Phase turbulence

When a is positive, (7.3.15) is consistent with the ordinary picture that locally convex fronts tend to be flattened. If a given front is concave, the flattening effect will ultimately be balanced with the sharpening effect (coming from the very fact that the front has a finite propagation velocity), so that formation of a shocklike structure is expected (Fig. 7.10). We already know, in fact, that the nonlinear phase diffusion equation admits a family of shock solutions (though in a different physical context see Sect. 6.2). In the present notation, the shock solutions (6.2.6) are expressed as 

Our principal concern in this section is the behavior of the numerical solutions of the phase turbulence equation (7.2.19) on a finite interval - /2 x /2, subject to the boundary conditions 

It is appropriate first to reduce the number of spurious parameters by suitable scaling. To achieve this, we should remember that the phase turbulence equation is valid only for small a, or for phenomena with a characteristic length of 

The system length in the new scale is given by. We are left with two parameters a and but the latter is only spurious the choice of its value is at our disposal 

Let b fixed to some value of 0(1), and r increased continuously, so that we can study the routes to chaos. For a given a, (7.4.4) can be integrated numerically by suitably discretizing x and t. The numerical results for (x, t) obtained are then Fourier-analyzed according to 

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J ct Spra.y, The mechanism that controls the breakup of a Hquid jet has been analy2ed by many researchers (22,23). These studies indicate that Hquid jet atomisation can be attributed to various effects such as Hquid—gas aerodynamic interaction, gas- and Hquid-phase turbulence, capillary pinching, gas pressure fluctuation, and disturbances initiated inside the atomiser. In spite of different theories and experimental observations, there is agreement that capillary pinching is the dominant mechanism for low velocity jets. As jet velocity increases, there is some uncertainty as to which effect is most important in causing breakup. 

The degree of turbulence would be classified as acceptable, but the unit must not be increased in capacity for fear of creating more water phase turbulence. 

Figure 10-113. The factor 4> for two-phase turbulent-turbulent flow. Note Reference number on chart is in Fair s article. (Used by permission Fair, J. R. Petroleum Refiner, Feb. 1960, p. 105. Gulf Publishing Company. All rights reserved.)...

The parameter p (= 7(5 ) in gas-liquid sy.stems plays the same role as V/Aex in catalytic reactions. This parameter amounts to 10-40 for a gas and liquid in film contact, and increases to lO -lO" for gas bubbles dispersed in a liquid. If the Hatta number (see section 5.4.3) is low (below I) this indicates a slow reaction, and high values of p (e.g. bubble columns) should be chosen. For instantaneous reactions Ha > 100, enhancement factor E = 10-50) a low p should be selected with a high degree of gas-phase turbulence. The sulphonation of aromatics with gaseous SO3 is an instantaneous reaction and is controlled by gas-phase mass transfer. In commercial thin-film sulphonators, the liquid reactant flows down as a thin film (low p) in contact with a highly turbulent gas stream (high ka). A thin-film reactor was chosen instead of a liquid droplet system due to the desire to remove heat generated in the liquid phase as a result of the exothermic reaction. Similar considerations are valid for liquid-liquid systems. Sometimes, practical considerations prevail over the decisions dictated from a transport-reaction analysis. Corrosive liquids should always be in the dispersed phase to reduce contact with the reactor walls. Hazardous liquids are usually dispensed to reduce their hold-up, i.e. their inventory inside the reactor. 

For a single-phase turbulent flow the ratio of the maximum to the average flow velocity is approximately 1.2, and the value of Co may also be close to 1.2 for a bubbly flow. Zuber and Findlay (1965) pointed out that, as the mixture velocity increases, the value of the exponent increases and flatter profiles result. 

The perimeter ib, representing the turbulent intensity at the bubble layer-core interface, is calculated as the product of the single-phase turbulent intensity at the bubble-layer edge and a two-phase enhancement factor. The resulting expression is... 

Chen, C. P., Studies in two-phase turbulence closure modeling, Ph.D. Thesis, Michigan State University, USA (1985). 

Precisely owing to the continuum description of the dispersed phase, in Euler-Euler models, particle size is not an issue in relation to selecting grid cell size. Particle size only occurs in the constitutive relations used for modeling the phase interaction force and the dispersed-phase turbulent stresses. 

