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.

Relationship between accuracy and number of velocity particles of the finite-difference lattice Boltzmann method in velocity slip simulations

2010 ◽  
Vol 132 (10) ◽  
Author(s):  
Minoru Watari
Keyword(s):  
Relaxation Process ◽  
Stokes Equations ◽  
Velocity Slip ◽  
Simulation Models ◽  
Knudsen Layer ◽  
Navier Stokes ◽  
Flow Area ◽  
Navier Stokes Equations ◽  
Conservation Of Momentum ◽  
Boltzmann Method

Relationship between accuracy and number of velocity particles in velocity slip phenomena was investigated by numerical simulations and theoretical considerations. Two types of 2D models were used: the octagon family and the D2Q9 model. Models have to possess the following four prerequisites to accurately simulate the velocity slip phenomena: (a) equivalency to the Navier–Stokes equations in the N-S flow area, (b) conservation of momentum flow Pxy in the whole area, (c) appropriate relaxation process in the Knudsen layer, and (d) capability to properly express the mass and momentum flows on the wall. Both the octagon family and the D2Q9 model satisfy conditions (a) and (b). However, models with fewer velocity particles do not sufficiently satisfy conditions (c) and (d). The D2Q9 model fails to represent a relaxation process in the Knudsen layer and shows a considerable fluctuation in the velocity slip due to the model’s angle to the wall. To perform an accurate velocity slip simulation, models with sufficient velocity particles, such as the triple octagon model with moving particles of 24 directions, are desirable.

Evolution of Generic Vanelets Applied to a High Pressure Compressor

2013 ◽  
Author(s):  
Andreas Loos ◽  
Tobias Mayenberger ◽  
Florian Danner ◽  
Hans-Peter Kau
Keyword(s):  
High Pressure ◽  
Stokes Equations ◽  
Main Flow ◽  
Navier Stokes ◽  
Stable Operation ◽  
Navier Stokes Equations ◽  
Surge Margin ◽  
Flow Phenomena ◽  
Negative Effect ◽  
Generic Design

The flow field of high pressure compressors is strongly influenced by secondary flow phenomena which lead to performance degradations. A significant fraction of the associated losses arises from tip as well as hub clearance vortices and their interaction with the main flow. In order to decrease the negative effect of clearance vortices, the application of vanelets, winglet-like structures attached to the tips of a cantilevered stator, is studied within the present paper. Different vanelets of generic design are applied to the stator and evaluated with respect to their aerodynamic effect by comparison against a datum configuration. The model comprises the investigated stator enclosed between two rotating blade rows. Detailed insight into the underlying phenomena is provided by numerical investigations with the compressible Reynolds-averaged Navier-Stokes equations. The structures led to an increased efficiency at the aerodynamic design point due to the suppression of the clearance mass flow in combination with a reduced vortex cross section. Under strongly throttled conditions a so called vanelet corner stall developed, which induced blockage near hub. Thus the main flow was displaced towards casing enhancing stable operation of the downstream rotor. Surge margin was consequently increased.

Inertial effect on gas squeeze film for large radius disc excited by standing waves with complex modal shapes

2019 ◽  
Vol 33 (24) ◽  
pp. 1950282 ◽  
Author(s):  
Yi Qiang Fan ◽  
M. Miyatake ◽  
S. Kawada ◽  
Bin Wei ◽  
S. Yoshimoto
Keyword(s):  
Reynolds Equation ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Standing Waves ◽  
Inertial Effect ◽  
Squeeze Film ◽  
Navier Stokes ◽  
Large Radius ◽  
Navier Stokes Equations ◽  
Sinusoidal Functions

In order to investigate the gas inertial effect on bearing capacity of acoustic levitation on condition of complex exciting shapes, a new kind of numerical model including inertial effect in cylindrical coordinates was proposed. The inertial terms in Navier–Stokes equations are packaged to derive modified Reynolds equations. The amplitudes of standing waves were tested by distance probe in experiment and film thickness equation were reconstructed by sum of the sinusoidal functions. The theoretical and experimental results implied that the inertial effect is strongly related to the exciting modal shapes. It is concluded that the proposal of modified Reynolds equation can provide more optimized numerical solutions to solve the problems about the deviation between theoretical and experimental data.

