The present article aims at optimising the spread of a bifurcating jet: a jet that combines axisymmetric and helical forcing to achieve increased mixing in a preferential plane. Parekh et al. ( Tech. Rep. TF-35, Stanford University, 1988) explained such a bifurcation as the result of nonlinear interaction between ring vortices (triggered by $m=0$ axisymmetric forcing), shifted off-axis in alternate directions (owing to $m=1$ helical forcing). Following this idea, we linearly optimise the periodic helical forcing to be applied at the inlet, in order to maximally displace the ring vortices of an axisymmetrically forced jet. Two norms are introduced for evaluating the effect of helical forcing onto the helical response: the standard ${\mathcal{L}}_{2}$ -norm and a semi-norm reflecting the off-axis vortex displacement. The linear results show one dominant forcing mode over the entire Strouhal band studied ( $0.35\leqslant St\leqslant 0.8$ ), with a large gain separation from suboptimals. The dominant forcing is mainly radial, independent of the chosen response norm, and provides a gain at least five times larger than what was achieved by previous ad hoc forcing strategies. Superposition of base flow and linear results show the alternate shifting and twisting provoked by the the small-amplitude helical forcing, which is an essential ingredient for triggering jet bifurcation. When tested in three-dimensional direct numerical simulations, low-amplitude helical forcing achieves efficient bifurcation at all Strouhal values studied. At high Strouhal numbers, an additional central branch emerges in the mean flow, leading to trifurcation. Across all frequencies, compared with ad hoc forcing strategies, the optimal forcing triggers a much stronger and robust spreading, by moving the bifurcation point upstream. As a result, bifurcating jets are observed over a much larger Strouhal band ( $0.35\leqslant St\leqslant 0.8$ ) compared with the band where ad hoc forcing achieves bifurcation in our setting ( $0.4\leqslant St\leqslant 0.5$ ).

中文翻译：

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射流分叉的最佳触发：应用于时间周期基流的最佳强迫示例

本文旨在优化分叉射流的扩散：一种结合轴对称和螺旋力的射流，以在优先平面上实现增加的混合。帕雷克等人。(Tech. Rep. TF-35, Stanford University, 1988) 将这种分叉解释为环涡流之间非线性相互作用的结果（由 $m=0$ 轴对称强迫触发），沿交替方向离轴偏移（由于 $ m=1$ 螺旋强迫）。遵循这个想法，我们线性优化了在入口处施加的周期性螺旋力，以最大程度地移动轴对称强制射流的环形涡流。引入了两个范数来评估螺旋力对螺旋响应的影响：标准 ${\mathcal{L}}_{2}$ -范数和反映离轴涡旋位移的半范数。线性结果显示了在研究的整个 Strouhal 波段上的一种主要强迫模式（$0.35\leqslant St\leqslant 0.8$），与次优的增益分离很大。占主导地位的强迫主要是径向的，与所选的响应范数无关，并且提供的增益至少是以前的临时强迫策略所获得的增益的五倍。基流和线性结果的叠加显示了由小幅度螺旋强迫引起的交替移动和扭曲，这是触发射流分叉的重要因素。在三维直接数值模拟中进行测试时，低振幅螺旋强迫在研究的所有 Strouhal 值下实现了有效的分叉。在高 Strouhal 数下，平均流中会出现一个额外的中央分支，导致三叉。在所有频率范围内，与临时强迫策略相比，最佳强迫通过将分叉点向上游移动来触发更强大和稳健的传播。结果，与在我们的设置中临时强迫实现分叉的波段（ $0.4\leqslant St\leqslant 0.5$ ）相比，在更大的 Strouhal 波段（ $0.35\leqslant St\leqslant 0.8$ ）上观察到分叉射流。