Direct numerical simulations

This chapter is devoted to methods for describing the turbulent transport of passive scalars. The basic transport equations resulting from Reynolds averaging have been derived in earlier chapters and contain unclosed terms that must be modeled. Thus the available models for these terms are the primary focus of this chapter. However, to begin the discussion, we first review transport models based on the direct numerical simulation of the Navier-Stokes equation, and other models that do not require one-point closures. The presentation of turbulent transport models in this chapter is not intended to be comprehensive. Instead, the emphasis is on the differences between particular classes of models, and how they relate to models for turbulent reacting flow. A more detailed discussion of turbulent-flow models can be found in Pope (2000). For practical advice on choosing appropriate models for particular flows, the reader may wish to consult Wilcox (1993). 

For homogeneous turbulent flows (no walls, periodic boundary conditions, zero mean velocity), pseudo-spectral methods are usually employed due to their relatively high accuracy. In order to simulate the Navier-Stokes equation, 

Using (4.2), the transformed Navier-Stokes equation becomes2 

At any fixed time t, the velocity field can be recovered by an inverse Fourier transform  

DNS of homogeneous turbulence thus involves the solution of a large system of ordinary differential equations (ODEs see (4.3)) that are coupled through the convective and pressure terms (i.e., the terms involving T). 

We briefly review the basic equations pertaining to a battery DNS model. The details of the various symbols are furnished in the Nomenclature section together with a brief but sufScient description in the text following the equations. 

---

Although direct numerical simulations under limited circumstances have been carried out to determine (unaveraged) fluctuating velocity fields, in general the solution of the equations of motion for turbulent flow is based on the time-averaged equations. This requires semi-... 

The engineer is offered a large variety of flow-modeling methods, whose complexity ranges from simple order-of-magnitude analysis to direct numerical simulation. Up to now, the methods of choice have ordinarily been experimental and semi-theoretical, such as cold flow simulations and tracer studies. 

L. Vervisch and T. Poinsot, Direct numerical simulation of non-premixed turbulent flames, Annu. Rev. Fluid Mech. 30 655-691,1998. 

Okong o, N. and J. Bellan, Consistent large-eddy simulation of a temporal mixing layer laden with evaporating drops. Rart 1. Direct numerical simulation, formulation and a priori analysis. J. Fluid Mech., 2004. 499 1-47. 

Hawkes, E.R., S. Sankaran, J.C. Sutherland, and J.H. Chen, Scalar mixing in direct numerical simulations of temporally-evolving plane jet flames with detailed CO/Hj kinetics. Proc. Combust. Inst., 2007. 31 1633-1640. 

Multiscale modeling is an approach to minimize system-dependent empirical correlations for drag, particle-particle, and particle-fluid interactions [19]. This approach is visualized in Eigure 15.6. A detailed model is developed on the smallest scale. Direct numerical simulation (DNS) is done on a system containing a few hundred particles. This system is sufficient for developing models for particle-particle and particle-fluid interactions. Here, the grid is much smaller... 

Mathpati, C.S. and Joshi, J.B. (2007) Insight into theories of heat and mass transfer at the solid/fluid interface using direct numerical simulation and large eddy simulation. Joint 6th International Symposium on Catalysis in Multiphase Reactors/5th International Symposium on Multifunctional Reactors (CAMURE-6/ISMR-5-), 2007, Pune. 

The spectral method is used for direct numerical simulation (DNS) of turbulence. The Fourier transform is taken of the differential equation, and the resulting equation is solved. Then the inverse transformation gives the solution. When there are nonlinear terms, they are calculated at each node in physical space, and the Fourier transform is taken of the result. This technique is especially suited to time-dependent problems, and the major computational effort is in the fast Fourier transform. 

The emphasis in this chapter is on the fruitful application of Large Eddy Simulations for reproducing the local and transient flow conditions in which these processes are carried out and on which their performance depends. In addition, examples are given of using Direct Numerical Simulations of flow and transport phenomena in small periodic boxes with the view to find out about relevant details of the local processes. Finally, substantial attention is paid throughout this chapter to the attractiveness and success of exploiting lattice-Boltzmann techniques for the more advanced CFD approaches. 

The term direct in Direct Numerical Simulations indicates that the flow is fully resolved by solving, without any modeling, the classical NS equations... 

B. The Promises of Direct Numerical Simulations and Large Eddy... 

The CFD models considered up to this point are, as far as the momentum equation is concerned, designed for single-phase flows. In practice, many of the chemical reactors used in industry are truly multiphase, and must be described in the context of CFD by multiple momentum equations. There are, in fact, several levels of description that might be attempted. At the most detailed level, direct numerical simulation of the transport equations for all phases with fully resolved interfaces between phases is possible for only the simplest systems. For... 

Pressure drop and dispersion were the focus of work by Magnico (2003) who simulated flow at lower Re by direct numerical simulation (DNS) in beds of spheres with an in-house code. Tobis (2000) simulated a small cluster of four spheres with inserts between them to compare to his experimental measurements of pressure drop. Gunjal et al. (2005) also focused on flow and pressure drop through a small cell of spheres, in order to validate the CFD approach by comparison to the MRI measurements in the same geometry made by Suekane... 

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). 

As discussed in Chapter 2, a fully developed turbulent flow field contains flow structures with length scales much smaller than the grid cells used in most CFD codes (Daly and Harlow 1970).29 Thus, CFD models based on moment methods do not contain the information needed to predict x, t). Indeed, only the direct numerical simulation (DNS) of (1.27)-(1.29) uses a fine enough grid to resolve completely all flow structures, and thereby avoids the need to predict x, t). In the CFD literature, the small-scale structures that control the chemical source term are called sub-grid-scale (SGS) fields, as illustrated in Fig. 1.7. 

Only direct numerical simulation (DNS) resolves all scales (Moin and Mahesh 1998). However, DNS is com-putationally intractable for chemical reactor modeling. 

Chasnov (1994) has carried out detailed studies of inert-scalar mixing at moderate Reynolds numbers using direct numerical simulations. He found that for decaying scalar fields the scalar spectrum at low wavenumbers is dependent on the initial scalar spectrum, and that this sensitivity is reflected in the mechanical-to-scalar time-scale ratio. Likewise, R is found to depend on both the Reynolds number and the Schmidt number in a non-trivial manner for decaying velocity and/or scalar fields (Chasnov 1991 Chasnov 1998). 

Direct numerical simulation studies of turbulent mixing (e.g., Ashurst et al. 1987) have shown that the fluctuating scalar gradient is nearly always aligned with the eigenvector of the most compressive strain rate. For a fully developed scalar spectrum, the vortexstretching term can be expressed as... 

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... 

---