An experimental investigation is performed to elucidate the variations of accelerations and pressure gradients in the external stream of retreating flow during the run-down phase of a non-breaking solitary wave, propagating over a 1:3 sloping beach. Two solitary waves that have the incident wave heights (H0) of 2.9 and 5.8 cm, with respective still water depths (h0) of 8.0 and 16.0 cm (Cases A and B), were generated in a wave flume, resulting in the incident wave-height to water-depth ratios (H0/h0) being identically equal to 0.363. The latter case was only used to highlight the non-dimensional features of the wave celerity, the time history of horizontal velocity and the breaker type, which all exhibit similarity to those of the former. Two flow visualization techniques such as particle trajectory method and fluorescent dye strip and a high-speed particle image velocimetry (HSPIV) were utilized to provide the flow images and velocity fields. Based on the ensemble-averaged velocity fields and profiles, the partially depth-averaged (i.e., excluding the part in the boundary layer) values of accelerations and pressure gradient at a specified measuring section are then smoothed by a symmetric five-point smoothing scheme. Eventually, the smoothed values of the accelerations and pressure gradient are used to highlight the dynamic features of the external stream of retreating flow. It is found that, at the section of the incipient flow separation, the non-dimensional local acceleration (with respect to the gravity acceleration) in the offshore direction keeps increasing from the moment at which the run-up motion ends to the counterpart at which the incipient flow separation occurs. Afterwards, growth of the primary vortex develops with its core translating offshore. The corresponding non-dimensional local acceleration at the (moving) core section increases to a maximum of around 1.0 at the instant for occurrence of the hydraulic jump with abrupt rise of the free surface; and then decreases to zero at time for transformation of the curling jet into the projecting jet. The results exhibit that the external stream of retreating flow is accelerated temporally in the offshore direction for the interval between the time for the end of run-up motion and that for the formation of projecting jet. However, for later time interval up to generation of the two-phase flow field, the non-dimensional local acceleration in the offshore direction varies from zero to a negative maximum of −2.117 at the moment for the projecting jet heading downward before the impingement. It then decreases in magnitude continuously. The trend reveals that the external stream is decelerated temporally in the offshore direction for this later time interval. Further, at the section of the incipient flow separation, the non-dimensional pressure gradient (also with reference to the gravity acceleration) in the offshore direction increases from 0.225 for the time at which the run-up motion ends to 0.721 for the instant at which the incipient flow separation takes place. The trend highlights the external stream being under increasing adverse pressure gradient and more decelerated spatially with the increasing time, thus resulting in occurrence of the incipient flow separation. Afterwards, the value of the non-dimensional pressure gradient keeps increasing and eventually reaches a positive maximum of 2.011 and then decreases consecutively until the two-phase flow field is generated. In addition, due to the influence of acceleration of the external stream in the offshore direction, the non-dimensional vorticity of primary vortex core increases with increasing time up to the moment for occurrence of the projecting jet. Nevertheless, the non-dimensional vorticity of primary vortex core keeps decreasing with increasing time T for the later time interval due to the influence of deceleration of the external stream in the offshore direction. Finally, considerably large magnitudes of the non-dimensional accelerations and pressure gradient greater than 1.5 taking place at two non-dimensional times are worthy of noting. The negative maximum value of the non-dimensional convective acceleration equal to −2.005 appears at the instant for the occurrence of hydraulic jump. In addition, the negative maximum values of the non-dimensional local acceleration, total acceleration and pressure gradient unexpectedly as high as −2.117, −1.694 and 2.011, respectively, appear simultaneously at time for the projecting jet heading towards the retreating free surface. Under such a situation, the external stream of retreating flow is highly decelerated in the offshore direction under the fairly large adverse pressure gradient, thus forcing the retreating flow to move upwards rapidly. Meanwhile, the non-dimensional local acceleration in the vertical direction is surprisingly found to be 3.37. The result strongly reconfirms the evident rise of the free surface in the vicinity of the core section and reveals very rapid change from negative, via nearly zero, to positive vertical velocity.
Period-Life of a Branching Process with Migration and Continuous Time
