Similitude and scaling laws for the rotating flow between concentric discs

In this article, a systematic dimensional analysis and similitude study for the three-dimensional rotating flow within the narrow spacing (usually of the order of 100 mu m) of multiple concentric discs is presented. An engineering application of this flow configuration is the Tesla disc turbine, and the dimensional analysis leads to proper scaling laws for such machines. Using the Buckingham Pi theorem, the list of non-dimensional numbers necessary for describing the incompressible flow through the concentric discs has been formulated and physical interpretation of the non-dimensional numbers has been provided. The complete criteria for achieving geometric, kinematic and dynamic similarity between a model and the prototype have been established. Computational fluid dynamics (CFD) solutions have been obtained for various geometries, fluid properties and flow conditions to demonstrate the validity of the similitude criteria developed. The CFD results show that, when the present similitude criteria are fulfilled, the values of all non-dimensional output parameters such as the power and pressure-drop coefficients remain unchanged for various combinations of input variables. What is more striking is that the three-dimensional variations of non-dimensional z component of velocity U-z (whose non-zero values are computed by the CFD solver even though the only physical boundary condition applied is Uz=0 at the inlet and on disc surfaces) are superposed on one another for all models and the prototype. Systematic methods for arriving at simplified conservation equations are discussed and it is shown how the corresponding list of non-dimensional numbers and similitude criteria evolve with such simplification of the conservation equations.