Scalar transport

The filtered transport equation for an inert, passive scalar has the form 

By analogy with the Smagorinsky model, the SGS scalar flux can be modeled using a gradient-diffusion model (Eidson 1985)  

The sub-grid-scale turbulent Schmidt number has a value of Scsgs % 0.4 (Pitsch and Steiner 2000), and controls the magnitude of the SGS turbulent diffusion. Note that due to the filtering process, the filtered scalar field will be considerably smoother than the original field. For high-Schmidt-number scalars, the molecular diffusion coefficient (T) will be much smaller than the SGS diffusivity, and can thus usually be neglected. 

As with the LES velocity PDF, a conditional PDF for the residual scalar field can be developed in terms of the LES composition PDF, denoted by / u OA lU, f).10 For a homogeneous scalar field with an isotropic filter, the conditional expected value of the scalar will have the property ( / U, p ) = p. Moreover, a transport equation can be derived for the residual scalar variance defined by11 

For turbulent reacting flows, LES introduces an additional closure problem due to filtering of the chemical source term (Cook and Riley 1994 Cook et al. 1997 Jimenez el al. 1997 Cook and Riley 1998 Desjardin and Frankel 1998 Wall et al. 2000). For the one-step 

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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. 

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 ... 

Via a passive scalar method [6] where or, denotes the volume fraction of the i-th phase, while T, represents the diffusivity coefiBcient of the tracer in the i-th phase. The transient form of the scalar transport equation was utilized to track the pulse of tracer through the computational domain. The exit age distribution was evaluated from the normalized concentration curve obtained via measurements at the reactor outlet at 1 second intervals. This was subsequently used to determine the mean residence time, tm and Peclet number, Pe [7]. 

The motions of the individual fluid parcels may be overlooked in favor of a more global, or Eulerian, description. In the case of single-phase systems, convective transport equations for scalar quantities are widely used for calculating the spatial distributions in species concentrations and/or temperature. Chemical reactions may be taken into account in these scalar transport equations by means of source or sink terms comprising chemical rate expressions. The pertinent transport equations run as... 

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. 

The CSTR model can be derived from the fundamental scalar transport equation (1.28) by integrating the spatial variable over the entire reactor volume. This process results in an integral for the volume-average chemical source term of the form ... 

The last term on the right-hand side of (1.28) is the chemical source term. As will be seen in Chapter 5, the chemical source term is often a complex, non-linear function of the scalar fields , and thus solutions to (1.28) are very different than those for the z nm-scalar transport equation wherein S is null. 

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... 

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 ... 

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

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,... 

Starting with the scalar transport equation ((1.28), p. 16), Reynolds averaging leads to the transport equation for the scalar means ... 

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

Like the Reynolds stresses, the scalar flux obeys a transport equation that can be derived from the Navier-Stokes and scalar transport equations. We will first derive the transport equation for the scalar flux of an inert scalar from (2.99), p. 48, and the governing equation for inert-scalar fluctuations. The latter is found by subtracting (3.89) from (1.28) (p. 16), and is given by... 

The transformed scalar transport equation then becomes... 

When the Schmidt number is greater than unity, addition of a scalar transport equation places a new requirement on the maximum wavenumber K. For Sc > 1, the smallest characteristic length scale of the scalar field is the Batchelor scale, 7b- Thus, the maximum wavenumber will scale with Reynolds and Schmidt number as... 

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. 

Note that the right-hand side of this expression is an E x I null matrix, and thus element conservation must hold for any choice of e e l,E and i e 1Moreover, since the element matrix is constant, (5.10) can be applied to the scalar transport equation ((1.28), p. 16) in order to eliminate the chemical source term in at least E of the K equations.9 The chemically reacting flow problem can thus be described by only K - E transport equations for the chemically reacting scalars, and E transport equations for non-reacting (conserved) scalars.10... 

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. 

Note that Nr = 2. Thus, by applying an appropriate linear transformation, it should be possible to rewrite the scalar transport equation in terms of two reacting and two conserved scalars. 

In order to find a linear transformation matrix to simplify the scalar transport equation, we will make use of the singular value decomposition (SVD) of Y ... 

At high Reynolds numbers, it is usually possible to assume that the mean scalar fields (e.g., (cc are independent of molecular-scale quantities such as the molecular-diffusion coefficients. In this case, it is usually safe to assume that all scalars have the same molecular diffusivity T. The conserved-scalar transport equation then simplifies to37... 

Also, by applying the linear transformation to (5.65), we can see that the conserved-variable scalar vector obeys a transformed scalar transport equation of the form... 

The interest in reformulating the conserved-variable scalars in terms of the mixture-fraction vector lies in the fact that relatively simple forms for the mixture-fraction PDF can be employed to describe the reacting scalars. However, if < /Vmf, then the incentive is greatly diminished since more mixture-fraction-component transport equations (Nmf) would have to be solved than conserved-variable-scalar transport equations (/V, << ). We will thus assume that N m = Nmf and seek to define the mixture-fraction vector only for this case. Nonetheless, in order for the mixture-fraction PDF method to be applicable to the reacting scalars, they must form a linear mixture defined in terms of the components of the mixture-fraction vector. In some cases, the existence of linear mixtures is evident from the initial/inlet conditions however, this need not always be the case. Thus, in this section, a general method for defining the mixture-fraction vector in terms of a linear-mixture basis for arbitrary initial/inlet conditions is developed. 

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. 

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... 

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. 

Biferale, L., A. Crisanti, M. Vergassola, and A. Vulpiani (1995). Eddy diffusivities in scalar transport. Physics of Fluids 7, 2725-2734. 

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