旋涡沿轴线的非线性分叉

Bifurcation of Vortex Motion Along Its Axis

本文研究了低速旋涡沿其轴向的演变，发现涡轴上λ＝（１／ρ＿ｏ）（ρω／Ｚ）＿０是决定其演变规律的重要参数。如果λ＞０，涡轴附近横截面的流线是由外向内转的稳定螺旋点形态；如果λ＜０，是由内向外转的不稳定螺旋点形态。如果沿涡轴，λ变号，则由变号点起出现Ｈｏｐｆ分叉，产生极限环。λ由正变为负，极限环是稳定的；λ由负变为正，极限环是不稳定的。一个稳定的旋涡要演变为破裂，一定要伴随先出现Ｈｏｐｆ分叉。旋涡破裂可能存在三种形态：第一种为泡状破裂，其破裂点为驻点，破裂泡的后缘一般是不封闭的；第二种为螺旋型破裂。Ｅｕｌｅｒ方程所描述的解指出，在反压区不可能出现速度为零的破裂点；第三种为泡形－螺旋型组合破裂形态，其破裂点为驻点，前面为小破裂泡，然后演变为螺旋流。文中还对上述理论发现与已有的实验和数值模拟结果作了对比分析。

The vortex evolution along its axis is investigated by using topological theory.It is pointed out thatλ=(1/ρ_o) at theaxis of vortex is an important parameter to determine the evolution mech-anism ofvortex.Ifλ>o along the axis,the sectional streamlines ncar the axis in the transversal plane possess a stable spiral(inward) pattern.Ifλ<o along the axis, the transversal streamlines are an unstable spiral(outward).As the sign ofλchanges along the axis from positive to neg-ative or from negative to positive, the flowpattern in the transversal plane will yield a Hopf bifurcation starting from the position where λ=o,meanwhile,theformer is a stable limit cycle and the later an unstable one. The vortex bursting is accompanied with an unstable trans-versal spiral pattern and the decrease of axial velocity. If the vortex is stablc initially,i.e.λ>o in theinitial region,the Hopfbifurcation certainly appears in advance of the bursting point.There are three kinds of vortex bursting which are the bubble type,bubble-spiral type,andspiral type. The bursting point for the bubble type and bubble-spiral type is a stagnation one,where the vortex flow possesses a saddle pat-tern in the longitudinal plane containing the axis.For the subsonic vortex moving forward along its axis with M_z<l at high Reynolds number,the sign ofλis the same as the and the opposite with .It means that in the adverse pressure region the unstable transversal spiral pattern is just accompanied with the decrease of axial velocity. As the vortex moves forward along the axis from the favora-ble pressure to the adverse pressure,the initially stable vortex will become bursting later through the Hopf bifurcation occuring in the ad-verse pressure region. If the subsonic vortex flow is described by Euler equations,the bubble type,bubble-spiral type of vortex bursting can not occur in the region of advcrse pressure gradient.In this case,the spiral burst occuring in the region of adverse pressure gradient possesses a bursting point where ω>o.In this meaning,the spiral bursting point may be no stagnation.