Dr. Graham Carey of the Institute for Computational and Engineering Sciences (ICES). The dissertation, "Simulation of Non-Newtonian Fluids on Workstation Clusters," earned Barth his doctorate in May 2004, but the studies began with a "benchmark contest" for a conference held in May 2001.

"Bill's interest in computational fluid dynamics and high-performance computing were well suited to this type of study," Carey says. Surprisingly, the behavior of shear-thinning fluids, while easy to describe qualitatively or observe empirically, is extraordinarily complex and difficult to capture in a computational fluid dynamics model. "Over the past several years, I found that I needed to pull together a number of recently developed computational techniques, and I needed a tremendous amount of computational power just to begin to attack some of these problems," Barth says.

As background to Barth's study, it is worth noting that fluid dynamics covers a lot of territory. Of the normal solid-liquid-gas phases of matter, the liquids and the gases are usually treated computationally as fluids. Some of the solids exhibit plastic, fluid behavior, too (ductile metals, for example). The formal mathematics of fluid dynamics even reaches out to encompass the behavior of plasmas (ionized gases) and the strange laboratory constructs called Bose-Einstein condensates: collections of atoms trapped at temperatures close to absolute zero.

Figure 1. Fully developed natural (buoyant) convection of a shear-thinning fluid in a box heated from below and cooled from above. Calculations made on Lonestar by William Barth and visualization done on Maverick by Karla Vega, both of TACC.

With such a wide range of targets comes a panoply of properties used to classify the kinds of fluids and their ways of flowing. An important property is viscosity, a measure of the resistance of a fluid to deformation under shear stress (any force that tends to change the shape of the material): the thicker the fluid, the higher its viscosity. If the viscosity does not change as the shear stress is applied, a fluid is said to be "Newtonian." Water and air are both Newtonian fluids.

Non-Newtonian fluids are those whose viscosity does change under a shear stress. The behavior of such fluids (and of Newtonian fluids as well) may depend also on whether other stresses include heating or cooling and on how long a stress is applied. The apparent viscosity of ketchup, for example, decreases with the duration of the stress--the harder and longer you pound on the bottom of the bottle, the more likely a spurt of runny ketchup will be your reward. Such fluids are classified as "shear-thinning" fluids. There are several varieties of these, classified by other properties like response to heating or applied pressure, or elastic behavior.

The classification of fluid types comes in the main from laboratory experiments, Barth notes. These are often aimed at deriving empirical ways of predicting fluid behavior or at describing fluid behavior in very special situations (e.g., lubricants for roller bearings). The computational simulation of fluid behavior has for the most part proceeded in parallel, but generally from physical first principles and without reference to experiment. One of the objectives of the Carey group and of Barth in his research was to find new ways to foster the interaction of experiment and simulation.

Thus Carey and Barth were surprised and pleased to note that the call for a major international conference on computational heat transfer included a "benchmark contest" that offered a chance to match experiment and simulation. "What was remarkable about it," Barth says, "was that the experiment that we were to match, involving natural, buoyancy-driven convection of a Newtonian fluid--air, actually--in a cubical box, heated from below and cooled from above, was carefully conducted to supply all the parameters a modeler would need to build a simulation of the same experiment." Instrumentation all over the experimental system supplied not only measurements of internal changes in the air mass but also "boundary conditions"--changes at the edges and sides of the system.

Figure 2. Geometry of lab experiment on natural convection in a cubical box, devised as a challenge for computational modelers.

Barth and several members of other groups brought computational solutions to the computational heat transfer conference (CHT01). "I used our finite-element fluid dynamics code called MGF," Barth says. MGF stands for "microgravity flow" and was originally written by the Carey group to validate space shuttle experiments, but it is widely applicable to fluid problems on Earth as well as in space.

The MGF simulation of the Newtonian benchmark was so successful that "the obvious next step was to simulate the way a non-Newtonian fluid would behave under the same experimental conditions," Barth says. This became the topic of his dissertation.

Professor Carey sums up the problem this way. "Studies of coupled heat and fluid flow have been almost exclusively confined to Newtonian fluid models," he says. "But many industrial and naturally occurring fluids of great interest to society require more complex non-Newtonian models and offer a rich area for fundamental phenomenological flow studies. The added complexity of these models and the need to capture detailed flow structures accurately also make this an area where parallel high performance computing is particularly appropriate."

