6.11 Turbulent mixing

Turbulent flow is characterised by significant mixing of fluid eddies as the Reynolds experiment in Sec. 6.1 shows. CFD simulations generally need to accommodate turbulent mixing since it influences the diffusion of mass, momentum and energy. While the fluid mixing by mass diffusion itself can be important, the effect on momentum diffusion is often critical because it impacts the calculation of viscous forces and, thus, the flow itself.

Eddy viscosity

Boussinesq was the first to devise a model for turbulence. He recognised the similarity between the random motion of both eddies in a turbulent fluid and particles at a molecular scale.

PICT\relax \special {t4ht=

By analogy to kinetic theory in Sec. 6.10 , shear stresses due to turbulence are caused by the net momentum, tangential to a plane, due to the motion of eddies. Boussinesq related this shear stress to the velocity gradient through an eddy viscosity eqn.14

He presented the turbulent stress eqn in tensor form, including a pressure. Kinetic theory relates pressure to fluctuations15 in particle velocity eqn by eqn; the kinetic energy associated with the fluctuations is eqn. Applying the same argument to velocity fluctuations eqn due to turbulence, leads to a turbulent “pressure” eqn, where eqn is the turbulent kinetic energy per unit mass.

By analogy with the Newtonian fluid model Eq. (2.41 ), the eddy viscosity model of Boussinesq, incorporating eqn, is

|-------------------------------| ☐☐☐t =|2 tdev D; ☐☐☐t = ☐☐☐t 2- kI | --------------------------3------ \relax \special {t4ht=
where eqn is the viscous component of Reynolds stress and eqn is the deformation rate tensor defined in Eq. (2.33 ). Inevitably Eq. (6.20 ) and Eq. (2.41 ) closely resemble one another.

The model of Eq. (6.20) requires some means of calculating eqn. Kinetic theory gives a quantitative prediction of eqn in Eq. (6.19 ) which led Boussinesq to hypothesise that eqn, where eqn and eqn are a representative speed and length, respectively, with the speed eqn relating to eqn due to turbulence.

The turbulent viscosity can also be expressed as eqn by absorbing the constant of proportionality within a characteristic speed eqn and a mixing length eqn, discussed in Sec. 6.12 .

Eddy viscosity and mixing length are useful concepts in turbulence modelling. However, it should be recognised that there are limitations in the analogy with kinetic theory, e.g.:

  • momentum is exchanged between submicroscopic particles through intermittent, discrete collisions, compared to the continuous interaction between eddies;
  • the magnitude of random particle motions is generally equal in all directions, whereas the level of turbulent fluctuations can sometimes vary significantly with direction.

14Joseph Boussinesq, Essai sur la théorie des eaux courantes, 1877.
15Daniel Bernoulli, Hydrodynamica, sive de viribus et motibus fluidorum commentarii, 1738.

Notes on CFD: General Principles - 6.11 Turbulent mixing