Interfacial Stability In Turbulent Pressure-Driven Channel Flow

We consider the motion of a deformable interface that separates a fully-developed turbulent gas flow from a thin layer of laminar liquid. We outline a linear model to describe the interaction between the turbulent gas flow and the interfacial waves; this consists of the Orr-Sommerfeld equation with the appropriate turbulent mean flow profile, together with a turbulent stress closure scheme. This approach permits us to determine numerically the growth rate of the wave amplitude, as a function of the relevant dimensionless system parameters and turbulence closure relations. It also extends previous work by accounting for the effects of the thin liquid layer on the dynamics. The growth rate of the wave amplitude depends sensitively on the choice of mean flow. Therefore, it is necessary to derive a mean-flow profile that incorporates the characteristics of the flow observed in experiments. The mean flow profile we obtain demonstrates the features of turbulence in the gas layer: it is linear near the channel wall and interface, and logarithmic in the core. By writing down the functional form of the profile, it is also possible to express the wall and interfacial shear stresses as a function of the applied pressure gradient. The other ingredient necessary to complete the model is a turbulent closure scheme. The simplest possible closure is the mixing-length model: since turbulent eddies are limited in size by the wall and interface, the eddy viscosity can be constituted as a simple function of the vertical coordinate. Using these inputs, we calculate the growth rate of the interfacial waves. We find that the incorporation of turbulent stresses through this model enhances the growth rate of the interfacial mode, while the growth rate of the internal mode is suppressed. The inclusion of the Reynolds stresses therefore gives rise to a significant correction in the wave growth rates. Previous work on this problem used a boundary-layer mean profile to model the mean flow, and we compare our results with this framework. (See, for example, [4], [1], and [3].) Although the models agree at shorter wavelengths, at long wavelengths it is necessary to take account of the bounded nature of the problem domain to obtain a correct picture of the wave growth rates.