Phase effect on the modal interaction of flow instabilities is investigated for laminar-to-turbulent transition in a flat-plate boundary-layer flow. Primary and secondary three-dimensional (3-D) oblique waves at various initial phase differences between these two instability modes. Three numerical methods are used for a systematic approach for the entire transition process, i.e. before the onset of transition well into fully turbulent flow. Floquet analysis predicts the subharmonic resonance where a subharmonic mode locally resonates for a given basic flow composed of the steady laminar flow and the fundamental mode. Because Floquet analysis is limited to the resonating subharmonic mode, nonlinear parabolised stability equation analysis (PSE) is conducted with various phase shifts of the subharmonic mode with respect to the given fundamental mode. The application of PSE offers insights on the modal interaction affected by the phase difference up to the weakly nonlinear stage of transition. Large-eddy simulation (LES) is conducted for a complete transition to turbulent boundary layer because PSE becomes prohibitively expensive in the late nonlinear stage of transition. The modulation of the subharmonic resonance with the initial phase difference leads to a significant delay in the transition location up to $\Delta Re_{x, tr} \simeq 4\times 10^5$ as predicted by the current LES. Effects of the initial phase difference on the spatial evolution of the modal shape of the subharmonic mode are further investigated. The mechanism of the phase evolution is discussed, based on current numerical results and relevant literature data.

中文翻译：

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不稳定模式之间的相位差对边界层转变的影响

针对平板边界层流中的层流到湍流过渡，研究了相位对流动不稳定性模态相互作用的影响。这两种不稳定模式之间不同初始相位差的初级和次级三维 (3-D) 斜波。三种数值方法用于整个过渡过程的系统化方法，即在完全湍流开始过渡之前。Floquet 分析可预测亚谐波共振，其中亚谐波模式对于由稳定层流和基本模式组成的给定基本流进行局部共振。由于 Floquet 分析仅限于共振次谐波模式，因此非线性抛物线稳定性方程分析 (PSE) 是在次谐波模式相对于给定基本模式的各种相移下进行的。PSE 的应用提供了对受相位差影响的模态相互作用的见解，直至弱非线性过渡阶段。大涡模拟 (LES) 用于完全过渡到湍流边界层，因为 PSE 在非线性过渡后期变得非常昂贵。具有初始相位差的次谐波谐振的调制导致过渡位置的显着延迟，直到 $\Delta Re_{x, tr} \simeq 4\times 10^5$ 正如当前 LES 所预测的那样。进一步研究了初始相位差对次谐波模式的模态形状空间演变的影响。基于当前的数值结果和相关文献数据，讨论了相演化的机制。