XXIV ICTAM, 21-26 August 2016, Montreal, Canada
UNSTEADY KUTTA CONDITION AND VORTEX-SHEET GENERATION
Xi Xia1 and Kamran Mohseni ∗1,2 1Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL, USA 2Department of Electrical and Computer Engineering, University of Florida, Gainesville, FL, USA
Summary Vortex method has been widely used for low-Reynolds-number flights as it reduces the computation domain from the entire flow field to only finite vortical structures. An essential challenge of the vortex method lies in the prediction of the vortex sheets separated from the solid body. To tackle this problem, this study extends the classical Kutta condition to unsteady situations based on the physical sense that flow cannot turn around a sharp edge. This unsteady Kutta condition can be readily applied to an airfoil with cusped trailing edge, where the forming vortex sheet is known to be tangential to the trailing edge. For a finite-angle trailing edge, previous studies indicate that the direction of the forming vortex sheet is ambiguous. Therefore, this study proposes a novel analytical formulation to determine the angle of the trailing-edge vortex sheet based on momentum conservation in the direction normal to the forming vortex sheet.
INTRODUCTION
Natural flies are observed to be superior than man-made aerial vehicles in terms of aerodynamic performance, especially the high lift-generation mechanisms. Early experimental investigations suggested that the existence of an attached leading edge vortex (LEV) contributes to enhanced lift generation for flapping wings [2]. To understand the fundamental mechanism of the LEV, recent analytical studies have been extensively focused on using vortex method (diagram shown in Fig. 1) to solve the unsteady aerodynamics of a flat plate wing by means of tracking the wake vortices that are shed from the wing [7]. The fundamental question associated with vortex methods is how to decide the rate at which vorticity is being created at the leading or trailing edge. In reality, the generation of vorticity is caused by the interaction between fluid and solid boundary that forms the shear layer, which is essentially a viscous process. Since viscosity is ignored in the Euler equa- tion, a typical solution to that is applying vorticity releasing conditions at the vortex shedding locations of the solid body, e.g. the steady-state Kutta condition at a sharp trailing edge, which requires a finite velocity at the trailing edge [5, 3]. Figure 1: Diagram showing the unsteady aerodynamic model For a Joukowski airfoil or a flat plate, the steady-state Kutta for a two-dimensional airfoil. The green contour line repre- condition is realized by setting the trailing edge to be a stag- sents the bound vortex sheet surrounding the airfoil. The red nation point in the mapped cylinder plane. The effect of this and blue structures are the leading and trailing edge vortices, implementation is that the flow from both sides of the trail- respectively. ing edge will be tangential to the edge, which guarantees the streamline emanating from this edge point to be inline with the plate, fulfilling the condition proposed in previous studies [1, 4]. In this study, the objective is to extend the steady-state Kutta condition to unsteady situations for cusped and finite- angle trailing edges.
UNSTEADY KUTTA CONDITION
The application of the steady-state Kutta condition for a flat plate or a Joukowski airfoil (with cusped trailing edge) has already been discussed in our previous work [7]. Basically, this condition is equivalent to enforcing a stagnation point at the trailing edge in the mapped cylinder plane. However, Xia & Mohseni [8] later pointed out that a stagnation point generally does not exist at the trailing edge of an unsteady flat plate that has rotary motion. To address this difficulty, they further proposed a modified Kutta condition which relaxes the trailing edge point from totally stagnant to only stagnant in the tangential direction of the surface in the cylinder plane. They have shown that this modified Kutta condition is consistent with the physical meaning of the classical Kutta condition that flow around the sharp edge should be prevented in a reference frame that is fixed to the flat plate. This condition can be generalized in the physical plane (the plane of the flat plate or airfoil) as
ug · n = 0, (1) where ug denotes the de-singularized flow velocity at the trailing edge, and n represents the vector perpendicular to the trailing-edge tangential vector (n · t = 0) as shown in Fig. 2.
∗Corresponding author. Email: mohseni@ufl.edu