The finite-element code MGF was used to solve Barth's non-Newtonian problems also. The finite-element method is a technique for obtaining solutions to a wide variety of problems in engineering and other disciplines. It proceeds by taking the domain under study, for example, a volume of fluid being heated in a box, and discretizing it: dividing it into little domains, often triangles (in two dimensions) or tetrahedra or hexahedra (in three). The variation of some field variable (fluid velocity in our example) across a single small element will have some simple form, and one can write a mathematical expression for that variation. Depending on the way in which the overall stresses are applied, each of the other elements in the problem will have similar or only slightly different expressions.

The computer can keep track of all of the unknowns for all of the variables in the whole system and work to solve it as a simultaneous-equation system, using linear algebra. In our example, what would emerge is a picture of the flow of the fluid in the box over time. Visualization techniques can bring out and emphasize the salient physical processes.

The art in using the finite-element method in a way that leads to straightforward and experimentally testable conclusions, particularly on massively parallel computational systems, consists in finding ways to do the linear algebra more efficiently. Barth worked to assemble a variety of solution strategies, including selecting a time-integration scheme for the time-dependent problems and determining the appropriate domain-decomposition approach to assign parts of problems to multiple processors. This required much in the way of mathematical derivation of the specific equations for solution. He then chose a solution scheme and appropriate preconditioners--computer codes that transform the algebraic matrices into equivalent but more easily solved matrices.

Finally, he simplified the input procedure by adding a "dial-an-operator" interface as a front end for initializing the MGF code. This last effort, a project Barth worked on for several years with a research scientist in the Carey group, Robert McLay, allows the MGF user to input linearized equations in a language resembling the LaTeX math-typesetting language that is widely used in scientific publications, and it greatly increased the versatility and ease of use of the MGF code.

"My main aim in solving a variety of different cases for my dissertation was to obtain high-quality computational results," Barth says, "because these could then be used as benchmarks for experiment and for other simulation codes. The good experimental results obtained for natural convection of a Newtonian fluid motivated a new attempt to increase confidence in computer simulations for coupled heat-transfer problems in non-Newtonian fluids."

Barth proceeded in careful stages. He focused on two non-Newtonian, shear-thinning fluids, one called a "Powell-Eyring fluid" and the other an "extended Williamson" fluid, which differ slightly in the way in which their viscosities change under stress. The two classes of shear-thinning fluid correspond to varieties of dilute suspensions, including biological fluids like blood.

He first solved two internal "pipe flow" problems to study the behavior of the fluids without the influence of thermal effects. These cases of flow through a straight, cylindrical pipe were readily comparable to the well known Newtonian cases, and they highlighted the shear-thinning aspects of the two fluids under a range of ratios of velocity to viscosity (Reynolds numbers). This also gave Barth an idea of how much more intensive the calculations would be for the non-Newtonian fluids, especially at higher Reynolds number; in these simple cases, solution required two to four times as much iteration.

Figure 3. Pressure-driven flow in a branched pipe, showing (left) contours of velocity for a Powell-Eyring fluid and (right) the twisting of the flow in one branch (see text below). Calculations performed on Longhorn by William Barth.

The next step was to calculate how the fluids responded to pressure-driven flow in a branched pipe (see Figure 3). "The geometry is of interest because of its relevance to branching pipe flows in engineering and also biology--think of blood flowing in branching veins and arteries," Barth says. Under increased pressure, the non-Newtonian Powell-Eyring fluid develops a strong internal twisting current, with some recirculation upstream. Similar recirculations develop in the extended Williamson fluid at even greater pressures. An ability to quantify and predict these behaviors could ultimately be of value in medical practice as well as in many civil engineering projects.

Now Barth was ready to tackle the CHT01 benchmark problem with the geometry given in Figure 2 above and a non-Newtonian fluid. The problem of natural convection has a long history in computational fluid dynamics. It was posed as a two-dimensional problem for comparison exercises as early as 1979. But there were no good experimental results to compare with calculations, for obvious reasons: it was extremely difficult to set up even a quasi-two-dimensional experiment from which good measurements could be taken.

