@Article{AAMM-3-2,
author = {Hua-Shu, Dou and Khoo, Cheong, Boo},
title = {Investigation of Turbulent Transition in Plane Couette Flows Using Energy Gradient Method},
journal = {Advances in Applied Mathematics and Mechanics},
year = {2011},
volume = {3},
number = {2},
pages = {165--180},
abstract = {<p style="text-align: justify;">The energy gradient method has been proposed with the aim of better
understanding the mechanism of flow transition from laminar flow to
turbulent flow. In this method, it is demonstrated that the transition
to turbulence depends on the relative magnitudes of the transverse gradient
of the total mechanical energy which amplifies the disturbance and the energy
loss from viscous friction which damps the disturbance, for given imposed
disturbance. For a given flow geometry and fluid properties, when the maximum
of the function $K$ (a function standing for the ratio of the gradient of total
mechanical energy in the transverse direction to the rate of energy loss due to
viscous friction in the streamwise direction) in the flow field is larger than a
certain critical value, it is expected that instability would occur for some
initial disturbances. In this paper, using the energy gradient analysis, the
equation for calculating the energy gradient function $K$ for plane Couette flow
is derived. The result indicates that $K$ reaches the maximum at the moving walls.
Thus, the fluid layer near the moving wall is the most dangerous position to generate
initial oscillation at sufficient high $\operatorname{Re}$ for given same level of
normalized perturbation in the domain. The critical value of $K$ at turbulent transition,
which is observed from experiments, is about 370 for plane Couette flow when two walls
move in opposite directions (anti-symmetry). This value is about the same as that for
plane Poiseuille flow and pipe Poiseuille flow (385-389). Therefore, it is concluded
that the critical value of $K$ at turbulent transition is about 370-389 for wall-bounded
parallel shear flows which include both pressure (symmetrical case) and shear driven
flows (anti-symmetrical case).</p>},
issn = {2075-1354},
doi = {https://doi.org/10.4208/aamm.10-m1017},
url = {https://global-sci.com/article/73587/investigation-of-turbulent-transition-in-plane-couette-flows-using-energy-gradient-method}
}