Approximation, Boussinesq

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... 

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 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. The Boussinesq Approximation Analyses of time-steady free convection usually assume that ... 

The adequaey of the Boussinesq approximations has been tested for natural eonveetion from a vertical plate (S31) and for mixed convection from a hori-... 

This example is motivated by a natural-convection problem (Fig. 3.13) where the body-force term is caused by slight density variations (often caused by temperature variations). Using the so-called Boussinesq approximation, the flow may be considered incompressible, but with the buoyant forces depending on slight density variations. 

Convection is accounted for by the Navier-Stokes equation in the Boussinesq approximation... 

The above discussion indicates that, in the analysis of free convective flows, the fluid properties can be assumed constant except for the density change with temperature which gives rise to the buoyancy force. This is, basically, the "Boussinesq approximation [6]. This approximation will be adopted in all the analyses given in this chapter. 

When the Boussinesq approximation discussed in the previous section is adopted, the... 

The Boussinesq approximation has, of course, been used in deriving the above equation. 

It will be assumed here that the flow in the enclosure is steady and remains laminar. It will also be assumed that the flow is two-dimensional. With these assumptions and using the Boussinesq approximation that was discussed earlier, the equations governing the flow in the enclosure are ... 

The boundary layer equations can, as previously discussed, only be applied to flows in which the Reynolds number is relatively large and in which there is no significant areas of reversed flow. This, in particular, severely limits the applicability of these equations in situations involving opposing flow. When these conditions are not satisfied, the solution must be obtained using the full governing equations. For example, if the flow can be assumed to be two-dimensional and if the Boussinesq approximations are applicable, the equations governing the flow are Eqs. (9.5) to (9.7). If the x-axis is vertical, these equations become ... 

Attention will be restricted to fully developed flow, i.e., to flow in which all the flow variables except temperature are not changing with distance, c. along the pipe. It will also be assumed that the wall heat flux is axially constant and the wall temperature is constant around the periphery although it of course varies with axial distance. Using the coordinate system shown in Fig. 9.31, the equations governing the flow are, if the Boussinesq approximation is adopted and if viscous dissipation... 

The turbulent kinetic energy equation was derived in Chapter 5 using the momentum equations and assuming buoyancy force effects were negligible. Re-derive this equation starting with momentum equations in which the buoyancy terms are retained. Assume a vertically upward flow and use the Boussinesq approximation. 

Using the Darcy flow model and the Boussinesq approximation, the governing equations are ... 

When Boussinesq approximation is adopted in full conservation equations, it is noted that the effect of buoyancy force appears in terms of GrjRe where Gr is the Grashof number and Re is the Reynolds number defined in terms of appropriate length, velocity and temperature scales. However, Leal et al. (1973) and Sparrow Minkowycz (1962) have shown that the equivalent buoyancy parameter with the boundary layer assump-... 

We consider the laminar two-dimensional motion of fluid past a hot semi-infinite plate, with the free stream velocity and temperature denoted by, Uoo and Too- We will focus our attention on the top of the plate, for which the temperature is T - that is greater than Too, while assuming the leading edge of the plate as the stagnation point. Governing equations are written in dimensional form (indicated by the quantities with asterisk), along with the Boussinesq approximation to represent the buoyancy effect,... 

The mean flow equations are obtained by invoking boundary layer approximation to the above conservation equations. For the two-dimensional steady incompressible flow with constant properties and Boussinesq approximation, the non-dimensional equations are written in a Cartesian coordinate system, fixed at the leading edge of the semi-infinite horizontal flat plate as,... 

The other canonical flow geometry considered in the first part consists of bluff-body flow instability dealt in chapter 5. This introduces the flow past a cylinder that actually suffers linear temporal instability moderated by nonlinear stabilization. This flow is different from that is discussed primarily in chapters 2 to 4, where the linear instability is via spatial growth. Also, for such flows nonlinearity leads to further destabilization, whereas for the flow past a cylinder, the nonlinearity stabilizes the linear instability and takes the flow to another equilibrium flow. In chapter 6, the effects of heat transfer via the restrictive condition of Boussinesq approximation for the canonical flow past flat plates is studied. This problem has been solved... 

The numerical convection model that is used to illustrate the visualization and quantification of mixing (Figures 1 -10) is based on the solution of the equations governing convection in the Earth s mantle, assuming that the mantle can be described as an anelastic and weakly compressible fluid at infinite Prandtl number. Under the extended Boussinesq approximation, we can write the equation of motion as... 

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