Bifurcation analysis, linear stability study, and direct numerical simulations of the
dynamics of a two-dimensional, incompressible, and laminar flow in a symmetric long
channel with a sudden expansion with right angles and with an expansion ratio D/d
(d is the width of the channel inlet section and D is the width of the outlet section)
are presented. The bifurcation analysis of the steady flow equations concentrates on
the flow states around a critical Reynolds number Rec(D/d)
where asymmetric states appear in addition to the basic symmetric states when
Re [ges ] Rec(D/d). The bifurcation
of asymmetric states at Rec has a pitchfork nature and the asymmetric perturbation
grows like √Re − Rec(D/d). The
stability analysis is based on the linearized equations of motion for the
evolution of infinitesimal two-dimensional disturbances imposed
on the steady symmetric as well as asymmetric states. A neutrally stable asymmetric
mode of disturbance exists at Rec(D/d)
for both the symmetric and the asymmetric equilibrium states. Using asymptotic methods, it is demonstrated that when
Re < Rec(D/d)
the symmetric states have an asymptotically stable mode of disturbance.
However, when Re > Rec(D/d),
the symmetric states are unstable to this mode of asymmetric disturbance. It is also shown that when
Re > Rec(D/d) the asymmetric
states have an asymptotically stable mode of disturbance. The direct numerical
simulations are guided by the theoretical approach. In order to improve the numerical
simulations, a matching with the asymptotic solution of Moffatt (1964) in the regions
around the expansion corners is also included. The dynamics of both small- and large-amplitude disturbances in the flow is described and the transition from symmetric to
asymmetric states is demonstrated. The simulations clarify the relationship between
the linear stability results and the time-asymptotic behaviour of the flow. The current
analyses provide a theoretical foundation for previous experimental and numerical
results and shed more light on the transition from symmetric to asymmetric states of
a viscous flow in an expanding channel. It is an evolution from a symmetric state,
which loses its stability when the Reynolds number of the incoming flow is above
Rec(D/d), to a stable asymmetric
equilibrium state. The loss of stability is a result of
the interaction between the effects of viscous dissipation, the downstream convection
of perturbations by the base symmetric flow, and the upstream convection induced
by two-dimensional asymmetric disturbances.
Flamelet Characteristics of Gaseous and Spray Lifted Flames on Two-dimensional Direct Numerical Simulations
