In the present article, a mathematical theory for the flow field within a Tesla disc turbine has been formulated in the appropriate cylindrical co-ordinate system. The basis of the theory is the Navier–Stokes equations simplified by a systematic order of magnitude analysis. The presented theory can compute three-dimensional variation of the radial velocity, tangential velocity and pressure of the fluid in the flow passages within the rotating discs. Differential equations as well as closed-form analytical relations are derived. The present mathematical theory can predict torque, power output and efficiency over a wide range of rotational speed of the rotor, in good agreement with recently published experimental data. The performance of the turbine is characterized by conceptualizing the variation of load through the non-dimensional ratio of the absolute tangential velocity of the jet and the peripheral speed of the rotor. The mathematical model developed here is a simple but effective method of predicting the performance of a Tesla disc turbine along with the three-dimensional flowfield within its range of applicability. A hypothesis is also presented that it may be possible to exploit the effects of intelligently designed and manufactured surface roughness elements to enhance the performance of a Tesla disc turbine.
In this article, the fluid dynamics of work transfer within the narrow spacing (usually of the order of 100 μm) of multiple concentric discs of a Tesla disc turbomachine (turbine or compressor) has been analysed theoretically and computationally. Both the overall work transfer and its spatial development have been considered. It has been established that the work transfer mechanism in a Tesla disc turbomachine is very different from that in a conventional turbomachine, and the formulation of the Euler's work equation for the disc turbomachine contains several conceptual subtleties because of the existence of complex, three dimensional, non-uniform, viscous flow features. A work equivalence principle has been enunciated, which establishes the equality between the magnitudes of work transfer determined rigorously from two different approaches—one based on the shear stress acting on the disc surfaces and the other based on the change in angular momentum of the fluid. Care is needed in identifying the shear stress components that are responsible for the generation or absorption of useful power. It is shown from the Reynolds transport theorem that mass-flow-averaged tangential velocities (as opposed to the normally used area-averaged values) must be used in determining the change in angular momentum; the calculation has to be carefully formulated since both radial velocity (that determines throughput) and tangential velocity (that generates torque) depend strongly on the coordinate perpendicular to the disc surfaces. The principle of work transfer has been examined both in the absolute and relative frames of reference, revealing the subtle role played by Coriolis force. The concept of a new non-dimensional quantity called the torque potential fraction ($\Delta \tilde H$ΔH̃) is introduced. The value of $\Delta \tilde H$ΔH̃ at any radial position increases with a decrease in inter-disc spacing. The computational fluid dynamic analysis shows that, for small value of inter-disc spacing and high value of tangential speed ratio, most of the angular momentum of the fluid is transferred to the surfaces of the discs in the inlet region and correspondingly, the value of the torque potential fraction is very high even in the inlet region. On the other hand, for larger inter-disc spacing, the change in angular momentum in the radial direction is more uniformly distributed between the inlet and the outlet, and the value of the torque potential fraction increases gradually with decreasing radius. The complex (sometimes continuous, sometimes disjointed) three-dimensional shapes of the iso-surfaces of Uθr (product of absolute tangential velocity and radius) have been shown, for the first time, which provide insight into the fluid dynamics of work transfer within corotating discs.
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