Discontinuities In Velocity Gradient Research Articles

Fully implicit and accurate treatment of jump conditions for two-phase incompressible Navier–Stokes equations

We present a numerical method for two-phase incompressible Navier–Stokes equation with jump discontinuities in the normal projection of the stress tensor and in the material properties. Although the proposed method is only first-order accurate, it does capture discontinuities sharply, not neglecting nor omitting any component of the jump condition. Discontinuities in velocity gradient and pressure are expressed using a linear combination of singular force and tangential derivatives of velocities to handle jump conditions in a fully implicit manner. The linear system for the divergence of the stress tensor is constructed in the framework of the ghost fluid method, and the resulting saddle-point system is solved via an iterative procedure using recently introduced techniques by Egan and Gibou [9]. Numerical results support the inference that the proposed method converges in L∞ norms even when velocities and pressures are not smooth across the interface and can handle a large density ratio that is likely to appear in a real-world simulation.

Geometric and induced effects in binocular stereopsis and motion parallax

This paper examines and contrasts motion-parallax analogues of the induced-size and induced-shear effects with the equivalent induced effects from binocular disparity. During lateral head motion or with binocular stereopsis, vertical-shear and vertical-size transformations produced ‘induced effects’ of apparent inclination and slant that are not predicted geometrically. With vertical head motion, horizontal-shear and horizontal-size transformations produced similar analogues of the disparity induced effects. Typically, the induced effects were opposite in direction and slightly smaller in size than the geometric effects. Local induced-shear and induced-size effects could be elicited from motion parallax, but not from disparity, and were most pronounced when the stimulus contained discontinuities in velocity gradient. The implications of these results are discussed in the context of models of depth perception from disparity and structure from motion.

Seismograms from Epstein Transition Zones

Spectral amplitudes and theoretical seismograms in the vicinity of caustic and critical points are evaluated numerically for acoustic waves propagating in Epstein media. The amplitude decay into the region beyond the caustic is compared with the asymptotic results and good agreement is found at high frequency. At low frequencies the partial reflection enhances the amplitudes at sub-critical angles and the amplitude increases with decreasing thickness of the transition zone. For a thin transition and low frequencies a head wave exists whose amplitude is inversely proportional to frequency. The study of amplitudes near caustics and critical point is of significant theoretical and experimental interest to seismologists. Improved coverage by stations of the USCGS worldwide network has provided better data and their interpretation has lead to more accurate estimates of elastic wave velocities within the Earth (Johnson 1967). In regions of high velocity gradient in the upper mantle caustics are formed and triplications occur on the travel-time curves. In this paper the amplitude properties near caustics and critical point are studied for the Epstein velocity transitions (Epstein 1930). Epstein profiles represent a group of models with transition zones for which solutions of the wave equation exist in terms of familiar tabulated functions. Wave propagation has also been studied analytically for linear transition layers (Nakamura 1964; Gupta 1966; Hirasawa & Berry 1971; Ward 1973) and exponential and parabolic layers (Merzer 1971 ; Rydbeck 1943). Linear layers have discontinuities in velocity gradient which may not be realistic. Merzer’s study is concerned only with head waves and their amplitude dependence on frequency, layer thickness and shape of the transition. Using Epstein models the caustics, head waves and shadow can be studied. Epstein (1930), Rawer (1939) and Phinney (1970) studied the modulus of the reflection coefficient in order to estimate the partial reflection. The modulus of the reflection coefficient itself, however, cannot give a full description of the energy propagation when the waves interact with the strong velocity gradient. That can only be studied when the spectral amplitudes and theoretical seismograms are evaluated. In this paper the exact spectral amplitudes are computed by contour integration using the method of Phinney & Cathles (1969). For evaluation of the theoretical seismograms the technique of Chapman & Phinney (1972) is employed. The amplitude curves are evaluated in the neighbourhood of both ends of the triplication. At high frequencies the results agree with the asymptotic results. At low

On the Interpretation of Pd

Summary Lehmann's observations of a large phase about 21°, lying a few seconds after the first arrival Pr, are discussed on two extreme hypotheses. On the first the distance varies as a quadratic function of sin e, so that dΔ/de vanishes at 21°, where there is a cusp in the time curve. The values of dt/dΔ for Pr and Pd are found to be such that there is no possible position for the nearer cusp. On the second, a cubic form for t is assumed up to 21° for Pd. Pr must enter between 17° and 18°. The Pd ray emerging at 21° must reach about 0.03 R below the Mohorovičić discontinuity. The times of Pr imply that at the transition there must be a considerable discontinuity of velocity gradient but only a small one of velocity. The nearer cusp would be near 15°, but as it lies less than a second after the first arrival it is probably undetectable.