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Research Papers

# Unsteady Flow and Dynamic Behavior of Ultrashort Lomakin Gas Bearings

[+] Author and Article Information
C. J. Teo, Z. S. Spakovszky, S. A. Jacobson

Gas Turbine Laboratory, Department of Aeronautics and Astronautics,  Massachusetts Institute of Technology, Cambridge, MA 02139

The axial flow-through time is much shorter than the flow change time around the circumference such that the reduced frequency $β=ΩL∕U¯∼0.05$.

Equation 12 has the form of the heat diffusion equation, the corresponding heat transfer problem being the sudden increase in temperature of a plate. The analogy is that vorticity spreads like heat (12).

The axial Mach number $Mz$ is of order 0.3. For the circumferential flow, numerical studies show that compressibility effects are insignificant for small radial eccentricities, as previously reported in Ref. 1.

An isotropic journal bearing has a circumferentially uniform direct-coupled radial stiffness.

These dimensions correspond to the geometric dimensions of the rotor and journal bearing of a novel gas-bearing-supported microturbine device (10).

J. Tribol 130(1), 011001 (Dec 06, 2007) (9 pages) doi:10.1115/1.2805403 History: Received April 03, 2006; Revised September 19, 2007; Published December 06, 2007

## Abstract

Ultrashort microscale high-speed gas bearings exhibit a whirl instability limit and dynamic behavior much different from conventional hydrostatic gas bearings. In particular, the design space for a stable high-speed operation is confined to a narrow region and involves a singular behavior. The previously developed ultrashort gas bearing theory (Liu (2005, “Hydrostatic Gas Journal Bearings for Micro-Turbomachinery  ,” ASME J. Vibr. Acoust., 127(2), pp. 157–164)) assumed fully developed flow in the journal bearing gap. There is experimental evidence that this assumption might not be fully applicable for the relatively short flow-through times in such bearings. This has an impact on the estimation of whirl instability onset, bearing operability and power requirements. In this paper, unsteady flow effects in the bearing gap are investigated with the goal to quantify their impact on the bearing dynamic behavior. It is shown that although three-dimensional flow calculations in the ultrashort journal bearing are necessary to quantify the onset of whirl instability, the underlying mechanisms can be qualitatively described by the impulsive starting of a Couette flow. Using this description, two time scales are identified that govern the journal bearing dynamic behavior: the viscous diffusion time and the axial flow-through time. Based on this, a reduced frequency parameter is introduced that determines the development of the flow field in the journal bearing and, together with bearing force models, yields a criterion for whirl instability onset. Detailed three-dimensional computational fluid dynamics calculations of the journal bearing flow have been conducted to assess the criterion. A singular behavior in whirl ratio as a function of the reduced frequency parameter is observed, verifying the refined stability criterion. Using high-fidelity flow calculations, the effects of unsteady journal bearing flow on whirl instability limit and bearing power loss are quantified, and design guidelines and implications on gas bearing modeling are discussed. The stability criterion is experimentally validated demonstrating repeatable, stable high-speed operation of a novel microbearing test device at whirl ratios of 35.

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## Figures

Figure 1

The diffusion of vorticity in an impulsively started Couette flow (top) and across the journal bearing gap in an ultrashort journal bearing (bottom)

Figure 2

Development of normalized circumferential velocity profile at various axial locations z∕L: impulsively started Couette flow (symbols) and numerical simulation results (lines)

Figure 3

(a) Whirl ratio as a function of journal bearing clearance. The points for each clearance correspond to different bearing DPs. (b) Singular behavior in whirl ratio at a reduced frequency βFD*≈0.43.

Figure 4

Rotor speed corrected cross-coupled hydrodynamic stiffness as a function of journal bearing clearance: fully developed flow model predictions (solid lines) compared to numerical simulation results (dashed lines)

Figure 5

Net rotor speed corrected cross-coupled hydrodynamic stiffness as a function of journal bearing clearance and hydrostatic DP

Figure 6

Comparison of experimental and analytical-CFD predictions of journal bearing stability boundary as a function of hydrostatic DP

Figure 7

Normalized local circumferential wall shear stress along the length of the journal bearing

Figure 8

Estimated journal bearing power requirements: fully developed (solid line) and developing analytical flow models (dashed line)

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