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RESEARCH PAPERS

# Navier–Stokes Analysis of a Regular Two-Dimensional Roughness Pattern Under Turbulent Flow Regime

[+] Author and Article Information
Frédéric Billy, Gérard Pineau

Université de Poitiers, SP2MI, Téléport 2, Boulevard Pierre et Marie Curie, Boîte Postale 30179, Futuroscope-Chasseneuil Cedex, 86962, France

Mihai Arghir

Université de Poitiers, SP2MI, Téléport 2, Boulevard Pierre et Marie Curie, Boîte Postale 30179, Futuroscope-Chasseneuil Cedex, 86962, Francemihai.arghir@lms.univ-poitiers.fr

The calculation cases are identified by indicating the driving effect (C ouette for the wall velocity and P oiseuille for the pressure gradient); the subscript is related to the Reynolds number based on the mass flow rate (for example $C2$ is $CRe=1930$, etc.)

In order to emphasize the link between this simplified analysis and the case of a damper seal, the upper moving wall will be designated as rotor and the lower stationary wall will be designated as stator.

The fact that $n0∕n1=1.2$ was of secondary importance in Hirs’ approach because this value never entered in the relation of the equivalent Couette-Poiseuille pressure gradient.

J. Tribol 128(1), 122-130 (Jun 22, 2005) (9 pages) doi:10.1115/1.2000271 History: Received February 24, 2004; Revised June 22, 2005

## Abstract

The present work deals with the flow characteristics induced by a two-dimensional textured surface. The texture consists of identical and equally spaced rectangles with characteristic lengths at least one order of magnitude larger than the clearance of the thin film. Periodic boundary conditions enable the analysis of a single groove and the complete Navier–Stokes analysis is carried on for turbulent flow Reynolds numbers. The analysis is performed for shear driven flows (Couette), pressure driven flows (Poiseuille), and combined Couette–Poiseuille flows. First, the presence of inertial forces generated by the groove is emphasized by the momentum balance performed for the computational cell. The peculiar effect of the groove is also shown by the rotor and the stator shear stresses variations. Finally, it is shown that despite the presence of fluid inertia forces, cell-averaged rotor, and stator shear stresses obtained for pure Couette or Poiseuille flows can be added or subtracted to obtain with good accuracy the characteristics of combined shear and pressure driven flows.

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## Figures

Figure 1

Geometry of the two-dimensional macrotexture

Figure 2

Dimensionless wall distance of the first grid point

Figure 3

Grid detail for case P2(Re=1930)

Figure 4

Dimensionless shear stress variations for Poiseuille flow, (a) Re=1960 and (b) Re=13,930

Figure 5

Dimensionless shear stress variations for Couette flow, (a) Re=1960 and (b) Re=13,930

Figure 6

Dimensionless drag effect vs Reynolds number

Figure 7

Dimensionless periodic static pressure on rotor surface for (a) Couette flow and (b) Poiseuille flow

Figure 8

Dimensionless lift effect (normal force) vs the Reynolds number

Figure 9

Dimensionless shear stress for combined Couette–Poiseuille flows

Figure 10

Correlation between (a) rotor shear stresses, (b) stator shear stresses, (c) additional drag ζ, and (d) additional lift W*, obtained from Couette–Poiseuille flows and the corresponding values obtained by superposing separated flows

Figure 11

Dimensionless pressure variation along the upper wall (S3) for case P2

Figure 12

Dimensionless shear stress variation along S3 (a) and S4+S5+S6 (b)

Figure 13

Contours of turbulent kinetic energy (m2∕s2)

Figure 14

Contours of the stream function (kg/s)

Figure 15

Contours of velocity magnitude (m/s)

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