On the Effect of the Plunging Velocity over the Aerodynamic Forces for an Airfoil in Subsonic Laminar Flow

Abstract

The objective of this work is the numerical simulation of the normal and axial forces resulting over a NACA 0012 airfoil with an imposed plunging motion in a subsonic laminar flow in order to asset the effect of the plunging velocity over these forces. The system of equations is written using a non-inertial frame of reference that is fixed to the airfoil. The effect of this motion is accounted by introducing pseudo-force and pseudo-work terms in the right-hand side of the momentum and energy equations, respectively. The compressible Navier-Stokes equations are numerically solved using a finite-volume discretization in combination with the skew-symmetric form of Ducros’ fourth-order numerical scheme for the calculation of the fluxes, while the time-marching process is achieved using a third-order Runge-Kutta scheme. The linear amplitude of the plunging motion is imposed as sinusoidal in time, and for all cases studied, the plunging frequency corresponds to the simulated vortex-emission frequency of the static airfoil at the same angle of attack. Three cases are studied. In the first case, the maximum plunging velocity of the airfoil has half the value of the free-flow velocity; in the second case, the maximum velocity is equal to the free-flow velocity; and in the third case it has twice the value of the free-flow velocity. All the results are obtained for a Reynolds number of 1,200 and a Mach number of 0.18. The numerical results show that a sequential twofold increment of the plunging velocity results in a proportional increment of the amplitude of the unsteady normal and axial force coefficients. For the mean coefficients, the response is not proportional. For the normal force, the sequential twofold increment results in a moderate increment of the mean normal force tending to a saturation value. On the other hand, the increment of the plunging velocity results in a strong increment of the negative axial force that strongly contributes to thrust, with no indication of saturation. Nomenclature A = nondimensional maximum amplitude of the plunging motion * A = maximum amplitude of the plunging motion * c = cord of the airfoil 1 C = nondimensional first gas constant in Sutherland’s formula * 1 C = first gas constant in Sutherland’s formula 2 C = nondimensional second gas constant in Sutherland’s formula * 2 C = second gas constant in Sutherland’s formula * p c = specific heat at constant pressure v c = nondimensional specific heat at constant volume * v c = specific heat at constant volume