What is it about?

THE turbulent Prandtl number variation within the near-wall region of fully developed boundary layers is obtained by using profile analysis. This is achieved by deriving expressions for the" eddy" diffusivities from Spalding's inner layer formula and its thermal analogue. The analysis is performed over two orders of magnitude of the molecular Prandtl number, encompassing most of the common, non-metallic fluids. The turbulent Prandtl number distribution is found to be complex and dependent on its molecular counterpart within the diffusive sublayer. It is compared with the distributions implied by several phenomenological models, and the consequences for the treatment of the near-wall region in numerical calculation procedures for thin shear layers are discussed.

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Why is it important?

The turbulent Prandtl number variation within the near-wall region of boundary lawyer flows, where the total stress/heat-flux remains essentially constant normal to the wall, has been obtained using profile analysis. This was achieved by deriving expressions for the corresponding distributions for the edcly diffusivities of momentum and heat from Spalding's inner layer velocity profile formula together with its thermal analogue. This approach ensures that the limiting conditions at high and low turbulence Reynolds numbers are satisfied explicitly. The turbulent Prandtl number variation was found to be complex, and molecular Prandtl number dependent within the diffusive sublayer, where the molecular and Reynolds fluxes are of comparable magnitude. These rather exotic distributions are a consequence of the inherent limitations imposed by the framework within which the concept of the turbulent Prandtl number was originally developed. It is therefore unlikely that any further insight into the transport mechanisms within the near-wall region can be obtained using this conceptual framework. Nevertheless, improved computations are possible via the development of turbulent Prandtl number models which better reflect the inner lawyer behaviour indicated by the present analysis.


The time-averaged momentum and thermal energy equations for turbulent thin shear layers contain apparent or 'Reynolds' shear stress and heat-flux terms that need to be modelled in order to close the equation set. Many and varied turbulence models have been devised for the Reynolds shear stress, whereas by far the most common practice for modelling the corresponding heat flux is to prescribe a value for the 'turbulent Prandtl number': the ratio of the turbulent or 'eddy' diffusivities for the momentum and heat. The latter diffusivities appear in the classical Boussinesq formulations for the total (molecular plus turbulent) shear stress and heat-flux which may be written, using conventional near-wall scaling. Thus, the present contribution seeks to resolve the uncertainty, outlined above, over the behaviour of the turbulent Prandtl number within the inner layer.

Professor Emeritus Geoffrey P Hammond
University of Bath

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This page is a summary of: Turbulent Prandtl number within a near-wall flow, AIAA Journal, November 1985, American Institute of Aeronautics and Astronautics (AIAA),
DOI: 10.2514/3.9149.
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