The motion of buoyant elements in turbulent surroundings
In this paper a theoretical model of the motion of isolated buoyant elements in turbulent surroundings is introduced, which takes into account both the growth due to turbulent entrainment and a loss of buoyant fluid to the environment. On dimensional grounds the outflow velocity is taken to be constant and proportional to some characteristic turbulent velocity in the environment, while the entrainment velocity is proportional to the upward velocity of the element. Numerical solutions of the resulting non-dimensional equations of motion are presented, corresponding to a wide range of stabilities. Typically, an element in stable, neutral or moderately unstable surroundings at first grows and then is eroded away, but at a certain value of a stability parameter γ elements become absolutely unstable and continue to grow and rise indefinitely. The value of γ is extremely sensitive to the level of turbulence in the environment, which could therefore exert a controlling influence on the growth of buoyant elements in unstable conditions; large elements are more likely to grow when the level of turbulence is low.
Laboratory experiments have been carried out in order to test one of the predictions of this theory, the form of the dependence of the height attained on total buoyancy and level of turbulence in uniform surroundings. The agreement is good, and numerical comparison of theory and experiment suggests that the assumed outflow velocity is of the same order as, but somewhat less than, the r.m.s. turbulent velocity.(Published Online March 28 2006)
(Received August 28 1962)
(Revised November 26 1962)