Coherent structures in countercurrent axisymmetric shear flows

Abstract

The dynamical behaviors of coherent structures in countercurrent axisymmetric shear flows are experimentally studied. The forward velocityU ₁ and the velocity rationR=(U ₁-U ₂)/(U ₁+U ₂), whereU ₂ denotes the suction velocity, are considered as the control parameters. Two kinds of vortex structures, i.e., axisymmetric and helical structures, were discovered with respect to different regimes in theR versusU ₁ diagram. In the case ofU ₁ ranging from 3 to 20m/s andR from 1 to 3, the axisymmetric structures play an important role. Based on the dynamical behaviors of axisymmetric structures, a critical forward velocityU ^cr₁ =6.8m/s was defined, subsequently, the subcritical velocity regime:U ₁>U ^cr₁ and the supercritical velocity regime:U ₁<U ^cr₁ . In the subcritical velocity regime, the flow system contains shear layer self-excited oscillations in a certain range of the velocity ratio with respect to any forward velocity. In the supercritical velocity regime, the effect of the velocity ratio could be explained by the relative movement and the spatial evolution of the axisymmetric structure undergoes the following stages: (1) Kelvin-Helmholtz instability leading to vortex rolling up, (2) first time vortex agglomeration, (3) jet column self-excited oscillation, (4) shear layer self-excited oscilation, (5) “ordered tearing”, (6) turbulence in the case ofU ₁<4m/s (the “ordered tearing” does not exist whenU ₁>4m/s), correspondingly, the spatial evolution of the temporal asymptotic behavior of a dynamical system can be described as follows: (1) Hopf bifurcation, (2) subharmonic bifurcation, (3) reversed superharmonic bifurcation, (4) superharmonic bifurcation, (5) chaos (“weak turbulence”) in the case ofU ₁<4m/s (superharmonic bifurcation does not exist whenU ₁>4m/s). The proposed new terms, superharmonic and reversed superharmonic bifurcations, are characterized of the frequency doubling rather than the period doubling. A kind of unfamiliar vortices referred to as the helical structure was discovered experimentally when the forward velocity around 2m/s and the velocity range from 1.1 to 2.3. There are two base frequencies contained in the flow system and they could coexist as indicated by the Wigner-Ville-Distribution and the temporal asymptotic behavior of the dynamical system corresponding to the helical vortex could be described as 2-torus as indicted by the 3D reconstructed phase trajectory and correlation dimension. The scenario of the spatial evolution of helical structures could be described as follows: the jet column is separated into two parts at a certain spatial location and they entangle each other to form the helical vortex until the occurrence of those separated jet columns to reconnect further downstream with the result that the flow system evolves into turbulence in a catastrophic form. Correspondingly, the dynamical system evolves directly into 2-torus through the supercritical Hopf bifurcation followed by a transition from a quasi-periodic attractor to a strange attractor. In the case ofU ₁=2m/s, the parametric evolution of the temporal asymptotic behavior of the dynamical system as the velocity ratio increases from 1 to 3 could be described as follows: (1) 2-torus (Hopf bifurcation), (2) limit cycle (reversed Hopf bifurcation), (3) strange attractor (subharmonic bifurcation).