Studies on secondary instabilities

Abstract

Significant advances in understanding early stages of transitional flows have been achieved by studying secondary instabilities in selected prototype flows. These secondary instabilities can be modeled as parametric instabilities of the nearly periodic flow that consists of the prototype velocity profile and a superposed finite-amplitude TS-wave (wavelength λ). The generally three dimensional secondary instabilities are governed by a linearized system of partial differential equations with periodic coefficients which are reduced to an algebraic eigenvalue problem through the application of a spectral collocation method

Following Floquet theory, previous analysis looked for subharmonic (wavelength 2 λ) and fundamental (wavelength λ) types of solutions. We extend the Floquet theory to solutions having arbitrary wavelengths, hence including the previous solutions as special cases. Modes with wavelength in between the subharmonic and fundamental values are called detuned modes. Detuned modes lead to combination resonance which has been observed in controlled transition experiments. Knowledge of the bandwidth of amplified detuned or (combination) modes is very important for clarification of the selectivity of the early stages of transition with respect to initial disturbances.

We have selected two flows: the Blasius boundary layer flow and the hyperbolic-tangent free-shear flow as prototypes of wall bounded Hows and unbounded Hows, respectively. In the Blasius flow we have concentrated on studying detuned modes. We found the growth rates of modes slightly detuned from the subharmonic wavelength to be almost as large as the growth rate of the subharmonic itself. This result is consistent with both the broadband spectra centered at subharmonic frequency observed in the "biased" experiment of Kachanov & Levchenko, wherein only the TS frequency was introduced, and with the large band-width of resonance in the "controlled" experiments, wherein a TS wave and the detuned modes were introduced simultaneously.

In the free-shear flow, our goals were three-fold. The first was to investigate whether the Floquet analysis based on the shape assumption for TS waves would provide results consistent with results for the stability of Stuart vortices. Second, we aimed at revealing the effect of viscosity on these results. Finally, we wanted to evaluate a group of spectral methods for the numerical treatment of the flow in an unbounded domain. We have made a detailed analysis of subharmonic, fundamental, and detuned modes. Results display the basically inviscid, convective character of the secondary instabilities, and their broadband nature in the streamwise and spanwise directions. In the inviscid limit, and for neutral TS waves, a detailed comparison is made with the closely related study on stability of Stuart vortices by Pierrehumbert & Widnall. Good quantitative agreement is obtained. For a wide range of Reynolds numbers and amplitudes of the 2-D primary wave, results reveal that the most unstable subharmonic modes are two-dimensional (vortex pairing). On the other hand, the most unstable fundamental modes are three-dimensional, with short spanwise wavelengths. Detuned modes have characteristics in between, being most unstable in the two-dimensional or three-dimensional form depending on the detuning value.

Comparisons of our results for a superposed TS wave of constant amplitude with results obtained by numerical simulations suggested that the growth of the TS wave may have a significant effect on the secondary disturbance growth. To check this hypothesis, we have developed a numerical method that accounts for small variations in the TS amplitude. However, the results indicate that the discrepancies are due to other yet concealed effects.