Reducing drag on a flat plate subjected to incompressible laminar flow

The idea behind this work comes from the question: What is the impact of plate corrugations on drag? In this context, a numerical study of laminar incompressible flow over a flat plate and over corrugated plates is carried out. Numerical analysis is performed for low Reynolds numbers (Re= 10, Re = 50, Re = 100, Re = 500, Re =1000) using the computational fluid dynamics (CFD) software ANSYS FLUENT. Simulations results are compared to each others and with those of the reference plate (flat plate (figure 4a)). Comparisons are made via drag coefficient Cd. This work is the beginning of a study that evaluates the impact of corrugations on drag reduction on a flat plate.


Introduction
Drag reduction is a very interesting topic thanks to the gain of energy that it allows. Several drag reduction techniques have been suggested by researchers in several fields (aeronautics, automotive ...). In that sense, adding agents to fluid, especially polymers, is a good solution to reduce drag. Then, surfactants with their four types (anionic, cationic, nonionic and amphoteric) are also good drag reducers [1]. Other solutions have been proposed by adding drag reducers such as devices added at wingtips of an aircraft (flyers, flaps ...). These devices act on marginal vortices to reduce drag [2]. In addition, there are several boundary layer control methods that have been developed to reduce frictional resistance of streamlined bodies, such as the injection of a different gas [3], the acceleration of the boundary layer [4], etc. All these devices are described in detail in scientific literature. Finally, the use of a liquid film on the surface of a body reduces its frictional resistance [5]. In this study, drag reduction is achieved by changing the shape of a flat plate in laminar flow regime. Simulations are performed by the use of the CFD software ANSYS FLUENT. This work is a preliminary study that assesses the influence of corrugations (shapes and orientations) and other patterns on drag force exerted on the plate.

Numerical modeling
This work deals with the study of a laminar incompressible fluid flow on a plate (dimensions: 1m×1m) (figure 3). The fluid (air) has a density ρ (kg m −3 ), a kinematic viscosity ν (m 2 s −1 ) and a pressure p (Pa). Non Dimensionalized Navier-Stokes equations in Cartesian coordinates for an incompressible fluid are: In these equations, variables are non-dimensionalized by choosing the reference scales L =1m and U =1m/s . Where L is the length of the flat plate and U is the fluid freestream velocity. The Reynolds number is defined by: The drag coefficient is defined by: where F t is the drag force and S is the surface area of the plate.

Tests configuration
The figure 1 represents the boundary conditions for the various tests. These conditions are detailed in the figure 2.

Results and discussion
The figure 4 represents the studied shapes of the plate. They have the same dimensions (1m×1m) and they are subject to the same flow conditions and for which drag coefficient C d is calculated.    • For Re=10, the inverted sinusoidal plate reduces remarkably drag coefficient of the flat plate C d from 2.305 to 2.071 (reduction of 10.2%).
• For the Reynolds numbers Re=500 and Re=1000, a slight increase of C d is observed compared to the flat plate.
This configuration will be used mainly for low Reynolds numbers since it reduces remarkably C d by 10.2% for Re=10, by 6.2% for Re=50 and by 3.4% for Re=100.

Conclusion
In this preliminary work, numerical simulations were carried out to study the effect of changing plate shape on drag reduction for a laminar incompressible air flow on this plate. Simulations were performed by the CFD software AN-SYS FLUENT. Drag coefficients are compared between the various plate shapes. The analysis of results leads to conclude that the inverted sinusoidal plate reduces drag coefficient of the plate for low Reynolds numbers. Future works will focus on the optimisation of flat plate shape to reduce drag coefficient for high Reynolds number flows and the extension to turbulent flows.