Thus, the three-dimensional experiment presented at CHT01 was not only a benchmark, but also a landmark in the field. The extension of this 3-D Newtonian problem to non-Newtonian fluids now opens up a broad space for further experimentation. Barth conducted hundreds of simulations to explore the complex parameter space with changing boundary conditions and varying internal distributions of temperature, pressure, viscosity, and other measurable quantities.

Figure 4. Another view of the evolution of natural convection in a non-Newtonian fluid.

As the picture here and at the top of this article illustrate, the behavior of the non-Newtonian fluids is complex. "I see it best by thinking in terms of the heat flux," Barth says. "The rising column of heated fluid is smaller in diameter and the upward velocity of the fluid is greater in the center of the box, with a broader return circulation along the sides. What is interesting is that, over time and under the higher temperatures, the non-Newtonian regime becomes first periodic, then metastable and aperiodic, not quite chaotic but never in equilibrium or a steady state." The simulations cover the evolution of the system over about thirty minutes of real time, and further investigation with longer simulations would be required to determine the ultimate evolution of the non-Newtonian fluid. Perhaps some cases may become fully chaotic, with turbulent flow throughout the domain.

"So we know that shear-thinning fluids transfer heat more rapidly but also more unevenly in convection than plain Newtonian fluids. At this point, to learn more of their secrets, we would need to improve the solver's performance and use upgraded or larger computers," Barth says. The simulations done for the dissertation "took more than a month of wall-clock time to compute," Barth notes, "which corresponds to about one CPU-year on the TACC Lonestar and Longhorn machines, both of which were used for the calculations."

Barth reported the results at the next meeting of the Computational Heat Transfer conference, held in May 2004, and articles are in preparation for journal publication. He and Carey are hoping that the simulations will stimulate complementary experimental work. "We know that the choice of fluid and appropriate experimental apparatus will present difficulties for our laboratory colleagues," Barth says, "but they should not be insuperable, and I think that whether our own results are verified or not, such experiments would be pathbreaking."

As a final exploration of the capacities of the MGF code and its new front end, Barth carried out some simulations of thermocapillary flows. They were first noted by Henri Bénard, a founder of fluid dynamics, at the turn of the 20th century. Bénard observed a pattern of hexagonal cells forming as he heated Newtonian fluids in shallow containers. Only in the mid-1950s, however, did other scientists identify the main force driving the formation of the cells. It was not buoyancy, but rather the variation in surface tension across the fluid. Experimenters found they could control the number and shapes of the cells in circular and square containers by changing the aspect ratio. Small numbers of cells with varying shapes (concentric circles, squares, wedges) form at lower aspect ratios (less than about 15), and as the aspect ratio is increased, the cells take on the classical hexagonal shape. Convection ultimately proves to be an intricate dance of buoyancy, viscosity, thermal diffusivity, and surface tension, and the thermocapillary flows are those in which surface tension is the driver.

Figure 5. Thermocapillary (surface-tension-driven) flow in a Powell-Eyring fluid as its viscosity changes from thick (left) to thin (right).

Barth simulated a single case that leads to a four-cell configuration for Newtonian fluids, again using the Powell-Eyring and extended Williamson models for non-Newtonian fluids. He used a 32 x 32 x 6 grid of uniformly spaced elements. Since the thermocapillary force acts to draw fluid from hot to cold as buoyancy causes warmer fluid to rise, each "spot" in the pictures corresponds to the top of an upwelling of fluid. "In these calculations, the changing viscosity of the non-Newtonian fluids does not appear to affect the number or overall geometry of the spots, but the shear-thinning does appear to increase the heat flux or transport of warm fluid from the bottom," Barth says.

"The studies I did for my dissertation helped to make MGF into a very general-purpose code," Barth says, "and they pushed not only the limits of our theories of non-Newtonian flows but also the limits of our computational power to investigate them." As a scientist at TACC, Barth is continuing his investigations as a way to benchmark the capacities of massively parallel cluster computers and to improve finite-element solution strategies. "It has given me a lot of experience I can use to collaborate in other projects in computational fluid dynamics using finite-element methods," Barth says. "I'm already involved in a number of these, with the Carey group and with others. Advancing the computational sciences is TACC's mission," he says, "and I'm excited about the opportunities we are exploring."