Motivation: The self-propelled wake is thought to be extremely sensitive to any momentum imbalance. How does acceleration affect the wake?
It is thought that a small amount of excess momentum injected into the wake during an accelerating maneuver significantly changes the behavior of the
wake leading to a significantly increased wake lifetime and the formation of larger coherent stuctures in the late wake. Support for this viewpoint
comes from an analytic prediction from Tennekes and Lumley in their 1972 textbook "A First Course in Turbulence", an experimental study by
Voropayev et. al.
on the large vortex structures generated by a maneuvering body in a stratified fluid
(link to paper)
as well as the difficulties encountered by numerous experimentalists in achieving wakes with zero net momentum as they attempted
to study momentumless wakes.
Direct numerical simulation was used to study the effect of increasing the amount of excess momentum and varying the shape of excess momentum in the
wake of a bluff body in a stratified fluid. Simulations were performed at a Reynolds number of 10,000 and a Froude number of 3 with varying amounts and shapes of excess momentum to describe the qualitatitive and quantitative evolution of a propelled wake with excess momentum in a stratified fluid.
Increasing the amount of excess momentum and/or decreasing the radial extent of the excess momentum was found to increase the defect velocity,
mean kinetic energy, shear in the velocity gradient and wake width. The increased shear in the mean profile results in increased
turbulent production which extracts energy from the mean profile to turbulent kinetic energy which results in increased dissipation.
The vorticity structures were found to increase slightly in size with increasing excess momentum although the difference in size was
significantly less than that suggested by previous experiments.
Buoyancy was found to preserve the doubly-inflected velocity profile in the vertical direction, and similarity for the mean velocity and
turbulent kinetic energy was found to occur in both horizontal and vertical directions.
Important conclusions: Adding a small to moderate amount of excess momentum, 40% or less, to a self-propelled wake does not produce qualitatively different behavior. Flow statistics retain the character of a self-propelled wake but with larger quantitative values.
For further details on this work, see the 2011 Journal of Fluid Mechanics paper by de Stadler and Sarkar.
For the unstratified analog to this work, see the 2011 AIAA Fluid Dynamics conference proceedings paper by de Stadler and Sarkar.
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| The wake behind a body moving to the left. (a) Towed body. (b) Self-propelled body with zero net momentum. (c) propelled body with excess momentum. | Increasing the amount of excess momentum. |
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| Vertical vorticity at the vertical centerplane. Left column: t=75. Right column: t=1400. (a) Self-propelled. (b) 5% excess momentum. (c) 20% excess momentum. (d) 40% excess momentum. | Mean velocity evolution for the 40% excess momentum case. (a) t=49. (b) t=125. (c) t=275. (d) t=1405. |