Instabilities and bifurcation of non-equilibrium two-phase flows

Abstract

New instabilites of unsteady transonic flows with non-equilibrium phase transition are presented including unsymmetric flow patterns with moving oblique shock systems in supersonic nozzles with perfectly symmetric shapes. The phenomena were first detected when performing experiments in our supersonic wind tunnel with atmospheric supply and could be perfectly reproduced by numerical simulations based on the Euler equations, i.e. neglecting the viscosity of the fluid. The formation of the liquid phase is modelled using the classical nucleation theory for the steady state together with the Hertz–Knudsen droplet growth law and yields qualitatively and quantitatively excellent agreement with experiments in the unsteady flow regime with high-frequency oscillations including the unstable transient change of the structure from symmetric to unsymmetric flow.For engineering applications the sudden increase or decrease of the frequency by a factor 2 or more and of the pressure amplitude at the bifurcation limits is of immediate practical interest, e.g. for flutter excitation of turbomachinery blading.

Instability in two-phase flows of steam

“…Some unsteady conditions are experienced in nucleating flows of steam in convergent-divergent nozzles, when the zone of rapid condensation occurs at a short distance downstream of the throat and the pressure rise associated with heat release is of a magnitude to propagate upstream (e.g. Barschdorff 1971;Yousif et al 1972;Saltanov 1976;Skillings & Jackson 1987;White & Young 1993;Adam & Schnerr 1997;Schnerr 2005), but these lead to steady oscillations. The unstable conditions considered in the present paper differ from these in character and form and occur over a much wider range of conditions.…”

On the stability of stationary shock waves in nozzle flows with homogeneous condensation

“…In this case flows with embedded normal shock waves occur ͑supercritical flows͒. Further increase in initial relative humidity can lead to the loss of stability of the stationary shock resulting in unsteady flow oscillations, as observed in experiments and unsteady two-dimensional numerical simulations ͑e.g., see Adam and Schnerr 14 ͒. In what follows we give a complete description of quasi-onedimensional steady-state flow patterns for condensing nozzle flows by studying the singularities of the differential system of equations ͑1͒-͑4͒.…”

“…We now compare the results for ( 0 ) cr of the quasi-onedimensional approximation obtained by the above algorithm for different nozzle configurations and different reservoir conditions with those of experiments and/or of unsteady twodimensional numerical simulations at the limit of stability, where bifurcation to unsteady flow with shock oscillations occurs ͑e.g., see Adam and Schnerr 14 and Lamanna 24 ͒. Table I shows such a comparison.…”

“…8 Two-dimensional numerical simulations of such flows were given by Schnerr 9 and Schnerr and Dohrmann. 13 If the heat addition to the flow is further increased ͑e.g., by increasing the initial relative humidity keeping appropriate reservoir conditions fixed͒, the flow becomes unsteady and different patterns of periodic flow structures with moving shock waves occur ͑e.g., see Adam and Schnerr 14 ͒. Unsteady flow phenomena with moving shocks were first discovered by Schmidt 15 in moist air expansions.…”

“…The first numerical calculations of such flows were presented by Saltanov and Tkalenko 16 for moist air and by Guha and Young 17 for wet-steam. Twodimensional numerical simulations have recently been carried out by White and Young 18 for wet-steam and by Adam and Schnerr 14 for moist air.…”