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Vortex breakdown phenomena in rotating fluids are investigated both theoretically and experimentally. The fluid is contained in a cone between two spherical surfaces. The primary swirling motion is induced ba the rotating lower boundary. The upper surface can be fixed with non-slip condition or can be a stress-free surface. Depending on these boundary conditions and on the Reynolds number, novel structures of recirculation zones are realized. The axisymmetric flow patterns are simulated numerically by a finite difference method. Experiments are done to visualize the topological structure of the flow pattern and to observe the existence ranges of the different recirculating flows. The comparison between theory and experiment shows good agreement with respect to the topological structure of the flow.
This mature textbook brings the fundamentals of fluid mechanics in a concise and mathematically understandable presentation. In the current edition, a section on dissipation and viscous potential flows has been added. Exercises with solutions help to apply the material correctly and promote understanding.
Jürgen Zierep passed away on July 29, 2021, at the age of 92. To him, science and education was not only a profession, but an affair of the heart. His impressive contributions in fluid mechanics comprise about 200 scientific publications in the fields of gas dynamics, similarity laws, flow instabilities, flows with energy transfer, and non-Newtonian fluids. In addition, he wrote eleven textbooks with great dedication. Those books by the “scientist who loves to teach” are nowadays available in different languages and regularly appear in new editions.
Flows in nature and technology are often associated with specific structures and pattern. This paper deals with the development and behaviour of such flow pattern. Flow structures are important for the mass, momentum and energy transport. The behaviour of different flow pattern is used by engineers to obtain an efficient mass and energy consumption. Mechanical power is transmitted via the momentum of rotating machine parts. Therefore the physical and mathematical knowledge of these basic concepts is important. Theoretical and experimental investigations of principle experiments are described in the following. We start with the classical problem of the flow between two concentric cylinders where the inner cylinder rotates. Periodic instabilities occur which are called Taylor vortices. The analogy between the cylindrical gap flow, the heat transfer in a horizontal fluid layer exposed to the gravity field and the boundary layer flow along concave boundaries concerning their stability behaviour is addressed. The vortex breakdown phenomenon in a cylinder with rotating cover is also described. A generalization to spherical sectors leads then to investigations with different boundary conditions. The spherical gap flow exhibits interesting phenomena concerning the nonlinear character of the Navier-Stokes equations. Multiple solutions in the nonlinear regime give rise to different routes during the laminar-turbulent transition. The interaction of two rotating spheres results in flow structures with separation and stagnation lines. Experimental results are confirmed by numerical simulations.
The structure of the separation bubble that appears in the secondary meridional flow between two coaxially rotating spheres at low and finite Reynolds number (Re) is considered. The low Re analytical study was motivated by recognizing some errors in the analytical work on this problem by Arunachalam and Majhi (1987, Q. Jl Mech. Appl. Math., 40, 47) whilst the finite Re experimental study was motivated by the desire to observe the separation bubble in the laboratory. Though the finite Re experiments were performed in a confined apparatus, they exhibit the qualitative features of the low Re theoretical predictions for the axisymmetric separation bubble that encloses two toroidal vortices symmetrically disposed above and below the mid‐plane of sphere separation, but strong effects of confinement are apparent. The flows observed include (i) a wall‐attached bubble symmetric about the mid‐plane at low Re, (ii) symmetric free‐standing bubbles at moderate Re, and (iii) an asymmetric bubble with flow separating from one sphere and attaching to the support shaft between the spheres at sufficiently high Re.
The free convection in a vertical gap is generalized to realize new analytical solutions of the Boussinesq-equations. The steady and time-dependent solutions for the temperature and velocity distribution are discussed in detail depending on the mass flux in vertical direction. The range of existence for flows with and without back flow is obtained. The transient behaviour of the solutions during the time-dependent development displays interesting physical behaviour.
Rotating flow systems are often used to study stability phenomena and structure developments. The closed spherical gap problem is generalized into an open flow system by superimposing a mass flux in meridional direction. The basic solutions at low Reynolds numbers are described by analytical methods. The nonlinear supercritical solutions are simulated numerically and realized in experiments. Novel steady and time-dependent modes of flows are obtained. The extensive results concern the stability behaviour, non-uniqueness of supercritical solutions, symmetry behaviour and transitions between steady and time-dependent solutions. The experimental investigations concern the visualization of the various instabilities and the quatitative description of the flow structures including the laminar-turbulent transition. A comparison between theoretical and experimental results shows good agreement within the limit of rotational symmetric solutions from the theory.
We generalize the fluid flow problem of an oscillating flat plate (II. Stokes problem) in two directions. We discuss first the oscillating porous flat plate with superimposed blowing or suction. The second generalization is concerned with an increasing or decreasing velocity amplitude of the oscillating flat plate. Finally we show that a combination of both effects is possible as well.
Shapes and structures of vortex breakdown phenomena in rotating fluids are visualized. We investigate the flow in a cylindrical container and in a cone between two spherical surfaces. The primary swirling flow is induced by the rotating upper disk in the cylindrical case and by the lower boundary in the spherical case. The upper surface can be fixed with a no slip condition or can be a stress-free surface. Depending on these boundary conditions and on the Reynolds number novel structures of recirculation zones are realized. Experiments are done to visualize the topological structure of the flow and to determine their existence range as function of the geometry and rotation rate. A comparison between the experimental and theoretical approach shows a good agreement in respect to the topological structures of the flows.
We herein present a topology design method based on local optimality criteria which has been implemented in an open source Navier-Stokes solver for turbulent flows. Our method aims for the fast generation of geometry proposals in the early conceptual phase. To the best of our knowledge, this is the first local criteria approach utilizing a wall function turbulence model in order to consider turbulent flows. In order to allow for the growth as well as the shrinkage, or even the formation or disappearance of structural features, a topological approach is chosen. By introducing a volume fraction parameter, we distinguish between fluid and solid properties in each control volume. The fluid-solid interface is represented by an immersed boundary method using a piecewise linear surface reconstruction.
This study focuses on the experimental and numerical investigations on a commercial Ranque-Hilsch vortex tube. Ranque-Hilsch vortex tubes have many applications in industry and production as they can generate a very cold flow just from pressurized air .e.g. machine tool cooling. Main objective of this study is the energy separation in the flow field which results in a temperature drop on the cold exit of the tube. This was investigated experimentally by measuring the outlet temperature on the cold exit and the pressure drop on the flow restrictor valve on the hot exit. At a pressure drop of 0.5 bar the vortex tube showed the best performance by reaching a cold exit temperature of –16.7 °C. The Inlet flow was pressurised air at 20 °C and 6 bar.<br /> The numerical analysis was carried out by full 3D steady state CFD-simulation using the commercial software ANSYS CFX 11.0. The three dimensional model represented a 120° sector of the tube using periodic boundary conditions. A comparison between different turbulence models (k – å, RNG k – å, k – ù, SST) was carried out. The classic k – å two layer turbulence model showed the best results compared to the experiment. The energy separation and the drop in cold exit temperature are highest when the viscous work term is included into the energy equation. These effects of including the viscous work term into the energy separation have also been investigated.