A discussion of modeling passive scalar transport in turbulent flows is given. Several methods employed to close the scalar-flux term ⟨u′ϕ′⟩ that arises during Reynolds averaging are provided. Alternatives and improvements to the gradient diffusion hypotheses are addressed, most notably, the need for an alternative to the global constant turbulent Schmidt and Prandtl numbers. The reader is given a brief history covering methods used to predict turbulent Schmidt and Prandtl numbers, along with recommendations for future research, based partially on studies by Professor Stuart Churchill. More detailed formulations of turbulent Schmidt or Prandtl numbers will enable better approximations of the influence of turbulence in models of passive scalar flows using the gradient diffusion hypothesis.
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Combest et al. (2011) studied this question.
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