Roughness on the surface of turbine blades induced by icing, dirt, erosion or manufacturing imperfections changes the aerodynamic configurations of wind turbines and reduces the power generation efficiency. In this work, a modified NACA0024 aerofoil is adopted to study effects of surface roughness on lift/drag forces. Three Reynolds numbers, 1000, 2000 and 5000 and a range of angles of attack [0°,20°] are studied. Since the magnitude of the roughness is small, it can be modelled as non-zero velocity boundary conditions imposed on the smooth surface without roughness. The flow with surface roughness can be therefore decomposed as the sum of a flow without roughness and a flow induced by roughness (or the velocity boundary conditions). The first flow can be obtained by solving the Navier-Stokes (NS) equation while the second one is governed by the linearized NS equation. Correspondingly the lift and drag forces acting on the aerofoil can be also decomposed as the sum of a force without considering roughness and a force induced by roughness. Instead of studying a particular type or distribution of roughness, we calculate the optimal roughness, which changes aerodynamic forces most effectively. This optimal roughness is obtained through a sensitivity study by solving an adjoint equation of the linearized NS equation. It is found that the optimal roughness with respect to both drag and lift forces is concentrated around the trailing edge and upper leading edge of the aerofoil and the lift is much more sensitive to roughness than the drag. Then the optimal roughness with a small magnitude is added to the smooth aerofoil geometry and this new geometry is tested through direct numerical simulations (DNS). It is found that the optimal roughness with a small magnitude (e-norm, defined as the square integration of the roughness around the surface, 0.001) induces over 10% change of the lift. Comparing the forces acting on the smooth surface and on the rough surface, it is noticed that the roughness changes the pressure force significantly while has little influence on the viscous forces. The pressure distribution is further inspected to study mechanisms of the effects of roughness on forces.

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