NONLINEAR STABILITY OF KOLMOGOROV FLOW WITH BOTTOM-FRICTION USING THEENERGY METHOD

Stability of Kolmogorov flow: U = - sin y to any finite disturbances i s treated by using the energy method. The linear damping term -lambda u due to the bottom friction is taken into account. The Euler-Lagrange equation is solved numerically and analytically to determine the crit ical Reynolds number, R(Ec), below which subcritical instability canno t occur. It is shown numerically that R(Ec) and the linear critical Re ynolds number, R(Lc) are of the same order in 0 < lambda < 200. The cr itical wavenumber, (alpha(Ec), beta(Ec)) is always (0, 0) when lambda < 49.1; otherwise alpha(Ec) not equal 0. By using a small wavenumber e xpansion, it is obtained that R(E) = [8 lambda(lambda + 1)](1/2) at al pha = 0. In the limit lambda --> infinity, numerical results suggest t hat R(Ec) --> 2 lambda and alpha(Ec)--> infinity. In this limit for ge neral parallel flow U(y) the relation: (2/M)lambda < R(Ec) < R(Lc) = ( 1/sigma(0))lambda is obtained analytically where M = max(y) \partial d erivative U/partial derivative y\, and sigma(0) is the inviscid maximu m growth rate.