Low-Re Model of Lien et al. (1996)
Based on series-expansion arguments, a general and coordinate invariant relationship between stresses and strains can be written as (Pope, 1975):
where and to , proposed by Shih et al. (1993) and applicable only to high-Re region, are
To examine the effect of streamline curvature on turbulence, a third-order correction suggested by Suga (1995) is also included, giving rise to
where and .
The value in Eq. (2) plays an important role in capturing flow separation from a curved surface, such as the suction side of aerofoils and compressor blades. This can be explained by introducing local equilibrium constraints into the model, yielding:
needs to be solved iteratively by, say, the Newton-Raphson method. In the original model, , leading to:
To slightly suppress the shear stress, is adjusted from 1.25 to 4 (without violating realizability), giving rise to:
Although this is, admittedly, an ad-hoc modification, its effect on compressor blades and aerofoils at off-design condition is significant. More importantly, one has to ensure that such a change does not affect the model's performance for attached flows.
The turbulent viscosity , arising from modeling framework, is
In order to account for the semi-viscous near-wall effect, a damping function is introduced into (10), which, by reference to Norris/Reynolds' (1975) one-equation model, can be derived as (Lien Leschziner, 1995):
The dissipation rate is obtained from the solution of a related transport equation:
and . The is introduced to ensure that the correct level of near-wall turbulence-energy dissipation is returned. Then, by neglecting the , and within the viscous sublayer,
To improve the model's performance for transitional flow, the model has been modified to return, very close to the wall,
where the underlined term is used to replace the original proposal, , which is identical to the former but only in the local equilibrium condition. Finally, a damping function in the form of is introduced to ensure vanishes as , with the coefficient C correlated with log-law of the wall in fully-developed channel flow. The result is:
It is appropriate to acknowledge that the use of the wall-normal distance in the present form of the -equation is an undesirable feature, with a view to its application to complex geometries. On the other hand, the equation does not include, in contrast to distance-free forms [e.g. Launder Sharma (1974)], terms of the type which are difficult to expand in general 3D coordinates and also provoke a high level of sensitivity to near-wall grid resolution.