This paper studies the global phase portraits of uniform isochronous centers system of degree six with polynomial commutator.Such systems have the form x=-y+xf(x,y),y=x+yf(x,y),where f(x,y)=a_(1)x+a_(2)xy+a_(3)xy^(2)+...This paper studies the global phase portraits of uniform isochronous centers system of degree six with polynomial commutator.Such systems have the form x=-y+xf(x,y),y=x+yf(x,y),where f(x,y)=a_(1)x+a_(2)xy+a_(3)xy^(2)+a_(4)xy^(3)+a_(5)xy^(4)=xσ(y),and any zero of 1+a_(1)y+a_(2)y^(2)+a_(3)y^(3)+a_(4)y^(4)+a_(5)y^(5),y=y is an invariant straight line.At last,all global phase portraits are drawn on the Poincare disk.展开更多
Uniform ventilation is important for the safety of long tunnel in underground buildings,it is difficult to install the large size air duct to ensure the centerline of each cross-section of the traditional variable cro...Uniform ventilation is important for the safety of long tunnel in underground buildings,it is difficult to install the large size air duct to ensure the centerline of each cross-section of the traditional variable cross-section ventila-tion system(VCVS)superimposed on the same horizontal axis,which is significant to the ventilation uniformity,energy consumption and installation convenience of the VCVS.On the contrary,each cross-section of the Equal Cross-section Ventilation System(ECVS)has the same horizontal axis,therefore,it is more convenient to manu-facture and install the large size air duct in the underground long tunnel and achieve uniform ventilation.This study proposes an ECVS,using computational fluid dynamics(CFD)numerical simulation analyzed the influences of main duct velocity,aspect ratio,and outlet numbers on uniformity and energy consumption per unit air vol-ume(ECPV).It revealed that when each ventilation equalizer’s valve Angle𝛽is given,uniformity of air supply decreases slightly with an increase in the inlet velocity.When the air supply main duct aspect ratio increases,the outlet velocity standard deviation range is from 0.22 to 0.34.When outlet numbers N=7~12 and𝛽are constant,air supply uniformity and resistance coefficient𝜉also decrease with the decrease of outlet numbers.The outlet number has a significant influence on the uniformity of system air supply,main duct velocity,and aspect ratio are relatively small.ECPV is positively correlated with the main duct velocity and outlet number,and is negatively correlated with aspect ratio.展开更多
基金supported by National Natural Science Foundation of China(No.12301197)Natural Science Foundation of Henan(No.232300420343)+2 种基金Science and Technology Research Project of Henan Province(No.232102210057)Scientific Research Foundation for Doctoral Scholars of Haust(No.13480077)Natural Science Foundation of Hunan(No.2021JJ30166)。
文摘This paper studies the global phase portraits of uniform isochronous centers system of degree six with polynomial commutator.Such systems have the form x=-y+xf(x,y),y=x+yf(x,y),where f(x,y)=a_(1)x+a_(2)xy+a_(3)xy^(2)+a_(4)xy^(3)+a_(5)xy^(4)=xσ(y),and any zero of 1+a_(1)y+a_(2)y^(2)+a_(3)y^(3)+a_(4)y^(4)+a_(5)y^(5),y=y is an invariant straight line.At last,all global phase portraits are drawn on the Poincare disk.
基金This work was supported by Scientific Research Fund of Hunan Provincial Education Department[19A001]Natural Science Foundation of Hunan Province[2019JJ40303]+1 种基金State Key Laboratory of Air-conditioning Equipment and System Energy Conservation[ACSKL2018KT18]Double-First class International Collaboration Fund of Changsha University of Science and Technology[2018IC16].
文摘Uniform ventilation is important for the safety of long tunnel in underground buildings,it is difficult to install the large size air duct to ensure the centerline of each cross-section of the traditional variable cross-section ventila-tion system(VCVS)superimposed on the same horizontal axis,which is significant to the ventilation uniformity,energy consumption and installation convenience of the VCVS.On the contrary,each cross-section of the Equal Cross-section Ventilation System(ECVS)has the same horizontal axis,therefore,it is more convenient to manu-facture and install the large size air duct in the underground long tunnel and achieve uniform ventilation.This study proposes an ECVS,using computational fluid dynamics(CFD)numerical simulation analyzed the influences of main duct velocity,aspect ratio,and outlet numbers on uniformity and energy consumption per unit air vol-ume(ECPV).It revealed that when each ventilation equalizer’s valve Angle𝛽is given,uniformity of air supply decreases slightly with an increase in the inlet velocity.When the air supply main duct aspect ratio increases,the outlet velocity standard deviation range is from 0.22 to 0.34.When outlet numbers N=7~12 and𝛽are constant,air supply uniformity and resistance coefficient𝜉also decrease with the decrease of outlet numbers.The outlet number has a significant influence on the uniformity of system air supply,main duct velocity,and aspect ratio are relatively small.ECPV is positively correlated with the main duct velocity and outlet number,and is negatively correlated with aspect ratio.
