On the bifurcation structure of axisymmetric vortex breakdown in a constricted pipe

Abstract

The bifurcation structure is presented for an axisymmetric swirling flow in a constricted pipe, using the pipe geometry of Beran and Culick [J. Fluid Mech. 242, 491 (1992)]. The flow considered has been restricted to a two-dimensional parameter space comprising the Reynolds number Re and the relative swirl V, of the incoming swirling flow. The bifurcation diagram is constructed by solving the time-dependent axisymmetric Navier-Stokes equations. The stability of the steady results presented by Beran and Culick, obtained from a steady axisymmetric Navier-Stokes code, has been confirmed. Further, the steady solution branch has also been extended to much larger V, values. At larger V,,, a stable unsteady solution branch has been identified. This unsteady branch coexists with the previously found stable steady solution branch and originates via a turning point bifurcation. The bifurcation diagram is of the type described by Benjamin [Proc. R. Sot. London Ser. A 359, 1 (1978)] as the canonical unfolding of a pitchfork bifurcation. This type of bifurcation structure in the two-dimensional parameter space (Re,V& suggests the possibility of hysteresis behavior over some part of parameter space, and this is observed in the present study. The implications of this on the theoretical description of vortex breakdown and the search for a criterion for its onset are discussed.

Effect of near-critical swirl on the Burke-Schumann reaction sheet

“…They show that large disturbances appear in the flow as the swirl level approaches a critical value ω c , signifying the first occurrence of vortex breakdown. Their results agree well with numerically obtained Navier-Stokes solutions of Beran and Culick [20], Beran [21], and Lopez [22]. We employ Wang and Rusak's [19] asymptotic description of the flow to determine the effect of the flow on the shape of the reaction zone.…”

“…This condition ignores the influence of wall boundary layers, with the expectation that these layers exert at most a quantitative effect on the phenomenon under study; see for example, the relevant discussion in Beran and Culick [20] and Lopez [22]. Also, along the pipe wall, the radial gradient of the mixture mass fraction vanishes.…”

Solitary waves with recirculation zones in axisymmetric rotating flows

“…Such a formulation does not exactly correspond to the experimental set-ups described in Sarpkaya (1971), Leibovich (1984) and Escudier (1988), or to the numerical simulations by Beran & Culick (1992), Lopez (1994) and Hanazaki (1996). In all of these numerical studies, breakdown of the approaching vortex was induced by a local contraction of the tube, so that the separation zone was observed in the lee of that contraction.…”

“…To this end they allow the radial velocity to be freely adjusted to reflect the upstream influence which is allowed to reach the inlet in their model. The model of Rusak et al (1998), although for a straight tube, was nevertheless claimed to be relevant to the studies of Beran & Culick (1992), Lopez (1994) and Darmofal (1996), where the condition of zero radial velocity at the inlet was used. However, the inlet conditions of Rusak et al (1998) do not contain the length scale of a local contraction (Beran & Culick 1992;Lopez 1994), or the length scale associated with a divergence of the tube (Darmofal 1996).…”

“…The model of Rusak et al (1998), although for a straight tube, was nevertheless claimed to be relevant to the studies of Beran & Culick (1992), Lopez (1994) and Darmofal (1996), where the condition of zero radial velocity at the inlet was used. However, the inlet conditions of Rusak et al (1998) do not contain the length scale of a local contraction (Beran & Culick 1992;Lopez 1994), or the length scale associated with a divergence of the tube (Darmofal 1996). These effects due to the tube profile are important in the problem studied by these authors, not only because these tube profiles prevent the changes of the inlet conditions (which the conditions of Rusak et al (1998) also do), but also because the properties of the generated wave are affected by the exact shape of the tube.…”