Impact of liquids with different densities

The collision of liquids of different densities is studied theoretically for the case of liquids having wedge-shaped configuration before the impact. Both liquids are assumed to be ideal and incompressible, and the velocity potential theory is used for the flow of each liquid. Surface tension and gravity effects are neglected. The problem is decomposed into two self-similar problems, one for each liquid. Across the interface between the liquids, continuity of the pressure and the normal component of the velocity is enforced through iteration. This determines the shape of the interface and other flow parameters. The integral hodograph method is employed to derive the solution consisting of analytical expressions for the complex-velocity potential, the complex-conjugate velocity, and the mapping function. They are all defined in the first quadrant of a parameter plane, in which the original boundary-value problem is reduced to a system of integro-differential equations in terms of the velocity magnitude and the velocity angle relative to the flow boundary. They are solved numerically using the method of successive approximations. The results are presented through streamlines, interface and free-surface shapes, the pressure and velocity distributions. Special attention is given to the structure of the splash jet rising as a result of the impact.

The Two-Phase Hell-Shaw Flow: Construction of an Exact Solution

We consider a two-phase Hele-Shaw cell whether or not the gap thickness is time-dependent. We construct an exact solution in terms of the Schwarz function of the interface for the two-phase Hele-Shaw flow. The derivation is based upon the single-valued complex velocity potential instead of the multiple-valued complex potential. As a result, the construction is applicable to the case of the time-dependent gap. In addition, there is no need to introduce branch cuts in the computational domain. Furthermore, the interface evolution in a two-phase problem is closely linked to its counterpart in a one-phase problem

Asymmetric impact between liquid and solid wedges

The hydrodynamic problem of impact between a solid wedge and a liquid wedge is analysed. The liquid is assumed to be ideal and incompressible; gravity and surface tension effects are ignored. The flow generated by the impact is assumed to be irrotational and therefore can be described by the velocity potential theory. The solution procedure is based on the analytical derivation of the complex-velocity potential in a parameter plane and the function mapping conformally the parameter plane onto the similarity plane. The mapping function is found as a combination of the derivatives of the complex potential in the similarity and parameter planes, through the integral equations for mixed and homogeneous boundary-value problems in terms of the velocity modulus and the velocity angle with the fluid boundary, together with the dynamic and kinematic boundary conditions. These equations are solved through a numerical method. The procedure is first verified through comparisons with some known results. Simulations are then made for a variety of cases, and detailed results are presented in terms of the free surface shape, streamlines, pressure distribution on the wetted solid surface, and contact angles between the free surface and the body surface.

THE STUDY OF THE UNSTEADY FLUID-DYNAMIC FORCES ACTING ON AN AUTO ROTATING ELLIPTIC CYLINDER

When a thin flat pate is released in the still air, the plate may fall aslant and automatically set into a rotational motion. This phenomenon is called autorotation. In this state, the unsteady fluid-dynamic forces act on the plate due to the vortex shedding and the rotation of the body. In this study, we focused on the auto-rotating phenomenon of an elliptic cylinder and investigated the effect of the cylinder rotation on the unsteady fluid-dynamic forces by the comparison between the result of the complex velocity potential theory and the one of numerical simulation. It was found that the low drag occurring at -160 degree phase and the high lift at zero degree phase are caused by the cylinder rotation.

A stationary potential-flow approximation for a breaking-wave crest

The flow in a breaking-wave crest is represented by a complex velocity potential on a Riemann surface, satisfying the Bernoulli condition on two free boundaries. The flow is assumed to be stationary in the reference frame which moves with the wave crest, and at large distances approximates Stokes corner flow in the main part of the fluid and a parabolic descending flow in the jet. The interaction of the jet with the rest of the fluid is neglected.The solution is obtained by means of a conformal transformation from a bounded, teardrop-shaped domain, using a Faber polynomial expansion. The Bernoulli condition is applied at a number of discrete points on the boundaries, and the resulting nonlinear equations for the expansion coefficients are solved iteratively. The resulting surface form is similar to that obtained by laboratory experiments and time-dependent numerical simulations of waves up to the point of breaking, with a stagnation point at the top of the crest, an overturning loop with major axis$\ap 8g^{-\frac{1}{3}}\Psi^{\frac{2}{3}}$, and a maximum acceleration of ≈ 5.4g, wheregis the gravitational acceleration and ψ is the flux in the jet.

The almost-highest wave: a simple approximation

The crest of a steep, symmetric gravity wave is shown to be closely approximated by the expression \[ x+iy = \frac{\alpha +\gamma i\chi}{(\beta + i\chi)^{\frac{1}{3}}}, \] where x, y are co-ordinates in the vertical plane, χ is the complex velocity potential and α, β, γ are certain constants. This expression is asymptotically correct both for small and for large values of |χ|; and the free surface agrees with the exact profile calculated by Longuet-Higgins & Fox (1977) everywhere to within 1·5 per cent. The pressure at the surface is constant to within 5 per cent.

The solitary wave of maximum amplitude

The maximum amplitude of the solitary wave of constant form is determined to be 0·83b, where b is the depth far from the crest. In the analysis it is assumed that the crest is pointed and the motion is two-dimensional and irrotational. The complex velocity potential is expressed in terms of known singularities and an infinite power series with unknown coefficients. Approximate solutions are obtained by truncating the power series after N terms, where N = 1, 3, 5, 7, and 9. The amplitude, a measure of the error, and several other pertinent quantities are computed for each value of N.