The full impulse response of two-dimensional jet/wake flows and implications for confinement

Abstract

In this theoretical study, a linear spatio-temporal analysis is performed on unconfined and confined inviscid jet/wake flows in order to determine whether they are absolutely or convectively unstable. The impulse response is evaluated in the entire outer fluid, rather than just at the point of impulse, over a wide range of density ratios. This confirms that the dominant saddle point can validly migrate into the plane of diverging eigenfunctions. This reveals that, at certain density ratios and shear numbers, the response can grow upstream in some directions with a cross-stream component, even though it decays directly upstream of, and at, the point of impulse. This type of flow is convectively unstable when unconfined, but becomes absolutely unstable when confined. Other effects of confinement are described in a previous paper. Together, these articles have important implications for the design of fuel injectors, which often employ confined shear flows at high Reynolds number and large density ratios to generate strong mixing in combustion chambers.

Stability analysis and breakup length calculations for steady planar liquid jets

“…In the present study we examine how the growth rate at these saddles and their positions are affected when the interface between the two fluids is placed within the shear layer. We find that this leads to important qualitative differences compared with previous studies in which the density interface was assumed to lie on only one side of the shear layer (Marmottant & Villermaux 2004;Juniper 2007). Our assumption to neglect viscosity in this study is valid because Yu & Monkewitz (1990) showed that the transition to absolute instability is caused by the interaction between the two shear layers and is not a viscous effect.…”

“…The expressions for c 3 to c 0 can also be found in the Appendix. For more information of the derivation of the dispersion relation, see Drazin & Reid (1981), Schmid & Henningson (2001), Healey (2007) or Juniper (2007.…”

“…We assume that a time-dependent forcing is turned on at t = 0 at the nozzle of the jet (x = 0) and that this can be written as the boundary conditionv(x, 0, t) = δ(x)f (t), wheref = 0 for t < 0 (Juniper 2007). The solution forv(x, y, t) can be written as the double-inverse Fourier transform…”

“…The density ratio q = 1/500 is considered because it corresponds to cold liquid diesel liquid fuel being injected into compressed air at about 5 atmospheres pressure, which is similar to the smallest density ratio in the experiments in figure 1. In this paper we adopt the notation of Juniper (2006Juniper ( , 2007 for the labelling of the saddle points. We denote the saddle whose position is controlled by the thickness of the shear layer and surface tension (Rees & Juniper 2009) as s 1 .…”

“…The saddle point on the contour with the largest growth rate gives the disturbance growth rate in the limit t → ∞. For a liquid jet there is one saddle point in the α-plane whose position is determined by the thickness of the shear layers and the value of the surface tension (known as the 's 1 ' saddle in Juniper 2007 and 'shear layer mode' in Lesshafft & Huerre 2007), and this saddle is located close to the real α-axis. Also in the α-plane is a set of saddle points close to the imaginary α-axis which are due to the interaction between the two shear layers (known as 's 2 ' saddles in Juniper 2007 and 'jet column modes' in Lesshafft & Huerre 2007).…”