Turbulent scalar transport

Nevertheless, despite all these remarkable achievements, some open questions still remain. Among them is the influence of the molecular transport properties, in particular Lewis number effects, on the structure of turbulent premixed flames. Additional work is also needed to quantify the flame-generated turbulence phenomena and its relationship with the Darrieus-Landau instability. Another question is what are exactly the conditions for turbulent scalar transport to occur in a coimter-gradient mode Finally, is it realistic to expect that a turbulent premixed flame reaches an asymptotic steady-state of propagation, and if so, is it possible, in the future, to devise an experiment demonstrating it ... 

A theoretical framework based on the one-point, one-time joint probability density function (PDF) is developed. It is shown that all commonly employed models for turbulent reacting flows can be formulated in terms of the joint PDF of the chemical species and enthalpy. Models based on direct closures for the chemical source term as well as transported PDF methods, are covered in detail. An introduction to the theory of turbulence and turbulent scalar transport is provided for completeness. 

Similarly, turbulent scalar transport models based on (1.28) for the case where the chemical source term is null have been widely studied. Because (1.28) in the absence... 

For example, the RTD can be computed from the results of a turbulent-scalar transport model, but not vice versa. 

There is no direct information on scalar transport due to velocity fluctuations. A PDF scalar-flux model is required to describe turbulent scalar transport.2... 

Kerstein, A. R. (1988). A linear-eddy model of turbulent scalar transport and mixing. 

Complex strain flame-front regime. Where the flame fronfs are still lamella-like but thickened due to enhanced turbulent diffusivity. Scalar transport is expected to be counter-gradient in this regime. 

Turbulent flame-front regime. Eddy-like contortions of fhe flame preheaf and burned gases zones give rise to "ouf of fronf" islands and peninsula sfructures of intermediate progress variable values. Scalar transport becomes gradient-like. 

The material covered in the appendices is provided as a supplement for readers interested in more detail than could be provided in the main text. Appendix A discusses the derivation of the spectral relaxation (SR) model starting from the scalar spectral transport equation. The SR model is introduced in Chapter 4 as a non-equilibrium model for the scalar dissipation rate. The material in Appendix A is an attempt to connect the model to a more fundamental description based on two-point spectral transport. This connection can be exploited to extract model parameters from direct-numerical simulation data of homogeneous turbulent scalar mixing (Fox and Yeung 1999). 

Despite the progress in CFD for inert-scalar transport, it was recognized early on that the treatment of turbulent reacting flows offers unique challenges (Corrsin 1958 Danckwerts 1958). Indeed, while turbulent transport of an inert scalar can often be successfully described by a small set of statistical moments (e.g., (U), k, e, (, and (scalar fields, which are strongly coupled through the chemical source term in (1.28). Nevertheless, it has also been recognized that because the chemical source term depends only on the local molar concentrations c and temperature T ... 

In Section 3.3, we will use (3.16) with the Navier-Stokes equation and the scalar transport equation to derive one-point transport equations for selected scalar statistics. As seen in Chapter 1, for turbulent reacting flows one of the most important statistics is the mean chemical source term, which is defined in terms of the one-point joint composition PDF +(+x, t) by... 

Owing to the complexity of multi-point descriptions, almost all scalar transport models for complex flows are based on one-point statistics. As shown in Section 2.1, one-point turbulence statistics are found by integrating over the velocity sample space. Likewise,... 

For turbulent mixing of an inert scalar mean scalar transport equation reduces to... 

In homogeneous turbulence, spectral transport can be quantified by the scalar cospectral energy transfer rate Tap(ic, t). We can also define the wavenumber that separates the viscous-convective and the viscous-diffusive sub-ranges nf by introducing the arithmetic-mean molecular diffusivity Tap defined by... 

In general, liquid-phase reactions (Sc > 1) and fast chemistry are beyond the range of DNS. The treatment of inhomogeneous flows (e.g., a chemical reactor) adds further restrictions. Thus, although DNS is a valuable tool for studying fundamentals,4 it is not a useful tool for chemical-reactor modeling. Nonetheless, much can be learned about scalar transport in turbulent flows from DNS. For example, valuable information about the effect of molecular diffusion on the joint scalar PDF can be easily extracted from a DNS simulation and used to validate the micromixing closures needed in other scalar transport models. 

