Semi-Empirical Flow Analysis

However, the analysis of experimental data on flow birefringence and the use of some theoretical (sometimes semi-empirical) concepts allows important conclusions... 

To proceed giving a thorough description of the equations used for turbulent flows, we need some results from the semi-empirical turbulent boundary layer flow analysis. 

In a comprehensive study. Dodge and Metzner [1959] carried out a semi-empirical analysis of the fiilly developed turbulent flow of power-law fluids in smooth pipes. They used the same dimensional considerations for snch flnids, as Millikan [1939] for incompressible Newtonian fluids, and obtained an expression which can be re-arranged in terms of the apparent power law index, (eqnation 3.26) as follows ... 

Mathematical formalism has been developed using semi-empirical considerations [36, 37]. Computer simulation smdies show that resulting equation predicts oscillations. Attempt has been made to provide justification on the bases of Navier-Stokes equation but it is open to question. Dimensional analysis has recently been employed for investigating the phenomena [31]. Flow dynamics and stability in a density oscillator have been examined by Steinbock and co-workers [38], They have related it to Rayleigh-Taylor instability of two different dense viscous liquids. A theoretical description has been presented which is based on a one-fluid model and a steady state approximation for a two-dimensional flow using Navier-Stokes equation. However, the treatment is quite complex and cannot explain the generation of electric potential oscillations. 

The semi-empirical analysis of many experimental data collected in two different pilot plants led to satisfactory correlations for the turbulent friction coefficient and to the prediction of laminar-turbulent transition velocity for the flow of viscoplastic suspensions in annuli. 

Originally, the concept of fluid boundary layer was presented by Prandtl [123]. Prandtl s idea was that for flow next to a solid boundary a thin fluid layer (i.e., a boundary layer) develops in which friction is very important, but outside this layer the fluid behaves very much like a ffictionless fluid. To define a demarcation line between these two flow regions the thickness of the boundary layer, 6, is arbitrarily taken as the distance away from the surface where the velocity reaches 99 % of the free stream velocity (e.g., [55], p. 192 [107], p. 12 [114], p. 545). To proceed giving a thorough description of the equations used for turbulent flows, we need some results from the semi-empirical turbulent boundary layer flow analysis. For a generalized shear flow in the vicinity of a flat horizontal solid wall, the boundary layer flow can be described in Cartesian coordinates. The stress, —Ojty.eff, associated with direction y normal to the wall is apparently dominant, thus the stream-wise Reynolds averaged momentum equation yields ... 

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