7.12 Heat transfer in turbulent flow

The initial focus of turbulence modelling is to capture the effect of mixing on momentum diffusion since it influences the overall flow solution. But other properties are also transported by the turbulent eddying motions, in particular heat.

The effects of turbulence on heat transfer can be described using the following equation for internal energy eqn, obtained by substituting the material derivative in Eq. (2.57 ) and ignoring eqn:

@ e ----+ r (u e) = r q + ☐☐☐ ru p(r u): @t \relax \special {t4ht=
In turbulent flow, internal energy can be decomposed into averaged and fluctuating components eqn, see Eq. (6.11 ). The ensemble average of the terms in eqn introduces a heat flux
 ---- qt = u0e0: \relax \special {t4ht=
This additional heat flux in the energy equation is equivalent to the Reynolds stress eqn, Eq. (6.14 ), in momentum. Boussinesq modelled eqn by Eq. (6.20 ), using the concept of an eddy viscosity eqn corresponding to turbulent mixing, analogous to viscosity due to molecular motion according to Newton’s law Eq. (2.41 ).

Similarly, eqn can be modelled using a turbulent thermal conductivity eqn due to turbulent mixing, by analogy with Fourier’s law Eq. (2.54 ) for conduction due to molecular interaction

q = rT: t t \relax \special {t4ht=
The total heat flux eqn in Eq. (7.44 ) is then expressed in terms of the combined turbulent mixing and molecular interaction, using an effective thermal conductivity eqn, as follows:
q = rT: eff \relax \special {t4ht=

Modelling turbulent heat transfer

Turbulent heat transfer can be incorporated into turbulence models based on eddy-viscosity and Reynolds-averaging, with additional thermal wall functions.

First, the calculation of eqn by Eq. (7.47) requires eqn from the turbulence model. A common approach to calculate eqn is from eqn based on an estimate of turbulent Prandtl number

 -t Prt = cp t: \relax \special {t4ht=
eqn provides a good estimate for many fluids, with eqn often chosen as a default value for CFD calculations. For some more unusual fluids, e.g. liquid metals, eqn.

Wall heat flux

The calculation of heat transfer through boundary walls is an important aspect of a many CFD simulations. Near walls, the distribution of eqn tends to mimic eqn.

PICT\relax \special {t4ht=

Consequently, the challenges of calculating wall heat flux eqn are similar to wall shear stress eqn. Cells close to the wall must be very thin to resolve the viscous sub-layer in eqn (when eqn).

Otherwise, wall functions can be used to adjust eqn to compensate for the under-prediction of eqn as described in Sec. 7.14 .

Notes on CFD: General Principles - 7.12 Heat transfer in turbulent flow