In terms of tensor analysis technique, the Navier-Stokes equations in helical coordinate system were derived. A steady incompressible flow of power-law fluid in helical pipes at low Reynolds number was investigated by...In terms of tensor analysis technique, the Navier-Stokes equations in helical coordinate system were derived. A steady incompressible flow of power-law fluid in helical pipes at low Reynolds number was investigated by the perturbation method. A second order solution of secondary flow was worked out. The secondary flow characteristics in helical pipes are analyzed. The effects of the number Dn, curvature and torsion on the secondary flow at different flow parameters were discussed. The results show that the secondary flow pattern changes from a single vortex to two vortices as the number Dn increases at a given curvature and a given torsion. Because of the effect of torsion, the secondary flow pattern changes from two almost symmetrical vortices to a single vortex as the torsion of the helical pipe increases while the Reynolds number and curvature hold constant. The secondary flow pattern cannot be affected by the curvature of the helical pipe at a given Dn number.展开更多
Without simplifying the N-S equations of Germano's[5], we study the flow in a helical circular pipe employing perturbation method. A third perturbation solution is fully presented. The first- second- and third-ord...Without simplifying the N-S equations of Germano's[5], we study the flow in a helical circular pipe employing perturbation method. A third perturbation solution is fully presented. The first- second- and third-order effects of curvature κ and torsion τ on the secondary flow and axial velocity are discussed in detail. The first-order effect of curvature is to form two counter-rotating cells of the secondary flow and to push the maximum axial velocity to the outer bend. The two cells are pushed to the outer bend by the pure second-order effect of curvature. The combined higher-order (second-, third-) effects of curvature and torsion, are found to be an enlargement of the lower vortex of the secondary flow at expense of the upper one and a clockwise shift of the centers of the secondary vortices and the location of maximum axial velocity. When the axial pressure gradient is small enough or the torsion is sufficiently larger than the curvature, the location of the maximal axial velocity is near the inner bend. The equation of the volume flux is obtained from integrating the perturbation solutions of axial velocity. From the equation the validity range of the perturbation solutions in this paper can be obtained and the conclusion that the three terms of torsion have no effect on the volume flux can easily be drawn. When the axial pressure gradient is less than 22.67, the volume flux in a helical pipe is larger than that in a straight pipe.展开更多
A study on the unsteady low-frequency oscillatory flow in a helical circular pipe is carried out based upon the blood flow in vessels, using the method of bi-parameter perturbation. The second order perturbation resul...A study on the unsteady low-frequency oscillatory flow in a helical circular pipe is carried out based upon the blood flow in vessels, using the method of bi-parameter perturbation. The second order perturbation results were obtained and the characteristics were analyzed at different time of the axial velocity, of the secondary flow, and of the wall shearing stress. Also done the analysis of above-mentioned variables that varied along with time and Womersley number. The results indicate that for a helical pipe, the torsion exerts the main influence on the distribution of secondary flow velocity, especially when the absolute value of axial press gradient is rather small. The severe variation of stream function takes place within a very short period, during which time the stream function develops from positive value to negative value and vice versa, while in most cases in a cycle, the variation is smooth. The wall shearing stress changes severely with theta too.展开更多
The low frequency oscillatory flow in a rotating curved pipe was studied by using the method of bi parameter perturbation. Perturbation solutions up to the second order were obtained and the effects of rotation on th...The low frequency oscillatory flow in a rotating curved pipe was studied by using the method of bi parameter perturbation. Perturbation solutions up to the second order were obtained and the effects of rotation on the low frequency oscillatory flow were examined in detail. The results indicated that there exists evident difference between the low frequency oscillatory flow in a rotating curved pipe and in a curved pipe without rotation. During a period, four secondary vortexes may exist on the circular cross section and the distribution of axial velocity and wall shear stress are related to the ratio of the Coriolis force to centrifugal force and the axial pressure gradient.展开更多
WT5”BZ]In this paper, the flow in a rotating curved annular pipe is examined by a perturbation method. A second order perturbation solution is presented. The characteristics of the secondary flow and the axial flow a...WT5”BZ]In this paper, the flow in a rotating curved annular pipe is examined by a perturbation method. A second order perturbation solution is presented. The characteristics of the secondary flow and the axial flow are studied in detail. The study indicates that the loops of the secondary flow are more complex than those in a curved annular pipe without rotation and its numbers depend on the ratio of the Coriolis force to centrifugal force F. As F≈-1, the secondary flow has eight loops and its intensity reaches the minimum value, and the distribution of the axial flow is like that of the Poiseuille flow. The position of the maximum axial velocity is pushed to either outer bend or inner bend, which is also determined by F. [WT5”HZ]展开更多
The fully developed laminar flow in helical elliptical pipes is influenced by curvature, torsion and aspect ratio of cross-section. With the aid of the symbolic manipulation technique, the governing equations were sol...The fully developed laminar flow in helical elliptical pipes is influenced by curvature, torsion and aspect ratio of cross-section. With the aid of the symbolic manipulation technique, the governing equations were solved by the Galerkin method, The procedures of implementing the Galerkin method for flows in curvilinear pipes were discussed. The effects of the aspect ratio and torsion on the flow structure, wall shear stress and flow ratio were examined in detail. The results show that the flow characteristic for aspect ratio larger than unit is quite different from those for the aspect ratio smaller than unit.展开更多
The combined effects of the system rotation (Coriolis force) and curvature (centrifugal force) on the flow in rotating curved circular pipe with small curvature are examined by perturbation method. A second order per...The combined effects of the system rotation (Coriolis force) and curvature (centrifugal force) on the flow in rotating curved circular pipe with small curvature are examined by perturbation method. A second order perturbation solution is presented. The secondary flow structure and the primary axial velocity distributions are studied in detail. The loops of the secondary flow are more complex than those in a curved pipe without rotation or a rotating straight pipe. Its numbers depend on the body force ratio F which represents the ratio of the Coriolis to the centrifugal force. The maximum of the axial velocity is pushed to either outer bend or inner bend, which is also determined by F. The results are confirmed by the results of other authors who studied the same problem by different methods.展开更多
文摘In terms of tensor analysis technique, the Navier-Stokes equations in helical coordinate system were derived. A steady incompressible flow of power-law fluid in helical pipes at low Reynolds number was investigated by the perturbation method. A second order solution of secondary flow was worked out. The secondary flow characteristics in helical pipes are analyzed. The effects of the number Dn, curvature and torsion on the secondary flow at different flow parameters were discussed. The results show that the secondary flow pattern changes from a single vortex to two vortices as the number Dn increases at a given curvature and a given torsion. Because of the effect of torsion, the secondary flow pattern changes from two almost symmetrical vortices to a single vortex as the torsion of the helical pipe increases while the Reynolds number and curvature hold constant. The secondary flow pattern cannot be affected by the curvature of the helical pipe at a given Dn number.
