Boussinesq

Bourdon tube Boussinesq assumption Bovatec Bovine albumin Bovine brain... 

Eddy Viscosity Models. A large number of closure models are based on the Boussinesq concept of eddy viscosity ... 

Closure Models Many closure models have been proposed. A few of the more important ones are introduced here. Many employ the Boussinesq approximation, simphfied here for incompressible flow, which treats the Reynolds stresses as analogous to viscous stresses, introducing a scalar quantity called the turbulent or eddy viscosity... 

Johnson, K.L., Contact Mechanics. Cambridge University Press, Cambridge, 1985. Boussinesq, J., Theory of Elasticity. McGraw Hill, New York, 1945, p. 338. 

The foundations for the seienee of modem partiele adhesion were laid in the 19th eentury with the work of Hertz, Boussinesq, and Cerruti. 

Boussinesq and Cerruti made use of potential theory for the solution of contact problems at the surface of an elastic half space. One of the most important results is the solution to the displacement associated with a concentrated normal point load P applied to the surface of an elastic half space. As presented in Johnson [49]... 

Implicit in all these solutions is the fact that, when two spherical indentors are made to approach one another, the resulting deformed surface is also spherical and is intermediate in curvature between the shape of the two surfaces. Hertz [27] recognized this concept and used it in the development of his theory, yet the concept is a natural consequence of the superposition method based on Boussinesq and Cerutti s formalisms for integration of points loads. A corollary to this concept is that the displacements are additive so that the compliances can be added for materials of differing elastic properties producing the following expressions common to many solutions... 

For other die geometries it is necessary to use the appropriate form of equation (4.12). The equations for a capillaiy and a slit die are derived in Chapter 5. For other geometries it is possible to use the empirical equation which was developed by Boussinesq. This has the form... 

According to Boussinesq (B12) the velocity of drop fall was expressed as... 

It is then assumed that due to this separation in scales, the so-called subgrid scale (SGS) modeling is largely geometry independent because of the universal behavior of turbulence at the small scales. The SGS eddies are therefore more close to the ideal concept of isotropy (according to which the intensity of the fluctuations and their length scale are independent of direction) and, hence, are more susceptible to the application of Boussinesq s concept of turbulent viscosity (see page 163). 

Usually, however, the stresses are modeled with the help of a single turbulent viscosity coefficient that presumes isotropic turbulent transport. In the RANS-approach, a turbulent or eddy viscosity coefficient, vt, covers the momentum transport by the full spectrum of turbulent scales (eddies). Frisch (1995) recollects that as early as 1870 Boussinesq stressed turbulence greatly increases viscosity and proposed an expression for the eddy viscosity. The eventual set of equations runs as... 

In some way, introducing an increased particle drag by means of Eq. (17) resembles the earlier proposal raised by Bakker and Van den Akker (1994b) to increase viscosity in the particle Reynolds number due to turbulence (in agreement with the very old conclusion due to Boussinesq, see Frisch, 1995) with the view of increasing the particle drag coefficient and eventually the bubble holdup in the vessel. Lane et al. (2000) compared the two approaches for an aerated stirred vessel and found neither proposal to yield a correct spatial gas distribution. 

The velocities and other solution variables are now represented by Reynolds-averaged values, and the effects of turbulence are represented by the Reynolds stresses, (—pu pTl) that are modeled by the Boussinesq hypothesis ... 

The Reynolds-averaged approach is widely used for engineering calculations, and typically includes models such as Spalart-Allmaras, k-e and its variants, k-co, and the Reynolds stress model (RSM). The Boussinesq hypothesis, which assumes pt to be an isotropic scalar quantity, is used in the Spalart-Allmaras model, the k-s models, and the k-co models. The advantage of this approach is the relatively low computational cost associated with the computation of the turbulent viscosity, fit. For the Spalart-Allmaras model, one additional transport equation representing turbulent viscosity is solved. In the case of the k-e and k-co models, two additional transport equations for the turbulence kinetic energy, k, and either the turbulence dissipation rate, s, or the specific dissipation rate, co, are solved, and pt is computed as a function of k and either e or co. Alternatively, in the RSM approach, transport equations can be solved for each of the terms in the Reynolds stress tensor. An additional scale-determining equation (usually for s) is also required. This means that seven additional transport equations must be solved in 3D flows. 

The plume is Boussinesq, or of constant density px except in the body force term. 

Later we shall include combustion and flame radiation effects, but we will still maintain all of assumptions 2 to 5 above. The top-hat profile and Boussinesq assumptions serve only to simplify our mathematics, while retaining the basic physics of the problem. However, since the theory can only be taken so far before experimental data must be relied on for its missing pieces, the degree of these simplifications should not reduce the generality of the results. We shall use the following conservation equations in control volume form for a fixed CV and for steady state conditions ... 

The mass flow rate in terms of the Boussinesq top-hat assumption is given by... 

The change in density or temperature is small. (This completes the full Boussinesq model.)... 

Boundary managing, in R D, 21 619-620 Boundary spanning, in R D, 21 619 Bound chloride formation, 10 358 Bound moisture, 9 96 Bourdon tube, 20 647-649 Boussinesq approximation, 11 779 Boutique fuels, 12 419 Bovatec, 20 136 Bovine hemoglobin, 4 125 Bovine insulin, 3 817 Bovine serum albumin (BSA), 20 573 properties of standard, 3 836t Bovine somatotropin (BST), 10 871 Bovine spongiform encephalitis/... 

We can illustrate the salient features of convective dispersal by choosing a simple velocity distribution in a rectangular convection cell (0associated with the onset of Benard instability in the conditions of Boussinesq approximations (e.g., Turcotte and Schubert, 1982). Let us make the calculation for the so-called free-slip conditions, which permit free movement along the boundaries, both vertical and horizontal, such as a convection cell which would be limited by no rigid boundary. From Turcotte and Schubert (1982), we take the velocity field to be... 

The z-direction velocity that is induced by a pressure gradient was first solved by Boussinesq [3]. Starting with Eq. A7.7 the pressure flow Eq. A7.19 can be developed as follows ... 

Boussinesq 255,744 Boyd 353 Bozzelll 238,374 Bremner 377 Brittin 374,523 Brizzolara 662 Brown 661,671 Broyer 92, 139, 142 Bruin 257 Broker 257, 258 Buchelll 661, 671 Buck 649... 

The derivation of the mixture-balance laws has been given by Chapman and Cowling for a binary mixture. Its generalization to multicomponent mixtures, as in Equation 5-1, uses a determination of the invariance of the Boltzmann equation. This development has been detailed by Hirschfelderet These derivations were summarized in the notes of Theodore von Karmin s Sorbonne lectures given in 1951-1952, and the results of his summaries were stated in Pinner s monograph. For turbulent flow, the species-balance equation can be represented in the Boussinesq approximation as ... 

A variety of statistical models are available for predictions of multiphase turbulent flows [85]. A large number of the application oriented investigations are based on the Eulerian description utilizing turbulence closures for both the dispersed and the carrier phases. The closure schemes for the carrier phase are mostly limited to Boussinesq type approximations in conjunction with modified forms of the conventional k-e model [87]. The models for the dispersed phase are typically via the Hinze-Tchen algebraic relation [88] which relates the eddy viscosity of the dispersed phase to that of the carrier phase. While the simplicity of this model has promoted its use, its nonuniversality has been widely recognized [88]. 

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