Modified Equation of Impulse for Solitons in an Open Water Channel

On the basis of the certain form of write of an impulse equation the modeling of the solitary waves in the water channel is examined at action of gravitation forces. It is shown that as against an existing method of modelling where the waves propagating from left to right turn out from the equation of an impulse, and from right to left from the continuity equation in the offered technique both waves turn out from the equation of impulse. It is marked that the given method is physically more correct. Calculation of a solitary wave, its velocity and geometrical characteristics is submitted.

Derivation of an Impulse Equation for Wind Turbine Thrust

Using the concept of impulse in control volume (CV) analysis, we derive a new equation for steady wind turbine thrust in a constant, spatially uniform wind. This equation contains the circumferential velocity and tip speed ratio. We determine the conditions under which the new equation reduces to the standard equation involving only the axial velocity. A major advantage of an impulse formulation is that it removes the pressure and introduces the vorticity, allowing the equation to be used unambiguously immediately behind the blades. Using two CVs with this downwind end – one rotating with the blades, and one stationary – highlights different aspects of the analysis. We assume that the vorticity is frozen relative to an observer rotating with the blades, so the vortex lines follow the local streamlines in the rotating frame and vorticity does not appear explicitly in the impulse equation for force. In the stationary frame, streamlines and vortex lines intersect, and one of the thrust terms can be interpreted as the effect of wake rotation. The impulse analysis also shows the significance of the radial velocity in wind turbine aerodynamics. By assuming both the radial and axial velocities are continuous through the rotor disk, their contributions to the final thrust expression cancel to leave an expression dependent on azimuthal velocity alone. The final integral equation for thrust can be viewed as a generalization of the Kutta–Joukowsky theorem for the rotor forces. We give a proof, for the first time, for the conditions under which the Kutta–Joukowsky equations apply. The analysis is then extended to the blade elements comprising the rotor. The new formulation gives a very simple, exact equation for blade element thrust which is the major contribution of this study. By removing the pressure partly through the kinetic energy contribution of the radial velocity, the new equation circumvents the long-standing concern over the role the pressure forces acting on the expanding annular streamtube intersecting each blade element. It is shown that the necessary condition for blade-element independence of the conventional thrust equation – that which involves the axial induction factor – is the constancy of the vortex pitch in the wake.

The toroidal bubble

It has been observed by Walters & Davidson (1963) that release of a mass of gas in water sometimes produces a rising toroidal bubble. This paper is concerned with the history of such a bubble, given that at the initial instant the motion is irrotational everywhere in the water. The variation of its overall radius a with time may be predicted from the vertical impulse equation, and it should be possible to make the same prediction by equating the rate of loss of combined kinetic and potential energy to the rate of viscous dissipation. This is indeed seen to be the case, but not before it is recognized that in a viscous fluid vorticity will continually diffuse out from the bubble surface, destroying the irrotationality of the motion, and necessitating an examination of the distribution of vorticity. The impulse equation takes the same form as in an inviscid fluid, but the energy equation is severely modified. Other results include an evaluation of the effect of a hydrostatic variation in bubble volume, and a prediction of the time which will have elapsed before the bubble becomes unstable under the action of surface tension.