Stationary Outer Cylinder Research Articles

Experimental study of convective heat transfer in the entrance region of an annulus with an external grooved surface

The aim of this experimental work is to study the effect of grooved surfaces in the entrance region of annular flows on local heat transfer. The outer stationary cylinder is grooved and the inner rotating cylinder is smooth. This configuration is applicable in industrial applications such as rotating heat pipes for cooling of superconducting machines or motors rotor, electrical generators where heat generates in the grooves containing wires, transient heating of axial compressor rotor drams, combustion chamber in turbojets, air-cooled axial-flux permanent-magnet machines. The experimental tests were performed based on aspect ratio of the groove, effective Reynolds number and Taylor number. It is observed that the dimensions of the groove, rotational speed as well as fluid axial velocity have significant effect on heat transfer rate. The present results show that the gradient of the averaged air temperature in the groove zone is larger than that in the gap between the two cylinders and temperature in this zone can be approached to the surface temperature if the channel length is long enough. In addition, a correlation for the local Nusselt number is proposed as a function of the effective Reynolds number, the groove aspect ratio and the local axial position from 270 experimental data. The correlation is capable to predict the local Nusselt number with a maximum error within 11% relative to the experimental data.

Electrochemical Reduction of Greenhouse Gas with Couette-Taylor Flow (CTF) Mixer

N2O, a greenhouse gas like CO2 or CH4, has 268 times greater global warming potential (GWP) than CO2. Even though the volumetric portion of N2O among the greenhouse gases is merely 6%, the potential of N2O to exert a greenhouse effect is equivalent to 3 billion ton of CO2, which is similar to 50% of total car emissions worldwide. In addition, N2O plays a significant role in depleting ozone layer. The concentration of such N2O has been increasing, due to manufacturing, transportation, and agricultural activities [1]. Various chemical methods such as a thermal decomposition, a high/medium temperature catalytic decomposition, and a selective catalytic reduction have been adopted in order to reduce N2O [2]. They have been applied to industrial fields: massive production of nitric acid, adipic acid, and caprolactam because wasted heat could be used to elevate temperature up to a point, whereupon thermal decomposition of N2O takes place. However, such high-temperature methods are inadequate for reduction of N2O, which is typically emitted from room-temperature process, such as biological or medical process. Furthermore, given that the high-temperature methods require an enormous amount of energy, the successful electrochemical reduction of N2O at room temperature with a simple apparatus could be an effective alternative. Although N2O is very stable in the atmosphere, it could be electrochemically reduced into N2 in an aqueous electrolyte solution [3]. The rate of electrochemical N2O reduction could be enhanced by adopting adequate catalysts and gas-dissolution system. In this study, Couette-Taylor flow (CTF) mixer with electrochemical system was developed for effective N2O reduction. It consists of a stationary outer cylinder and a rotating inner cylinder, which are working and counter electrodes used for an electrochemical reaction. The gas injected into the solution present between two cylinders could form micro bubbles with great gas pressure and gas-liquid interfacial area than bulk gas due to the Taylor vortices in the CTF mixer [4]. The application of the CTF mixer resulted in the enhanced N2O solubility and rapid dissolution. Based on the improved N2O dissolution, high N2O conversion over 90% was obtained. In this presentation, we will focus on the optimization of operating condition of CTF mixer such as catalyst, rotational speed of inner cylinder, gas flow rate, and liquid circulation rate for efficient N2O reduction. The optimized system was also applicable to electrochemical reduction of CO2 not only N2O.

Inertial migration of particles in Taylor-Couette flows

An experimental study of inertial migration of neutrally buoyant particles in the circular Couette flow (CCF), Taylor vortex flow (TVF) and wavy vortex flow (WVF) is reported. This work considers a concentric cylinder Taylor-Couette device with a stationary outer cylinder and rotating inner cylinder. The device has a radius ratio of η = ri/ro = 0.877, where ri and ro are the inner and outer radii of the flow annulus. The ratio of the annular width between the cylinders (δ = ro − ri) and the particle diameter (dp) is α = δ/dp = 20. For η = 0.877, the flow of a Newtonian fluid undergoes transitions from CCF to TVF and TVF to WVF at Reynolds numbers Re = 120 and 151, respectively, and for the dilute suspensions studied here, these critical Reynolds numbers are almost unchanged. In CCF, particles were observed to migrate, due to the competition between the shear gradient of the flow and the wall interactions, to an equilibrium location near the middle of the annulus with an offset toward the inner cylinder. In TVF, the vortex motion causes the particles to be exposed to the shear gradient and wall interactions in a different manner, resulting in a circular equilibrium region in each vortex. The radius of this circular region grows with increase in Re. In WVF, the azimuthal waviness results in fairly well-distributed particles across the annulus.

