Inertial flow

The first term in this equation is important if laminar, or viscous, flow predominates in a system, while the second term is important if turbulent, or inertial, flow predominates. Equation (1) can be rearranged to the form shown in Eq. (2). This form expresses Umfm terms of known system parameters. 

Ret < 150 Steady laminar inertial flow in which pressure drop depends nonlinearly on interstitial velocity and and boundary layers in the pores become pronounced with an inertial core appearing in the pores ... 

Ret < 300 Unsteady laminar inertial flow in which laminar wake oscillations appear in the pores and vortices form at around Ret — 250 ... 

The dependence of the relative pressure drop on the porosity at otherwise equal conditions in the viscous and inertial flow regimes are shown in Figure 8.5, where AP0A is the pressure drop at eb = 0.4. All other variables in the equation remaining constant a change in the void fraction from 0.4 to 0.5 reduces the pressure drop more than 2.8 times in viscous flow regime and more than 2.3 times in the inertial regime. 

CDn viscous to inertial flow transitional parameter for curved duct flow, dimensionless... 

With theyl-vortex we postulated that the velocity in axial direction in the core is limited and a constant value equal to the velocity v at the end of the inertial flow period. This value can be measured fairly easily since in the equilibrium zone, which we find at a level far enough away from the walls that its influence is not disturbing the measurements. 

The ratio tangential flow rate over tangential flow rate for a Newtonian liquid calculated by Anne-Archard [14] is plotted in function of the Deborah number for different Reynolds numbers in Figure 13. It can be noticed that in the case of a low inertial flow, the increase of the elasticity generates a decrease of the flow rate and, thus, of the mixing effectiveness the opposite effect happens when the inertial effects are more important. 

CONSTANTINESCU,V.N. and GALETUSE,S. Operating Characteristics of Journal Bearings in Turbulent Inertial Flow ASME Journal of Lubrication Technology, April 1982 vol 104pl73-179... 

Here the inlet region to a fluid film thrust bearing will be analysed using Che B.E.M., the first time that such an approach has been adopted. The present formulation of the method limits application to conditions of slow viscous flow. This is clearly a restriction, however, the primary purpose of the study reported was the implementation of the B.E.M. analysis to the problem and the identification of the attendant advantages. Enhancement of the technique to cover inertial flow situations is one of a number of important developments for the future. 

T.G. Drake and R.L. Shreve. High-speed motion pictures of nearly steady, uniform, two-dimensional, inertial flows of granular material. Journal of Rheology (1978-present), 30(5) 981-993,1986. 

Values of Rep above about 500 represent the inertial flow regime, for which the drag coefficient has been found to be approximately constant ... 

The tube-flow analogy inertial flow conditions... 

The same procedure described above for low velocity, viscous flow has been applied to the other extreme of high velocity, inertia-dominated flow. In this case the tube-flow equation is expressed in terms of the dimensionless friction factor f, which in the inertial flow regime remains essentially constant for a given tube ... 

The inertial flow regime revised tube-flow analogy... 

A convenient effect of the above changes to the conventional pressure loss expressions for the viscous and inertial flow regimes has been the unification of the dependencies on void fraction these are now the same in eqns (3.21) and (3.23). There is, however, another factor to consider regarding the general applicabihty of these relations. 

The results of these three investigations are shown in Figure 3.6, the first as broken lines representing the published correlations, the other two as raw data points. The range covered is enormous six orders of magnitude in both dimensionless unrecoverable pressure loss and particle Reynolds number from well inside the viscous to deep into the inertial flow regimes. 

The parameter n correlates with the terminal particle Reynolds number Rer. it acquires constant values in both the creeping flow and inertial flow regimes ( 4.8 and 2.4 respectively), changing progressively with Ret in the intermediate regime between these limiting values. The following convenient relation (Khan and Richardson, 1989) enables n to be... 

Once again we have arrived, quite independently, at a result for the expansion characteristics of homogeneous fluidized beds which is in compete agreement with the Richardson-Zaki relation, this time for inertial flow conditions eqn (4.4), = 2.4. The unhindered-particle limit, 1, yields, as it must, the inertial regime relation of eqn (2.13) for Uf... 

This form is identical to the Richardson-Zaki equation, eqn (4.4). It therefore relates the parameter n in that empirical relation to the ratio of the void fraction and fluid flux exponents in the expression for unrecoverable pressure loss, eqn (4.8) n = —b/a. Note that under both viscous and inertial flow conditions (a=l, = 4.8 and a = 2, =2.4, respectively), the void fraction exponent b assumes the value of —4.8. This unexpected coincidence will now be put to effective use. 

The fluid flux exponent, 4.8/n, in eqns (4.13) and (4.14) converges to the correct limits of 1 and 2 for the viscous and inertial flow regimes, for which n has values of 4.8 and 2.4 respectively. In the intermediate regime it serves to provide a convenient, if approximate, interpolation between these two extremes. The value of —4.8 for the void fraction exponent also represents an approximation for intermediate flow conditions values as different as —4.2 have been reported for Reynolds numbers of around 50 where the maximum deviation appears to occur (Khan and Richardson, 1990 Di Felice, 1994). These reservations are of secondary relevance, however, pointing only to the possibility of minor quantitative inaccuracies in predictions arising from analyses in which the relations of eqns (4.13) and (4.14) are applied. 

Suppose there are two canonical types of inertial flow around a particle with each its own fluid-particle interaction force, such as a steady-state drag and a history related force. Each of these forces is due to a particular fluid flow field around the particle in question. Unless the principle of separation of scales applies and/or inertial effects can safely be ignored, the Hnear addition of two flow fields each described by its own Navier—Stokes equation does not yield the total flow field, just because of the convective or inertial terms in the Navier—Stokes equation, e.g.. 

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