Swirling Laminar Pipe Flow of Suspensions

1973 ◽  
Vol 40 (2) ◽  
pp. 331-336 ◽  
Author(s):  
S. K. Tung ◽  
S. L. Soo
Keyword(s):  
Pipe Flow ◽  
Stokes Equations ◽  
Fluid Phase ◽  
Navier Stokes ◽  
Swirl Ratio ◽  
Particulate Phase ◽  
Flow Properties ◽  
Navier Stokes Equations ◽  
Flow Reynolds Number ◽  
Laminar Pipe Flow

Vortex pipe flow of suspensions with laminar motion in the fluid phase is treated. The pipe consists of two smoothly joined sections, one stationary and the other rotating with a constant angular velocity. The flow properties of the fluid phase are determined by solving the complete Navier-Stokes equations numerically. The governing parameters are the flow Reynolds number and swirl ratio. Subsequent numerical solution to the momentum equations governing the particulate phase provides for both particle velocity and concentration distributions.

scholarly journals Generalized enthalpy model of a high-pressure shift freezing process

2012 ◽  
Vol 468 (2145) ◽  
pp. 2744-2766 ◽  
Author(s):  
Nadia A. S. Smith ◽  
Stephen S. L. Peppin ◽  
Ángel M. Ramos
Keyword(s):  
High Pressure ◽  
Stokes Equations ◽  
Freezing Point ◽  
Food Samples ◽  
Navier Stokes ◽  
Freezing Process ◽  
Frozen Foods ◽  
Navier Stokes Equations ◽  
Pressure Shift ◽  
Pressure Shift Freezing

High-pressure freezing processes are a novel emerging technology in food processing, offering significant improvements to the quality of frozen foods. To be able to simulate plateau times and thermal history under different conditions, in this work, we present a generalized enthalpy model of the high-pressure shift freezing process. The model includes the effects of pressure on conservation of enthalpy and incorporates the freezing point depression of non-dilute food samples. In addition, the significant heat-transfer effects of convection in the pressurizing medium are accounted for by solving the two-dimensional Navier–Stokes equations. We run the model for several numerical tests where the food sample is agar gel, and find good agreement with experimental data from the literature.

scholarly journals Effect of Inlet Temperature Non-Uniformity on High-Pressure Turbine Performance

2010 ◽  
Author(s):  
Craig I. Smith ◽  
Dongil Chang ◽  
Stavros Tavoularis
Keyword(s):  
High Pressure ◽  
Heat Load ◽  
Stokes Equations ◽  
Computation Time ◽  
Radial Pressure ◽  
Navier Stokes ◽  
Inlet Temperature ◽  
High Pressure Turbine ◽  
Navier Stokes Equations ◽  
Hot Streak

The temperature of the flow entering a high-pressure turbine stage is inherently non-uniform, as it is produced by several discrete, azimuthally-distributed combustors. In general, however, industrial simulations assume inlet temperature uniformity to simplify the preparation process and reduce computation time. The effects of a non-uniform inlet field on the performance of a commercial, transonic, single-stage, high-pressure, axial turbine with a curved inlet duct have been investigated numerically by performing URANS (Unsteady Reynolds-Averaged Navier-Stokes equations) simulations with the SST (Shear Stress Transport) turbulence model. By adjusting the alignment of the experimentally-based inlet temperature field with respect to the stator vanes, two clocking configurations were generated: an aligned case, in which each hot streak impinged on a vane and a misaligned case, in which each hot streak passed between two vanes. In the aligned configuration, the hot streaks produced higher time-averaged heat load on the vanes and lower heat load on the blades. As the aligned hot streaks impinged on the stator vanes, they also spread spanwise due to the actions of the casing passage vortices and the radial pressure gradient; this resulted in a stream entering the rotor with relatively low temperature variations. The misaligned hot streaks were convected undisturbed past the relatively cool vane section. Relatively high time-averaged enthalpy values were found to occur on the pressure side of the blades in the misaligned configuration. The non-uniformity of the time-averaged enthalpy on the blade surfaces was lower in the aligned configuration. The flow exiting the rotor section was much less non-uniform in the aligned case, but differences in calculated efficiency were not significant.

