A multiple-scales solution of the undular hydraulic jump problem

The propagation of unsteady disturbances in a slowly varying cylindrical duct carrying mean swirling flow is described. A consistent multiple-scales solution for the mean flow and disturbance is derived, and the effect of finite-impedance boundaries on the propagation of disturbances in mean swirling flow is also addressed.Two degrees of mean swirl are considered: first the case when the swirl velocity is of the same order as the axial velocity, which is applicable to turbomachinery flow behind a rotor stage; secondly a small swirl approximation, where the swirl velocity is of the same order as the axial slope of the duct walls, which is relevant to the flow downstream of the stator in a turbofan engine duct.The presence of mean vorticity couples the acoustic and vorticity equations and the associated eigenvalue problem is not self-adjoint as it is for irrotational mean flow. In order to obtain a secularity condition, which determines the amplitude variation along the duct, an adjoint solution for the coupled system of equations is derived. The solution breaks down at a turning point where a mode changes from cut on to cut off. Analysis in this region shows that the amplitude here is governed by a form of Airy's equation, and that the effect of swirl is to introduce a small shift in the location of the turning point. The reflection coefficient at this corrected turning point is shown to be exp (iπ/2).The evolution of axial wavenumbers and cross-sectionally averaged amplitudes along the duct are calculated and comparisons made between the cases of zero mean swirl, small mean swirl and O(1) mean swirl. In a hard-walled duct it is found that small mean swirl only affects the phase of the amplitude, but O(1) mean swirl produces a much larger amplitude variation along the duct compared with a non-swirling mean flow. In a duct with finite-impedance walls, mean swirl has a large damping effect when the modes are co-rotating with the swirl. If the modes are counter-rotating then an upstream-propagating mode can be amplified compared to the no-swirl case, but a downstream-propagating mode remains more damped.

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Multiple scales solution for free vibrations of a rotating shaft with stretching nonlinearity

Scientia Iranica ◽

10.1016/j.scient.2012.12.004 ◽

2013 ◽

Vol 20 (1) ◽

pp. 131-140 ◽

Cited By ~ 19

Author(s):

S.A.A. Hosseini ◽

M. Zamanian

Keyword(s):

Multiple Scales ◽

Free Vibrations ◽

Rotating Shaft ◽

Multiple Scales Solution

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Conditions under which a supercritical turbidity current traverses an abrupt transition to vanishing bed slope without a hydraulic jump

Journal of Fluid Mechanics ◽

10.1017/s0022112007006738 ◽

2007 ◽

Vol 586 ◽

pp. 119-145 ◽

Cited By ~ 16

Author(s):

SVETLANA KOSTIC ◽

GARY PARKER

Keyword(s):

Hydraulic Jump ◽

Turbidity Current ◽

Field Application ◽

Coarse Grained ◽

Turbidity Currents ◽

Jump Problem ◽

Fall Velocity ◽

Supercritical Flow ◽

Subcritical Flow ◽

Fine Grained

Turbidity currents act to sculpt the submarine environment through sediment erosion and deposition. A sufficiently swift turbidity current on a steep slope can be expected to be supercritical in the sense of the bulk Richardson number; a sufficiently tranquil turbidity current on a mild slope can be expected to be subcritical. The transition from supercritical to subcritical flow is accomplished through an internal hydraulic jump. Consider a steady turbidity current flowing from a steep canyon onto a milder fan, and then exiting the fan down another steep canyon. The flow might be expected to undergo a hydraulic jump to subcritical flow near the canyon–fan break, and then accelerate again to critical flow at the fan–canyon break downstream. The problem of locating the hydraulic jump is here termed the ‘jump problem’. Experiments with fine-grained sediment have confirmed the expected behaviour outlined above. Similar experiments with coarse-grained sediment suggest that if the deposition rate is sufficiently high, this ‘jump problem’ may have no solution with the expected behaviour, and in particular no solution with a hydraulic jump. In such cases, the flow either transits the length of the low-slope fan as a supercritical flow and shoots off the fan–canyon break without responding to it, or dissipates as a supercritical flow before exiting the fan. The analysis presented below confirms the existence of a range associated with rapid sediment deposition where no solution to the ‘jump problem’ can be found. The criterion for this range is stated in terms of an order-one dimensionless parameter involving the fall velocity of the sediment. The criterion is tested and confirmed against the experiments mentioned above. A sample field application is presented.

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