Tribological Study of a Slider Bearing in the Supersonic Regime

2016 ◽  
Vol 138 (4) ◽  
Author(s):  
Florence Dupuy ◽  
Benyebka Bou-Saïd ◽  
Mathieu Garcia ◽  
Grégory Grau ◽  
Jérôme Rocchi ◽  
...  
Keyword(s):  
High Speed ◽  
Reynolds Equation ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Slider Bearing ◽  
Navier Stokes Equations ◽  
Supersonic Regime ◽  
Two Dimensional Flow ◽  
Modified Reynolds Equation ◽  
The One

Aerodynamic slider bearings are currently used in various types of turbomachinery. Many such systems perform at increasingly faster speeds and may operate in the supersonic regime. Although there is extensive research on compressible lubrication extrapolated to high-speeds, very little of it addresses the potential supersonic nature of the flow. It is well known in compressible flow that many of the tendencies of subsonic flow actually reverse themselves as the singularity at Mach one is traversed. Thus, examination of this high-speed regime may yield some unanticipated results. The behavior of a thin film of air in the supersonic regime is studied in the two-dimensional flow case with rigid sliding surfaces. The one-dimensional bearing studied has a dual profile consisting of an inlet region converging wedge of constant slope and an exit region of constant gap. Two approaches are compared: the solution of a modified Reynolds equation, and the solution to a version of Navier–Stokes equations adapted to thin films. The results show that the modified Reynolds equation approach, which is useful to describe the behavior of lubricating fluids at high subsonic speeds may be inadequate in the supersonic regime. The present studies show the absence of shock and expansion wave phenomena for cases in which the film thickness ratio does not exceed 0.01.

scholarly journals A rigorous derivation of the stationary compressible Reynolds equation via the Navier–Stokes equations

2018 ◽  
Vol 28 (04) ◽  
pp. 697-732 ◽  
Author(s):  
I. S. Ciuperca ◽  
E. Feireisl ◽  
M. Jai ◽  
A. Petrov
Keyword(s):  
Reynolds Equation ◽  
Stokes System ◽  
Stokes Equations ◽  
A Priori ◽  
Limit Problem ◽  
Navier Stokes ◽  
Rigorous Derivation ◽  
Navier Stokes Equations ◽  
One Dimensional ◽  
The One

We provide a rigorous derivation of the compressible Reynolds system as a singular limit of the compressible (barotropic) Navier–Stokes system on a thin domain. In particular, the existence of solutions to the Navier–Stokes system with non-homogeneous boundary conditions is shown that may be of independent interest. Our approach is based on new a priori bounds available for the pressure law of hard sphere type. Finally, uniqueness for the limit problem is established in the one-dimensional case.

Singularities

2018 ◽  
Author(s):  
S. G. Rajeev
Keyword(s):  
Stokes Equations ◽  
Negative Answer ◽  
Three Dimensions ◽  
Picard Iteration ◽  
Navier Stokes ◽  
Scale Invariant ◽  
Navier Stokes Equations ◽  
L2 Norm ◽  
Two Dimensional Flow ◽  
Physical Consequences

The initial value problem of the incompressible Navier–Stokes equations is explained. Leray’s classic study of it (using Picard iteration) is simplified and described in the language of physics. The ideas of Lebesgue and Sobolev norms are explained. The L2 norm being the energy, cannot increase. This gives sufficient control to establish existence, regularity and uniqueness in two-dimensional flow. The L3 norm is not guaranteed to decrease, so this strategy fails in three dimensions. Leray’s proof of regularity for a finite time is outlined. His attempts to construct a scale-invariant singular solution, and modern work showing this is impossible, are then explained. The physical consequences of a negative answer to the regularity of Navier–Stokes solutions are explained. This chapter is meant as an introduction, for physicists, to a difficult field of analysis.

An Exact Solution of the Unsteady Navier-Stokes Equations Resulting From a Stretching Surface

1994 ◽  
Vol 61 (3) ◽  
pp. 629-633 ◽  
Author(s):  
S. H. Smith
Keyword(s):  
Exact Solution ◽  
Stokes Equations ◽  
Limited Range ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Stretching Surface ◽  
Navier Stokes Equations ◽  
Short Period ◽  
Two Dimensional Flow ◽  
Surrounding Fluid

When a stretching surface is moved quickly, for a short period of time, a pulse is transmitted to the surrounding fluid. Here we describe an exact solution in terms of a similarity variable for the Navier-Stokes equations which represents the effect of this pulse for two-dimensional flow. The unusual feature is that this solution is only valid for a limited range of the Reynolds number; outside this domain unbounded velocities result.

The turning couples on an elliptic particle settling in a vertical channel

1994 ◽  
Vol 271 ◽  
pp. 1-16 ◽  
Author(s):  
Peter Y. Huang ◽  
Jimmy Feng ◽  
Daniel D. Joseph
Keyword(s):  
Shear Stress ◽  
Vortex Shedding ◽  
Stokes Equations ◽  
Vertical Channel ◽  
Navier Stokes ◽  
Front Face ◽  
Two Dimensional ◽  
Navier Stokes Equations ◽  
Back Face ◽  
The One

We do a direct two-dimensional finite-elment simulation of the Navier–Stokes equations and compute the forces which turn an ellipse settling in a vertical channel of viscous fluid in a regime in which the ellipse oscillates under the action of vortex shedding. Turning this way and that is induced by large and unequal values of negative pressure at the rear separation points which are here identified with the two points on the back face where the shear stress vanishes. The main restoring mechanism which turns the broadside of the ellipse perpendicular to the fall is the high pressure at the ‘stagnation point’ on the front face, as in potential flow, which is here identified with the one point on the front face where the shear stress vanishes.

