Some of the difficulties associated with this last problem can be presented as follows. When determining the velocity field of an unstable flow without a free surface, time only appears as a parameter. Consequently, the unstable motion can completely change the character of the flow, for example by creating surface waves in the flow.
As mentioned at the beginning, the theoretical investigation of the cavity currents is based on proposed artificial models. The centrifugal force due to the curvature of the free surface in the base flow now plays the role of a. We will assume that the pressure in the cavity of the unsteady flow will be kept at the same constant value Pc'.
For now, assume that the basic steady flow solution is given. Due to the assumed symmetry of the flow, it suffices to consider only the flow in the left half of the physical z-plane. The considered part of the flow in the z, f and Q planes is shown in Figure 3.
Moreover, due to the assumed symmetry, only the left half of the z-plane of the flow needs to be considered. Due to the presence of the imaging plate, an additional assumption is needed for this model in the study of its small time behavior. For ease of later application, the basic flow solution is briefly reproduced below.
For ease of derivation, instead of using the Cartesian coordinates (x, y) we will use the intrinsic coordinates (s, n) of the extended steady base flow in the following formulation, where s is the distance measured along a streamline in the base flow and n is the measured distance from a streamline such that it is directed at the boundary of the cavity in the flow, as shown in Figure 10. The pressure field of the resulting flow, p(s, n, t). is related to the velocity potential through the Bernoulli equation which can be written in the form. where p is the density of the fluid under consideration, p and q are. the cavity pressure and the constant velocity of the fluid particles on the free surface of the cavity in the base flow, respectively. which gives the pressure field of the disturbed flow. There are two boundary conditions on the free surface of the cavity, the kinematic boundary condition and the pressure condition.
The vorticity, /;" of a steady two-dimensional flow can be expressed in terms of internal coordinates as [cfo Ref. Also, we note that, since the velocity of the fluid particles at the free surface of the base flow has been normalized to unity, the complex velocity at the free surface of the base flow can be written as , which implies that the streamlines of the base flow are seen convex by the fluid, which is the free surface boundary condition expressed in complex variable form .
If we denote the displacement of the wetted side of the solid body from its stable position S by Given the complexity of the free surface boundary condition, the details involved in solving the problem formulated above will depend on the underlying flow and will therefore vary from case to case. Because the wetted side of the solid is together with the free surface •. f plane, it is always possible to transform the entire current field into o. upper half of a given s-plane such that stream:mline ~ = 0 coin- o. cides with the real axis of the s-plane.
With the boundary conditions expressed in the form of equations 133, 138 and 142, the solution of f. can be obtained in the s-plane by applying any of the :methods described below:. With the boundary conditions on o. the free surface and the solid body expressed in the form of equations 133 and 144 A Hilbert problem can be constructed in the s-plane by the method described in section 2, Part 1. The solution of the thus constructed Hilbert problem will we lead to a second order linear partial differential equation with variable coefficients, the solution of which gives f. ii). This relatively simple problem was chosen to demonstrate the application of this general theory to a very special case; this application is partly intended for the verification of the complicated expression of the free surface boundary condition, since the problem was already solved by Lord Kelvin (28) in a completely different way (the solution is obtained by using the axial symmetry of the basic flow field).
The solution of the disturbance current is obtained using the first solution method mentioned in the last section. We assume that at time t = 0 a small perturbation is exerted on the base current, so that the resulting unstable rotational motion can be regarded as a perturbation of the base current. Then the position of a fluid particle and the complex velocity potential of the resulting flow can be written as.
In the following we proceed to express the quantities in equation 166 in terms of the basic flow and the perturbation variables. To express the kinematic boundary condition (ii) stated previously, we can use the relation dz/dt = w(z, t) when evaluated at S (In the following derivation, instead of the form of equation 163 to use, it will be more convenient to write the position of the perturbed free surface in the .
