In this publication the focus lies on the design process of the full supersonic double throated wind tunnel. Starting with the fundamental equations of gas dynamics in combination with an analytical model of the pressure reservoir, the area of the throat at the nozzle and the runtime of the blowdown wind tunnel were computed. Based on these results, the shape of a shock free nozzle was calculated by the method of characteristics. For this purpose, a nozzle design program was developed using Python. In order to validate the results of the method of characteristics program, these results were compared with the area-Mach number relation, which is the exact analytical solution of the isentropic flow through supersonic nozzles. The convergent part of the nozzle, which initially accelerates the flow to sonic speed, cannot be calculated by the method of characteristics, since it applies to supersonic flows only. Hence the subsonic convergent section of the nozzle was designed directly with 2D CFD using CD Adapco Star-CCM+ v. 10.06. A parametric model of the convergent nozzle section was used to find the optimum nozzle shape, i.e. a nozzle which results in a maximum mass flow rate in order to have an undisturbed flow field and Mach number in the following test section. In order to decelerate the flow again from supersonic to subsonic flow after the test section and minimize the total pressure losses, an oblique shock diffuser was used [1]. As for the convergent subsonic nozzle, the optimum shape of a diffusor was found by 2D CFD analysis. Putting all these elements together, i.e. nozzle, test section and diffuser the optimum supersonic wind tunnel shape was found. Finally, a full 3D simulation of the supersonic wind tunnel was performed in order to validate the complete design procedure and computations and also to include the viscous effect of the side walls. These results and the whole design process are presented and analyzed in the paper.

The efflux velocity equation from Torricelli is well known in fluid mechanics. It can be derived analytically applying Bernoulli’s equation. Bernoulli’s equation is obtained integrating the momentum equation on a stream line. For verification purposes the efflux velocity for a large tank or vessel was also computed analytically applying the momentum equation, delivering, however, a different result as the Torricelli equation. In order to validate these theoretical results the vertical and the horizontal efflux velocity case was simulated with computational fluid dynamics CFD. Furthermore, simple experiments for the horizontal and vertical efflux equation were performed.

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The logarithmic law of the wall is usually derived for the flat plate assuming stationary, two-dimensional fully developed flow with no external pressure gradient. The Prandtl mixing length model for the turbulence is applied, which assumes homogeneous turbulence and two empirical constants, and the logarithmic wall law is derived. It is than stated in the textbooks that it is universally valid without a proof. As a justification experimental evidence is shown. First this proof will be shown in detail. Than a more general approach based on similarity considerations is made to show the universal validity of the logarithmic law of the wall. Starting from the Navier-Stokes equation a general non dimensional form of this equation is derived showing its dependency from four non-dimensional numbers, the Strouhal, Euler, Reynolds and the Froude number. Then wall bounded laminar flows are analyzed by dimensional analysis. The laminar boundary length and time scales are derived and used to non-dimensionalize the Navier-Stokes equation. With this specific non-dimensionalization for the laminar boundary layer a more specific non dimensional Navier-Stokes equation is derived. Then the high Reynolds limit is taken with considerations of orders of magnitude and the boundary layer equations are derived. Finally, for turbulent near wall flows a dimensional analysis is made and the corresponding near wall non-dimensional velocities and coordinates y+ and u+ are derived from the Buckingham-Π theorem. Using these variables to non-dimensionalize the Navier-Stokes equations in the near wall turbulent region the third author Malcherek showed that the so derived non-dimensional Navier-Stokes equations do not depend on any non-dimensional number and has a unique solution. Hence, the logarithmic law of the wall must be universally valid, without any simplification, any turbulence model, empirical constant or further assumptions. In such a way the students do not have to believe anymore in the universality of the logarithmic law of the wall based on empirical evidence only, now this fact has been proven by the third author Malcherek and the larger context has been elaborated by all authors for an advanced teaching of wall bounded flows.