Two-dimensional flow of a viscous fluid in a channel with porous walls

1991 ◽  
Vol 227 ◽  
pp. 1-33 ◽  
Author(s):  
Stephen M. Cox
Keyword(s):  
Finite Time ◽  
Stokes Equations ◽  
Pitchfork Bifurcation ◽  
Time Dependent ◽  
High Rate ◽  
Navier Stokes ◽  
Recent Theory ◽  
Navier Stokes Equations ◽  
Two Dimensional Flow ◽  
Steady Solutions

We consider the flow of a viscous incompressible fluid in a parallel-walled channel, driven by steady uniform suction through the porous channel walls. A similarity transformation reduces the Navier-Stokes equations to a single partial differential equation (PDE) for the stream function, with two-point boundary conditions. We discuss the bifurcations of the steady solutions first, and show how a pitchfork bifurcation is unfolded when a symmetry of the problem is broken.Then we describe time-dependent solutions of the governing PDE, which we calculate numerically. We analyse these unsteady solutions when there is a high rate of suction through one wall, and the other wall is impermeable: there is a limit cycle composed of an explosive phase of inviscid growth, and a slow viscous decay. The inviscid phase ‘almost’ has a finite-time singularity. We discuss whether solutions of the governing PDE, which are exact solutions of the Navier-Stokes equations, may develop mathematical singularities in a finite time.When the rates of suction at the two walls are equal so that the problem is symmetrical, there is an abrupt transition to chaos, a ‘homoclinic explosion’, in the time-dependent solutions as the Reynolds number is increased. We unfold this transition by perturbing the symmetry, and compare direct numerical integrations of the governing PDE with a recent theory for ‘Lorenz-like’ dynamical systems. The chaos is found to be very sensitive to symmetry breaking.

scholarly journals Flow Pressure Behavior Downstream of Ski Jumps

Fluids ◽  
2020 ◽  
Vol 5 (4) ◽  
pp. 168 ◽  
Author(s):  
Agostino Lauria ◽  
Giancarlo Alfonsi ◽  
Ali Tafarojnoruz
Keyword(s):  
High Speed ◽  
Stokes Equations ◽  
Dynamic Pressure ◽  
Pressure Head ◽  
Navier Stokes ◽  
Maximum Pressure ◽  
Navier Stokes Equations ◽  
Free Jet ◽  
Jet Characteristics ◽  
The Impact

Ski jump spillways are frequently implemented to dissipate energy from high-speed flows. The general feature of this structure is to transform the spillway flow into a free jet up to a location where the impact of the jet creates a plunge pool, representing an area for potential erosion phenomena. In the present investigation, several tests with different ski jump bucket angles are executed numerically by means of the OpenFOAM® digital library, taking advantage of the Reynolds-averaged Navier–Stokes equations (RANS) approach. The results are compared to those obtained experimentally by other authors as related to the jet length and shape, obtaining physical insights into the jet characteristics. Particular attention is given to the maximum pressure head at the tailwater. Simple equations are proposed to predict the maximum dynamic pressure head acting on the tailwater, as dependent upon the Froude number, and the maximum pressure head on the bucket. Results of this study provide useful suggestions for the design of ski jump spillways in dam construction.

Influence of Inertia Terms on High Pressure Gap Flow Applications in Hydraulics

2019 ◽  
Author(s):  
Felix Fischer ◽  
Andreas Rhein ◽  
Katharina Schmitz
Keyword(s):  
High Pressure ◽  
Reynolds Equation ◽  
Stokes Equations ◽  
Fluid Phase ◽  
Simulation Models ◽  
Rigid Bodies ◽  
Navier Stokes ◽  
Dependent Manner ◽  
Fluid Inertia ◽  
Navier Stokes Equations

Abstract Hydraulic pumps, which reach pressures up to 3000 bar, are often realized as plunger-piston type pumps. In the case of a common-rail pump for diesel injection systems, the plunger is driven by a cam-tappet construction and the contact during suction stroke is maintained by a helical spring. Many hydraulic piston-based high pressure pumps include gap seals, which are formed by small clearances between the two surfaces of the piston and the bushing. Usually the gap height is in the magnitude of several micrometers. Typical radial gaps are between 0.5 and 1 per mil of the nominal diameter. These gap seals are used to allow and maintain pressure build up in the piston chamber. When the gap is pressurized, a special flow regime is reached. For the description of this particular flow the Reynolds equation, which is a simplification of the Navier-Stokes equations, can be used as done in the state of the art. Furthermore, if the pressure in the gap is high enough — 500 bar and above — fluid-structure interactions must be taken into account. Pressure levels above 1500 or 2000 bar indicate the necessity for solving the energy equation of the fluid phase and the rigid bodies surrounding it. In any case, the fluid properties such as density and viscosity, have to be modelled in a pressure dependent manner. This means, a compressible flow is described in the sealing gap. Viscosity changes in magnitudes while density remains in the same magnitude, but nevertheless changes about 30 %. These facts must be taken into account when solving the Reynolds equation. In this paper the authors work out that the Reynolds equation is not suitable for every piston-bushing gap seal in hydraulic applications. It will be shown that remarkable errors are made, when the inertia terms in the Navier-Stokes equations are neglected, especially in high pressure applications. To work out the influence of the inertia terms in these flows, two simulation models are built up and calculated for the physical problem. One calculates the compressible Reynolds equation neglecting the fluid inertia. The other model, taking the fluid inertia into account, calculates the coupled Navier-Stokes equations on the same geometrical boundaries. Here, the so called SIMPLE (Semi-Implicit Method for Pressure Linked Equations) algorithm is used. The discretization is realized with the Finite Volume Method. Afterwards, the solutions of both models are compared to investigate the influence of the inertia terms on the flow in these specific high pressure applications.

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