Dynamic Plumes and Solid Wall Interactions: Transient Levitation of Falling Bodies
Animation:
As an extreme example in which the injected quantity cannot mix with the ambient fluid, consider recently obtained experimental results concerning the motion of falling bodies through stratified fluids (similar to
Figure 2. Vertical buoyant jets through a strong stable density step: Left is oil (0.8 g/cm3), right is alcohol-water mixture (0.8 g/cm3). (Thanks to former UNC undergraduates Ryan McCabe and Daniel Healion for assistance with the experiment.)
the tank setup in Figure 2) (Abaid, Adalsteinsson, Agyapong, McLaughlin, 2004). This study has focused upon the effect of self-generated plumes upon the falling body and has documented situations in which a falling body may generate a dynamic plume that through hydrodynamic coupling, may temporarily arrest the body. Of course, any body moving through a fluid experiences a hydrodynamic drag (which sets terminal velocities of falling bodies) in which the viscous boundary condition of vanishing fluid flow at the solid boundary necessarily drags a blob of ambient fluid along the moving body. In a constant density fluid, there is no potential energy cost associated with moving such a parcel of ambient fluid vertically. However, in strongly stratified fluids, a parcel of fluid moved from one altitude to another may develop a potential energy (buoyancy), as when the body falls through a sharp density transition layer. The momentum of the attached blob of fluid thrusts it into the lower (heavier) fluid, at which point the blob becomes a density anomaly and rises sharply. This motion in turn drags the falling body along with it.
Figure 3 shows three montages of a descending sphere at uniformly spaced times. The (5 mm radius) sphere in this case has a density of 1.04 g/cm3 and is falling in a stratified tank whose top is fresh water (0.997 g/cm3) and whose bottom is salt water (1.039 g/cm3), again with a transition thickness of approximately 1 in. The top montage demonstrates the arrest and transient rise of the initially falling bead, and subsequent return to slow descent, each image uniformly spaced 1.5 s apart. The bead ultimately comes to rest at the tank bottom. The middle montage is the same as the top, only uniformly spaced at 0.1 s intervals. The lower montage has the same time sequence as the middle row, only focusing upon the shadow on the back of the tank, which highlights the entrained, plume-forming fluid.The nature of this phenomenon is both nonlinear and dynamic. The nonlinear effect of such plumes upon
Figure 3. Top: Digital snapshots of bead position on uniform 1.5 s intervals, Middle: uniformly spaced on 0.1 s intervals, Bottom: shadowgraph depicting the dynamic plume on same time interval as middle row (Abaid, Adalsteinsson, Agyapong, McLaughlin, 2004). (Thanks to David Adalsteinsson for help with formatting the collage in his DataTank program and thanks to former UNC undergraduate Nicole Abaid for assistance with the experimental effort.)
the motion of solid bodies has been incorporated in a reduced system of ordinary differential equations in which the drag law for the falling body is modified to account for the dynamics of the plume which may modify the relative velocity of the falling sphere (Abaid, Adalsteinsson, Agyapong, McLaughlin, 2004). To describe the detailed dynamics of such transient plumes is quite difficult. Historically, there has been more success in the modeling of plume geometries under steady-state geometries. In pioneering work, Morton, Turner, and Taylor (Morton et al., 1956; Morton, 1967; Turner, 1995) were the first to model maintained plumes using an entrainment hypothesis which has become a standard in many fields (Fisher et al., 1979; Sparks et al., 1997).