Study on Non-Uniqueness of Taylor Vortex Flow Changing Inner Cylinder Acceleration Time

In concentrically rotating double cylinders consisting of a stationary outer cylinder a rotating inner cylinder, Taylor vortex flow appears. Taylor vortex flow occurs in journal bearings, various fluid machineries, containers for chemical reaction, and other rotating components. Therefore, the analysis of the flow structure of Taylor vortex flow is highly effective for its control. The main parameters that determine the modes of Taylor vortex flow of a finite length are the aspect ratio, Reynolds number Re. Aspect ratio is defined as the ratio of the cylinder length to the gap length between cylinders, and Re is determined on the basis of the angular speed of the inner cylinder. Aspect ratio was set to be 4.0, and Re to be values in the range from 100 to 1000 at intervals of 100. Thus far, a large number of studies on Taylor vortex flow have been carried out; however, the effects of the differences in initial conditions have not yet been sufficiently clarified. In this study, we changed the acceleration time of the inner cylinder in a numerical analysis, and examined the resulting changes in the mode formation and bifurcation processes. The acceleration time was changed from 1.0 s to 10.0 s. As a result, a difference was observed in the final mode depending on the difference in the acceleration time. From this finding, non-uniqueness, which is a major characteristic of Taylor vortex flow, was confirmed. However, no regularities regarding the difference in mode formation were found and the tendency of the mode formation process was not specified. Moreover, the processes of developing the vortex resulting in different final modes were monitored over time by visual observation. Similar flow behaviors were initially observed after the start of the calculation. Then, a bifurcation point, at which the flow changed to a mode depending on the acceleration time observed, and finally the flow became steady. In addition, there was also a difference in the time taken for the well-developed flow to reach the steady state. Both EFD (Experimental Fluid Dynamics) and CFD (Computational Fluid Dynamics) results show good agreement qualitatively.

Continuous counter-current centrifugal extraction column with high throughput using a spiral inner cylinder

This article presents a continuous counter-current extraction column with a high throughput that relies on a modified Taylor–Couette vortex flow in a centrifugal force field. The extraction column consists of a top settler, bottom settler, stationary outer cylinder connecting the two settlers, and rotating inner cylinder with a spiral guide channel. The organic and aqueous phases flow continuously and counter-currently through the annulus between the outer and inner cylinders. The high-speed inner cylinder breaks up the organic phase into small droplets in the continuous aqueous phase, and three flow patterns, i.e., a disordered droplet flow, banded droplet flow, and full droplet flow, can be formed at different rotating speeds. The spiral rotating inner cylinder was proved to facilitate the formation of the banded droplet flow, which enhances the mass transfer and increases the operating flexibility. A high back-extraction efficiency of approximately 95.6% was achieved using a spiral rotating inner cylinder of 25 cm in contact height with a rotating speed of 800 rpm, even at a high total throughput of up to 7.3 L h−1 cm−2.

