Boussinesq formulation

To render Eq. (5.62) solvable, it is necessary to provide an expression for the turbulent Reynolds stress. For isotropic turbulent flows, similar to the transport processes in laminar flows, a scalar turbulent viscosity /xT is defined using the Boussinesq formulation... 

The correlation of two fluctuating components may be best formulated using Boussinesq s approach [Boussinesq, 1877] by introducing a transport coefficient. Thus, the last term in Eq. (5.153) can be expressed by... 

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... 

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. ... 

In the following we resume the notations used by Maxey and Riley (1983) and write the Basset, Boussinesq, Oseen, and Tchen (BBOT) equations in a form that is quasi-identical to that of Maxey and Riley. This formulation is interesting insofar as it highlights the relative movement of the particle with respect to the fluid. The three components of the particle s relative velocity with respect to the fluid are obtained by solving the following differential equations ... 

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