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The fluid dynamics of work transfer in the non-uniform viscous rotating flow within a Tesla disc turbomachine

### Abstract

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 ( ) is introduced. The value of 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 U θ r (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.

© 2014 AIP Publishing LLC

Received 31 May 2013
Accepted 20 January 2014
Published online 03 March 2014

Article outline:

I. INTRODUCTION
II. FORMULATION OF EULER’S WORK EQUATION AND WORK EQUIVALENCE PRINCIPLE FOR TESLA DISC TURBOMACHINE
A. Determination of angular momentum
B. Determination of viscous torque
C. Work equivalence principle
D. Turbine versus pump or compressor
III. ANALYTICAL DEMONSTRATION OF THE WORK EQUIVALENCE PRINCIPLE, EQ. (11)
A. Synopsis of an analytical theory for determining the flow field
B. Equivalence of Euler torque (Γ_{ Euler }) and torque calculated from shear stress (Γ_{ shear })
IV. INTERPRETATION OF TORQUE IN ABSOLUTE AND RELATIVE FRAMES: ROLE OF CORIOLIS FORCE
A. Work equivalence principle in the relative frame
V. COMPUTATIONAL FLUID DYNAMICS SOLUTION
VI. EXACT THEORY VERSUS AREA-AVERAGING APPROACH
VII. FURTHER PHYSICAL REFLECTIONS FOR DISC TURBINE
VIII. CONCLUSION

/content/aip/journal/pof2/26/3/10.1063/1.4866263

http://aip.metastore.ingenta.com/content/aip/journal/pof2/26/3/10.1063/1.4866263

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/content/aip/journal/pof2/26/3/10.1063/1.4866263

2014-03-03

2016-10-22

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