The evolution of the two-phase turbulence depends on the initial random position of the particles, the motion of which modifies the turbulent-flow field directly. These DNS are therefore a nice example of two-way coupling between the two phases see Fig. 12. From these DNS, detailed knowledge can be derived as to the frequency of the particle-particle collisions and the forces involved... 

Leaving aside the difficult question of whether this model holds for multiphase flows, we still have the problem of determining in terms of the computed properties of the flow. The reader should appreciate that choosing an effective viscosity for a multiphase flow is much more complicated than just adding a turbulence model as done in single-phase turbulent flows. Indeed, even for a case involving two fluids (e.g., two immiscible liquids) for which the molecular viscosities are constant, the choice of the effective viscosities is not obvious. For example, even if the mass-average velocity defined by... 

Second, due to the difficulty of accessing multiphase flows with laser-based flow diagnostics, there is very little experimental data available for validating multiphase turbulence models to the same degree as done in single-phase turbulent flows. For example, thanks to detailed experimental measurements of turbulence statistics, there are many cases for which the single-phase k- model is known to yield poor predictions. Nevertheless, in many CFD codes a multiphase k-e model is used to supply multiphase turbulence statistics that cannot be measured experimentally. Thus, even if a particular multiphase turbulent flow could be adequately described using an effective viscosity, in most cases it is impossible to know whether the multiphase turbulence model predicts reasonable values for... 

With the assumption of two-phase turbulent flow, a simplified method has been developed [195] for estimating emergency vent sizes which is discussed in Section 3.3.4.7. 

Dowling, D. R. (1991). The estimated scalar dissipation rate in gas-phase turbulent jets. 

Dengler and Addoms 8 measured heat transfer to water boiling in a 6 m tube and found that the heat flux increased steadily up the tube as the percentage of vapour increased, as shown in Figure 14.4. Where convection was predominant, the data were correlated using the ratio of the observed two-phase heat transfer coefficient (htp) to that which would be obtained had the same total mass flow been all liquid (hi) as the ordinate. As discussed in Volume 6, Chapter 12, this ratio was plotted against the reciprocal of Xtt, the parameter for two-phase turbulent flow developed by Lockhart and Martinelli(9). The liquid coefficient hL is given by ... 

Authors efforts in this part of the work have been concentrated on developing turbulence closures for the statistical description of two-phase turbulent flows. The primary emphasis is on development of models which are more rigorous, but can be more easily employed. The main subjects of the modeling are the Reynolds stresses (in both phases), the cross-correlation between the velocities of the two phases, and the turbulent fluxes of the void fraction. Transport of an incompressible fluid (the carrier gas) laden with monosize particles (the dispersed phase) is considered. The Stokes drag relation is used for phase interactions and there is no mass transfer between the two phases. The particle-particle interactions are neglected the dispersed phase viscosity and pressure do not appear in the particle momentum equation. 

Mashayek, R., D.B. Taulbee, and P. Givi. 1998. Modeling and simulation of two-phase turbulent flow. In Propulsion combustion Fuels to emissions. Ed. G. D. Roy. Washington, DC Taylor Francis. 241-80. 

Pozorski, J., and J. P. Minier. 1999. Probability density function modeling of dispersed two-phase turbulent flow. Phys. Rev. E 59 855-63. 

Mashayek, F. 1999. Simulation and modeling of two-phase turbulent flows for prediction and control of combustion systems. 12th ONR Propulsion Meeting Proceedings. Eds. G. D. Roy and S. L. Anderson. Salt Lake City, UT. 88-95. 

The liquid-phase turbulence, which enhances the particle-liquid drag force thereby increasing the value of the masstransfer coefficient, is characterized by the turbulence intensity (uT), as defined by the following relation ... 

The "correlative" multi-scale CFD, here, refers to CFD with meso-scale models derived from DNS, which is the way that we normally follow when modeling turbulent single-phase flows. That is, to start from the Navier-Stokes equations and perform DNS to provide the closure relations of eddy viscosity for LES, and thereon, to obtain the larger scale stress for RANS simulations (Pope, 2000). There are a lot of reports about this correlative multi-scale CFD for single-phase turbulent flows. Normally, clear scale separation should first be distinguished for the correlative approach, since the finer scale simulation need clear specification of its boundary. In this regard, the correlative multi-scale CFD may be viewed as a "multilevel" approach, in the sense that each span of modeled scales is at comparatively independent level and the finer level output is interlinked with the coarser level input in succession. 

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