Low-Re 
Model of Lien et al. (1996)
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
and
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:
where
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):
where 
.
The dissipation rate 
is obtained from the solution of a related transport equation:
where
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.
