Please use this identifier to cite or link to this item: http://dx.doi.org/10.14279/depositonce-223
Main Title: Schwach nichtlineare Stabilitätsuntersuchung der gekrümmten freien Scherschichten
Translated Title: Weakly Nonlinear Stability Analysis on the Curved Free Shear Layers
Author(s): Kuo, Chung-Che
Advisor(s): Fiedler, Heinrich
Granting Institution: Technische Universität Berlin, Fakultät V - Verkehrs- und Maschinensysteme
Type: Doctoral Thesis
Language: German
Language Code: de
Abstract: Ein theoretisches Modell, das mit der modifizierten schwach nichtlinearen Stabilitätstheorie und der Energiemethode aufgebaut ist, wurde auf eine gekrümmte, turbulente freie Scherschicht angewendet. Die Ansätze früherer schwach nichtlinearer Theorien, wonach die Amplituden der Wellen 2. Ordnung streng proportional dem Quadrat der Amplitude der Wellen 1. Ordnung sein sollen, und deren Anfachungsraten folglich proportional zueinander sein müssen, wurde in der vorliegenden Arbeit durch eine Energiemethode ersetzt, die die individuelle Entwicklung der Impulsverlustdicke und die jeweilige Amplitude von Wellen verschiedener Ordnung zu berechnen erlaubt. Ebenso konnte in der vorgestellten Theorie auf die "shape assumption" verzichtet werden. In diesem Fall ändern sich die Formfunktionen der Wellen verschiedener Ordnungen in Strömungsrichtung nach den zuständigen Entwicklungsgleichungen. Die Verformung des Geschwindigkeitsprofils wegen der Nichtlinearität ("mean flow distortion") wurde ebenfalls berücksichtigt. Im Vergleich mit dem ebenen Fall befinden sich in einer gekrümmten freien Scherschicht viel kompliziertere dreidimensionale kohärente Strukturen, insbesondere die Görtler-Wirbeln ähnlichen Längsstrukturen, die bei einer instabilen Krümmung durch die Zentrifugalkraft hervorgerufen werden. Bei stabiler und instabiler Krümmung, wobei die ebene Scherschicht als Sonderfall für Vergleichszwecke ebenfalls enthalten ist, sind sowohl die lokale Stabilitätscharakteristik als auch die räumliche Entwicklung der Strömung untersucht worden. Die lokale Stabilitätsanalyse wurde mit drei Kontrollparametern systematisch durchgeführt, nämlich der Krümmung, der Wellenzahl in Querrichtung und dem Geschwindigkeitsverhältnis der Scherströmung. Das Ergebnis hat gezeigt, daß die lokale Stabilitätscharakteristik der Scherschicht sehr empfindlich auf die Variation der Kontrollparameter reagiert, insbesondere im Niedrigfrequenzbereich. Ergebnisse, die unserer intuitiven Vorstellung über die stabil oder instabil gekrümmte Scherschicht nicht vollständig entsprechen, sind dargestellt und diskutiert worden. Demzufolge kann eine Krümmung nur unter Berücksichtigung des Frequenzbereiches als stabilisierend oder destabilisierend charakterisiert werden. Als Nebenaspekt ist auch die Möglichkeit der Entstehung von absoluter Instabilität in gekrümmter freier Scherschicht untersucht worden. Daraus hat sich ergeben, daß das kritische Geschwindigkeitsverhältnis, oberhalb dessen mit absoluter Instabilität gerechnet werden muß, mit der zunehmenden Krümmung sinkt. Zur Untersuchung der Scherschichtentwicklung wurde die modifizierte, schwach nichtlineare Theorie zusammen mit der Energiemethode und den aus mehreren Messungen gemittelten Anfangsbedingungen angewendet. Die schwach nichtlineare Theorie zeigt eine sehr gute Übereinstimmung mit dem Experiment: das Wachstum einer instabil gekrümmten Scherschicht ist stärker als die bei den anderen zwei Konfigurationen, aber das Scherschichtwachstum bei stabiler Krümmung ist nicht kleiner als das im ebenen Fall. Ein Erklärungsmodell auf der Basis der lokalen linearen Ergebnisse kann diese Beobachtung auch qualitativ unterstützen. Zusätzliche Berechnungen, in denen alle nichtlinearen Terme in den Gleichungen nicht berücksichtigt wurden, hat im Fall mit Krümmung ein überschätztes Scherschichtwachstum geliefert, welches eine große Abweichung von den Messdaten zeigten. Das spricht dafür, daß die Nichtlinearität eine unvernachlässigbare Rolle in der Entwicklung der gekrümmten freien Scherschichten spielt. Die Theorie hat auch bestätigt und erklärt, daß die stationären Längsstrukturen nur bei instabiler Krümmung deutlich zu erkennen sind.