Finite-Reynolds-Number Effects in Steady, Three-Dimensional Airway Reopening

2006 ◽  
Vol 128 (4) ◽  
pp. 573-578 ◽  
Author(s):  
Andrew L. Hazel ◽  
Matthias Heil
Keyword(s):  
Reynolds Number ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Elastic Tube ◽  
Navier Stokes ◽  
Fluid Inertia ◽  
Navier Stokes Equations ◽  
Airway Reopening ◽  
Reynolds Number Effects ◽  
Steady Propagation

Motivated by the physiological problem of pulmonary airway reopening, we study the steady propagation of an air finger into a buckled elastic tube, initially filled with viscous fluid. The system is modeled using geometrically non-linear, Kirchhoff-Love shell theory, coupled to the free-surface Navier-Stokes equations. The resulting three-dimensional, fluid-structure-interaction problem is solved numerically by a fully coupled finite element method. Our study focuses on the effects of fluid inertia, which has been neglected in most previous studies. The importance of inertial forces is characterized by the ratio of the Reynolds and capillary numbers, Re∕Ca, a material parameter. Fluid inertia has a significant effect on the system’s behavior, even at relatively small values of Re∕Ca. In particular, compared to the case of zero Reynolds number, fluid inertia causes a significant increase in the pressure required to drive the air finger at a given speed.

Determination of Dynamic Coefficients in a Hydrodynamic Journal Bearing Based on the 3-D Navier-Stokes Equations and Considering Cavitation Effects

2012 ◽  
Author(s):  
Changhu Xing ◽  
Minel J. Braun
Keyword(s):  
Pressure Distribution ◽  
Reynolds Equation ◽  
Journal Bearing ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Maximum Magnitude ◽  
Navier Stokes Equations ◽  
Dynamic Coefficients ◽  
Finite Perturbation ◽  
Hydrodynamic Journal Bearing

Dynamic coefficients are very important for the stability of a hydrodynamic journal bearing and therefore for its design. In order to determine the stiffness, damping and added mass coefficients of the hydrodynamic bearing, the finite perturbation method around its stabilization position was employed. Based on the Reynolds equation with Gumbel cavitation algorithm, the maximum magnitude of the perturbation was judged by comparing results from finite perturbation (numerical way) to those from infinitesimal perturbation (additional analytical equations need to be derived based on order analysis), as well as theoretical analysis. Using the determined perturbation amplitude, the full three-dimensional Navier-Stokes equations in CFD-ACE+ were used to evaluate coefficients from an actual lubricant and compare to those obtained with Reynolds equation. Finally, a homogeneous gaseous cavitation algorithm is coupled with the Navier-Stokes equation to establish the pressure distribution in the bearing. When gas concentration was varied, the pressure distribution as well as the dynamic coefficients changed significantly.

