On the preferred flapping motion of round twin jets

TL;DR

This work analyzes the dominant motion in closely spaced twin round jets using linear stability theory (LST). It employs two complementary formulations—(i) a complete compressible Navier–Stokes discretization in Cartesian coordinates for finite-thickness jets and (ii) an inviscid vortex-sheet model—to compute the eigenmodes and the dispersion relations, focusing on the $m=1$ family. The results demonstrate that jet coupling shifts the eigenvalues away from helical motion, promoting non-helical flapping (varicose and sinuous) modes, with a stripe-like dependence on Strouhal number $St$ and jet separation $s/D$; a parametric map shows the leading mode changing with $(St,s/D)$ and with jet Mach number $M_j$ and temperature ratio $T_R$. The findings offer a framework for understanding twin-jet screech and downstream nozzle-plane pulsations, and suggest that SA1 and SS1 will dominate under significant coupling, while caution is warranted for shock-containing jets due to resonance effects.

Abstract

Linear stability theory (LST) is often used to model the large-scale flow structures in the turbulent mixing region and near pressure field of high-speed jets. For perfectly-expanded single round jets, these models predict the dominance of $m=0$ and $m = 1$ helical modes for the lower frequency range, in agreement with empirical data. When LST is applied to twin-jet systems, four solution families appear following the odd/even behaviour of the pressure field about the symmetry planes. The interaction between the unsteady pressure fields of the two jets also results in their coupling. The individual modes of the different solution families no longer correspond to helical motions, but to flapping oscillations of the jet plumes. In the limit of large jet separations, when the jet coupling vanishes, the eigenvalues corresponding to the $m=1$ mode in each family are identical, and a linear combination of them recovers the helical motion. Conversely, as the jet separation decreases, the eigenvalues for the $m=1$ modes of each family diverge, thus favouring a particular flapping oscillation over the others and preventing the appearance of helical motions. The dominant mode of oscillation for a given jet Mach number $M_j$ and temperature ratio $T_R$ depends on the Strouhal number $St$ and jet separation $s$. Increasing both $M_j$ and $T_R$ independently is found to augment the jet coupling and modify the $(St,s)$ map of the preferred oscillation mode. Present results predict the preference of two modes when the jet interaction is relevant, namely varicose and especially sinuous flapping oscillations on the nozzles plane.

Figures (8)

• Figure 1: Twin-jet configuration and geometry, showing the different coordinate systems employed.
• Figure 2: LST eigenspectra corresponding to $M_j=1.5, T_R = 1, R/\theta = 12.5$ and $St=0.3$ and the four solution families: SS ($+$), AS ($\circ$), SA ($\times$), AA ($\square$). ($a$) Single jet; ($b$) Twin jet with separation $s/D = 2.2$.
• Figure 3: Pressure eigenfunctions corresponding to a single jet at $M_j=1.5, T_R = 1, R/\theta = 12.5$ and $St = 0.3$. The streamwise dependence is obtained from the corresponding eigenvalue, eliminating the spatial growth. Left column: iso-contours of the real pressure component. The grey circle shows the nozzle circumference. Right column: phase angle $\phi$ at a cylinder of radius 0.75$D$. ($a,d$) SA1; ($b,e$) AS1; ($c,f$) Linear combination of SA1 and AS1 to produce the $m=1$ helical mode.
• Figure 4: Dependence of the $m=1$ eigenvalues on the jet separation $s/D$: ($a$) In the complex $k$ plane. The eigenvalues spiral outwards with $s/D$ decreasing from 5 to 1.8. ($b$) Real and ($c,d$) imaginary parts. Panel ($d$) is a zoom in of panel ($c$). $M_j=1.5, T_R = 1, R/\theta = 12.5$ and $St= 0.3$. SS ($+$), AS ($\circ$), SA ($\times$), AA ($\square$). The horizontal dashed line corresponds to the $m=1$ mode of the single jet.
• Figure 5: Pressure eigenfunctions corresponding to twin jets with $s/D = 2.2$ at $M_j=1.5, T_R = 1, R/\theta = 12.5$ and $St = 0.3$. The streamwise dependence is obtained from the corresponding eigenvalue, eliminating the spatial growth corresponding to SA1. Left column: iso-contours of the real pressure component. The grey circles shows the nozzle circumference. Right column: phase angle $\phi$ at a cylinder of radius 0.75$D$ centred on one jet.($a$) SA1; ($b$) AA1; ($c$) Linear combination of SA1 and AA1.