Fluid Dynamics Research at CNLD

Pattern Formation in Fluid Systems

Couette-Taylor Vortices Over a century ago, Maurice Couette created a simple device to measure the viscosity of a fluid. He placed the fluid between two concentric cylinders and measured the amount of torque required to turn the inner cylinder. As one increases the rotation rate of the inner cylinder, more torque is required. However, at a certain rotation rate, the variation of torque with cylinder speed changes abruptly. This occurs at the onset of an instability, the sudden change of the pattern of fluid flow as a parameter is varied. In this case, a series of circulating cells known as Couette-Taylor vortices emerge in the fluid flow . The presence of this instability places limits on the utility of Couette's viscometer. Many fluid instabilities have such practical consequences; for instance, the presence of fingering instabilities during oil extraction can mean that half of the oil in a reservoir is left in the ground. At the Center for Nonlinear Dynamics, we study fluid instabilities and pattern formation in many fluid systems.

Turbulent patterns in 2D soap films
Because soap films are very thin (about 0.0001 cm thick), they are often used as experimental models of two-dimensional fluid flows. Furthermore, the vibrant colors on a soap film can conveniently be used to track the flow. We study the flows in a flat, horizontal soap film driven into motion by conveyer belts running near, but not actually touching the film. These belts induce an air flow which in turn drives the film, just as you can drive a flow on a soap bubble by blowing on it. We observe the resulting flow by watching the swirling variations in film thickness. When the film is forced using belts moving in alternating directions, a transition from large flow loops to an array of small vortices is seen (Fig. 1). Our work shows that flows in a soap film cannot be simply treated as two-dimensional flows, due to the drag of the surrounding air on the film. [John Burgess, Chris Bizon]

Figure 1: A black and white picture of vortices in a horizontal, flat soap bubble driven beyond the onset of flow instability. The variations of brightness in the image correspond to variations in film thickness.

Figure 2: Couette-Taylor Pictures
Flow in a Couette-Taylor Geometry
In addition to primary instability mentioned above, the Couette-Taylor fluid system exhibits many secondary instabilities, creating a small zoo of fluid patterns. The Couette-Taylor system can also exhibit turbulence at high rotation rates, creating an ideal system for study of this difficult problem.

Figure 3: A snapshot of viscous fingers (also known as Saffman-Taylor fingers) created from experimental data. Air is flowing from the
left, displacing the more viscous oil. At this moderately high flow rate, secondary branching of the original finger occurs frequently.

Evolution of Viscous Fingers
New types of instabilities can occur when we consider systems with two immiscible fluids. When one fluid is driven into another in a small gap between two plates (a quasi-2D geometry known as a Hele-Shaw cell), a moving interface is created. If the invading fluid is more viscous than the displaced fluid (e.g., oil displacing air), this interface becomes flat as it moves. However, when the invading fluid is less viscous than the displaced fluid (e.g., air displacing oil), the moving interface is unstable to small disturbances. These disturbances grow into "fingers" which evolve in different ways, depending on the fluid forcing velocity. For small forcing, a single uniformly moving finger eventually dominates the system. As the forcing is increased, this finger narrows, but never decreases below one-half of the width of the system. As the forcing is increased further, the finger begins to split and create side branches, as shown in Fig. 3. The complexity of these secondary fingers increases as the finger velocity is further increased. [Mitchell Moore]

Concentration Driven Convection in 2D
When a fluid is heated from below, the lower fluid becomes less dense than the higher fluid. The light fluid rises, leading to a convective flow pattern. For large differences in temperature, this flow pattern becomes extremely complicated, i.e. turbulent. Turbulent convection is important in both solar and planetary atmospheres. We studied turbulent convection in between two flat vertical glass planes. This allowed us to completely visualize the flow patterns. Experimentally, this pattern was driven using salt, rather than heat, to induce the density difference that causes the patterns. In addition, we performed high resolution numerical simulations to compare to the experimental results. In both cases, the transport of concentration (or heat) was due to powerful plumes. Fig. 4 shows a comparison between experiment and simulation of these concentration plumes. [Alexei Predtechensky, Chris Bizon]

Figure 4: A false color image of turbulent convection plumes in an experiment and in a computer simulation.


Turbulent patterns in 2D soap films Because soap films are very thin (about 0.0001 cm thick), they are often used as experimental models of two-dimensional fluid flows. Furthermore, the vibrant colors on a soap film can conveniently be used to track the flow. We study the flows in a flat, horizontal soap film driven into motion by conveyer belts running near, but not actually touching the film. These belts induce an air flow which in turn drives the film, just as you can drive a flow on a soap bubble by blowing on it. We observe the resulting flow by watching the swirling variations in film thickness. When the film is forced using belts moving in alternating directions, a transition from large flow loops to an array of small vortices is seen (Fig. 1). Our work shows that flows in a soap film cannot be simply treated as two-dimensional flows, due to the drag of the surrounding air on the film. [John Burgess, Chris Bizon]	Figure 1: A black and white picture of vortices in a horizontal, flat soap bubble driven beyond the onset of flow instability. The variations of brightness in the image correspond to variations in film thickness.

Figure 2: Couette-Taylor Pictures	Flow in a Couette-Taylor Geometry In addition to primary instability mentioned above, the Couette-Taylor fluid system exhibits many secondary instabilities, creating a small zoo of fluid patterns. The Couette-Taylor system can also exhibit turbulence at high rotation rates, creating an ideal system for study of this difficult problem.

Figure 3: A snapshot of viscous fingers (also known as Saffman-Taylor fingers) created from experimental data. Air is flowing from the left, displacing the more viscous oil. At this moderately high flow rate, secondary branching of the original finger occurs frequently.
Evolution of Viscous Fingers New types of instabilities can occur when we consider systems with two immiscible fluids. When one fluid is driven into another in a small gap between two plates (a quasi-2D geometry known as a Hele-Shaw cell), a moving interface is created. If the invading fluid is more viscous than the displaced fluid (e.g., oil displacing air), this interface becomes flat as it moves. However, when the invading fluid is less viscous than the displaced fluid (e.g., air displacing oil), the moving interface is unstable to small disturbances. These disturbances grow into "fingers" which evolve in different ways, depending on the fluid forcing velocity. For small forcing, a single uniformly moving finger eventually dominates the system. As the forcing is increased, this finger narrows, but never decreases below one-half of the width of the system. As the forcing is increased further, the finger begins to split and create side branches, as shown in Fig. 3. The complexity of these secondary fingers increases as the finger velocity is further increased. [Mitchell Moore]

Concentration Driven Convection in 2D When a fluid is heated from below, the lower fluid becomes less dense than the higher fluid. The light fluid rises, leading to a convective flow pattern. For large differences in temperature, this flow pattern becomes extremely complicated, i.e. turbulent. Turbulent convection is important in both solar and planetary atmospheres. We studied turbulent convection in between two flat vertical glass planes. This allowed us to completely visualize the flow patterns. Experimentally, this pattern was driven using salt, rather than heat, to induce the density difference that causes the patterns. In addition, we performed high resolution numerical simulations to compare to the experimental results. In both cases, the transport of concentration (or heat) was due to powerful plumes. Fig. 4 shows a comparison between experiment and simulation of these concentration plumes. [Alexei Predtechensky, Chris Bizon]	Figure 4: A false color image of turbulent convection plumes in an experiment and in a computer simulation.