We investigate the onset of temporal dynamics of a fluidic oscillator (FO) in the laminar regime at very low Reynolds number ($Re=U{h}_i/\nu ,$ where $U$ and $h_i$ are the velocity and channel height at inlet, and $\nu$ is the kinematic viscosity of the fluid). Both two- and spanwise-periodic-three-dimensional simulations are performed using high-order spectral element methods to characterise the flow inside the FO cavity and the outcoming jet. While the flow remains steady and symmetric for sufficiently low $Re,$ the two-dimensional FO undergoes a symmetry-breaking Hopf bifurcation at $R{e}_{{\mathrm{H}}_2}=75.7$ that results in a space-time symmetric periodic solution. The remnant space-time symmetry is later broken in a pitchfork bifurcation of limit cycles at $R{e}_{{\mathrm{P}}_2}=294.2$. Taking three-dimensionality into consideration results in an early spanwise-invariance disruption at about $R{e}_3\simeq 68$ of wavelength $13h_i$ that retards the onset of the oscillation to $R{e}_{{\mathrm{H}}_3}=82.1$. This effect is little representative of actual FO implementations, which are typically much narrower than the wavelength of the spanwise instability observed at these low $Re$. Contrary to what happens with FOs in the turbulent oscillatory regime, the oscillation frequency of the output jet in the laminar regime is found to decrease with $Re$. The mechanism that drives the oscillation, however, remains the same: as the high speed flow inside the cavity attaches to one of the internal walls through the Coandă effect, the feedback channels divert part of the momentum back, which pushes the incoming flow against the opposite wall. The alternate deviation of the flow inside the cavity to one or the other side results in the periodic flipping of the output jet. The pressure contribution to net momentum at the end of the feedback channels is found to be double that of the advective momentum flux at the early time-periodic regime but both become comparable as $Re$ is increased. The jet sweeping angle amplitude is more pronounced for the two-dimensional FO as compared to three-dimensional at a fixed given $Re,$ the Coandă effect being only partially fulfilled in the latter case. In both cases the sweep amplitude increases with $Re$. The instability of the output jet, which becomes slightly chaotic already at very low $Re,$ is responsible for triggering the cavity instability that drives the oscillation slightly earlier than it would, should outside noise be suppressed altogether.

我们研究了在非常低的雷诺数（$\mathrm{[R}Ë=üH_{\mathrm{一世}}/\nu ，$ 在哪里 $ü$ 和 $H_{\mathrm{一世}}$ 是入口处的速度和通道高度，以及 $\nu$是流体的运动粘度）。使用高阶谱元素方法进行了二维和展向周期三维模拟，以表征FO腔和射流内部的流动。当流量保持稳定且对称时，流量足够低$\mathrm{[R}Ë，$ 二维FO在 $\mathrm{[R}{Ë}_{{\mathrm{H}}_{\mathrm{2个}}}=75.7$导致时空对称周期解。剩余的时空对称性随后在极限环的干草叉分叉处被打破$\mathrm{[R}{Ë}_{{\mathrm{P}}_{\mathrm{2个}}}=294.2$。考虑到三维度，将导致大约在大约30年代早期的跨度不变性破坏。$\mathrm{[R}{Ë}_3\simeq 68$ 波长 $13H_{\mathrm{一世}}$ 延迟了振荡的开始 $\mathrm{[R}{Ë}_{{\mathrm{H}}_3}=82.1$。这种影响几乎不能代表实际的FO实现，它们通常比在这样低的温度下观察到的展向不稳定性的波长要窄得多。$\mathrm{[R}Ë$。与在湍流振荡状态下FO发生的情况相反，发现层流状态下输出射流的振荡频率随着$\mathrm{[R}Ë$。但是，驱动振荡的机制保持不变：由于腔内的高速流通过柯安德效应附着到内壁之一上，反馈通道将部分动量转移回去，从而将进入的流推向反方向。对面的墙。腔体内流向一侧或另一侧的交替偏移会导致输出射流的周期性翻转。发现在反馈通道末端，对净动量的压力贡献是早期时间周期内对流动量通量的两倍，但两者可与$\mathrm{[R}Ë$增加。与二维FO相比，二维FO的喷射扫掠角幅度更为明显$\mathrm{[R}Ë，$在后一种情况下，Coandă效应只能部分实现。在这两种情况下，扫描幅度都随$\mathrm{[R}Ë$。输出射流的不稳定性，在非常低的温度下已经变得有些混乱$\mathrm{[R}Ë，$ 如果完全抑制外部噪声，则引起腔不稳定的原因是腔的不稳定性会比引起振荡的时间提前一点。