Henning Bunzel, Bent J. Christensen, Nicholas M. Kiefer and Lars Korsholm
Numerical simulations have been performed for three‐dimensional natural convection of water near its maximum‐density (cold water) inside rectangular enclosures with differential…
Abstract
Numerical simulations have been performed for three‐dimensional natural convection of water near its maximum‐density (cold water) inside rectangular enclosures with differential heating at the vertical (left and right) walls. The horizontal (top and bottom) walls and the lateral (front and rear) walls are taken as insulated. Computations are performed for the buoyancy‐driven convection of cold water with density inversion parameter θm = 0.5 in the enclosures with aspect ratio (height/width) Ay = 8 and depth ratios (depth/width) Az = 0.5, 1, and 2. The influence of the depth ratio on the onset of oscillatory convection in a cold‐water‐filled enclosure is investigated. The presence of the lateral walls tends to suppress the onset of unsteadiness in the convective flow. The main features of the oscillatory convection flow and temperature fields as well as the instability mechanism in the three‐dimensional enclosure were similar to those found in the two‐dimensional model. However, there exists a strong three‐dimensionality in the spatial distribution of the fluctuation amplitude. With the decrease of the depth ratio, the damping effect of the lateral walls becomes increasingly pronounced, leading to a reduced heat transfer rate.
Details
Keywords
Lian-Li Feng and Tian-Tian Zhang
The purpose of this paper is to find homoclinic breather waves, rogue waves and soliton waves for a (3 + 1)-dimensional generalized Kadomtsev–Petviashvili (gKP) equation, which…
Abstract
Purpose
The purpose of this paper is to find homoclinic breather waves, rogue waves and soliton waves for a (3 + 1)-dimensional generalized Kadomtsev–Petviashvili (gKP) equation, which can be used to describe the propagation of weakly nonlinear dispersive long waves on the surface of a fluid.
Design/methodology/approach
The authors apply the extended Bell polynomial approach, Hirota’s bilinear method and the homoclinic test technique to find the rogue waves, homoclinic breather waves and soliton waves of the (3 + 1)-dimensional gKP equation.
Findings
The results imply that the gKP equation admits rogue waves, homoclinic breather waves and soliton waves. Moreover, the authors also find that rogue waves can come from the extreme behavior of the breather solitary wave. The authors analyze the propagation and interaction properties of these solutions to better understand the dynamic behavior of these solutions.
Originality/value
These results may help us to further study the local structure and the interaction of waves in KP-type equations. It is hoped that the results can help enrich the dynamic behavior of such equations.
Details
Keywords
Jin-Jin Mao, Shou-Fu Tian and Tian-Tian Zhang
The purpose of this paper is to find the exact solutions of a (3 + 1)-dimensional non-integrable Korteweg-de Vries type (KdV-type) equation, which can be used to describe the…
Abstract
Purpose
The purpose of this paper is to find the exact solutions of a (3 + 1)-dimensional non-integrable Korteweg-de Vries type (KdV-type) equation, which can be used to describe the stability of soliton in a nonlinear media with weak dispersion.
Design/methodology/approach
The authors apply the extended Bell polynomial approach, Hirota’s bilinear method and the homoclinic test technique to find the rogue waves, homoclinic breather waves and soliton waves of the (3 + 1)-dimensional non-integrable KdV-type equation. The used approach formally derives the essential conditions for these solutions to exist.
Findings
The results show that the equation exists rogue waves, homoclinic breather waves and soliton waves. To better understand the dynamic behavior of these solutions, the authors analyze the propagation and interaction properties of the these solutions.
Originality/value
These results may help to investigate the local structure and the interaction of waves in KdV-type equations. It is hoped that the results can help enrich the dynamic behavior of such equations.
Details
Keywords
Pingping He, Feng Gao, Yan Li, Wenwu Wu and Dongya Zhang
Under fix-position preload, the high rotation speed of the angular contact ball bearing exacerbates the frictional heat generation, which causes the increase of the bearing…
Abstract
Purpose
Under fix-position preload, the high rotation speed of the angular contact ball bearing exacerbates the frictional heat generation, which causes the increase of the bearing temperature and the thermal expansion. The high rotation speed also leads to the centrifugal expansion of the bearing. Under the thermal and centrifugal effect, the structural parameters of the bearing change, affecting the mechanical properties of the bearing. The mechanical properties of the bearing determine its heat generation mechanism and thermal boundary conditions. The purpose of this paper is to study the effect of centrifugal and thermal effects on the thermo-mechanical characteristics of an angular contact ball bearing with fix-position preload.
Design/methodology/approach
Because of operating conditions, elastic deformation occurs between the ball and the raceway. Assuming that the surfaces of the ball and channel are absolutely smooth and the material is isotropic, quasi-static theory and thermal network method are used to establish the thermo-mechanical coupling model of the bearing, which is solved by Newton–Raphson iterative method.
Findings
The higher the rotation speed, the greater the influence of centrifugal and thermal effects on the bearing dynamic parameters, temperature rise and actual axial force. The calculation results show that the effects of thermal field on bearing dynamic parameters are more significant than the centrifugal effect. The temperature rise and actual axial force of the bearing are measured. Comparing the calculation and the experimental results, it is found that the temperature rise and the actual axial force of the bearing are closer to reality considering thermal and centrifugal effects.
Originality/value
In the past studies, the thermo-mechanical coupling characteristics research and experimental verification of angular contact ball bearing with fix-position preload are not concerned. Research findings of this paper provide theoretical guidance for spindle design.
