Modeling and Simulation of Turbulent Mixing and Reaction by Daniel Livescu & Arash G. Nouri & Francine Battaglia & Peyman Givi

Author:Daniel Livescu & Arash G. Nouri & Francine Battaglia & Peyman Givi
Language: eng
Format: epub
ISBN: 9789811526435
Publisher: Springer Singapore

2 Governing Equations

The details of the derivation of the transport equations governing fluid flows in thermodynamic equilibrium are documented in many undergraduate and graduate texts [2, 40–45] and will not be repeated here. Flows in thermodynamic non-equilibrium and multiphase flows are not considered, but common practical approaches to treating these flows are discussed by Gnoffo et al. [46], Park [47] and Faeth [48]. The derivations of the governing equations for the motion of a fluid in thermodynamic equilibrium lead to a set of elegant nonlinear partial differential equations (PDEs) governing the transport of several conserved quantities: species mass, momentum, and energy. One of the earliest complete discussions of these equations with an application to high speed reacting flows is offered by Drummond [49]. These equations can also be further manipulated to obtain other transport equations for quantities such as vorticity, enthalpy, or (combining with the second law of thermodynamics) entropy that have been found useful in elucidating physical behaviors of flows [50].

The governing conservation equations encompass a wide range of physical fluid flow phenomena. One particularly complex phenomena is that of a turbulent flow. The discussion of turbulence physics is beyond the scope of this text, but several excellent texts are available [51–54] with many more describing computational [2, 49, 55–60] and modeling [61–65] treatments. Nevertheless, a few words relevant to the current discussion are warranted. Foremost, it should be stated that one characteristic of turbulent flows, which is responsible for the difficulty encountered in theoretical and numerical analysis, is the multiscale nature of turbulence. That is, the fluid motions in a turbulent flow occur over a wide range of both time and length scales with the ratio of large to small turbulence flow scales proportional to the 3/4 power of the Reynolds number [54]. For a problem of practical interest, this leads to the required number of computational cells for direct numerical simulation (DNS) to be on the order of (i.e., three orders of magnitude in each of the three spatial dimensions). By contrast, the grid resolutions that are used in simulations of practical interest on capacity cluster hardware and in the amount of time required to make a programmatic impact are typically on the order of . That is, the typical current capability for numerical simulations of turbulent flows is almost two to three orders of magnitude smaller than that required for the corresponding direct simulations. Therefore, even before the flamelet model is introduced to reduce the computational cost of combustion, any numerical simulation involving turbulent flow must be set up to utilize the existing computer hardware in a reasonable amount of time and still be able to investigate and analyze turbulent reacting flows of interest.

One approach is to reduce the effective dynamic range of turbulent length and time scales to that which can be reasonably considered for simulations on a current computer. Unfortunately, this constrains the simulations to only a portion of the turbulence length scales with the removed portions requiring a mathematical model for the effects they have on those being simulated.

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