An axisymmetric laminar swirling jet of a viscous incompressible fluid flowing from a rotating semi-infinite tube in space filled with the same fluid is explored. The inner surface of the tube rotates with a constant angular velocity, and the outer surface is stationary or rotates with the same angular velocity. It is shown that in the first case, the flow field far from the tube orifice is described by the Loitsyansky asymptotic solution, and in the second case (with weak coflow), it is described by the Long-Gol’dshtik-Zubtsov self-similar solution. The Gol’dshtik hidden invariant is generalized to arbitrary axisymmetric swirling jets, and its influence on the jet asymptotics is studied. Strongly swirling jets are calculated, and the dependence of the parameters of the recirculation zone (vortex breakdown in a swirling jet) on the swirl number and the Reynolds number is examined.

中文翻译：

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轴对称旋流浸没射流的不变量和渐近性

探索了从旋转的半无限管中流出的粘性不可压缩流体的轴对称层流旋流射流，该流体充满相同流体的空间。管内表面以恒定角速度旋转，外表面静止或以相同角速度旋转。结果表明，在第一种情况下，远离管口的流场由 Loitsyansky 渐近解描述，在第二种情况下（弱协流），它由 Long-Gol'dshtik-Zubtsov 自解描述。类似的解决方案。将Gol'dshtik隐藏不变量推广到任意轴对称旋流射流，研究了其对射流渐近性的影响。计算出强烈旋转的射流，