To characterize contactless seals in turbo machinery, their discharge behavior, the development of the circumferential velocity (swirl) and the loss induced total temperature increase (windage heating) are of special interest for the designer. For the discharge behavior of non-rotating labyrinth seals, a well established set of non-dimensional numbers already exists: the discharge coefficient of two seals with different sizes but similar geometry is identical, if pressure ratio, axial Reynolds number, fluid properties and turbulence level are also identical. In this paper, the set of non-dimensional numbers is extended to cover swirl and windage heating using the well established Buckingham-π theorem to derive possible candidates. First, as a proof of concept, the known set of numbers for the non-rotating case was redeveloped and subsequently the influence of rotation was included. To validate the candidates, a comprehensive numerical parametric study was conducted. A variety of convergent and divergent stepped labyrinth seals was scaled from laboratory to typical engine conditions such that the dimensionless numbers stayed constant. Then, simulations at different rotational speeds, radii, and inlet circumferential velocities were performed to investigate the effects of rotation while maintaining nearly constant discharge behavior. The numerical data were used to validate the new non-dimensional numbers and to derive laws for the scaling of labyrinth seals. The non-dimensional numbers can also be applied to other seal types, such as brush or finger seals, because their theoretical deduction does not imply a specific geometry.

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