Thermally induced low-frequency oscillations

Summary This paper discusses nonlinear thermally induced oscillations in quasi-one dimensional flows in ducts. In practice such oscillations are frequently observed in furnaces and combustion chambers. The problem considered involves entropy disturbances which are convected through a nozzle at the end of a tube, thereby producing an acoustic wave which propagates upstream and leads to a modulation of the mass flow at the inlet of the tube. Alternatively, if the rate of heat addition (i. e. the rate of fuel addition in the case of a combustion chamber) responds only weakly or not at all to the oscillating pressure at the inlet, the modulated air flow produces an entropy oscillation (due to the oscillating equivalence ratio in the case of a combustor) downstream of the zone of heat addition (reaction zone). To obtain general stability limits for this kind of self-induced oscillation, a second-order analysis is developed which leads to a nonlinear wave equation. The convection of entropy disturbances introduces nonlinear memory effects which are responsible for a non-local character of the wave equation. The wave equation is solved with the help of a numerical evolution scheme, making use of a suitable scaling transformation which does not change the form of the equation.

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Zusammenfassung Es werden nichtlineare thermisch getriebene Schwingungen in quasieindimensionalen Rohren untersucht. In der Praxis kann dieser Schwingungstyp häufig in Feuerungsanlagen und Brennkammern beobachtet werden. Die betrachteten Schwingungen werden von Entropiestörungen angetrieben, welche die Austrittsdüse eines Rohres durchqueren und dabei Schall erzeugen. Eine der erzeugten Schallwellen läuft stromaufwärts und verursacht eine Massenstrommodulation am Eintritt des Rohres. Falls die Wärmezufuhrrate (Brennstoffzufuhrrate in einer Brennkammer) nur schwach oder gar nicht auf Druckschwankungen am Eintritt reagiert, dann verursacht die Luftmassenstromschwankung eine Entropieschwankung (aufgrund des schwankenden Brennstoff/LuftVerhältnisses) am Ende der Zone (Reaktionszone), in welcher Wärme zugeführt wird. Um allgemeine Stabilitätsgrenzen für solche selbsterregten Schwingungen zu ermitteln, wird eine Theorie zweiter Ordnung entwickelt, die zu einer nichtlinearen Wellengleichung führt. Die Konvektion von Entropiestörungen führt zu nichtlinearen Gedächtniseffekten, die für den nicht-lokalen Charakter der Wellengleichung verantworlich sind. Die Gleichung wird mit Hilfe eines numerischen Evolutionsschemas gelöst, wobei von einer Skalierungstransformation Gebrauch gemacht wird, welche die Form der Gleichung nicht verändert.

     

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