Modeling for Turbulent Flows

The most widely adopted method for the turbulent flow analysis is based on time-averaged equations using the Reynolds decomposition concept. In the following, we discuss the Reynolds decomposition and time-averaging method. There are other methods such as direct numerical simulation (DNS), large-eddy simulation (LES), and discrete-vortex simulation (DVS) that are being developed and are not included here. 

In using the Reynolds decomposition, closure of the time-averaged Navier-Stokes equations cannot readily be realized because of the unknown correlation terms such as turbulent 

Reynolds stresses generated by time averaging. Thus, additional equations are needed to correlate these terms with time-averaged quantities. These additional equations may come from turbulence models. The two most commonly used turbulence models, the mixing length model and the k-e model, are introduced. 

It is assumed that the instantaneous Navier-Stokes equations for turbulent flows have the exact form of those for laminar flows. From the Reynolds decomposition, any instantaneous variable, (j , can be divided into a time-averaged quantity and a fluctuating part as 

The integral time period to should be short compared to the characteristic time scale of the system. At the same time, t0 must also be long enough so that 

---

All models for turbulent flows are semiempirical in nature, so it is necessary to rely upon empirical observations (e.g., data) for a quantitative description of friction loss in such flows. For Newtonian fluids in long tubes, we have shown from dimensional analysis that the friction factor should be a unique function of the Reynolds number and the relative roughness of the tube wall. This result has been used to correlate a wide range of measurements for a range of tube sizes, with a variety of fluids, and for a wide range of flow rates in terms of a generalized plot of/ versus /VRe- with e/D as a parameter. This correlation, shown in Fig. 6-4, is called a Moody diagram. 

Durbin, P. A., N. N. Mansour, and Z. Yang (1994). Eddy viscosity transport model for turbulent flow. Physics of Fluids 6, 1007-1015. 

Haworth, D. C. and S. B. Pope (1986). A generalized Langevin model for turbulent flows. 

Minier, J.-P. and J. Pozorski (1997). Derivation of a PDF model for turbulent flows based on principles from statistical physics. Physics of Fluids 9, 1748-1753. 

Pope, S. B. and Y. L. Chen (1990). The velocity-dissipation probability density function model for turbulent flows. Physics of Fluids A Fluid Dynamics 2, 1437-1449. 

Taylor (T4, T6), in two other articles, used the dispersed plug-flow model for turbulent flow, and Aris s treatment also included this case. Taylor and Aris both conclude that an effective axial-dispersion coefficient Dzf can again be used and that this coefficient is now a function of the well known Fanning friction factor. Tichacek et al. (T8) also considered turbulent flow, and found that Dl was quite sensitive to variations in the velocity profile. Aris further used the method for dispersion in a two-phase system with transfer between phases (All), for dispersion in flow through a tube with stagnant pockets (AlO), and for flow with a pulsating velocity (A12). Hawthorn (H7) considered the temperature effect of viscosity on dispersion coefficients he found that they can be altered by a factor of two in laminar flow, but that there is little effect for fully developed turbulent flow. Elder (E4) has considered open-channel flow and diffusion of discrete particles. Bischoff and Levenspiel (B14) extended Aris s theory to include a linear rate process, and used the results to construct comprehensive correlations of dispersion coefficients. 

In addition, for highly nonlinear cases, the mesh size needed to avoid spurious solutions may be so small that this approach is not feasible, (ii) The CFD approach also uses averaged models (e.g., k-s model for turbulent flows) with closure schemes that are not always justified and contain adjustable constants. 

The important result is that the two-mode models for a turbulent flow tubular reactor are the same as those for laminar flow tubular reactors. The two-mode axial dispersion model for turbulent flow tubular reactors is again given by Eqs. (130)—(134), while the two-mode convection model for the same is given by Eqs. (137)—(139), where the reaction rate term r((c)) is replaced by the Reynolds-averaged reaction rate term rav((c)). The local mixing time for turbulent flows is given by... 

W. P. Jones, Models for Turbulent Flows with Variable Density and Combustion, in Prediction Methods for Turbulent Flows, W. Kollmann, ed.. New York Hemisphere Publishing Corp., 1980, 379-421. 

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. 

Saffman, P.G. (1977), Results of a two-equation model for turbulent flows and development of relaxation stress model for application to staining and rotating flows. Proceedings of Project SQUID Workshop on Turbulence in Internal Flows, Hemisphere, New York, p. 191. 

A simple conceptual model for turbulent flow deals with eddies, small portions of fluid in the boundary layer that move about for a short time before losing their identity [8], The transport coefficient, which is defined as eddy diffusivity for momentum transfer eM, has the form... 

P.A. Durbin and B.A. Pettersson Reif, Statistical Theory and Modeling for Turbulent Flows, 2nd Edition, 2010, Wiley and Sons Inc., Chichester, U.K., 372pp. 

Axial Dispersion Model for Turbulent Flow in Round Tubes An empirical correlation given by Wen and Fan (1975) for turbulent flow in a round tube is ... 

Jones WP (1980) Models for turbulent flows with variable density and combustion. In Koll-man W (ed) Prediction methods for turbulent flow. Hemisphere, Washington, DC... 

Menter, F. R. Multiscale model for turbulent flows in 24th fluid dynamic conference. Am. Inst. Aeronaut. Astronaut. 1993. 

---