The last term on the right-hand side is unclosed and represents scalar transport due to velocity fluctuations. The turbulent scalar flux ( , varies on length scales on the order of the turbulence integral scales Lu, and hence is independent of molecular properties (i.e., v and T).17 In a CFD calculation, this implies that the grid size needed to resolve (4.70) must be proportional to the integral scale, and not the Batchelor scale as required in DNS. In this section, we look at two types of models for the scalar flux. The first is an extension of turbulent-viscosity-based models to describe the scalar field, while the second is a second-order model that is used in conjunction with Reynolds-stress models. 

In this case, if the boundary and initial conditions allow it, either ej or c can be used to define the mixture fraction. The number of conserved scalar transport equations that must be solved then reduces to one. In general, depending on the initial conditions, it may be possible to reduce the number of conserved scalar transport equations that must be solved to min(Mi, M2) where M = K - Nr and M2 = number of feed streams - 1. In many practical applications of turbulent reacting flows, M =E and M2 = 1, and one can assume that the molecular-diffusion coefficients are equal thus, only one conserved scalar transport equation (i.e., the mixture fraction) is required to describe the flow. 

The reduction of the turbulent-reacting-flow problem to a turbulent-scalar-mixing problem represents a significant computational simplification. However, at high Reynolds numbers, the direct numerical simulation (DNS) of (5.100) is still intractable.86 Instead, for most practical applications, the Reynolds-averaged transport equation developed in... 

Of all of the methods reviewed thus far in this book, only DNS and the linear-eddy model require no closure for the molecular-diffusion term or the chemical source term in the scalar transport equation. However, we have seen that both methods are computationally expensive for three-dimensional inhomogeneous flows of practical interest. For all of the other methods, closures are needed for either scalar mixing or the chemical source term. For example, classical micromixing models treat chemical reactions exactly, but the fluid dynamics are overly simplified. The extension to multi-scalar presumed PDFs comes the closest to providing a flexible model for inhomogeneous turbulent reacting flows. Nevertheless, the presumed form of the joint scalar PDF in terms of a finite collection of delta functions may be inadequate for complex chemistry. The next step - computing the shape of the joint scalar PDF from its transport equation - comprises transported PDF methods and is discussed in detail in the next chapter. Some of the properties of transported PDF methods are listed here. 

Relative to velocity, composition PDF codes, the turbulence and scalar transport models have a limited range of applicability. This can be partially overcome by using an LES description of the turbulence. However, consistent closure at the level of second-order RANS models requires the use of a velocity, composition PDF code. 

Similarity states of passive scalar transport in isotropic turbulence. Physics of Fluids 6, 1036-1051. 

Janicka, J., W. Kolbe, and W. Kollmann (1979). Closure of the transport equation for the probability density function of turbulent scalar fields. Journal of Non-Equilibrium Thermodynamics 4, 47-66. 

Vedula, P. (2001). Study of Scalar Transport in Turbulent Flows Using Direct Numerical Simulations. Ph. D. thesis, Georgia Institute of Technology, Atlanta. 

Lagrangian characteristics of turbulence and scalar transport in direct numerical simulations. Journal of Fluid Mechanics 427, 241-274. 

Garrick, S. C., F. A. Jaberi, and P. Givi. 1999. Large-eddy simulation of scalar transport in a turbulent jet flow. In Recent advances in DNS and LES, Buid mechanics and its applications. Eds. D. Knight and L. Sakkel. The Netherlands Kluwer Academic Publ. 54 155-66. 

Moin, P., K. Squires, W. Cabot, and S. Lee. 1991. A dynamic subgrid-scale model for compressible turbulence and scalar transport. J. Physics Fluids 3(ll) 2746-57. 

Martinez-Bazan C, Montanes JL, Lasheras JC (1999) On the breakup of an air bubble injected into a fully developed turbulent flow. Part 2. Size PDF of the resulting daughter bubbles. J Fluid Mech 401 183-207 Melzak ZA (1957) A scalar transport equation. Transactions of the American Mathematical Society 85 547-560... 

The remaining scalar transport equations, like the k-e turbulence model, are solved in the same wav as the discrete scalar transport equation (12.183). 

The particular scalar transport equation for the liquid phase turbulent kinetic energy is written ... 

Eckhardt B, Schumacher J (2004) Turbulence and passive scalar transport in a free slip surface. Physical Review E, 64 016314 Foss M (2001) Wave damping and momentum transfer. Doctoral thesis, University of Tromso, Norway... 

---