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 ($\Delta \tilde H$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.

Analytical and Numerical Solutions of the Rotor Flow in Tesla Turbines

“…Around the turn of the millennium, first CFD approaches came up. Currently, it is still an ongoing topic [7,8]. Some researchers also deal with microscale friction turbines, such as Romanin et al [4] or Krishnan et al [5].…”

“…Several Tesla rotor flow models like the ones presented in [3,7,9], as well as the models described before [1] assume that the third velocity component is zero compared to the radial and tangential velocity distribution. The axial velocity distribution over the inter-disk spacing for different radial positions are shown in Fig.…”

The physics of pressure variation in microchannels within corotating or static discs

“…(iv) The inter-disc spacing b most strongly affects the viscous component ∆p io,viscous ; its effects on the other three components are mainly through the velocity field (although the inertia component ∆p io,inertia contains the velocity gradient ∂V r /∂z, the gradient is multiplied by a small quantity V z ). (v) The radial variation in V θ can be complex, 27,30 and consequently there can be subtle variation in the magnitude and sign of the Coriolis component ∆p io,Coriolis , which depends on the cross-product of the relative tangential velocity V θ and angular velocity of the discs. The sign of V θ does not bring similar complexity in the variation of ∆p io,inertia , since it appears as a squared quantity (i.e., as V 2 θ ).…”

“…The grids are constructed carefully as per Ref. 30. The grids are distributed in r and z directions in accordance with the difference in the flow physics in the two directions.…”

“…The grid distribution in the z-direction is non-uniform with very small grid size close to the surfaces of the two discs (to capture the velocity gradient on the surface accurately) and with progressively larger grid size as one moves away from the surfaces to the middle of the inter-disc gap (with a successive ratio of 1.05). Guha and Sengupta 30 have shown that when there is no z-dependence in the applied boundary conditions at inlet for the radial and tangential velocities, both velocities change very rapidly within a very short radial distance from the inlet, within which the z-profiles of both velocities are created from their uniform values at inlet. In order to capture this effect properly, the grids in the radial direction are divided into two zones-non-uniform and uniform.…”