Jürgen Zierep

Asymptotic solution of transonic nozzle flows with homogeneous condensation. I. Subcritical flows

A detailed analysis of supercritical transonic nozzle flows with stationary normal shock waves is presented. A classification scheme based on the normal shock location is obtained using asymptotic methods, and four distinct supercritical flow regimes are distinguished. A simple shock fitting technique that determines the shock location within any desired precision is introduced. Consequently, an algorithm that exhibits the asymptotic solution in each supercritical flow regime is developed for the expansion of moist air in nozzles with atmospheric supply conditions by utilizing the classical nucleation theory and the Hertz–Knudsen droplet growth law. Good agreement with the recent static pressure measurements and visualized shock locations is achieved in relatively slender nozzles when the condensed phase is assumed to consist purely of water drops.

The mathematical theory of thermal choking in nozzle flows

The mathematical theory of sub- and supercritical nozzle flows is presented by a unified description of integro-algebraic and differential formulations of the flow equations. The critical amount of heat necessary for a thermally choked flow is defined and models which approximate this critical amount of heat are constructed for nozzle flows with both given internal heat source distributions and nonequilibrium condensation. In particular a cubic equation for an estimate of the limiting condensate mass fraction for shock free condensing flows is derived and a criterion for the existence of supercritical condensing flows based on this estimate is established. The necessary and sufficient conditions for thermal choking are then stated. It is shown that the commonly accepted view, which asserts that the flow Mach number reaches unity at thermal choking (known to be not always true in condensing flows), only exhibits a necessary condition for a thermally choked flow.

Der senkrechte Verdichtungsstoss an der gekrümmten Wand unter Berücksichtigung der Reibung

ZusammenfassungDas Aufsetzen eines senkrechten Verdichtungsstoßes auf einer gekrümmten Wand ist sowohl für die Theorie als auch für die Anwendungen von Interesse. Experimentelle Untersuchungen von Ackeret, Feldmann und Rott zeigten erstmals, daß am Rand der turbulenten Grenzschicht die Druckverteilung hinter dem Stoß eine singuläre Nachexpansion aufweist. Dies konnte durch eine theoretische Betrachtung der reibungsfreien Strömung von Oswatitsch und Zierep bestätigt werden. Wir berichten über eigene neue Ergebnisse, dieses Problem unter Berücksichtigung der Reibungseinflüsse zu behandeln. In Analogie zu dem von Hilden in seiner von Oswatitsch angeregten Dissertation (Aachen 1961) behandelten Problem der einfallenden und der reflektierten Welle wird auch bei uns das Stromfeld ausgehend von der Wand unterteilt in1)eine wandnahe Reibungsschicht, der der Druck von außen aufgeprägt wird,2)eine reibungsfreie, kompressible Scherschicht,3)eine reibungsfreie, kompressible schallnahe Außenströmung. Dieses Modell läßt wesentliche Vereinfachungen in den Grundgleichungen zu und gestattet anschließend das Randwertproblem für einen aufsitzenden senkrechten Stoß zu lösen. Es ergeben sich Druckverteilungen, wie sie auch im Experiment beobachtet werden. Insbesondere tritt auch hier das singuläre Verhalten in der schallnahen Außenströmung auf, das im Innern der Grenzschicht aber sofort geglättet wird.SummaryThe problem of the ending of a normal shock at a curved wall is of interest for theoretical and applied reasons. Experimental investigations first made by Ackeret, Feldmann and Rott indicate a singular behaviour of the pressure gradient behind the shock (expansion) at the outer edge of the turbulent boundary layer which was confirmed by calculations for inviscid flow by Oswatitsch and Zierep. We present new results which we have obtained for this problem by studying the viscous case. According to the thesis of Hilden (Aachen 1961) suggested by Oswatitsch we divide the flow beginning at the wall in1)a viscous layer near the wall with pressure induced from outside,2)a frictionless, compressible shear-layer,3)a frictionless, compressible transsonic potential flow. This model allows essential simplifications in the equation and gives a solution for the boundary value problem of the normal shock at the wall. We get pressure distributions similar to experimental ones. Especially we get the same singular behaviour in the potential flow which is immediately smoothed out in the boundary layer.