Inertial flow transitions of a suspension in Taylor–Couette geometry

Experiments on the inertial flow transitions of a particle–fluid suspension in the concentric cylinder (Taylor–Couette) flow with rotating inner cylinder and stationary outer cylinder are reported. The radius ratio of the apparatus was $\unicode[STIX]{x1D702}=d_{i}/d_{o}=0.877$, where $d_{i}$ and $d_{o}$ are the diameters of inner and outer cylinders. The ratio of the axial length to the radial gap of the annulus $\unicode[STIX]{x1D6E4}=L/\unicode[STIX]{x1D6FF}=20.5$, where $\unicode[STIX]{x1D6FF}=(d_{o}-d_{i})/2$. The suspensions are formed of non-Brownian particles of equal density to the suspending fluid, of two sizes such that the ratio of annular gap to the mean particle diameter $d_{p}$ was either $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FF}/d_{p}=30$ or $100$. For the experiments with $\unicode[STIX]{x1D6FC}=100$, the particle volume fraction was $\unicode[STIX]{x1D719}=0.10$ and for the experiments with $\unicode[STIX]{x1D6FC}=30$, $\unicode[STIX]{x1D719}$ was varied over $0\leqslant \unicode[STIX]{x1D719}\leqslant 0.30$. The focus of the work is on determining the influence of particle loading and size on inertial flow transitions. The primary effects of the particles were a reduction of the maximum Reynolds number for the circular Couette flow (CCF) and several non-axisymmetric flow states not seen for a pure fluid with only inner cylinder rotation; here the Reynolds number is $Re=\unicode[STIX]{x1D6FF}d_{i}\unicode[STIX]{x1D6FA}\unicode[STIX]{x1D70C}/2\unicode[STIX]{x1D707}_{s}$, where $\unicode[STIX]{x1D6FA}$ is the rotation rate of the inner cylinder and $\unicode[STIX]{x1D70C}$ and $\unicode[STIX]{x1D707}_{s}$ are the density and effective viscosity of the suspension. For purposes of maintaining uniform particle distribution, the rotation rate of the inner cylinder (or $Re$) was decreased slowly from a state other than CCF to probe the transitions. When $Re$ was decreased, pure fluid transitions from wavy Taylor vortex flow (WTV) to Taylor vortex flow (TVF) to CCF occurred. The suspension transitions differed. For $\unicode[STIX]{x1D6FC}=30$ and $0.05\leqslant \unicode[STIX]{x1D719}\leqslant 0.15$, with reduction of $Re$, additional non-axisymmetric flow states, namely spiral vortex flow (SVF) and ribbons (RIB), were observed between TVF and CCF. At $\unicode[STIX]{x1D719}=0.30$, the flow transitions observed were only non-axisymmetric: from wavy spiral vortices (WSV) to SVF to CCF. The values of $Re$ corresponding to each flow transition were observed to reduce with increase in particle loading for $0\leqslant \unicode[STIX]{x1D719}\leqslant 0.30$, with the initial transition away from CCF, for example, occurring at $Re\approx 120$ for the pure fluid and $Re\approx 75$ for the $\unicode[STIX]{x1D719}=0.30$ suspension. When the particle size was reduced to yield $\unicode[STIX]{x1D6FC}=100$, at $\unicode[STIX]{x1D719}=0.10$, only the RIB (and no SVF) was observed between TVF and CCF.

Radial buoyancy effects on momentum and heat transfer in a circular Couette flow

The numerical investigation of a circular Couette flow with a radial temperature gradient is performed to elucidate the influence of the radial buoyancy on flow and heat transfer for different values of the Prandtl number when the gravitational acceleration is neglected. We consider an infinite-length cylindrical annulus of radius ratio 0.8 with the inner rotating cylinder and the outer stationary cylinder. The flow is stabilized with the outward heating while it is destabilized with the inward heating. A weakly nonlinear analysis shows that the transition to stationary axisymmetric modes is supercritical while the oscillatory axisymmetric modes occur via a subcritical bifurcation. The effect of the centrifugal buoyancy on the transfer of angular momentum (i.e., torque) is quite weak while the effect on the heat transfer is significant.

The numerical simulation of Taylor-Couette flow with radial temperature gradient

The Taylor-Couette flow with radial temperature gradient is a canonical problem for the study of heat transfer in engineering issues. However, gaining insight into the transitional Taylor-Couette flow with temperature gradient still requires detailed experimental and numerical investigations. In the present paper we have performed computations for the cavity of aspect ratio Γ= 3.76 and radii ratios η= 0.82 and 0.375 with the heated rotating bottom disk and stationary outer cylinder. We analyse the influence of the end-wall boundary conditions and the thermal conditions on the flow structure, and on the distributions of the Nusselt number and torque along the inner and outer cylinders. The averaged values along the inner cylinder of the Nusselt number and torque obtained for different Re are analysed in the light of the results published in [2, 16, 17].

Experimental Characterization of Magnetorheological Fluids Using a Custom Searle Magnetorheometer: Influence of the Rotor Shape

In this paper, we investigated the behavior of magnetorheological fluids (MRFs) by means of a custom Searle magnetorheometer. The Searle measuring system is composed of an outer stationary cylinder and an inner cylinder driven by a motor. The MRF is placed between the two concentric cylinders. While the bob is rotating, the fluid transmits a torque to the outer cylinder, which is related to the velocity profile through the gap and the magnetic field applied. In particular, we wanted to assess the influence of different bob shapes on the shear stress–shear rate curves. Indeed, a non-circular cross section causes cyclic variations on the gap thickness and, consequently, can squeeze the active MRFs, causing the formation of thicker columns and higher values of shear stresses. This phenomenon known as a squeeze-strengthen effect can be exploited to achieve higher energy absorption on magnetorheological devices like brakes and clutches. Two different cross sections were taken into account and then compared with the cylindrical one: 1) elliptical and 2) quadrilobate. In both cases, the gap size varies from 0.25 to 0.75 mm. The experimental results showed that with the new bob shapes, the squeeze-strengthen effect does not occur. However, using the elliptical and quadrilobate bobs, changes in the slope of the flow curves have been measured. This may suggest that the post-yield viscosity can be influenced by more complex factors like non-uniform magnetic field or hydrodynamic phenomena. Further studies, such as 3-D magnetic simulation and CFD analysis, will be taken into account to explain the new phenomena brought up in this paper.

