Beyond secondary instability: on the emergence of finite-amplitude waves in Görtler vortices

TL;DR

This paper addresses predicting finite-amplitude waves on Görtler vortices in a boundary layer over a concave wall, going beyond traditional secondary-instability analysis. It develops and applies the Parabolised Coherent Structures (PCS) method, which couples nonlinear vortex–wave interaction to a parabolised, spatially marching framework by integrating the Reynolds-stress feedback from traveling waves into the mean flow. The PCS reproduces the SB87 experimental observations of wave amplitude and displacement thickness up to about $x^*\approx110$ cm and provides growth-rate predictions that align better with experiments than linear secondary-instability theory, demonstrating a viable route to non-parallel VWI in boundary layers. The work also connects exact coherent structures with practical transition pathways, suggesting PCS as an efficient tool for non-parallel boundary-layer analyses and potential extensions to receptivity and bypass-transition scenarios.

Abstract

Görtler vortices developing over a concave wall support rapidly oscillating wavelike disturbances through secondary instabilities. Although experiments indicate that the finite-amplitude evolution of these waves acts as a precursor to turbulence transition, accurate and efficient prediction has remained out of reach. We overcome this limitation by using the Parabolised Coherent Structures (PCS) method of Song & Deguchi (2025), which incorporates the nonlinear vortex-wave interaction into a standard spatial-marching approach. Our computations successfully reproduce the wave amplitude and displacement thickness observed in the widely known experiments of Swearingen & Blackwelder (1987).

Figures (8)

• Figure 1: Displacement boundary layer thickness $\delta^*_{\text{disp}}$ measured at the spanwise locations corresponding to the peaks (up triangle, solid lines) and valleys (down triangle, dashed lines) of the Görtler vortices. The symbols are the experimental results taken from figure 9 of SB87, while the lines show our computational results. The inset shows the streamwise velocity from the BRE computation at $x^*=90$[cm] (in the same format as figure \ref{['fig:fig4_new']}). The magenta lines and symbols indicate the positions of the peaks and valleys of the mushroom-shaped vortices.
• Figure 2: (a) Neutral curve in the $x^*$--$f^*$ plane resulting from the secondary instability analysis. The filled circles are the linear critical points used in the PCS computations in panel (b). The open circle corresponds to the analysis in figure \ref{['fig:high_low_different_R_combine']}-(b). (b) Local wavelength of the finite-amplitude wave obtained using the PCS method (the magenta, red, and green lines correspond to $f^* = 157$, 170, and 185 [Hz], respectively). The black line shows the secondary instability analysis results for the second odd mode.
• Figure 3: A snapshot of the flow field computed using the PCS. The colourmap at the selected streamwise positions shows the steady streak field $\overline{u}$. Red/blue isosurfaces are 20% maximum/minimum of the streamwise vorticity of the wave component, $\partial_y\tilde{w}-\partial_z \tilde{v}$.
• Figure 4: The flow field at $x^*=100$ [cm]. The black lines show contours of $\overline{u}$ at $0.1,0.2,\dots,0.9$, with the thick line indicating 0.8. The red lines are contours of $\tilde{u}_{\text{rms}}=0.01,0.02,$ and 0.03, with the thick line highlighting 0.02. (a) BRE, (b) PCS, (c) experimental results from figures 11 and 16 of SB87.
• Figure 5: Downstream growth of the wave amplitude. (a) Amplitude of the wave field. Circles denote the experimental results taken from figure 17 in SB87. Lines are the PCS results shown in figure \ref{['fig:fig2']}-(b). (b) Growth rate $\sigma^*=\sigma/L^*$, where $\sigma(X)=\frac{1}{\tilde{u}_{\text{max}}}\frac{d \tilde{u}_{\text{max}}}{dX}$. For a fair comparison, finite-difference approximations are applied to both the experimental (circles) and PCS (diamonds) results. The line shows the growth rate of the second odd mode predicted by the secondary instability analysis. Both computational results are for$f^*=170$ [Hz].