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Research Papers: Fundamental Issues and Canonical Flows

Power Scaling of Radial Outflow: Bernoulli Pads in Equilibrium

[+] Author and Article Information
Kristina M. Kamensky

Department of Mechanical Engineering,
Michigan State University,
East Lansing, MI 48824
e-mail: kmk@egr.msu.edu

Aren M. Hellum

Vehicle Dynamics and Control Group,
Naval Undersea Warfare Center,
Newport, RI 02841
e-mail: aren.hellum@navy.mil

Ranjan Mukherjee

Fellow ASME
Department of Mechanical Engineering,
Michigan State University,
East Lansing, MI 48824
e-mail: mukherji@egr.msu.edu

1Corresponding author.

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received September 28, 2018; final manuscript received February 28, 2019; published online April 15, 2019. Assoc. Editor: Ioannis K. Nikolos.This work is in part a work of the U.S. Government. ASME disclaims all interest in the U.S. Government's contributions.

J. Fluids Eng 141(10), 101201 (Apr 15, 2019) (9 pages) Paper No: FE-18-1655; doi: 10.1115/1.4043061 History: Received September 28, 2018; Revised February 28, 2019

A Bernoulli pad uses an axial jet to produce radial outflow between the pad and a proximally located parallel surface. The flow field produces a force between the surfaces, which depends upon their spacing h. The direction of this force is repulsive as h approaches zero and becomes attractive as h increases. This yields a stable equilibrium point heq, where the force is equal to zero. The present computational work indicates that a power-law relationship exists between heq and the inlet fluid power required to sustain this equilibrium spacing when each is appropriately scaled. This scaling is derived principally from the wall shear; an additional term incorporating the inlet Reynolds number is used to account for the force applied to the system. The relationship is valid over a range of forces acting on the system, geometric, and material properties.

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Figures

Grahic Jump Location
Fig. 1

(a) Parameters characterizing radial outflow between a Bernoulli pad and a proximal parallel surface. The embodiments (b)–(e) correspond to a workpiece being lifted, a workpiece being levitated, the pad being levitated, and the pad being lifted, respectively. The applied force Fa is the sum of all forces acting on the freely moving component (pad or surface) which are not produced by the fluid flow between the pad and the surface. The convention used here is that Fa > 0 (Fa < 0) attempts to increase (decrease) the gap between the pad and the proximal surface.

Grahic Jump Location
Fig. 2

Comparison of CFD simulation results with experimental results in Ref. [6]: (a) pressure p as a function of radius r for d =4 mm, D =40 mm, h =0.4 mm, and m˙=5.1×10−4 kg/s, analytical results available for r =0 mm [6] are also presented; (b) fluid force on workpiece F as a function of gap height h for a mass flow rate of m˙=5.1×10−4 kg/s. Stable and unstable equilibrium points are found at h =0.29 mm and h =5.1 mm, respectively. The insets are provided to indicate the magnitude of the error bars where it would not be visible otherwise.

Grahic Jump Location
Fig. 3

Example of interpolation to find equilibrium gap height heq for zero and a nonzero value of the applied force Fa

Grahic Jump Location
Fig. 4

Fluid power W˙in required to maintain an equilibrium height heq for Fa = 0. The nominal dataset is d =4 mm, D =40 mm, and εs = 0.04 mm, based on air as the working fluid; each other dataset varies one parameter from this baseline. The dotted line is the best fit to the nominal dataset.

Grahic Jump Location
Fig. 5

Ratio of inlet fluid pressure to inlet momentum as a function of equilibrium height heq for Fa = 0. The nominal dataset is d =4 mm, D =40 mm, and εs = 0.04 mm, based on air as the working fluid; each other dataset varies one parameter from this baseline. The dotted line is the best fit to the nominal dataset.

Grahic Jump Location
Fig. 6

Nondimensional fluid power W˙* required to maintain a given nondimensional equilibrium height h*. The nominal dataset is d =4 mm, D =40 mm, and εs = 0.04 mm, based on air as the working fluid; each other dataset varies one parameter from this baseline. The dotted line is the best fit to all data.

Grahic Jump Location
Fig. 7

Nondimensional fluid power W˙* required to maintain a given nondimensional equilibrium height h* for different values of F* when the force correction term in Eq. (6) is not used. The nominal dataset is d =4 mm, D =40 mm, and εs = 0.04 mm, using air as the working fluid. The dotted line is the best fit tothe F* = 0 dataset.

Grahic Jump Location
Fig. 8

Nondimensional fluid power W˙corr* required to maintain a given nondimensional equilibrium height h* for different values of F* when the force correction term in Eq. (6) is used with C =2/5. The nominal dataset is d =4 mm, D =40 mm, and εs = 0.04 mm, using air as the working fluid. The dotted line is the best fit to the F* = 0 dataset. Points associated with F* = +6 at h* > 0.25 are not visible in the present spread.

Grahic Jump Location
Fig. 9

Fluid power W˙in required to maintain an equilibrium height heq for the nominal data set: d =4 mm, D =40 mm, εs = 0.04 mm, and air as the working fluid with Fa = 0. The GCI error bars are included with sample calculations shown in Table 3.

Tables

Errata

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