A technique to measure wavenumber mismatch between quadratically interacting modes

Nonlinear energy cascade by means of three-wave resonant interactions is a characteristic feature of transitioning and turbulent flows. Resonant wavenumber mismatch between these interacting modes can arise from the dispersive characteristics of the interacting waves and from spectral broadening due to random effects. In this paper, a general technique is presented to estimate the average level of instantaneous wavenumber mismatch, k=k_m-k_i-k_j, between components whose frequencies obey the resonant selection condition, f_m-f_i-f_j=o. Cross-correlation of the auto-bispectrum is used to quantify the level of mismatch. The concept of bispectrum coupling coherency is introduced to determine the confidence level in the wavenumber mismatch estimates. These techniques are then applied to measure wavenumber mismatch in the transitioning field of a plane wake. The results show that the average of the instantaneous mismatch between the actual interacting modes k_m-k_i-k_j is in general not equal to the mismatch between the average wavenumbers of each interacting mode k_m-k_i-k_j.The authors wish to express their gratitude to Dr. Christoph P. Ritz of the Swiss National Science Foundation for his comments in the preparation of this work. This paper is based in part upon work supported by the Texas Advanced Research Program under Grant No. ARP 3280, and in part by the National Science Foundation under Grant No. MSM-82112O5. The digital signal processing techniques were developed under the Office of Naval Research under contract number N00014-88-k-0638