Steady thermocapillary-buoyant convection in a shallow annular pool. Part 1:Single layer fluid

This paper examines the steady thermocapillarybuoyant convection in a shallow annular pool subjected to a radial temperature gradient. A matched asymptotic theory is used to obtain the asymptotic solutions of the flow and thermal fields in the case of small aspect ratios,which is defined as the ratio of the layer thickness to the gap width. The flow domain is divided into the core region away from the cylinder walls and two end regions near each cylinder wall. Asymptotic solutions are obtained in the core region by solving the core and end flows separately and then joining them through matched asymptotic expansions. For the system of silicon melt,the asymptotic solutions are compared with the results of numerical simulations. It is found that the two kinds of solutions have a good agreement in the core region for a small aspect ratio. With the increase of aspect ratio,the applicability of the present asymptotic solutions decreases gradually.

Determination of the height of Mount Everest using the shallow layer method

Shallow layer method(SLM)based on the definition of the geoid can determine the gravity field inside the shallow layer.In this study,the orthometric height of Mount Everest(HME)is calculated based on SLM,in which the key is to construct the shallow layer model.The top and bottom boundaries of the shallow layer model are the natural surface of the Earth and the surface at a certain depth below the reference geoid,respectively.The model-combined strategies to determine the geoid undulation(N)based on SLM are applied to calculate the HME by two approaches:(1)direct calculation by combining N and geodetic height(h);(2)calculation by the segment summation approach(SSA)using the gravity field inside the shallow layer.On December 8,2020,the Chinese and Nepalese governments announced an authoritative value of 8848.86 m,which is referred to a geoid determined by the International Height Reference System(IHRS)(i.e.,the geopotential is 62636853.4 m^(2) s^(-2)).Here,our results(combined strategies(1)EGM2008 and CRUST1.0,(2)EGM2008 and CRUST2.0,(3)EIGEN-6 C4 and CRUST1.0,and(4)EIGEN-6 C4 and CRUST2.0)are referred to the geoid defined by WGS84(i.e.,the geopotential is 62636851.7 m^(2) s^(-2)).The differences between our results and the authoritative value(8848.86 m)are 0.448 m,-0.009 m,-0.295 m,and -0.741 m by the first approach,and 0.539 m,0.083 m,-0.214 m,and -0.647 m by the second approach.When the reference surface WGS84 geoid is converted to the IHRS geoid,the differences are 0.620 m,0.163 m,-0.123 m,and -0.569 m by the first approach,and0.711 m,0.225 m,-0.042 m,and -0.475 m by the second approach.

THE CAUCHY PROBLEM FOR THE TWO LAYER VISCOUS SHALLOW WATER EQUATIONS

In this paper,the Cauchy problem for the two layer viscous shallow water equations is investigated with third-order surface-tension terms and a low regularity assumption on the initial data.The global existence and uniqueness of the strong solution in a hybrid Besov space are proved by using the Littlewood-Paley decomposition and Friedrichs'regularization method.

Experimental Study of Turbulent Wake Behind a Sine Shaped Island in a Shallow-Water Layer

A series of experiments is conducted to study shallow-water flow in the wake of a sine shaped island. Digital particle imaging velocimetry (DPIV) is used to measure velocities in the turbulent wake behind a sine shaped island for different characteristic coefficients S. Flow streamlines are given for the wake flows. The measured results show that the characteristic coefficient S is uniquely related to the flow pattern around a sine shaped island in a shallow water layer. An S value of approximately 0.20 is the critical value for transition from a vortex street to unsteady flow and a value of approximately 0.40 is the critical value for transition from unsteady flow to steady flow.