Details
Keywords
Zhe Liu, Wei Chen, Desheng Li and Wenjing Zhang
In high-speed processing, the influence on the machining accuracy of a machine tool is greatly caused by the thermal deformation of the motorized spindle; a further study on the…
Abstract
Purpose
In high-speed processing, the influence on the machining accuracy of a machine tool is greatly caused by the thermal deformation of the motorized spindle; a further study on the thermal characteristics of the spindle is given in this paper. This study aims to reduce the thermal error and improve the performance of the machine tool by discussing the relationships between the temperature distributions and rotating accuracy caused by the thermal deformations of the spindle.
Design/methodology/approach
The paper opted for a method combining the theoretical analysis and the experimental study to study the thermal stability of the high-speed motorized spindle. First of all, a finite element model of the spindle was built with ANSYS, whereby temperature distributions and the thermal deformations were successively obtained at different speeds. And then, both the temperature field and the rotating accuracy of the motorized spindle were measured simultaneously by the thermal stability experiment. Finally, the experimental and theoretical results were compared and validated.
Findings
The thermal stability of the motorized spindle was studied in this paper, and some findings from the study were as follows: the spindle’s rotating accuracy maintained good in X direction but bad in Y and Z directions in terms of the deformations; the higher front-end temperature of the spindle which can significantly affect the rotating accuracy is needed to be controlled mainly; the recovery speed of the spindle deformation lagged behind the temperature’s fallback speed; the vibration graph about radial rotating sensitivity synthesized by X1 and X2 presented a trifoliate shape.
Originality/value
Based on a built test-bed which can synchronously measure the motorized spindle’s temperature distribution and rotating accuracy with five-point method, the coupling effects of the thermal deformation and temperature are embodied, and not only the vibration graph but also the thermal tilt angles can be gained. Therefore, considering the influence of the thermal deformation on the heat generated by the bearings, the paper fulfilled a study by which it was obtained that the front-end temperature of the spindle, which was higher and could significantly affect the rotating accuracy, needed to be controlled mainly.
Details
Keywords
Hui Wang, Shou-Fu Tian and Yi Chen
The purpose of this paper is to study the breather waves, rogue waves and solitary waves of an extended (3 + 1)-dimensional Kadomtsev–Petviashvili (KP) equation, which can be used…
Abstract
Purpose
The purpose of this paper is to study the breather waves, rogue waves and solitary waves of an extended (3 + 1)-dimensional Kadomtsev–Petviashvili (KP) equation, which can be used to depict many nonlinear phenomena in fluid dynamics and plasma physics.
Design/methodology/approach
The authors apply the Bell’s polynomial approach, the homoclinic test technique and Hirota’s bilinear method to find the breather waves, rogue waves and solitary waves of the extended (3 + 1)-dimensional KP equation.
Findings
The results imply that the extended (3 + 1)-dimensional KP equation has breather wave, rogue wave and solitary wave solutions. Meanwhile, the authors provide the graphical analysis of such solutions to better understand their dynamical behavior.
Originality/value
These results may help us to further study the local structure and the interaction of solutions in KP-type equations. The authors hope that the results provided in this work can help enrich the dynamic behavior of such equations.
Details
Keywords
The stability of two‐dimensional natural convection of water near its density maximum (cold water) inside a vertical rectangular enclosure with an aspect ratio of eight is…
Abstract
The stability of two‐dimensional natural convection of water near its density maximum (cold water) inside a vertical rectangular enclosure with an aspect ratio of eight is investigated via a series of direct numerical simulations. The simulations aim to clarify, under the influence of density inversion, the physical nature of the instability mechanism responsible for the laminar buoyancy‐driven flow transition from a steady state to an oscillatory state in the enclosure filled with cold water. Two values of the density inversion parameter, m= 0.4 and 0.5, where the density inversion of cold water may exert strong influence on the flow, are considered in the present study. The results show that the transition from steady state to periodically oscillatory convection arises in the cold‐water‐filled enclosure through a Hopf bifurcation. The oscillatory convection in the water‐filled enclosure for both values of m is found to feature an oscillatory multicellular structure within the contra‐rotating bicellular flow regions. A traveling wave motion accordingly results along the maximum density contour, which demarcates the contra‐rotating bicellular flows in the enclosure. For both cases the nature of transition into unsteadiness is found to be buoyancy‐driven. The critical Rayleigh number for the bifurcation at m = 0.4 is found to be markedly higher than that at m = 0.5.
Details
Keywords
De-xing Zheng, Weifang Chen and Miaomiao Li
Thermal performances are key factors impacting the operation of angular contact ball bearings. Heat generation and transfer about angular contact ball bearings, however, have not…
Abstract
Purpose
Thermal performances are key factors impacting the operation of angular contact ball bearings. Heat generation and transfer about angular contact ball bearings, however, have not been addressed thoroughly. So far, most researchers only considered the convection effect between bearing housings and air, whereas the cooling/lubrication operation parameters and configuration effect were not taken into account when analyzing the thermal behaviors of bearings. This paper aims to analyze the structural constraints of high-speed spindle, structural features of bearing, heat conduction and convection to study the heat generation and transfer of high-speed angular contact ball bearings.
Design/methodology/approach
Based on the generalized Ohm’s law, the thermal grid model of angular contact ball bearing of high-speed spindle was first established. Next Gauss–Seidel method was used to solve the equations group by Matlab, and the nodes temperature was calculated. Finally, the bearing temperature rise was tested, and the comparative analysis was made with the simulation results.
Findings
The results indicate that the simulation results of bearing temperature rise for the proposed model are in better agreement with the test values. So, the thermal grid model established is verified.
Originality/value
This paper shows an improved model on forecasting temperature rise of high-speed angular contact ball bearings. In modeling, the cooling/lubrication operation parameters and structural constraints are integrated. As a result, the bearing temperature variation can be forecasted more accurately, which may be beneficial to improve bearing operating accuracy and bearing service life.