Phase turbulence equation

There may be an additional value in studying spatio-temporal chemical turbulence, in connection with its possible relevance to some biological problems. This is expected from the fact that the fields of coupled limit cycle oscillators (or nonoscillating elements with latent oscillatory nature) are often met in living systems. In some cases, such systems show orderly wavelike activities much the same as those observed in the Belousov-Zhabotinsky reaction. There seems to be no reason why we should not expect such organized motion to become unstable and hence show turbulent behavior. The recent work by Ermentrout (1982) who derived a Ginzburg-Landau type equation for neural field seems to be of particular interest in this connection. 

The stability problem of the uniform time-periodic solution o(0 to the reaction-diffusion equations is formally developed as follows. Here the system size is assumed to be infinitely large. Let the deviation u(t) about 0(0 be expressed as a Fourier series  

Since u is real, Ua = u a. The reaction-diffusion equations are then linearized in 

This equation is a simple generalization of (3.4.1). Analogously to (3.4.2), the general solution of (7.2.1) becomes 

The stability of the uniform oscillations to small-amplitude non-uniform fluctuations is determined from the eigenvalues of Ag. The eigenvectors / and uf in Sect. 3.4 are now generalized to include dependence on the wavevector q. Thus, 

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In what follows, we always assume y>0. The solution of (7.2.19) turns out chaotic for sufficiently large system size, and will be analyzed in Sect. 7.4. This equation may be called the phase turbulence equation. Recently, the same partial differential equation was derived by Sivashinsky in connection with the dynamics of combustion, and was used in discussing the turbulization of flame fronts (Sivashinsky, 1977, 1979 Michelson and Sivashinsky, 1977). 

We derived in Chap. 4 an evolution equation for slowly varying wavefronts in two-dimensional reaction-diffusion systems. Quite analogously to the dynamics of oscillatory systems with a slowly varying phase pattern, we obtained an asymptotic expansion (4.3.28). If a happened to be small and negative, while the other parameters were of ordinary magnitude and y positive, then the same reasoning advanced in Sect. 7.2 applies, and we get the one-dimensional phase turbulence equation... 

Although some physical implications of the nonlinear phase diffusion equation with positive a have been discussed in Sect. 6.2, we have not yet discussed the same equation in relation to the wavefront dynamics this should be done before going into the phase turbulence equation. Let the wavefront form a straight line which is slightly non-parallel to the y direction (Fig. 7.9). Then the nonlinear phase diffusion equation becomes... 

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[Pg.127]

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... 

Fig. 7.13. Snapshot of a chaotic phase distribution obtained for the phase turbulence equation (7.4.4) with (7=1...

It should be noted that the Ginzburg-Landau equation subject to the no-flux boundary conditions is invariant under the spatial inversion -x, which was also the case for the phase turbulence equation. Although this kind of symmetry property was not very important for the onset of phase turbulence, the same property is crucial to the understanding of the peculiar bifurcation structure in the present case. It is appropriate to make use of the system s symmetry by introducing a complex variable W(x, t) via... 

An integral form of Equation (3.15) was used to derive the pressure ratio for scaleup in series of a turbulent liquid-phase reactor, Equation (3.34). The integration apparently requires ji to be constant. Consider the case where ii varies down the length of the reactor. Define an average viscosity... 

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. 

Theoretical investigations of the problem were carried out on the base of the mathematical model, combining both deterministic and stochastic approaches to turbulent combustion of organic dust-air mixtures modeling. To simulate the gas-phase flow, the k-e model is used with account of mass, momentum, and energy fluxes from the particles phase. The equations of motion for particles take into account random turbulent pulsations in the gas flow. The mean characteristics of those pulsations and the probability distribution functions are determined with the help of solutions obtained within the frame of the k-e model. 

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. 

Fuvpi, %uvP2/ and Vuvp3 are the average liquid velocities for transducers 1, 2, and 3, respectively, from channel 0 to the channel where the gas-liquid interface is located. The constant 0.7 is obtained in the region 0single-phase turbulent flow the assumption made here is that the gas phase is located in the upper part of the pipe and the liquid velocity, not disturbed by the gas phase, develops in the lower part of the pipe as it does in single-phase turbulent flow. 

Turbulence is the most complicated kind of fluid motion. There have been several different attempts to understand turbulence and different approaches taken to develop predictive models for turbulent flows. In this chapter, a brief description of some of the concepts relevant to understand turbulence, and a brief overview of different modeling approaches to simulating turbulent flow processes is given. Turbulence models based on time-averaged Navier-Stokes equations, which are the most relevant for chemical reactor engineers, at least for the foreseeable future, are then discussed in detail. The scope of discussion is restricted to single-phase turbulent flows (of Newtonian fluids) without chemical reactions. Modeling of turbulent multiphase flows and turbulent reactive flows are discussed in Chapters 4 and 5 respectively. 

Balance equations listed here are before time averaging. For more details of time-averaged two-phase balance equations, the reader is referred to Ranade and van den Akker (1994) and the FLUENT manual. Turbulence was modeled using a standard k-s turbulence model. Governing equations for turbulent kinetic energy, k and... 

The turbulent gas/liquid flow in baffled tanks with turbine stirrer can be predicted. A mathematical model has been developed for turbulent, dispersed G/L flow. The time-averaged two phase momentum equations were solved by using a finite volume algorithm. The turbulent stresses were simulated with a K-fi-model. The distribution of gas around the stirrer blades is predicted for the first time. This model also enables an a priori prediction of the drop in the power dissipated by the stirrer in the presence of gas. Predicted flow characteristics for the gas/liquid dispersion show good agreement with the experimental data. 

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