Positivity and Negativity of Solutions to a Schrödinger Equation in ℝ N

Weak L ² -solutions u of the Schrödinger equation, –u + q(x) u – u = f(x) in L ² , are represented by a Fourier series using spherical harmonics in order to prove the following strong maximum and anti-maximum principles in (N 2): Let ₁ denote the positive eigenfunction associated with the principal eigenvalue ₁ of the Schrödinger operator . Assume that the potential q(x) is radially symmetric and grows fast enough near infinity, and f is a `sufficiently smooth' perturbation of a radially symmetric function, f 0 and 0 f / C const a.e. in . Then u is ₁-positive for - < < ₁ (i.e., u c ₁ with c const > 0) and ₁-negative for ₁ < < ₁ + (i.e., u –c₁ with c const > 0), where > 0 is a number depending on f. The constant c > 0 depends on both and f.     

Unsteadiness of Flow Separation and End-Effects Regime in a Thrust-Optimized Contour Rocket Nozzle

Turbulent flow separation in over-expanded rocket nozzles is investigated experimentally in a sub-scale model nozzle fed with cold air and having a thrust-optimized contour. Depending upon the pressure ratio either a free shock separation (FSS) or a restricted shock separation (RSS) is observed with a significant hysteresis between these two flow regimes. It is shown that the RSS configuration may involve several separated regions. Analysis of wall pressure fluctuations give quantitative information on the fluctuating pressure field directly connected with the occurrence of significant side loads. Direct measurements of the evolution of the side loads with respect to the pressure ratio show the occurrence of three distinct peaks which are explained by the wall pressure fluctuations measurements.     

Influence of roughness on attachment line boundary layer transition in hypersonic flow

 Attachment line boundary layer transition on swept cylinders is studied in a low enthalpy hypersonic wind tunnel at M _∞=7.14. Sweep angles of 60° and 70° are used and transition is detected by means of heat flux measurements. The influence on attachment line transition of single 2D-roughness elements, in the form of tripwires or slots, as well as 3D obstacles is determined and the results are analyzed with respect to Poll’s criterion. Received: 16 January 1996 / Accepted: 12 July 1996     

Smooth leading edge transition in hypersonic flow

The boundary layer transition along the attachment line of a smooth swept circular cylinder in hypersonic flow is investigated in a blowdown wind tunnel. A wide range of spanwise Mach numbers Me (3.28 to 6.78) is covered with the help of different models at several sweep angles (60°?Λ?80°). The transition is indirectly detected by means of heat flux measurements. The influence of the wall to stagnation temperature ratio is investigated by cooling the model with liquid nitrogen.