Dynamical system approach to instability of flow past a circular cylinder

The main aim of this paper is to relate instability modes with modes obtained from proper orthogonal decomposition (POD) in the study of global spatio-temporal nonlinear instabilities for flow past a cylinder. This is a new development in studying nonlinear instabilities rather than spatial and/or temporal linearized analysis. We highlight the importance of multi-modal interactions among instability modes using dynamical system and bifurcation theory approaches. These have been made possible because of accurate numerical simulations. In validating computations with unexplained past experimental results, we noted that (i) the primary instability depends upon background disturbances and (ii) the equilibrium amplitude obtained after the nonlinear saturation of primary growth of disturbances does not exhibit parabolic variation with Reynolds number, as predicted by the classical Stuart–Landau equation. These are due to the receptivity of the flow to background disturbances for post-critical Reynolds numbers (Re) and multi-modal interactions, those produce variation in equilibrium amplitude for the disturbances that can be identified as multiple Hopf bifurcations. Here, we concentrate on Re = 60, which is close to the observed second bifurcation. It is also shown that the classical Stuart–Landau equation is not adequate, as it does not incorporate multi-modal interactions. To circumvent this, we have used the eigenfunction expansion approach due to Eckhaus and the resultant differential equations for the complex amplitudes of disturbance field have been called here the Landau–Stuart–Eckhaus (LSE) equations. This approach has not been attempted before and here it is made possible by POD of time-accurate numerical simulations. Here, various modes have been classified either as a regular mode or as anomalous modes of the first or the second kind. Here, the word anomalous connotes non-compliance with the Stuart–Landau equation, although the modes originate from the solution of the Navier–Stokes equation. One of the consequences of multi-modal interactions in the LSE equations is that the amplitudes of the instability modes are governed by stiff differential equations. This is not present in the traditional Stuart–Landau equation, as it retains only the nonlinear self-interaction. The stiffness problem of the LSE equations has been resolved using the compound matrix method.

Effects of surface film on the linear stability of an air–sea interface

The linear stability of turbulent shear flow over a film-covered sea surface is studied theoretically. A compound matrix method (Wheless & Csanady 1993), is used to solve the eigenvalue problem numerically. The numerical method has been adjusted to a coupled air–sea system. In the stability problem the vertical component of the turbulent Reynolds stress has been taken into account. As pointed out by Wheless & Csanady, the second derivative of the traditional log–linear wind profile has a rather extreme behaviour near the matching point of the linear and logarithmic part. To improve the model, a new profile is calculated based on an eddy viscosity distribution for channel flow (Quarmby & Anand 1969), which has continuous derivatives all the way down to the surface. Calculations of the wave growth rates corresponds well with earlier theoretical results as well as laboratory measurements. The energy flux from the air to the sea caused by the pressure work at the surface has been calculated. An intriguing result obtained here is that this flux seems to be strongly dependent on the elastic property of the surface film. The flux attains a maximum for finite values of the film elasticity parameter.

Instability waves on the air–sea interface

We used a compound matrix method to integrate the Orr–Sommerfeld equation in an investigation of short instability waves (λ < 6 cm) on the coupled shear flow at the air–sea interface under suddenly imposed wind (a gust model). The method is robust and fast, so that the effects of external variables on growth rate could easily be explored. As expected from past theoretical studies, the growth rate proved sensitive to air and water viscosity, and to the curvature of the air velocity profile very close to the interface. Surface tension had less influence, growth rate increasing somewhat with decreasing surface tension. Maximum growth rate and minimum wave speed nearly coincided for some combinations of fluid properties, but not for others.The most important new finding is that, contrary to some past order of magnitude estimates made on theoretical grounds, the eigenfunctions at these short wavelengths are confined to a distance of the order of the viscous wave boundary-layer thickness from the interface. Correspondingly, the perturbation vorticity is high, the streamwise surface velocity perturbation in typical cases being five times the orbital velocity of free waves on an undisturbed water surface. The instability waves should therefore be thought of as fundamentally different flow structures from free waves: given their high vorticity, they are akin to incipient turbulent eddies. They may also be expected to break at a much lower steepness than free waves.