Adjoint sensitivity and optimal perturbations of the low-speed jet in cross-flow

Abstract

The tri-global stability and sensitivity of the low-speed jet in cross-flow are studied using the adjoint equations and finite-time horizon optimal disturbance analysis at Reynolds number $Re=2000$, based on the average velocity at the jet exit, the jet nozzle exit diameter and the kinematic viscosity of the jet, for two jet-to-cross-flow velocity ratios $R=2$ and $4$. A novel capability is developed on unstructured grids and parallel platforms for this purpose. Asymmetric modes are more important to the overall dynamics at $R=4$, suggesting increased sensitivity to experimental asymmetries at higher $R$. Low-frequency modes show a connection to wake vortices. Adjoint modes show that the upstream shear layer is most sensitive to perturbations along the upstream side of the jet nozzle. Lower frequency downstream modes are sensitive in the cross-flow boundary layer. For $R=2$, optimal analysis reveals that for short time horizons, asymmetric perturbations dominate and grow along the counter-rotating vortex pair observed in the cross-section. However, as the time horizon increases, large transient growth is observed along the upstream shear layer. When $R=4$, the optimal perturbations for short time scales grow along the downstream shear layer. For long time horizons, they become hybrid modes that grow along both the upstream and downstream shear layers.

Effect of tabs on transverse jet instabilities, structure, vorticity dynamics and mixing

“…Also in agreement with this prior study, we observed a decrease in the severity of CVP alteration as the tab was moved further from the upstream region of the jet, where tab positions past L60/R60, towards the downstream region of the jet, showed few perceptible differences in the cross-section and centreplane as compared with the non-tabbed case. These findings on the sensitivity of tab placement to its proximity to the upstream region also were consistent with the proposed wavemaker region of an AU JICF from adjoint sensitivity analysis of the DNS by Regan & Mahesh (2019), incorporated the same geometry as in the original nozzle in our experiments.…”

“…For low momentum flux ratios, especially at but also at , the structures were contained entirely within the USL of the jet, while at higher momentum flux ratios, as shown by modes one and two at (figure 19 a (1)– a (2)), the dominant mode structures spanned the upstream and downstream shear layers, and appeared to be merged. This was analogous to the expansion of the wavemaker region to the downstream shear layer as documented in the DNS by Regan & Mahesh (2019), and the origination of the dominant eignenmode on the downstream side of the jet as shown in Regan & Mahesh (2017), both for a CU jet at and . It is also noted that the wake structures were more pronounced in figures 17( a )–19( a ) as the momentum flux ratio was decreased and as the jet became more bent over.…”

“…Yet it is interesting that the USL for J = 7 was also influenced by tab positioning away from the upstream, even at the downstream location, in contrast to a lack of significant influence for such tab positions on the naturally CU case of J = 61. These behaviours in fact agree with the computationally predicted wavemaker regions of both convectively and AU low speed jets in crossflow, that is, the regions of greatest sensitivity to perturbations within the flow field (Regan & Mahesh 2019). These DNS indicate that a naturally AU transverse jet has a wavemaker region in the upstream near-field region of the jet, whereas the wavemaker region for a naturally CU JICF extends along the entire USL of the jet and wraps around to the downstream side of the jet.…”

“…Yet the effects of the tab position on the jet's cross-sectional structure, even when there was little obvious change in the USL, were quite significant (figure 8), leading to apparent flipping of the asymmetric CVP-like structure, though the specific downstream location at which the image slice was taken could influence this orientation. While there was a logical connection between tab-generated shear layer characteristics and jet structural alteration, a more consistent connection was observed between the predicted wavemaker regions for AU and CU jets from the work of Regan & Mahesh (2019) and the tab orientations having the greatest effects. For naturally AU shear layers such as with J = 7, the wavemaker region in the upstream region was consistent with the effectiveness of tab placement in that region (e.g.…”

“…While our findings cannot be explained by USL spectral changes alone, they are supported by the proposed wavemaker region of a CU JICF. Adjoint sensitivity analysis of the DNS by Regan & Mahesh (2019) for a JICF with indicates that the wavemaker extends along the entire USL of the jet, and even wraps around towards the downstream side of the jet as the flow is deflected and reoriented with the crossflow. In light of this, it may be that the wavemaker region continues to grow around the periphery of the jet nozzle exit as the momentum flux ratio is increased until the downstream side of the jet is enveloped.…”