Continuation of modelling of perforated plate aerodynamics performance

Abstract This paper continues a physical modelling of a transpiration flow, presented in [P.P. Doerffer, R. Bohning, Aerosp. Sci. Technol. 4 (2000) 525–534], by developing a system of gasdynamic equations for the viscous flow in tubes of a perforated plate. Asymptotic expansions give interesting analytical expressions for the pressure loss in the tubes as a function of Mach number in a perforation hole.

Asymptotic solution of shock tube flows with homogeneous condensation

The asymptotic solution of shock tube flows with homogeneous condensation is presented for both smooth, or subcritical, flows and flows with an embedded shock wave, or supercritical flows. For subcritical flows an analytical expression, independent of the particular theory of homogeneous condensation to be employed, that determines the condensation wave front in the rarefaction wave is obtained by the asymptotic analysis of the rate equation along pathlines. The complete solution is computed by an algorithm which utilizes the classical nucleation theory and the Hertz-Knudsen droplet growth law. For supercritical flows four distinct flow regimes are distinguished along pathlines intersecting the embedded shock wave analogous to supercritical nozzle flows. The complete global solution for supercritical flows is discussed only qualitatively owing to the lack of a shock fitting technique for embedded shock waves. The results of the computations obtained by the subcritical algorithm show that most of the experimental data available exhibit supercritical flow behaviour and thereby the predicted onset conditions in general show deviations from the measured values. The causes of these deviations are reasoned by utilizing the qualitative global asymptotic solution of supercritical flows

Bedingung für das Einsetzen der Ablösung der turbulenten Grenzschicht an der gekrümmten Wand mit senkrechtem Verdichtungsstoss

ZusammenfassungDie vorliegende Arbeit setzt eine frühere Untersuchung [1] der Verfasser fort. Daselbst wird das Aufsetzen eines senkrechten Verdichtungsstoßes auf einer gekrümmten Wand in turbulenter Grenzschicht untersucht. Aufgrund dieser Lösung läßt sich ein Kriterium für die durch einen senkrechten Stoß verusachte Ablösung der Grenzschicht angeben. Das zugehörige Diagramm enthält die drei Größen: Mach-Zahl, Reynolds-Zahl und Wandkrümmung und wird ausführlich diskutiert.SummaryThis publication continues earlier investigations [1] of the authors. There had been studied the ending of a normal shock at a curved wall in turbulent boundary layer. According to this solution we get a criterium for the starting of a separation of the boundary layer due to a normal shock. The diagram contains the three quantities: Mach-number, Reynolds-number and curvature of the wall. This diagram is discussed in detail.

PERFORATED PLATE AERODYNAMICS FOR PASSIVE SHOCK CONTROL

Transpiration flow through perforated walls plays ever increasing role in aerodynamics due to a frequent application of the flow control by means of blowing or suction. An experimental study is presented here which allowed to determine a transpiration flow characteristic in a form of a simple function. It is very useful for modelling of such flows. Additionally a method of “aerodynamic porosity” determination was proposed which is much more reliable than geometrical description of the porosity. A theoretical analysis of the flow through a perforation hole was also carried out. The flow is considered as compressible and viscous. The gasdynamic analysis led us to a very similar result as obtained from experiment.

Gleichzeitiger Viskositäts- und Kompressibilitätseinfluß

In diesem Kapitel werden Kompressibilitat und Reibung in einfacher Form gleichzeitig berucksichtigt. Wir benutzen ein Modell, bei dem die Reibung allein im Impulssatz uber die Wandschubspannung τw = (λ/4)(ϱ/2)w 2 eingeht. Fur die Widerstandszahl λ gilt hierin im allgemeinen

Hydro- und Aerostatik

{\rm{\lambda }}\,{\rm{ = }}\,f\,{\rm{(}}Re{\rm{,}}\,M{\rm{),}}\,\,\,\,\,Re\,{\rm{ = }}\,\frac{{w{d_{\rm{h}}}}}{v}\,{\rm{ = }}\,\frac{{\varrho w\, \cdot \,{\rm{4}}A}}{{\eta U}}{\rm{.}}