A Rayleigh-quotient model for identifying hydrodynamic eigenvalues

The previous paper computed the hydrodynamic spectrum of a flexible rectangular plate, confirmed that the eigenvalues organise into branches, and found that within a branch the chordwise wavenumber varies almost linearly with the imaginary part of the reciprocal eigenvalue. It established the framework and an empirical scaling law, and left two questions unanswered: why the spectrum is organised into branches at all, and whether the eigenvalue and the shape of its eigenvector can each be predicted from the other.

This paper answers both. It also corrects the earlier reading. Spanwise variation was previously interpreted as occurring from mode to mode along each branch; the present analysis shows the spanwise order is constant within a branch and changes only from one branch to the next.

The computed mode shapes factor, to a good approximation, into a chordwise part times a spanwise part. Along the chord they are travelling waves with an exponentially varying amplitude envelope running from leading to trailing edge. Across the span they are standing waves.

That gives three descriptors and nothing else: a chordwise amplitude variation rate, positive for amplification and negative for attenuation; a chordwise travelling wavenumber; and an integer counting spanwise half-waves. Together they form a complex chordwise wavenumber and a spanwise mode index.

The complex chordwise wavenumber comes out of the eigenvector by projecting the discrete chordwise differentiation operator onto the mode, a Rayleigh quotient in its own right. Equivalently it can be recovered by least-squares fits along the chord: the log magnitude against position gives the amplitude rate, the phase against position gives the wavenumber. Both fits run station by station across the span and are averaged with weights set by the energy at each station.

The spanwise index needs more care. Multiplying an eigenvector by a complex constant of unit magnitude leaves the eigenproblem unchanged but flips signs throughout the real part, so counting sign changes directly is unreliable. The count is therefore taken only after the chordwise factor has been divided out, by projecting the spanwise profile at the most energetic chordwise station onto a half-sine basis and taking the harmonic with the largest projection.

Two operators appear in the governing equations. The kinematic one maps the prescribed surface displacement to the normal velocity it requires, and it acts on the separable ansatz in closed form. The hydrodynamic one maps circulation to the normal velocity it induces and carries the memory of the shed wake; it is dense and has no such closed form.

The move is to represent that dense operator by a single scalar: its Rayleigh quotient evaluated on the trial wave. Combining the two then gives the eigenvalue directly as a Rayleigh quotient of the trial wave, turning a large eigenproblem into an expression in three numbers.

The explanation falls out of the parameterisation. Two of the three descriptors are continuous and one is an integer. Within a branch the spanwise order is held at a single integer value while the chordwise rate and wavenumber vary continuously along it; moving to the next branch means stepping the integer.

A continuous sweep in two parameters, indexed by a discrete third, is precisely a family of curves rather than a filled region. That is why the eigenvalues do not fill the complex plane uniformly, and the previously observed linear relationship within a branch is recovered analytically rather than fitted.

The reduced model reproduces the computed spectrum to within about two per cent across a range of chord-to-span ratios and reduced frequencies, and it recovers the real and imaginary parts of each eigenvalue separately rather than only getting the magnitude right.

The approximation enters in exactly one place and it is worth being precise about where. The projection and the least-squares fits recover the descriptors exactly from an exact eigenvector; the error appears only when those descriptors are used to rebuild a model mode and that mode is fed back through the quotient.

A second result follows from the model, and it is the one with the clearest physical consequence. The reciprocal eigenvalue has the dimensions of a velocity and can be read as a modal phase speed, which is why the spectrum is most naturally examined in those terms.

The sign of its imaginary part then separates the modes that produce thrust from those that extract energy from the flow, with the boundary lying at the downstream-travelling wave whose phase speed equals the free-stream speed. Across the whole spectrum there is no exception in the sign of the power. The eigenvalue alone therefore determines the energetic role of its mode, which makes the route run both ways: from the shape of a mode to its eigenvalue, and from the eigenvalue back to the mode.

With Amanda Smyth and Luca Di Mare, Department of Engineering Science, University of Oxford. In preparation.

Where the earlier work introduced and exploited the eigenmode framework, this explains and predicts the structure of the spectrum that framework produces.