A Multiscale Method Modeling Surface Texture Effects

2006 ◽  
Vol 129 (2) ◽  
pp. 221-230 ◽  
Author(s):  
Alex de Kraker ◽  
Ron A. J. van Ostayen ◽  
A. van Beek ◽  
Daniel J. Rixen
Keyword(s):  
Surface Texture ◽  
Reynolds Equation ◽  
Journal Bearing ◽  
Stokes Equations ◽  
Multiscale Method ◽  
Operating Conditions ◽  
Contact Load ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Inertia Effects

In this paper a multiscale method is presented that includes surface texture in a mixed lubrication journal bearing model. Recent publications have shown that the pressure generating effect of surface texture in bearings that operate in full film conditions may be the result of micro-cavitation and/or convective inertia. To include inertia effects, the Navier–Stokes equations have to be used instead of the Reynolds equation. It has been shown in earlier work (de Kraker et al., 2006, Tribol. Trans., in press) that the coupled two-dimensional (2D) Reynolds and 3D structure deformation problem with partial contact resulting from the soft EHL journal bearing model is not easy to solve due to the strong nonlinear coupling, especially for soft surfaces. Therefore, replacing the 2D Reynolds equation by the 3D Navier–Stokes equations in this coupled problem will need an enormous amount of computing power that is not readily available nowadays. In this paper, the development of a micro–macro multiscale method is described. The local (micro) flow effects for a single surface pocket are analyzed using the Navier–Stokes equations and compared to the Reynolds solution for a similar smooth piece of surface. It is shown how flow factors can be derived and added to the macroscopic smooth flow problem, that is modeled by the 2D Reynolds equation. The flow factors are a function of the operating conditions such as the ratio between the film height and the pocket dimensions, the surface velocity, and the pressure gradient over a surface texture unit cell. To account for an additional pressure buildup in the texture cell due to inertia effects, a pressure gain is introduced at macroscopic level. The method also allows for microcavitation. Microcavitation occurs when the pressure variation due to surface texture is larger than the average pressure level at that particular bearing location. In contrast with the work of Patir and Cheng (1978, J. Lubrication Technol., 78, pp. 1–10), where the microlevel is solved by the Reynolds equation, and the Navier–Stokes equations are used at the microlevel. Depending on the texture geometry and film height, the Reynolds equation may become invalid. A second pocket effect occurs when the pocket is located in the moving surface. In mixed lubrication, fluid can become trapped inside a pocket and squeezed out when the pocket is running into an area with higher contact load. To include this effect, an additional source term that represents the average fluid inflow due to the deformation of the surface around the pocket is added to the Reynolds equation at macrolevel. The additional inflow is computed at microlevel by numerical solution of the surface deformation for a single pocket that is subject to a contact load. The pocket volume is a function of the contact pressure. It must be emphasized that before ready-to-use results can be presented, a large number of simulations to determine the flow factors and pressure gain as a function of the texture parameters and operating conditions have yet to be done. Before conclusions can be drawn, regarding the dominanant mechanism(s), the flow factors and pressure gain have to be added to the macrobearing model. In this paper, only a limited number of preliminary illustrative simulation results, calculating the flow factors for a single 2D texture geometry, are shown to give insight into the method.

The Influence of Fluid Inertia in Unsteady Lubrication Films

1982 ◽  
Vol 104 (2) ◽  
pp. 180-186 ◽  
Author(s):  
A. Sestieri ◽  
R. Piva
Keyword(s):  
Computational Model ◽  
Incompressible Flows ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Geometrical Shape ◽  
Fluid Inertia ◽  
Navier Stokes Equations ◽  
Film Boundary ◽  
Simple Boundary ◽  
Complete Set

The influence of the inertial forces in steady and unsteady lubrication films has been analyzed by means of an accurate computational model of the complete set of Navier Stokes equations for incompressible flows. A coordinate transformation accounts for the geometrical shape of the film boundary and its variation in time, reducing the numerical integration to a rectangular constant domain in the transformed plane. A dimensional analysis of the equations, both in steady and unsteady conditions, is performed to deduce the most significant nondimensional variables and parameters to be considered for a proper evaluation of the inertial effects. The applications are restricted to two-dimensional fields and simple boundary conditions to test the simulation capabilities of the computational model, by comparison with analytical solutions available in literature. Several geometrical and kinematic conditions are discussed and the effect of the convective and time dependent inertial terms is quantitatively evaluated.

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