An objective of the present paper is to experimentally clarify the torsion effect on the flow in helical circular pipes. We have made six helical circular pipes having different pitches and common non-dimensional curv...An objective of the present paper is to experimentally clarify the torsion effect on the flow in helical circular pipes. We have made six helical circular pipes having different pitches and common non-dimensional curvature δ of about 0.1. The torsion parameter β0, which is defined by β0 = τ/（2δ）1/2 with non-dimensional torsion r, are taken to be 0.02, 0.45, 0.69, 1.01, 1.38 and 1.89 covering from small to very large pitch. The velocity distributions and the turbulence of the flow are measured using an X-type hot-wire anemometer in the range of the Reynolds number from 200 to 20000. The results obtained are summarized as follows： The mean secondary flow pattern in a cross section of the pipe changes from an ordinary twin-vortex type as is seen in a curved pipe without torsion （toroidal pipe） to a single vortex type after one of the twin-vortex gradually disappears as β0 increases. The circulation direction of the single vortex is the same as the direction of torsion of the pipe. The mean velocity distribution of the axial flow is similar to that of the toroidal pipe at small β0, but changes its shape as β0 increases, and attains the shape similar to that in a straight circular pipe when ,β0 = 1.89. It is also found that the critical Reynolds number, at which the flow shows a marginal behavior to turbulence, decreases as ,β0 increases for small ,β0, and then increases after taking a minimum at ,β0 ≈ 1.4 as ,β0 increases. The minimum of the critical Reynolds number experimentally obtained is about 400 at ,β0 ≈ 1.4.展开更多
In this paper,we extend Su-Zhang’s Cheeger-Mller type theorem for symmetric bilinear torsions to manifolds with boundary in the case that the Riemannian metric and the non-degenerate symmetric bilinear form are of pr...In this paper,we extend Su-Zhang’s Cheeger-Mller type theorem for symmetric bilinear torsions to manifolds with boundary in the case that the Riemannian metric and the non-degenerate symmetric bilinear form are of product structure near the boundary.Our result also extends Brning-Ma’s Cheeger-Mller type theorem for Ray-Singer metric on manifolds with boundary to symmetric bilinear torsions in product case.We also compare it with the Ray-Singer analytic torsion on manifolds with boundary.展开更多
文摘An objective of the present paper is to experimentally clarify the torsion effect on the flow in helical circular pipes. We have made six helical circular pipes having different pitches and common non-dimensional curvature δ of about 0.1. The torsion parameter β0, which is defined by β0 = τ/（2δ）1/2 with non-dimensional torsion r, are taken to be 0.02, 0.45, 0.69, 1.01, 1.38 and 1.89 covering from small to very large pitch. The velocity distributions and the turbulence of the flow are measured using an X-type hot-wire anemometer in the range of the Reynolds number from 200 to 20000. The results obtained are summarized as follows： The mean secondary flow pattern in a cross section of the pipe changes from an ordinary twin-vortex type as is seen in a curved pipe without torsion （toroidal pipe） to a single vortex type after one of the twin-vortex gradually disappears as β0 increases. The circulation direction of the single vortex is the same as the direction of torsion of the pipe. The mean velocity distribution of the axial flow is similar to that of the toroidal pipe at small β0, but changes its shape as β0 increases, and attains the shape similar to that in a straight circular pipe when ,β0 = 1.89. It is also found that the critical Reynolds number, at which the flow shows a marginal behavior to turbulence, decreases as ,β0 increases for small ,β0, and then increases after taking a minimum at ,β0 ≈ 1.4 as ,β0 increases. The minimum of the critical Reynolds number experimentally obtained is about 400 at ,β0 ≈ 1.4.
基金supported by National Natural Science Foundation of China(Grant No.11101219)
文摘In this paper,we extend Su-Zhang’s Cheeger-Mller type theorem for symmetric bilinear torsions to manifolds with boundary in the case that the Riemannian metric and the non-degenerate symmetric bilinear form are of product structure near the boundary.Our result also extends Brning-Ma’s Cheeger-Mller type theorem for Ray-Singer metric on manifolds with boundary to symmetric bilinear torsions in product case.We also compare it with the Ray-Singer analytic torsion on manifolds with boundary.
