Recent developments in applied mathematics and mechanics: theory, experiment and practice. Devoted to the 80th anniversary of academician N.N.Yanenko

2 Institute for Aero- and Gasdynamics, Stuttgart University, Germany

To advance boundary-layer (BL) stability, receptivity and transition research and to ultimately provide reliable predictions of transition for supersonic flight vehicles, a wind tunnel is required with very low stream disturbance levels comparable to free flight conditions. Experiments show that high noise levels in conventional supersonic wind tunnels at Mach numbers M>2 is produced by acoustics radiated from turbulent BL developing on nozzle walls [1]. It has been shown that size of quiet core of test section flow is defined by the transition location on nozzle walls [2]. The main point of efforts to develop so called ‘quiet’ supersonic wind tunnels [3] during last 30 years [1-3] consists in maintaining nozzle BL laminar as long as possible at high flow unit Reynolds numbers. Linear stability theory of compressible flows has been successfully applied for this purpose.

Institute of Aero- and Gasdynamics at Stuttgart University, Germany, has a very large short duration supersonic wind tunnel SWK [4]. Due to its size (dimensions of the test section exit are 1.2´ 0.8 m²) and other design peculiarities SWK nozzles have properties which promote sufficiently low noise level in the test section [4]. However theoretical analysis of SWK nozzle BL development and stability has not been accomplished up to now and is a subject of present investigation.

Theoretical analysis of Tollmien-Schlichting (TS) instability of SWK contour wall boundary layer has been performed. Computations revealed that most part of the BL is stable. TS instability is possible only in a very short area in the vicinity of the nozzle head and in the region of adverse pressure gradient downstream of the throat. The location of laminar-turbulent transition (LTT) in the SWK nozzle BL due to TS instability was estimated by means of e^N-method. It was found that transition is possible in the convex wall region only for flow with high unit Reynolds number R₁» 50× 10⁶ m^-1. For lower R₁ contour BL remains laminar (N<10) which has an experimental confirmation. Pitot pressure fluctuation measurements in combination with wave patterns computed by means of method-of-characteristics revealed that the origin of test section noise is situated rather on sidewalls than on the contour walls. It seems that instabilities of 3D sidewall BL (cross-flow and corner flow instability) lead to an early transition and are the main sources of noise, limiting a quiet core of the test section flow, while the transition by Goertler instability on contour walls is only of a secondary meaning. To reduce the danger of TS instability and to eliminate irregular shimmering Mach waves it is proposed to design (by means of an improved method) new nozzle blocks with smaller subsonic contraction angle and consequently with a more flat sonic surface. Filling of nozzle corners is recommended to avoid corner vortex instability. Further detailed theoretical study of Goertler, cross-flow and corner instabilities is recommended for better understanding of noise sources in the SWK test section flow. Computations should be escorted by corresponding experiments to obtain further information about the dominant instability modes.

This work has been supported by the German Research Foundation (DFG) within research center SFB-259 established at Stuttgart University, subproject C5.