Richardson Annular Overshoot

Consider the transient flow in a circular duct where the pressure gradient can vary periodically in time, but at any instant in time is uniform axially. The axial momentum equation, for a constant-viscosity fluid, can be written as 

This problem can be cast in nondimensional form using the same variables as the impulsively started flow problem, Section 4.6. The nondimensional equation is 

The nondimensional oscillation frequency is u = cor /v. If is small compared to one, then the oscillation would be expected to have little effect. If is large, then the effect is large, including flow reversals. Averaged over a full period, the mean flow must follow the Poiseuille parabolic velocity distribution. At any instant in time, however, the velocity profile can be very different. 

From the governing equation itself, it is clear that the sinusoidal contribution to the pressure gradient averages to zero over a full cycle, leaving the average Hagen-Poiseuille flow. 

The classic Richardson overshoot problem has no mean flow. Hence there is no constant contribution in the pressure-gradient term, 

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This problem can be solved analytically, but it is complicated to do so. In any case, an interesting attribute of result is called the Richardson annular overshoot. The numerical solution is shown in Fig. 4.11. In this illustration, fi — I and the frequency is [Pg.176]

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