Prandtl number

The Prandtl number (Pr) is a dimensionless number that expresses the relationship between kinematic diffusivity (momentum diffusivity) and thermal diffusivity, for a viscous fluid.

In other words, it allows to evaluate the amplitude of the mechanically perturbed thermodynamic system area, with respect to the thermally perturbed area and therefore, to understand the relative importance between the influence of the mechanical action and the thermal one.

The Prandtl number is also defined as the ratio between the mechanical boundary layer and the thermal boundary layer:

\[\mathrm{Pr}=\dfrac{\delta_m}{\delta_t}\simeq\dfrac{\nu}{\alpha}\]

where \(\nu\) is the kinematic viscosity and \(\alpha\) the thermal diffusivity.

Turbulent Prandtl number

The turbulent Prandtl number (Prt) is a non-dimensional term defined as the ratio between the momentum eddy diffusivity and the heat transfer eddy diffusivity. The introduction of eddy diffusivity and subsequently the turbulent Prandtl number works as a way to define a simple relationship between the extra shear stress and heat flux that is present in a turbulent flow. If the momentum and thermal eddy diffusivities are zero (no apparent turbulent shear stress and heat flux), then the turbulent flow equations reduce to the laminar equations.

\[\mathrm{Pr}_{\mathrm{t}}=\dfrac{\varepsilon_M}{\varepsilon_H}\]

Where \(\varepsilon_M\) is the eddy diffusivities for momentum transfer and \(\varepsilon_H\) the heat transfer. It is useful for solving the heat transfer problem of turbulent boundary layer flows. The simplest model for Prt is the Reynolds analogy, which yields a turbulent Prandtl number of 1. From experimental data, Prt has an average value of 0.85 but ranges from 0.7 to 0.9 depending on the Prandtl number of the fluid in question.

Magnetic Prandtl number

The Magnetic Prandtl number (Prm) is the magnetic analog of the Prandtl number, a dimensionless quantity occurring in magneto-hydro-dynamics which approximates the ratio of momentum diffusivity (viscosity) and magnetic diffusivity. It is defined as:

\[\mathrm{Pr}_\mathrm{m} = \dfrac{\mathrm{Re_m}}{\mathrm{Re}} = \dfrac{\nu}{\eta} = \dfrac{\mbox{viscous diffusion rate}}{\mbox{magnetic diffusion rate}}\]

where: Rem is the magnetic Reynolds number; Re is the Reynolds number; \(\nu\) is the momentum diffusivity (kinematic viscosity); \(\eta\) is the magnetic diffusivity.

References

  1. Turbulent Prandtl number. Wikipedia. https://en.wikipedia.org/wiki/Turbulent_Prandtl_number
  2. Magnetic Prandtl number. Wikipedia. https://en.wikipedia.org/wiki/Magnetic_Prandtl_number

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