Experimental study of expansion and compression effects on the stability of Taylor vortex flow

The aim of the present experimental work is to determine the stability limits of Taylor cells by expanding and compressing the cells. The investigation was performed under laminar flow condition with a wide gap between an inner rotating cylinder and outer stationary cylinder. In order to allow the expansion and compression of the cells, the test rig was designed with a sliding upper end plate and a fixed lower end plate. The objectives are to determine the maximum and minimum size limits of each number of cells as well as the stability margin of them. Since, unstable cells have various oscillations and time dependent structures which change the behavior of the flow; the investigations of the stability limits is quite necessary to avoid the possible generation of unstable cells. In addition, the results are used to detect the number of cells that can be generated in the gap at different fluid column lengths. A stability map, locating the stability state of all possible numbers of cells, is assigned in the results. The map provides overlap zones between stable cells, in which the operating conditions will always lead to stable cells, even if the number of cells is changed by changing the initial conditions. Moreover, a rare phenomenon was observed during the compression process when the cells jumps unusually from six to two cells without passing through the four-cell mode.

The Three-Dimensional Velocity Distribution of Wide Gap Taylor-Couette Flow Modelled by CFD

A numerical investigation is conducted for the flow between two concentric cylinders with a wide gap, relevant to bearing chamber applications. This wide gap configuration has received comparatively less attention than narrow gap journal bearing type geometries. The flow in the gap between an inner rotating cylinder and an outer stationary cylinder has been modelled as an incompressible flow using an implicit finite volume RANS scheme with the realisablek-εmodel. The model flow is above the critical Taylor number at which axisymmetric counterrotating Taylor vortices are formed. The tangential velocity profiles at all axial locations are different from typical journal bearing applications, where the velocity profiles are quasilinear. The predicted results led to two significant findings of impact in rotating machinery operations. Firstly, the axial variation of the tangential velocity gradient induces an axially varying shear stress, resulting in local bands of enhanced work input to the working fluid. This is likely to cause unwanted heat transfer on the surface in high torque turbomachinery applications. Secondly, the radial inflow at the axial end-wall boundaries is likely to promote the transport of debris to the junction between the end-collar and the rotating cylinder, causing the build-up of fouling in the seal.

Experimental and Numerical Study of Axial Turbulent Fluid Flow and Heat Transfer in a Rotating Annulus

The present work is an experimental and a numerical investigation of the turbulent fluid flow and heat transfer in an annular channel between two concentric cylinders with heated stationary outer cylinder (constant heat flux) and adiabatic rotating inner cylinder. Numerically, the governing equations are discretized in a finite volume fashion using a non-staggered (collocated) arrangement of the variables. The solutions were obtained using the SIMPLE algorithm with upwind scheme. A computer program in FORTRAN 90 was build to solve a set of partial differential equations that govern the fluid flow and heat transfer in annular channels. The experimental results are obtained for an inlet air velocity range of 2–6 m/s, for a wall heat flux range of 600–1200 W/m2 and a rotational speed range of inner cylinder of 0–1500 rpm with a gap width of 1.5 cm. Finally, the relationships between the average Nusselt number and the effective Reynolds number for experimental and numerical results were proposed and compared with those in the existing literature.

