Numerical Prediction of Drag Reduction and Interfacial Shapes Over Superhydrophobic Surfaces

Abstract

Two different numerical methodologies are developed to study the drag-reducing effects of superhydrophobicity for realistic surfaces. A multiphase Navier--Stokes volume-of-fluid based approach is developed to investigate the effect of interface curvature, and turbulence on drag reduction in the presence of realistic roughness. A Gibbs energy minimizer is developed using a level set framework to determine interfacial equilibrium locations over realistic rough surfaces given an external pressure and surface properties. Both laminar and turbulent regimes are investigated. Direct numerical simulation (DNS) is used to study the drag reduction by superhydrophobic surfaces (SHS) in laminar channel flow. Resolved multiphase simulations using the volume-of-fluid (VOF) methodology are performed to study the effects of groove geometry, interface shear rate, and meniscus penetration independently. The trapped gas is simulated as both flat and meniscal interfaces. The drag reduction initially increases with interface deflection into the groove and then decreases for large deflections as the interface velocity approaches zero due to the proximity to the bottom of the groove. DNS is also performed for two wall-bounded flow configurations: laminar Couette flow at $Re=740$ and turbulent channel flow at $Re_{\tau}=180$, where $\tau$ is the shear stress at the wall. The top wall is smooth and the bottom wall is a realistically rough superhydrophobic surface (SHS), generated from a three-dimensional surface profile measurement. The air-water interface, which is assumed to be flat, is simulated using the volume-of-fluid (VOF) approach. The two flow regimes are studied for varying interface height $h$ to understand its effect on slip and drag reduction ($DR$). For the laminar Couette flow, the presence of the surface roughness is felt up to $40\%$ of the channel height in the wall-normal direction. The nonlinear dependence of $DR$ on $h$ is observed to have three distinct regions. A nonlinear curve fit is obtained for a gas fraction $\phi_g$ as a function of $h$, where $\phi_g$ determines the amount of slip area exposed to the flow. A power-law fit is obtained from the data for the effective slip length as a function of $\phi_g$ and is compared to those derived for structured geometry. For the turbulent channel flow, statistics of the flow field are compared to that of a smooth wall to understand the effects of roughness and $h$. Four cases are simulated ranging from fully wetted to fully covered, and two intermediate regions in between. Scaling laws for slip length, slip velocity, roughness function, and $DR$ are obtained for different penetration depths and are compared to past work for structured geometry. $DR$ is shown to depend on the competing effects of slip velocity and turbulent losses due to the Reynolds shear stress contribution. The presence of trapped air in the cavities significantly alters near-wall flow physics where we examine near-wall structures and propose a physical mechanism for their behavior. The fully wetted roughness increases the peak value of turbulent intensities, whereas the presence of the interface suppresses them. The pressure fluctuations have competing contributions between turbulent pressure fluctuations and stagnation pressure due to asperities, the near-wall structure is altered and breaks down with increasing slip. Overall, there exists a competing effect between the interface and the asperities, the interface suppresses turbulence whereas the asperities enhance them. The present work demonstrates DNS over a realistic SHS for the first time, to the best of our knowledge. A robust numerical methodology to predict equilibrium interfaces over arbitrary solid surfaces is also developed. The kernel of the proposed method is the distance regularized level set equations (DRLSE) with techniques to incorporate the no-penetration and volume-conservation constraints. In this framework, we avoid reinitialization that is typically used in traditional level set methods. This allows for a more efficient algorithm since only one advection equation is solved, and avoids numerical error associated with the re-distancing step. A novel surface tension distribution, based on a harmonic mean, is prescribed such that the zero level set has the correct liquid-solid surface tension value. This leads to a more accurate prediction of the triple contact point location. The method uses second-order central difference schemes that facilitate easy parallel implementation and is validated by comparing it to traditional level set methods using canonical problems. The application of the method, in the context of Gibbs free energy minimization, is to obtain the equilibrium location of liquid-gas interfaces. The method is validated against existing analytical solutions, and its capability to predict equilibrium shapes over both structured and realistic rough surfaces is demonstrated. The solid boundaries are user-prescribed; to enable a Gibbs energy minimization for a real surface, a method was developed that allows implicit nonparametric shape reconstruction from an unorganized set of data points. The central idea is to take the point cloud data from a real surface scan and reconstruct a solid boundary for the Gibbs energy minimization. This is achieved in two steps: $(i)$ reconstruct a distance field by solving the Eikonal equation using fast sweeping methods, and $(ii)$ minimize surface energy based on the distance potential to reconstruct the surface. The physical process can be thought of as an elastic membrane that covers the data set which evolves under a gradient flow until it shrink-wraps the point cloud data, thereby reconstructing the shape like a mold. This becomes a pre-processing step by which a point cloud data is implicitly reconstructed by a level set that represents the solid surface, which is then read into the Gibbs energy minimizer to obtain the liquid-gas equilibrium interface.