Turbulence frequency

Sleiched278 has indicated that this expression is not valid for pipe flows. In pipe flows, droplet breakup is governed by surface tension forces, velocity fluctuations, pressure fluctuations, and steep velocity gradients. Sevik and Park 279 modified the hypothesis of Kolmogorov, 280 and Hinze, 270 and suggested that resonance may cause droplet breakup in turbulent flows if the characteristic turbulence frequency equals to the lowest or natural frequency mode of an... 

The integral-scale turbulence frequency is the inverse of the turbulence integral time scale. The turbulence time and length scales are defined in Chapter 2. 

For large Reynolds numbers, the right-hand side of this expression will be large, thereby forcing the scalar dissipation rate to attain a stationary solution quickly. Thus, for a fully developed scalar spectrum, the scalar mixing rate is related to the turbulent frequency by... 

Other two-equation models have been developed using various combinations of k and e to derive alternative transport equations to replace (4.47). The most popular is, perhaps, the k-co model (see Wilcox (1993) for a detailed discussion of its advantages), where the turbulence frequency co = e/k is used in place of s. The standard transport equation for... 

In the joint velocity, composition PDF description, the user must supply an external model for the turbulence time scale r . Alternatively, one can develop a higher-order PDF model wherein the turbulence frequency > is treated as a random variable (Pope 2000). In these models, the instantaneous turbulence frequency is defined as... 

Although it is possible to derive a PDF transport equation for stochastic model for the Fagrangian turbulence frequency a> (t) is developed along the lines of those discussed in Section 6.7. The goal of these models is to reproduce as many of the relevant one-point, two-time statistics of the Fagrangian fluid-particle turbulence frequency, o>+(t), as possible. Examples of two such models (log-normal model (Jayesh and Pope 1995) and gamma-distribution model (Pope and Chen 1990 Pope 1991a Pope 1992)) can be found in Pope (2000). Here we will... 

Given a stochastic model for the turbulence frequency, it is natural to enquire how fluctuations in co will affect the scalar dissipation rate (Anselmet and Antonia 1985 Antonia and Mi 1993 Anselmet et al. 1994). In order to address this question, Fox (1997) extended the SR model discussed in Section 4.6 to account for turbulence frequency fluctuations. The resulting model is called the Lagrangian spectral relaxation (LSR) model. The LSR model has essentially the same form as the SR model, but with all variables conditioned on the current and past values of the turbulence frequency [ /(. ),. v < r. In order to simplify the notation, this conditioning is denoted by ( , e.g.,... 

In these equations, the mean velocity and turbulence frequency are available from the FV code at (for example) grid-cell centers, and must be interpolated to the particle positions. 

The algorithms discussed earlier for time averaging and local time stepping apply also to velocity, composition PDF codes. A detailed discussion on the effect of simulation parameters on spatial discretization and bias error can be found in Muradoglu et al. (2001). These authors apply a hybrid FV-PDF code for the joint PDF of velocity fluctuations, turbulence frequency, and composition to a piloted-jet flame, and show that the proposed correction algorithms virtually eliminate the bias error in mean quantities. The same code... 

The PDF codes presented in this chapter can be (and have been) extended to include additional random variables. The most obvious extensions are to include the turbulence frequency, the scalar dissipation rate, or velocity acceleration. However, transported PDF methods can also be applied to treat multi-phase flows such as gas-solid turbulent transport. Regardless of the flow under consideration, the numerical issues involved in the accurate treatment of particle convection and coupling with the FV code are essentially identical to those outlined in this chapter. For non-orthogonal grids, the accurate implementation of the particle-convection algorithm is even more critical in determining the success of the PDF simulation. 

The same test cases can be used for velocity, composition PDF codes, but with the turbulent kinetic energy and turbulence frequency specified in place of the turbulent diffusivity. 

Jayesh and S. B. Pope (1995). Stochastic model for turbulent frequency. Technical Report FDA 95-05, Cornell University. 

Sevik and Park (S9) suggested that resonance can cause bubble and drop breakup in turbulent flow fields when the characteristic turbulence frequency matches the lowest or natural frequency mode of an entrained fluid particle. Breakup in turbulent flow fields is discussed below. 

Most codes also include a stochastic model for the turbulence frequency. 

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