Model Boussinesq

Surface-tension forces act in a direction tangent to the interface. Boussinesq (B6) assumed, for a surface undergoing dilatation, that there must also be another force acting normal to the interface. Utilizing the same assumptions as Hadamard, he arrived at a Stokes-law correction factor of 

The derivations of Hadamard and of Boussinesq are based on a model involving laminar flow of both drop and field fluids. Inertial forces are deemed negligible, and viscous forces dominant. The upper limit for the application of such equations is generally thought of as Re 1. We are here considering only the gross effect on the terminal velocity of a drop in a medium of infinite extent. The internal circulation will be discussed in a subsequent section. 

Most drop situations in extraction are far above the upper limit of application of the preceding equations. A drop moving through a liquid at a velocity such that the viscous forces could be termed negligible can not exist. It will break up into two or more smaller droplets (HIO, K5). Most real situations involve both viscous and inertial terms, and the Navier-Stokes equations can not then be solved. 

Correlations have been developed to relate terminal velocity, drop size, peak size, peak velocity, and maximum drop size to the physical properties of the system (El, HIO, K5). Some of the accumulated information in these areas is given below. 

At steady-state terminal velocity, the acceleration term in the force balance (B9) on a moving submerged body is zero, and the balance may be written so that gravity forces are balanced by the sum of buoyancy and resistance forces. 

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

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. 

K. Hirayama and T. Hiraishi, A Boussinesq model for wave breaking and run-up in a coastal zone, ID, Fifth Int. Symp. Ocean Waves Measurement Analysis, WAVES 2005, CD-ROM, No. 151 (2005), pp. 1-10. 

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... 

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... 

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. 

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]. 

Boussinesq (B4) proposed that the lack of internal circulation in bubbles and drops is due to an interfacial monolayer which acts as a viscous membrane. A constitutive equation involving two parameters, surface shear viscosity and surface dilational viscosity, in addition to surface tension, was proposed for the interface. This model, commonly called the Newtonian surface fluid model (W2), has been extended by Scriven (S3). Boussinesq obtained an exact solution to the creeping flow equations, analogous to the Hadamard-Rybczinski result but with surface viscosity included. The resulting terminal velocity is... 

Pure pressure flow was first formulated and solved by Joseph Boussinesq in 1868, and combined pressure and drag flow in 1922 by Rowell and Finlayson (19) in the first mathematical model of screw-type viscous pumps. The detailed solution by the method of separation of variables is given elsewhere (17c), and the resulting velocity profile is given by... 

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

The third term of Eq. (3) contains eji, which is generally modeled in terms of turbulent dispersion in a manner analogous to the well-known gradient hypothesis of Boussinesq, as proportional to the gradient of holdup in the z direction, the constant of proportionality being referred to as the turbulent dispersion coefficient ... 

In the previous section, stability criteria were obtained for gas-hquid bubble columns, gas-solid fluidized beds, liquid-sohd fluidized beds, and three-phase fluidized beds. Before we begin the review of previous work, let us summarize the parameters that are important for the fluid mechanical description of multiphase systems. The first and foremost is the dispersion coefficient. During the derivation of equations of continuity and motion for multiphase turbulent dispersions, correlation terms such as esv appeared [Eqs. (3) and (10)]. These terms were modeled according to the Boussinesq hypothesis [Eq. (4)], and thus the dispersion coefficients for the sohd phase and hquid phase appear in the final forms of equation of continuity and motion [Eqs. (5), (6), (14), and (15)]. However, for the creeping flow regime, the dispersion term is obviously not important. 

All of these models require some form of empirical input information, which implies that they are not general applicable to any type of turbulent flow problem. However, in general it can be stated that the most complex models such as the ASM and RSM models offer the greatest predictive power. Many of the older turbulence models are based on Boussinesq s (1877) eddy-viscosity concept, which assumes that, in analogy with the viscous stresses in laminar flows, the Reynolds stresses are proportional to the gradients of the time-averaged velocity components ... 

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... 

The k,E-model is based on a first order turbulence model closure according to Boussinesq. In analogy to laminar flows, the Reynolds stresses are assumed to be proportional to the gradients of the mean velocities. Transport equations for the turbulent kinetic energy and the turbulent dissipation are developed from the Navier-Stokes equations assuming an isotropic turbulence. The implementation of this model and the parameters used can be found in [10],... 

Although numerous turbulence models are reported in the literature,1113 by far the most popular is the two-equation k-e model, first proposed by Jones and Launder.14 In this model, the turbulent stresses are recast in a form similar to the molecular stress tensor with mean velocity gradients, an assumption generally known as the Boussinesq hypothesis ... 

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... 

Prandtl s model derivation can then be briefly sketched, introducing the Boussinesq [19] [20] approximation for the turbulent viscosity. Starting out with the simple kinetic theory relation that the molecular viscosity equals the molecular velocity times the mean free path, an analogous relation can be formulated for the turbulent viscosity in terms of the turbulent mixing length and a suitable velocity scale, Ut Iv. ... 

The first-order closure models are all based on the Boussinesq hypothesis [19, 20] parameterizing the Reynolds stresses. Therefore, for fully developed turbulent bulk flow, i.e., flows far away from any solid boundaries, the turbulent kinetic energy production term is modeled based on the generalized eddy viscosity hypothesis , defined by (1.380). The modeled fc-equation is... 

In the context of reactor modeling, it is important to notice that this model rely on the Boussinesq eddy-viscosity concept which is based on the assump)-tion that turbulence is isotropic. This means that the normal Re3molds stresses are considered equal and that the eddy viscosity is approximately isotropic. Therefore, the k-e model cannot reproduce secondary flows which arise due to unequal normal Re3molds stresses. Unfortunately, the non-isotropic effects... 

For LES performed in physical space, the basic sub-grid stress model is the eddy-viscosity model proposed by Smagorinsky . The Smagorinsky model is based on the gradient transport hypothesis and the sub-grid viscosity concept, just as the Reynolds stress models based on the Boussinesq eddy viscosity hypothesis, and expressed as ... 

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