Boussinesq Postulation

Momentum conservation equation Eq. (1.4) ujuj equation Eq. (1.8) (Boussinesq postulate) k equation Eq. (l.lla)... 

Since the k — e model is isotropic due to applying the Boussinesq postulate Eq. (1.7), the RNG model modifies it to approximate the anisotropic character of some flow process and gives better result The governing equations are similar to k E model as shown below ... 

The feature of this model is to solve the Reynolds stress Eq. (1.6) directly instead of using the Boussinesq postulate. 

Similar to the Boussinesq postulate or Pick s law, the m c and u f on the right side of Eq. (3.10) can be considered proportional to the corresponding negative gradients as shown below... 

As shown in preceding chapters, on the basis of Boussinesq postulate, the Reynolds stress —pu Uj, Reynolds heat flux —piifT, and Reynolds mass flux —u f (or —pujc ) can be expressed respectively as proportional to their gradients of average velocity, temperature, and concentration ... 

In order to reduce the computer load of standard Reynolds mass flux model, the complicated Eq. (1.23a) for expressing m-m can be replaced by the simpler Eq. (1.8). Then, the model becomes the combination of Reynolds mass flux and the Boussinesq postulate (two-equation model). It is called hereafter as hybrid Reynolds mass flux model. The model equations are given below. 

The unknown quantities in this model are (/, Uj, Uu, P, K , Pi, C, u[c, Ujc, u d, totally eleven versus eleven model equations available. Since this model employ Boussinesq postulate, it is isotropic. 

The Boussinesq-Scriven constitutive law postulates that = rj id. The surface viscosity, however, being a property of the adsorption layer itself, should be related to the relative displacement of the adsorbed molecules. As a result the dilatational surface viscous stress should be proportional to the rate of local deformation, i.e.. For insoluble... 

In 1877, Boussinesq [1] postulated that the Reynolds stress in turbulent flow was proportional to the time-averaged velocity gradient. Mathematically, it can be expressed as ... 

Besides applying the postulation similar to the Boussinesq s (or Pick s law) to solve the Reynolds mass flux — mJc in terms of isotropic turbulent mass diffusivity Dt as described in preceding Sect. 3.2 by c — Sc two-equation model, another model has been recently developed to solve the anisotropic Reynolds mass flux —M-c directly instead of using D, to close the turbulent species mass conservation equation. The Reynolds mass flux model discussed in this section could be known as a result following the turbulence closure postulations for the second-order closure turbulence model in the book of Chen and Jaw [23]. 

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