ON THE RIGOROUS MATHEMATICAL DERIVATION FOR THE VISCOUS PRIMITIVE EQUATIONS WITH DENSITY STRATIFICATION

In this paper, we rigorously derive the governing equations describing the motion of a stable stratified fluid, from the mathematical point of view. In particular, we prove that the scaled Boussinesq equations strongly converge to the viscous primitive equations with density stratification as the aspect ratio goes to zero, and the rate of convergence is of the same order as the aspect ratio. Moreover, in order to obtain this convergence result, we also establish the global well-posedness of strong solutions to the viscous primitive equations with density stratification.

Baroclinic pressure gradient difference schemes of subtracting the local averaged density stratification in sigma coordinates models

Much has been written of the error in computing the baroclinic pressure gradient (BPG) with sigma coordinates in ocean or atmospheric numerical models. The usual way to reduce the error is to subtract area-averaged density stratification of the whole computation region. But if there is great difference between the area-averaged and the local averaged density stratification, the error will be obvious. An example is given to show that the error from this method may be larger than that from no correction sometimes. The definition of local area is put forward. Then, four improved BPG difference schemes of subtracting the local averaged density stratification are designed to reduce the error. Two of them are for diagnostic calculation (density field is fixed), and the others are for prognostic calculation (density field is not fixed). The results show that the errors from these schemes all significantly decrease.

A Study on Improvement of RANS Analysis for Erosion of Density Stratified Layer of Multicomponent Gas by Buoyant Jet in a Containment Vessel

The analysis on a density stratification layer consisting of multiple gases in the reactor containment vessel is important for the safety assessment of sever accidents. The JAEA （Japan Atomic Energy Agency） has started the project on the containment thermal hydraulics. We carried out CFD （computational fluid dynamics） analyses in order to investigate the erosion of the density stratification layer by a vertical buoyant jet under this project. We used the RANS （Reynolds averaged numerical simulation） and LES （large eddy simulation） models to analyze the erosion of a density stratification layer by a vertical buoyant jet in a small vessel which represents a containment vessel. This numerical study calculates the turbulent mixing of a two-component （air and helium） gas mixture. The turbulence models used for the RANS analyses are two types of k-ε models. The first model is the low Reynolds number k-ε model developed by Launder and Sharma. The second model is modified from the first model in order to accurately consider the turbulent production and damping in a stratification layer. The results indicated while the erosion rate calculated by the low-Re k-ε model was much faster than that of the LES model, the modified k-ε model could calculate the erosion rate similar to the LES result.

潜艇运动激发的表面波

A moving submarine can generate internal waves,as well as extremely small free surface waves,in a fluid with density stratification.In this study,the internal and free surface wave wakes caused by a moving submarine in two layers of constant density fluid were studied numerically using the commercial software STAR-CCM+.The realizable k–εturbulence model was used to solve the Reynolds-averaged Navier–Stokes equation,and the volume of the fluid method was used to monitor the fluctuations of the internal interface and free surface.Different cases of a moving submarine with different cruising speeds and relative diving depths were studied.Results showed that the maximum fluctuation amplitude of the free surface increased as the speed of the submarine increased;however,the maximum fluctuation amplitude of the internal interface first decreased and then increased.When the submarine moved at the maximum cruising speed,the maximum fluctuation amplitude of the free surface decreased as the diving depth increased,while the wavelength of the free surface wave was basically the same.If the submarine moved at the minimum cruising speed,then the wave elevation in the free surface was extremely small,but the internal surface had obviously large-amplitude internal waves,and the relative diving depth had a great influence on internal waves.

Observations and modelling of the travel time delay and leading negative phase of the 16 September 2015 Illapel,Chile tsunami

