Abstract The modal and non-modal linear stability analyses of a three-dimensional plane Couette–Poiseuille flow through a porous channel are studied based on the two-domain approach, where fluid and porous layers are treated as distinct layers separated by an interface. The unsteady Darcy–Brinkman equations are used to describe the flow in the porous layer rather than the unsteady Darcy’s equations. In fact, the Brinkman viscous diffusion terms are necessary to capture the momentum boundary layer developed close to the fluid-porous interface. The modal stability analysis is performed under the framework of the Orr–Sommerfeld boundary value problem. On the other hand, the non-modal stability analysis is performed under the framework of the time-dependent initial value problem in terms of normal velocity and normal vorticity components. The Chebyshev spectral collocation method along with the QZ algorithm is implemented to solve the boundary value problem numerically for disturbances of arbitrary wavenumbers. The convergence test of spectrum demonstrates that more Chebyshev polynomials are required to arrest the flow dynamics in the momentum boundary layer once the Couette flow component is turned on. Two different types of modes, so-called the fluid layer mode and the porous layer mode are identified in the modal stability analysis. The most unstable fluid layer mode intensifies while the most unstable porous layer mode attenuates in the presence of the Couette flow component. Further, the mechanism of modal instability is deciphered by using the method of the energy budget. It is found that the energy production term supplies energy from the base flow to the disturbance via the Reynolds stress, and boosts the disturbance kinetic energies for the fluid layer and the porous layer. Moreover, the non-modal stability analysis demonstrates that short time energy growth exists in the parameter space and becomes significant in the presence of the Couette flow component and the permeability of the porous medium.

中文翻译：

---

覆盖多孔层的平面 Couette-Poiseuille 流的线性稳定性

摘要 基于双域方法研究了三维平面 Couette-Poiseuille 流通过多孔通道的模态和非模态线性稳定性分析，其中流体层和多孔层被视为由界面分隔的不同层。非定常 Darcy-Brinkman 方程用于描述多孔层中的流动，而不是非定常 Darcy 方程。事实上，Brinkman 粘性扩散项对于捕捉靠近流体-多孔界面发展的动量边界层是必要的。模态稳定性分析是在 Orr-Sommerfeld 边值问题的框架下进行的。另一方面，非模态稳定性分析是在法向速度和法向涡量分量的瞬态初值问题框架下进行的。实现了切比雪夫谱搭配方法和QZ算法，以数值求解任意波数扰动的边值问题。谱的收敛性测试表明，一旦打开 Couette 流动分量，就需要更多的 Chebyshev 多项式来阻止动量边界层中的流动动力学。在模态稳定性分析中识别了两种不同类型的模态，即所谓的流体层模式和多孔层模式。在存在库埃特流动分量的情况下，最不稳定的流体层模式会增强，而最不稳定的多孔层模式会减弱。进一步，利用能量收支的方法破译了模态不稳定性的机制。发现能量产生项通过雷诺应力将能量从基流提供给扰动，增加了流体层和多孔层的扰动动能。此外，非模态稳定性分析表明，短时间能量增长存在于参数空间中，并且在存在 Couette 流动分量和多孔介质渗透率的情况下变得显着。