The near-wall region of highly turbulent Taylor–Couette flow

Direct numerical simulations of the Taylor–Couette (TC) problem, the flow between two coaxial and independently rotating cylinders, have been performed. The study focuses on TC flow with mild curvature (small gap) with a radius ratio of ${\it\eta}=r_{i}/r_{o}=0.909$, an aspect ratio of ${\it\Gamma}=L/d=2{\rm\pi}/3$, and a stationary outer cylinder. Three inner cylinder Reynolds numbers of $1\times 10^{5}$, $2\times 10^{5}$ and $3\times 10^{5}$ were simulated, corresponding to frictional Reynolds numbers between $Re_{{\it\tau}}\approx 1400$ and $Re_{{\it\tau}}\approx 4000$. An additional case with a large gap, ${\it\eta}=0.5$ and driving of $Re=2\times 10^{5}$ was also investigated. Small-gap TC was found to be dominated by spatially fixed large-scale structures, known as Taylor rolls (TRs). TRs are attached to the boundary layer, and are active, i.e. they transport angular velocity through Reynolds stresses. An additional simulation was also conducted with inner cylinder Reynolds number of $Re=1\times 10^{5}$ and fixed outer cylinder with an externally imposed axial flow of comparable strength to the wind of the TRs. The axial flow was found to convect the TRs without any weakening effect. For small-gap TC flow, evidence was found for the existence of logarithmic velocity fluctuations, and of an overlap layer, in which the velocity fluctuations collapse in outer units. Profiles consistent with a logarithmic dependence were also found for the angular velocity in large-gap TC flow, albeit in a very reduced range of scales. Finally, the behaviour of both small- and large-gap TC flow was compared to other canonical flows. Small-gap TC flow has similar behaviour in the near-wall region to other canonical flows, while large-gap TC flow displays very different behaviour.

Numerical simulation of mixed convection in vertical concentric cylinders

We conduct numerical simulation of mixed convection in a gap between rotating hot inner cylinder and stationary cold outer cylinder to investigate flow regimes at various intensities of rotational speed and buoyancy in terms of Reynolds number and Grashof number. For the current study, a projection method with explicit Adams-Bashforth and implicit Crank-Nicolson schemes was adopted. Flows with various Reynolds numbers are investigated at fixed Grashof number, and flow pattern is examined through streamlines and iso-surfaces of helicity. Phase plots constructed by velocity components, Nusselt number and torque reveal the transition between different flow regimes. With Grashof number less than 3000, the flow go through a sequence of axisymmetric, spiral and Taylor-Couette flow as Reynolds number increases, while the flow with Grashof number 4000 shows complex pattern including quasi-periodic flows and chaos in a wide range of Reynolds number. Transition to chaos is investigated using the budget analysis of centrifugal and buoyant forces, indicating that competition between the two driving forces causes chaos.

Cylindrical Couette flow of a rarefied gas: Effect of a boundary condition on the inverted velocity profile.

The cylindrical Couette flow of a rarefied gas between a rotating inner cylinder and a stationary outer cylinder is investigated under the following two kinds of kinetic boundary conditions. One is the modified Maxwell-type boundary condition proposed by Dadzie and Méolans [J. Math. Phys. 45, 1804 (2004)] and the other is the Cercignani-Lampis condition, both of which have separate accommodation coefficients associated with the molecular velocity component normal to the boundary and with the tangential component. An asymptotic analysis of the Boltzmann equation for small Knudsen numbers and a numerical analysis of the Bhatnagar-Gross-Krook model equation for a wide range of the Knudsen number are performed to clarify the effect of each accommodation coefficient as well as of the boundary condition itself on the behavior of the gas, especially on the flow-velocity profile. As a result, the velocity-slip and temperature-jump conditions corresponding to the above kinetic boundary conditions are derived, which are necessary for the fluid-dynamic description of the problem for small Knudsen numbers. The parameter range for the onset of the velocity inversion phenomenon, which is related mainly to the decrease in the tangential momentum accommodation, is also obtained.

CFD simulation of shear flow and mixing in a Taylor–Couette reactor with variable cross-section inner cylinders

A Taylor–Couette (TC) reactor has been considered as a promising technology to produce uniform ultra-fine particles with controlled morphology and size because of its rather narrow shear rate distribution in comparison to a conventional stirred tank. However, small particles in the TC reactor are likely entrapped by the Taylor vortices and concentrated more in the core of the vortices, thus having an adverse or a positive impact on producing the uniformly fine particles. To mitigate this disadvantage, one approach is to adopt an inner cylinder with a variable cross-section through the proper design to obtain a non-constant gap between the inner and outer cylinders in the TC reactor. This study investigates the use of specific profiles of the cross section for the inner cylinder to deform the Taylor vortices, effectively reducing the regions of low velocity and shear rate in the TC reactor. The profiles are composed of a series of curves. The flow patterns and shear rate distributions in the modified TC reactors were investigated using CFD modelling. The sliding mesh method and the Reynolds stress model were employed in the simulation to account for highly turbulent rotating flows. The simulated results compared with those for the classical TC reactor clearly indicate that the regions of low shear rate are effectively reduced in the modified TC reactors because the Taylor vortices occurring in the gap becomes temporal-periodic with respect to the stationary outer cylinder. The obtained shear rate distributions are remarkably narrower than those of the classical TC reactor. It was revealed that the curvature of curves for the cross-section profiles can significantly affect the shear rate distributions, and there exists an optimal profile for the inner cylinder cross-section.

