A simple variational criterion for turbulent flow in pipe
Chemical Engineering Science, 1999, 54(8): 1151-1154
Jinghai Li*, Zhongdong Zhang, Wei Ge, Qicheng Sun, Jie Yuan
Fig. 1 compares the calculated radial velocity profiles from Model 1 at different Re for both air and water flow with experimental data correlated by Karman (1939) and Bejan (1982). The subfigures at the top are calculated with α = 0, as solved analytically from Model 2, showing the parabolic velocity distribution well known for laminar flow, that is, laminar flow in pipe distributes its volume flux according to W ̅ v= min which is now an absolute minimum without any constraint of inertial effect. With α = 1, a uniform velocity distribution is calculated from Model 1 for any Re, as shown in the subfigures at the bottom for both air and water, leading to an absolute maximum dissipation without any constraint of viscous effect.
The three subfigures in the middle of Fig. 1 for both air and water show radial velocity profiles for turbulent flow in pipe calculated from Model 1 at different Re with 0<α<1 (constant velocity gradient in the wall region from r = 0.9999R to r = R is assumed). With increasing Re, the boundary layer becomes thinner and thinner due to increasing inertial effect. It should be noted that as long as Re is identical the velocity profiles are the same no matter whether the fluid is air or water, and whether pipe diameters are the same or not.
Fig. 2 shows the dependence of α on Re, showing identical α for both air and water at the same Re, which increases with increasing Re in the turbulent regime, indicating increasing inertial dominance. The values of α are determined by comparison with experimental data. Further work is needed to formulate this parameter.