Under-expanded jets have been the subject of intense research since they were first described theoretically by Prandtl and his Göttingen group (Prandtl 1904, 1907, 1913; Ackeret 1927). The acoustic-flow instability associated with an under-expanded supersonic impinging jet is known to cause self-sustained oscillations (Henderson 1966; Donaldson et al. 1971). Such oscillations have been observed in other configurations, for instance, subsonic impinging jets (Ho & Nosseir 1981; Tam & Ahuja 1990), a resonance tube, an edge-tone and a plate with a cavity (Raman & Srinivasan 2009).
The accepted conceptual theory describing the feedback oscillations in supersonic impinging jets is due to Powell (1988), which is an analogue of the Rossiter mechanism found in open-cavity flow (Rossiter 1964). The feedback loop is based on the interactions between the aero-acoustic field, which in under-expanded supersonic jet flows has a number of sources, and the jet flows as evidenced in the ultrahigh-speed flow visualisation of Risborg & Soria (2009) and Soria & Risborg (2019). As can be clearly evidenced in these ultrahigh-speed schlieren visualisations, shear-layer disturbances develop in the form of Kelvin–Helmholtz (K–H) instabilities, which subsequently interact with the oblique shock, Mach disk and stand-off shock. These interactions displace the shocks, creating high-intensity acoustic waves at the shock locations and impinging region, which travel upstream towards the jet orifice via the quiescent flow domain outside of the supersonic jet. As the circumferential edge of the jet expands upon exit from the nozzle in an under-expanded supersonic jet, some or all of these acoustic waves are blocked from directly reaching the jet nozzle lip. However, the acoustic waves are also reflected by the surrounding structures and the reflected waves can reach the nozzle lip and can be internalised into the initial conditions.
In this project, high-fidelity large-eddy simulations have performed to address fundamental research questions associated with supersonic under-expanded impinging jets.
Receptivity mechanism at nozzle lip:
Receptivity has been studied since the 1970s with the main focus on receptivity in the simple configuration of the mixing layer formed by a splitter plate. Receptivity in the configuration of an under-expanded supersonic jet is crucially important. It is the main component of self-sustained oscillations. An innovative approach of an impulse response analysis is used to characterise the receptivity process in the under-expanded supersonic impinging jets. It was shown that receptivity is a function of the location of the incident pulse, frequency and azimuthal wavenumber of the input signal and nozzle-to-wall distance.
Karami, S., Stegeman, P., Ooi, A., Theofilis, V. and Soria, J. “Receptivity characteristics of supersonic under-expanded impinging jets”, Journal of Fluid Mechanics (Journal Front Cover), 889, p. A27 (2020)
Formation of coherent structures in under-expanded supersonic impinging jets
Large-eddy simulations of under-expanded impinging jets are performed to study the mechanism by which the initial high-frequency instabilities change to low-frequency coherent structures within a short distance. The spectral characteristic of the Mach energy norm is utilised to obtain the spatial growth of instabilities. Linear spatial instability analysis with streamwise varying mean flow profiles is also performed.
Cross-correlation of velocity and pressure show that hydrodynamic wavepackets form approximately one jet diameter downstream of the nozzle lip. No evidence has been found to support the 'collective interactive' mechanism of Ho & Nosseir (JFM, Vol. 105, p. 119-142, 1981). The 'vortex pairing' of Winant & Browand (JFM, Vol. 63, p. 237-255, 1974) is observed near the nozzle; however, it has an insignificant role in the sharp reduction of the most unstable frequency of disturbances. Nonetheless, both Mach energy norm and linear spatial instability analyses show that the most unstable frequency of disturbances decreases rapidly in a very short distance from the nozzle lip in the near-nozzle region through the spatial growth of instabilities where linear instability analysis over-predicts the frequency of the most unstable instabilities.
Ref: Karami, S., Edgington-Mitchell, D., A., Theofilis, V. and Soria, J. “Characteristics of acoustic and hydrodynamic waves in under-expanded supersonic impinging jet”, Journal of Fluid Mechanics, 905, p. A34 (2020).