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Linear stability analysis of variable-density jets
Student: Greg Rodebaugh (rodebaugh: jhu.edu)
(Figures: sample eigenvalue results instability regimes) (Publications)
Hydrodynamic stability examines how a base, laminar fluid flow responds to a given imposed disturbance. If the flow returns to its base state, then the flow is called stable, while if it undergoes a transition to a different state, the flow is called unstable. Instability does not automatically imply turbulent flow, although stability analysis does provide an analytical tool to investigate the transition to turbulence of a given flow configuration. In linear stability analysis one studies a laminar base flow subjected to infinitesimal perturbations. The governing equations are then linearized by discarding products of perturbations.
The goal of this work is to determine the linear stability properties of three-dimensional, variable density, non-reacting jets with radial density profiles similar to those seen in combustion. This is one part of a combined approach, involving analytical, numerical, and experimental studies, to further the understanding of the fluid mechanical properties of reacting flow systems. For high-Reynolds number, nearly parallel flows, such as jets, the main instability mechanism is Kelvin-Helmholtz instability, which is inherently an inviscid process. For this reason, the study is restricted to the incompressible Euler equations, with variable density. The linearized equations in cylindrical coordinates are
where the velocity, pressure, and density in the base flow are
and the perturbation variables are
Next, we assume the disturbances take a sinusoidal form,
with complex valued frequency and wave number, (beta) and (alpha), and integer valued azimuthal wave number, n. The system of linear equations is reduced to a single second-order equation for pressure,
The pressure equation can be written as an eigenvalue problem for (beta), with the resulting equation being solved using a Chebyshev expansion that recovers the leading frequency eigenvalues. A representative frequency spectrum is plotted in Figure 1. If the imaginary part (beta)i of any eigenmode is greater than zero, then the flow is linearly unstable.
We are interested in relating the local flow instability computed numerically to the global instability observed in experiments. This is done using the locally defined concepts of absolute and convective instability. An unstable flow is called convectively unstable if the instability only propagates downstream with time; the flow is absolutely unstable if the the instability propagates both downstream and upstream with time, thus rendering the entire flow unstable for a sufficiently long time interval (Figure 2).
To determine this behavior we evaluate the impulse response of the linearized flow equations at large time. Practically, this involves finding saddle points in the complex (alpha) plane that satisfy certain constraints, and then using a steepest descent argument to determine the long time behavior of the disturbance. For a given parameter set, we then compute the transition boundary between absolute and convective instability. This provides an upper bound for where global instability will occur. These local concepts also provide insight into the development of global instability modes, since it has been shown that finite regions of absolute instability play a role in the creation of global modes.
Our current work involves computing the absolute/convective boundary for a range of flow parameters, including jet density profiles characteristic of diffusion flames, and conducting complementary imaging experiments that will help to clarify the realistic instability modes.
Publications (inquiries: email Greg Rodebaugh, rodebaugh: jhu.edu)
All materials © 2008 Applied Fluid Imaging Laboratory. Last page update 7.25.08.
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