The systematic discrepancies in both tsunami arrival time and leading negative phase(LNP)were identified for the recent transoceanic tsunami on 16 September 2015 in Illapel,Chile by examining the wave characteristics from the tsunami records at 21 Deep-ocean Assessment and Reporting of Tsunami(DART)sites and 29 coastal tide gauge stations.The results revealed systematic travel time delay of as much as 22 min(approximately 1.7%of the total travel time)relative to the simulated long waves from the 2015 Chilean tsunami.The delay discrepancy was found to increase with travel time.It was difficult to identify the LNP from the near-shore observation system due to the strong background noise,but the initial negative phase feature became more obvious as the tsunami propagated away from the source area in the deep ocean.We determined that the LNP for the Chilean tsunami had an average duration of 33 min,which was close to the dominant period of the tsunami source.Most of the amplitude ratios to the first elevation phase were approximately 40%,with the largest equivalent to the first positive phase amplitude.We performed numerical analyses by applying the corrected long wave model,which accounted for the effects of seawater density stratification due to compressibility,self-attraction and loading(SAL)of the earth,and wave dispersion compared with observed tsunami waveforms.We attempted to accurately calculate the arrival time and LNP,and to understand how much of a role the physical mechanism played in the discrepancies for the moderate transoceanic tsunami event.The mainly focus of the study is to quantitatively evaluate the contribution of each secondary physical effect to the systematic discrepancies using the corrected shallow water model.Taking all of these effects into consideration,our results demonstrated good agreement between the observed and simulated waveforms.We can conclude that the corrected shallow water model can reduce the tsunami propagation speed and reproduce the LNP,which is observed for tsunamis that have propagated over long distances frequently.The travel time delay between the observed and corrected simulated waveforms is reduced to<8 min and the amplitude discrepancy between them was also markedly diminished.The incorporated effects amounted to approximately 78%of the travel time delay correction,with seawater density stratification,SAL,and Boussinesq dispersion contributing approximately 39%,21%,and 18%,respectively.The simulated results showed that the elastic loading and Boussinesq dispersion not only affected travel time but also changed the simulated waveforms for this event.In contrast,the seawater stratification only reduced the tsunami speed,whereas the earth’s elasticity loading was responsible for LNP due to the depression of the seafloor surrounding additional tsunami loading at far-field stations.This study revealed that the traditional shallow water model has inherent defects in estimating tsunami arrival,and the leading negative phase of a tsunami is a typical recognizable feature of a moderately strong transoceanic tsunami.These results also support previous theory and can help to explain the observed discrepancies.

Transient flexural-and capillary-gravity waves due to disturbances in two-layer density-stratified fluid

Generation of the transient flexural- and capillary-gravity waves by impulsive disturbances in a two-layer fluid is investi- gated analytically. The upper fluid is covered by a thin elastic plate or by an inertial surface with the capillary effect. The density of each of the two immiscible layers is constant. The fluids are assumed to be inviscid and incompressible and the motion be irrotational. A point force on the surface and simple mass sources in the upper and lower fluid layers are considered. A linear system is establi- shed within the framework of potential theory. The integral solutions for the surface and interracial waves are obtained by means of the Laplace-Fourier transform. A new representation for the dispersion relation of flexural- and capillary-gravity waves in a two- layer fluid is derived. The asymptotic representations of the wave motions are derived for large time with a fixed distance-to-time ratio with the Stokes and Scorer methods of stationary phase. It is shown that there are two different modes, namely the surface and interracial wave modes. The wave systems observed depend on the relation between the observer＇s moving speed and the intrinsic minimal and maximal group velocities.

The sloshing of stratified liquid in a two-dimensional rectangular tank

The sloshing of inviscid liquid of stratified density in a rectangular tank is analyzed.As the flow is no longer irrotional,the governing equation is found to be quite different from the Laplace equation used for the liquid of constant density.In particular it contains terms of mixed temporal and spatial derivatives.The problem is solved based on the variable separation method and Laplace transform for the constant Vaisala-Brunt frequency.It is found that the stratification of density may have small effects on those natural frequencies associated with the constant density,but many new natural frequencies have appeared as a result of its effect.

A general two-phase mixture model for sediment-laden flow in open channel

This work extends the sediment-laden mixture model with consideration of the turbulence damping and particle wake effects under the framework of improved efficiency and accuracy.The mixture model consists of the continuity and momentum equations for the sediment-laden mixture,and the continuity equation for the sediment.A theoretical formula is derived for the relative velocity between the water and sediment phases,with consideration of the effects of the pressure gradient,the shear stress and the lift force.A modified expression of the particle wake effect,inducing the local turbulence enhancement around the sediment particle,is employed to improve the turbulent diffusion of the coarse sediment.The k_(m)-ε_(m) model is proposed to close the mixture turbulence,with the turbulence damping effect due to the high sediment concentration expressed by the density-stratification term without an empirical parameter.The k_(m)-ε_(m) turbulence model requires smaller computational work and offers better results than an empirical density-stratification turbulence model in high sediment concentration cases.Consequently,with the proposed mixture model,the sediment transport in the open channel under a wide range of sediment sizes and concentrations can be revealed with the results in good agreement with experimental data for the velocity,the sediment concentration and the turbulent kinetic energy.