Curved mixing layers exist in many problems in nature and technology, such as the jet-stream in the atmosphere, the flows past a backward facing step, or over the separation bubble (zone) on an airfoil with high angle of attack. The stability characteristics of such flows depend primarily on two dominating instability mechanisms, which can interact with each other. Those are the Kelvin-Helmholtz-instability and the centrifugal instability. Generally speaking, because of the inflection point in the velocity profile of a free shear layer, the spanwise structures (the "rolls") are initiated as a consequence of Kelvin-Helmholtz-instability, while the centrifugal instability is responsible for streamwise counter-rotating vortex pairs in a manner similar to Görtler-vortices in a boundary layer along a concave wall. In a curved free shear layer, where both of these two instability mechanisms are present, they are so to say competing against each other, since the centrifugal instability may either enhance or stabilize the disturbances in the flow. According to the Rayleigh circulation criterion, the centrifugal force may destabilize a curved free mixing layer if the inner stream is faster, and on the other side, it may stabilize a curved free mixing layer if the outer stream is faster. The coexistence of these two instability mechanisms and their resulting structures has been observed in earlier experiments in curved mixing layers. Unexpectedly, the streamwise counter-rotating vortex pairs, which appear in the unstably curved shear layers, are found to be very stable, regular and defined. This kind of vortex pairs influences strongly the evolution of the shear layer and the energy transport within it. The study of the coexistence of the spanwise and streamwise vortices in a curved mixing layer serves as a basis for understanding the complicate behavior of the flows in the separation zone, for example. Making use of the linear stability theory, curved free shear layers with stable or unstable curvatures are studied here for the first time in a systematic way according to the three parameters: curvature, spanwise wave numbers and velocity ratio. The results of the local linear analysis show that the amplification rate and the associating phase velocity are strongly affected by the centrifugal instability. Essentially physically different results as presented by Liou [1994] were found. We compared also our results with those by Monkewitz and Huerre [1982], in view of the coexistence of the control parameters: shear (velocity ratio) and curvature. Liou showed a decrease of the amplification rates over all frequencies in the stably curved flow and an overall increase due to unstable curvature. Somehow different, our results reveal that under a stable curvature the waves with low frequency, including the most unstable mode, are actually damped, having a smaller amplification rate as in the plane case. But some disturbances with high frequencies show, however, even higher excitation rates in spite of the stable curvature of the flow. Inversely, the same is true for the unstable curvature, where only the waves with low frequency are more strongly excited - again including the most unstable mode -, while waves in the high frequency region are suppressed. These results are quite different from our previous intuitive understanding about the effects of stable and unstable curvature. The reason for this disagreement between our results and those given by Liou is believed to lie in Lious application of the theoretical considerations from the Görtler-flows to the mixing layer, and his discarding too many "small" curvature terms, which were however found to have a large effect on the amplification-rate and the associated phase velocity at high frequency. Furthermore, it is found in our work that there are strong nonlinear effects between these three control parameters, especially between smaller velocity ratio and/or larger curvature and large spanwise wave numbers. Generally speaking, the observable coherent structures depend on the competition between the centrifugal forcing and the shear stress due to the velocity difference. For plane mixing layers, Huerre and Monkewitz [1985] calculated the critical velocity ratio lambda=1.315, beyond which absolute instability must be expected. They found also that a global instability mode can be brought about by a locally absolute unstable flow. In such case, one finds a self-exciting, or rather, an oscillating flow, which has a certain characteristic frequency. A global influence on