Pressure Boussinesq equation

Development of numerical models for time-domain wave transformation has been tried by many researchers, but random wave-breaking process has not been well reproduced in these models. One of the exceptions is the Boussinesq-type model developed by Hirayama et and Hirayama and Hiraishi, who employed the breaking criterion of the vertical pressure gradient by Nadaoka et alJ They raised the threshold gradient from 0 to 0.5 to compensate the insufficiency in numerical accuracy due to the features of weak nonlinearity inherent to the Boussinesq equation. They have succeeded in reproducing the pc /variation across the surf zone. 

The BBO equation delineates a sphere in linear motion at low Reynolds numbers [Basset, 1888 Boussinesq, 1903 Oseen, 1927], For a spherical particle, the BBO equation, based upon Eq. (3.37) with the replacement of the buoyancy force with the pressure gradient force, can be expressed as [Soo, 1990]... 

Circulation models are based on the equations of motion of the geophysical fluid dynamics and on the thermodynamics of seawater. The model area is divided into finite size grid cells. The state of the ocean is described by the velocity, temperature, and salinity in each grid cell, and its time evolution can be computed from the three-dimensional model equations. To reduce the computational demands, the model ocean is usually incompressible and the vertical acceleration is neglected, the latter assumption is known as hydrostatic approximation. This removes sound waves in the ocean from the model solution. In the horizontal equations, the Boussinesq approximation is applied and small density changes are ignored except in the horizontal pressure gradient terms. This implies that such models conserve... 

Note that the turbulent viscosity parameter has an empirical origin. In connection with a qualitative analysis of pressure drop measurements Boussinesq [19] introduced some apparent internal friction forces, which were assumed to be proportional to the strain rate ([20], p 8), to fit the data. To explain these observations Boussinesq proceeded to derive the same basic equations of motion as had others before him, but he specifically considered the molecular viscosity coefficient to be a function of the state of flow and not only on the system properties [135]. It follows that the turbulent viscosity concept (frequently referred to as the Boussinesq hypothesis in the CFD literature) represents an empirical first attempt to account for turbulence effects by increasing the viscosity coefficient in an empirical manner fitting experimental data. Moreover, at the time Boussinesq [19] [20] was apparently not aware of the Reynolds averaging procedure that was published 18 years after the first report by Boussinesq [19] on the apparent viscosity parameter. 

We have already noted that the general class of flows driven by buoyancy forces that are created because the density is nonuniform is known as natural convection. If we examine the Boussinesq approximation of the Navier-Stokes equations, (12-170), we can see that there are actually two types of natural convection problems. In the first, we assume that a fluid of ambient temperature 71, is heated at a bounding surface to a higher temperature I. This will produce a nonuniform temperature distribution in the contiguous fluid, and thus a nonuniform density distribution too. Let us suppose that the heated surface is everywhere horizontal. Then there is a steady-state solution of (12-170) with u = 0, and the body-force terms balanced by a modification to the hydrostatic pressure distribution, such that... 

Equations (16.26), (16.27), and (16.29) can be simplified for a shallow atmospheric layer next to the ground using the Boussinesq approximations. The conditions of their validity have been examined by Spiegel and Veronis (1960), Calder (1968), and Dutton (1976). The fundamental idea is of these approximations is to first express the equilibrium profiles of pressure, density, and temperature as functions of x3 only as follows ... 

Thus, in order to solve the hydrodynamic problem of liquid motion in view of the change of 2 at the interface, we should first And out the distribution of substance concentration, temperature and electric charge over the surface. These distributions, in turn, are influenced by the distribution of hydrodynamic parameters. Therefore the solution of this problem requires utilization of conservation laws - the equations of mass, momentum, energy, and electric charge conservation with the appropriate boundary conditions that represent the balance of forces at the interface the equality of tangential forces and the jump in normal forces which equals the capillary pressure. In the case of Boussinesq model, it is necessary to know the surface viscosity of the layer. From now on, we are going to neglect the surface viscosity. 

The tidal current fields at the project sea area are simulated by the three-dimensional numerical model (MIKE3 FM) which developed by Danish Hydraulics Research Institute. The model is based on the solution of three-dimensional incompressible Reynolds Navier-Stokes equations, subject to the assumption of Boussinesq and hydrostatic pressure. 

The solution to this equation is more difficult than the solution to Eq. 7.194. The case of pure pressure flow was first solved by Boussinesq [130] in 1868. The solution to the combined drag and pressure flow was first published in 1922 [98] the authorship of this publication remains a question. Since the 1922 publication, numerous workers have presented solutions to this problem. Meskat [131] reviewed various solutions and demonstrated that they were equivalent. The velocity profile resulting from the drag flow can be written as ... 

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