文摘Without simplifying the N-S equations of Germano's[5], we study the flow in a helical circular pipe employing perturbation method. A third perturbation solution is fully presented. The first- second- and third-order effects of curvature κ and torsion τ on the secondary flow and axial velocity are discussed in detail. The first-order effect of curvature is to form two counter-rotating cells of the secondary flow and to push the maximum axial velocity to the outer bend. The two cells are pushed to the outer bend by the pure second-order effect of curvature. The combined higher-order (second-, third-) effects of curvature and torsion, are found to be an enlargement of the lower vortex of the secondary flow at expense of the upper one and a clockwise shift of the centers of the secondary vortices and the location of maximum axial velocity. When the axial pressure gradient is small enough or the torsion is sufficiently larger than the curvature, the location of the maximal axial velocity is near the inner bend. The equation of the volume flux is obtained from integrating the perturbation solutions of axial velocity. From the equation the validity range of the perturbation solutions in this paper can be obtained and the conclusion that the three terms of torsion have no effect on the volume flux can easily be drawn. When the axial pressure gradient is less than 22.67, the volume flux in a helical pipe is larger than that in a straight pipe.
文摘A study on the unsteady low-frequency oscillatory flow in a helical circular pipe is carried out based upon the blood flow in vessels, using the method of bi-parameter perturbation. The second order perturbation results were obtained and the characteristics were analyzed at different time of the axial velocity, of the secondary flow, and of the wall shearing stress. Also done the analysis of above-mentioned variables that varied along with time and Womersley number. The results indicate that for a helical pipe, the torsion exerts the main influence on the distribution of secondary flow velocity, especially when the absolute value of axial press gradient is rather small. The severe variation of stream function takes place within a very short period, during which time the stream function develops from positive value to negative value and vice versa, while in most cases in a cycle, the variation is smooth. The wall shearing stress changes severely with theta too.
文摘The low frequency oscillatory flow in a rotating curved pipe was studied by using the method of bi parameter perturbation. Perturbation solutions up to the second order were obtained and the effects of rotation on the low frequency oscillatory flow were examined in detail. The results indicated that there exists evident difference between the low frequency oscillatory flow in a rotating curved pipe and in a curved pipe without rotation. During a period, four secondary vortexes may exist on the circular cross section and the distribution of axial velocity and wall shear stress are related to the ratio of the Coriolis force to centrifugal force and the axial pressure gradient.
文摘WT5”BZ]In this paper, the flow in a rotating curved annular pipe is examined by a perturbation method. A second order perturbation solution is presented. The characteristics of the secondary flow and the axial flow are studied in detail. The study indicates that the loops of the secondary flow are more complex than those in a curved annular pipe without rotation and its numbers depend on the ratio of the Coriolis force to centrifugal force F. As F≈-1, the secondary flow has eight loops and its intensity reaches the minimum value, and the distribution of the axial flow is like that of the Poiseuille flow. The position of the maximum axial velocity is pushed to either outer bend or inner bend, which is also determined by F. [WT5”HZ]
基金Project supported by the National Natural Science Foundation of China (Grant No :10272096)
文摘The fully developed laminar flow in helical elliptical pipes is influenced by curvature, torsion and aspect ratio of cross-section. With the aid of the symbolic manipulation technique, the governing equations were solved by the Galerkin method, The procedures of implementing the Galerkin method for flows in curvilinear pipes were discussed. The effects of the aspect ratio and torsion on the flow structure, wall shear stress and flow ratio were examined in detail. The results show that the flow characteristic for aspect ratio larger than unit is quite different from those for the aspect ratio smaller than unit.
文摘The combined effects of the system rotation (Coriolis force) and curvature (centrifugal force) on the flow in rotating curved circular pipe with small curvature are examined by perturbation method. A second order perturbation solution is presented. The secondary flow structure and the primary axial velocity distributions are studied in detail. The loops of the secondary flow are more complex than those in a curved pipe without rotation or a rotating straight pipe. Its numbers depend on the body force ratio F which represents the ratio of the Coriolis to the centrifugal force. The maximum of the axial velocity is pushed to either outer bend or inner bend, which is also determined by F. The results are confirmed by the results of other authors who studied the same problem by different methods.