the flow behavior is then possible. This can be very important for the flow control by a backward facing step or the separation bubble on an airfoil. At the end of the linear stability analyses in this paper, the influence of centrifugal forces on this limiting velocity ratio for the absolute instability is studied. The stability calculation of the spatial-temporal evolution in the present work indicates that this critical velocity ratio is strongly influenced by the streamline curvature. Based on the modified weakly nonlinear theory suggested by Zhou [1995], together with the energy method, a theoretical model for the nonlinear spatial evolution of large scale coherent structures in a curved mixing layer is proposed in the second part of the present work. As the basic disturbance we associate the most unstable mode or - if present - the artificial excitation with the wave of 1st order, and those waves as of 2nd order, which originate from the nonlinear interactions of the waves of 1st order. The "shape assumption&" for the waves of 2nd order, as was done in previous investigations, is not considered necessary in this work. The form functions of the waves of 2nd order are not considered constant in downstream direction as in the shape assumption, but change with the development of the shear layer. Based on the results of the wave of 1st order, which can be calculated with the linear theory on the basis of local parameters, the form functions of the 2nd order can be calculated locally from their appropriate governing evolution equations. Furthermore, it is not a necessary condition of our model, that - as according to the Landau equation, which was often applied in the previous nonlinear analysis - the square of amplitude and the associated amplification rate of the waves of 2nd order must be strictly proportional to the of amplitude and amplification rate of the waves of 1st order, respectively. The individual downstream evolution of the shear layer thickness and the amplitudes of the different waves are calculated by means of the energy method under the assumption of locally parallel flow. The energy exchange between the mean flow and the waves of different scales is also calculated. The mean flow distortion, originating from the nonlinear effects, is also considered. In order to study the evolution of the large-scale coherent structures in a curved mixing layer, a three dimensional, spatial analysis has been employed. For the numerical evaluation, a forward difference procedure of 4th order accuracy, suggested by Malik[1990], is applied. The Muller-method is used to locally solve the eigenvalue problem in the linear theory. After applying the theoretical model in the present work to a non-excited curved mixing layer a very good agreement between the theoretical and experimental results was achieved. * Results show that the growth-rate of the shear layer with unstable curvature is much larger than that of a plane mixing layer, while the growth-rate under the stable curvature is not at all much smaller than the growth-rate of a mixing layer without curvature. This confirms the prediction with the model based on the results of the linear theory. * The calculation with the 'mean flow distortion' describes very well not only the downstream evolution of the velocity profile, but also helps to explain better why the streamwise counter rotating vortex structures will be found only under the unstable curvature, not in a plane or stably curved mixing layer. This different observation under a stable or unstable curvature would not be achieved without the considering of nonlinearity. * The energy exchange between the mean flow, the waves of different scales and the stochastic turbulent fluctuation is found to be strongly affected by the centrifugal effects, so that the downstream evolution of the amplitude of different waves is also strongly influenced. * Many other theoretical results of the linear and nonlinear theory are well confirmed by the experimental observations.
URI: urn:nbn:de:kobv:83-opus-1259
http://depositonce.tu-berlin.de/handle/11303/520
http://dx.doi.org/10.14279/depositonce-223
Exam Date: 17-Jul-2000
Issue Date: 11-Jan-2001
Date Available: 11-Jan-2001
DDC Class: 500 Naturwissenschaften und Mathematik
Subject(s): Krümmung
Nichtlinear
Scherschicht
Stability
Usage rights: Terms of German Copyright Law
Appears in Collections:Technische Universität Berlin » Fakultäten & Zentralinstitute » Fakultät 5 Verkehrs- und Maschinensysteme » Publications

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