The Persistence of Vortex Structures Between Rotating Cylinders in the 106 Taylor Number Range

The flow in the annular gap d = Ro - Ri between a stationary outer cylinder of radius Ro and a co-axial rotating inner cylinder of radius Ri is characterised at large gap, over the radius ratio (η) range 0.44 ≤ η ≤ 0.53 and aspect ratio (Γ) range 7.81 ≤ Γ ≤ 11.36. These configurations are more representative of turbomachinery bearing chambers and large rotating machinery than the journal bearing geometries of narrow gap commonly reported in the literature. Particle Image Velocimetry measurements are taken across the full meridional plane over the Taylor number (Ta) range 258 × 10 3 ≤ Ta ≤ 10.93 × 10 6 , which is 1000 times higher than the critical Taylor number for the onset of the first axisymmetric Taylor instability. Well-defined azimuthal vortex structures persist over this high Taylor number range, characterised by a constant number of vortices at a given aspect ratio, vortex core axial and radial motion over time, and mixing between neighbour vortices. This regime, with wavy Taylor vortex flow like features, is defined over the range 1.18 × 10 6 ≤ Ta ≤ 6.47 × 10 6 . These results form a body of experimental evidence from which further progress in the understanding of the wavy Taylor vortex dynamics can be sought, through advanced flow dynamic models that reproduce the persistence of the observed flow features.

Effects of the computational domain size on direct numerical simulations of Taylor-Couette turbulence with stationary outer cylinder

In search for the cheapest but still reliable numerical simulation, a systematic study on the effect of the computational domain (“box”) size on direct numerical simulations of Taylor-Couette flow was performed. Four boxes with varying azimuthal and axial extents were used. The radius ratio between the inner cylinder and the outer cylinder was fixed to η = ri/ro = 0.909. The outer cylinder was kept stationary, while the inner rotated at a Reynolds number Rei = 105. Profiles of mean and fluctuation velocities are compared, as well as autocorrelations and velocity spectra. The smallest box is found to accurately reproduce the torque and mean azimuthal velocity profiles of larger boxes, while having smaller values of the fluctuations than the larger boxes. The axial extent of the box directly reflects on the Taylor-rolls and plays a crucial role on the correlations and spectra. The azimuthal extent is found to play a minor role in the simulations, as the boxes are large enough. For all boxes studied, the spectra do not reach a box independent maximum.

0815 規則的粗さをもつ壁面近傍の流れ計測に基づく流体摩擦評価式の検討

PIV measurements of flows near the wall roughness have been carried out. The flow is induced by the concentric cylinder, which consists of rotating inner cylinder and stationary outer cylinder. The inner rotating cylinder has the regularly distributed semi-spherical roughness on the surface. Flow construction, such as longitudinal vortices, low-velocity region in the wake, and high-velocity linear region corresponding to the vortices were clarified by the PIV measurements. From the measured velocity distribution, a simple estimation equation for the flow resistance was proposed. The validity of this proposed equation was confirmed by comparing with the friction measured directly by torque of the inner cylinder.

The homogenization method of roughness analysis in turbulent lubrication

The hydrodynamic lubrication theory founded by Osborne Reynolds in 1886 is based on several assumptions, including that the contact surfaces are perfectly smooth, and the lubricant flow regime is laminar. However, in certain special cases these hypotheses may not be valid, and it is seen that surface roughness and turbulence affect bearing performance. The approach adopted in the present work is based on the application of the homogenization method to the turbulent Reynolds equation. In homogenization, multiscale expansion of the fluid pressure terms leads to a system of four partial differential equations governing two types of problems (local and global). The solutions are, respectively, periodic functions, and the homogenized pressure. As an illustrative example, a journal bearing with a rough surface operating in the turbulent regime is analyzed. Numerical simulations are performed by imposing periodic isotropic roughness patterns over the stationary outer cylinder surface. The homogenized solutions are compared to direct solutions of the deterministic problem. Homogenization is shown to be a powerful but practical method to attack problems of rough surfaces and is applied to turbulent lubrication for the first time. In the plain journal bearing example, the effect of roughness is to increase the load significantly in many cases. The effect of turbulence is to greatly increase the load, in both the rough and smooth cases.