Steady free-surface vortical flows parallel to the horizontal bottom

Abstract

Steady, free-surface, vortical flows of an inviscid, incompressible, heavy fluid over a horizontal, rigid bottom are considered. All flows of constant depth are described for any Lipschitz vorticity distribution. It is shown that the values of Bernoulli's constant, for which such flows exist, are greater than or equal to some critical value depending on the vorticity. For the critical value, only one flow exists and it is unidirectional. Supercritical flows exist for all values of Bernoulli's constant greater than the critical one; every such flow is also unidirectional and its depth is smaller than that of the critical flow. Furthermore, at least one flow other than supercritical does exist for every value of Bernoulli's constant greater than the critical one. It is found that for some vorticity distributions, the number of constant depth flows increases unrestrictedly as Bernoulli's constant tends to infinity. However, all these flows, except for one or two, have counter-currents; their number depends on Bernoulli's constant and increases by at least two every time when this constant becomes greater than a critical value (the above mentioned is the smallest of them), belonging to a sequence defined by the vorticity. A classification of vorticity distributions is presented; it divides all of them into three classes in accordance with the behaviour of some integral of the distribution on the interval [0, 1]. For distributions in the first class, a unidirectional subcritical flow exists for all admissible values of Bernoulli's constant. For vorticity distributions belonging to the other two classes such a flow exists only when Bernoulli's constant is less than a certain value. If Bernoulli's constant is greater than this value, then at least one flow with counter-currents does exist along with the unidirectional supercritical flow. The second and third classes of vorticity distributions are distinguished from one another by the character of the counter-currents. If a distribution is in the second class, then a near-bottom counter-current is always present for sufficiently large values of Bernoulli's constant. For distributions in the third class, a near-surface counter-current is always present for such values of the constant. Several examples illustrating the results are considered. © The author 2011. Published by Oxford University Press all rights reserved.