Boussinesq equations

As we have stated, the Boussinesq equations are based on the assumption that the material dealt with is elastic and isotropic. Of course, this... 

Before starting to discuss (116), we make an observation. The fast time evolution (116) is also observed in driven systems that cannot be described on the level Ath- For example, let us consider the Rayleigh-Benard system (i.e., a horizontal layer of a fluid heated from below). It is well established experimentally that this externally driven system does not reach thermodynamic equilibrium states but its behavior is well described on the level of fluid mechanics (by Boussinesq equations). This means that if we describe it on a more microscopic level, say the level of kinetic theory, then we shall observe the approach to the level of fluid mechanics. Consequently, the comments that we shall make below about (116) apply also to driven systems and to other types of systems that are prevented from reaching thermodynamical equilibrium states (as, e.g., glasses where internal constraints prevent the approach to Ath)-... 

Equations have been derived to define the vertical and shear stresses at any depth below and any radial distance from a point load. The best known and probably the most used are the Boussinesq equations, which assume an elastic, isentropic material, a level surface and an infinite surface extension in all directions. Although these conditions cannot be met by soils, the equation for vertical stress is used with reasonable accuracy with soils whose stress-strain relationship is linear. This normally precludes the use of the equation for stresses approaching failure. In its most useful form the equation reduces to ... 

Equations (12—168)—(12 170) are known as the Boussinesq equations of motion and will form the basis for the natural convection stability analyses in this chapter. In fact, the Boussinesq approximation has been used in much of the published theoretical work on natural convection flows. Although one should expect quantitative deviations from the Boussinesq predictions for systems in which the temperature differences are large (greater than 10°C-20°C), it is likely that the Boussinesq equations remain qualitatively useful over a considerably larger range of temperature differences. In any case, although the Boussinesq equations represent a very substantial simplification of the exact equations, the essential property of coupling between the thermal and velocity fields is preserved, and, even in the Boussinesq approximation, the solution of natural convection problems is more complicated than the forced convection heat transfer problems that we encountered earlier. 

The Rayleigh Benard configuration is sketched in Fig. 12 4. We assume that the fluid is between two infinite plane surfaces that are separated by a distance d. The lower surface is at a constant temperature i and the upper one is at a lower temperature T0. Our starting point is the time-dependent Boussinesq equations, (12-160), (12 169), and (12-170), but now in dimensionless form,... 

The governing equations for the disturbance variables are just the Boussinesq equations, (12 172), (12 176), and (12 177). However, in view ofthe fact that e 1, these equations... 

So far, the general point of departure for the nonlinear analyses attempted has been not the set of equations of motion in their complete form, but rather the set that results by neglecting the temperature dependence of all fluid properties except density, and also the dissipation of kinetic energy. This is known as the Boussinesq approximation. Using the Boussinesq equations, Woronetz (Wl) was able in 1934 to obtain expressions for the velocity perturbations in a fluid confined between coaxial rotating cylinders or spheres which were maintained at different temperatures. Volkovisky (V3) used the... 

Boussinesq equations in 1939 to obtain an expression for the velocity along a filament of fluid rising above a point or line heat source placed at the bottom of a fluid layer. 

Gor kov (G2), in 1957, and Malkus and Veronis (Ml), in 1958, succeeded in calculating the steady-state amplitude, as a function of the Rayleigh number, for a given cellular pattern and wavelength using the time-independent Boussinesq equations. A perturbation technique was employed in which the variables were expanded in terms of an amplitude parameter, e, such as... 

In 1960, Stuart and Watson (cf. S12) examined the same problem using the time-dependent Boussinesq equations and obtained a solution which converged to that of Malkus and Veronis as time approached infinity, thus demonstrating that under unstable conditions a differential disturbance can indeed lead to finite amplitude steady convection. 

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. 

For trans-oceanic propagation of tsunamis, the nonlinearity of waves can be neglected because the free surface displacement is much smaller in comparison with the water depth. However, the dispersion effect of tsunami waves should be considered properly for the far-held tsunamis. Thus, the following linear Boussinesq equations can be used as governing equations. 

To develop an efficient and relatively accurate numerical model for the propagation of dispersive tsunamis over slowly varying topography, Yoon et al derived the following linear Boussinesq-type wave equation (LBTWE) from the linear Boussinesq equations (10.1)-(10.3) as... 

In order to test the applicability of the present model, the propagation of tsunamis is first simulated with an initial Gaussian shape of water surface for the case of various constant water depths, and the computed free surface displacements are compared with the analytical solutions of the linear Boussinesq equations. The initial free surface profile and the velocity of free surface movement are described by (10.10) and (10.11), respectively, as... 

The analytical solution of the linear Boussinesq equations is given by Carrier as... 

O. Nwogu, Alternative form of Boussinesq equations for nearshore wave propagation, J. Waterway, Port, Coastal Ocean Eng. 119(6), 618-638 (1993). 

Boussinesq equations numerical integrator (Karambas and Koutitas ). It is a phase resolving model representing both wave and currents at the same time. 

As consequence of (4.1) the buoyancy term in equation (2.1 lb) is negligible for a weakly expansible liquid, and as consequence, in place of Boussinesq equations (3.5a,b,c), we derive from the dominant dimensionless equations (2.11a,b,c) the following incompressible model equations ... 

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