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Unsteady Kutta Condition and Vortex-Sheet Generation

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Unsteady Kutta Condition and Vortex-Sheet Generation 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 n n Bound vortex sheet γ1 Bound vortex sheet γ 1 ∆θ2 ∆θ0 u2 u o o 1 t u1 t ∆θ1 Trailing edge vortex sheet γg u1 ug ug u u2 Bound vortex sheet γ2 2 Bound vortex sheet γ2 Trailing edge vortex sheet γ Flat plate Trailing edge Airfoil Trailing edge g Figure 2: The formation of vortex sheet for a cusped trailing edge (left) and a finite-angle trailing edge (right). The unsteady Kutta condition (Eq. 1) have been shown to yield promising results in estimating vortex-sheet formation at cusped trailing edges, yet, it was only validated for the situation where a flat plate or a Joukowski airfoil can be mapped to a circle. For a general airfoil, solving Eq. 1 might be challenging because of possible finite-angle trailing edge. As shown in Fig. 2, the two bound vortex sheets at a finite-angle trailing edge are at a certain angle to each other, which is different from the case of a flat plate or a cupsed airfoil where the two bound vortex sheets share the same tangent. This inconsistency of the flow direction at the trailing edge creates an ambiguity in deciding the angle of the streamline emanating from the trailing edge, especially for unsteady cases. In this study, we propose that the angle of the trailing-edge vortex sheet should be decided in a momentum conservation sense. As a result, the angle of the forming vortex sheet can be decided by γ2 sin ∆θ2 + γ1 sin ∆θ1 = 0 with ∆θ1 · ∆θ2 ≥ 0; (2) and ∆θ1 + ∆θ2 = ∆θ0; (3) where ∆θ0 is the finite angle of the trailing edge. The unsteady Kutta condition for an airfoil is validated through the simulation of a NACA 0012 airfoil with a combined pitching and heaving motion, the experi- ment of which was conducted by Schouveiler et al. [6]. The chord length, c, and the towing speed, U, are 0.1 m and 0.4 m/s, respectively. This cor- responds to a Reynolds number of 4 × 104. The phase difference angle, Figure 3: Comparison between flow visualization and , between the pitching and heaving motions is set to 90◦ in all circum- simulation. The NACA 0012 airfoil is towed from right ◦ stances. Fig. 3 shows the comparison of the flow structures between this to left with St = 0:45 and αmax = 30 . The flow simulation and the flow visualization image of the original experiment visualization image is adapted from Schouveiler et al. ◦ [6]. for a sample case with St = 0:45 and αmax = 30 . The good matching of the wake patterns provides qualitative support for the unsteady Kutta condition for a finite-angle trailing edge. CONCLUSIONS The unsteady Kutta condition is extended from the classical Kutta condition to study the vortex sheet formation at a sharp edge. Applying the unsteady Kutta condition to an airfoil with finite-angle trailing edge is not straightforward, as the angle of the forming vortex sheet is ambiguous. This study proposes to determine this angle based on the momentum conservation in the direction normal to the forming vortex sheet. The good agreement between simulation and experiment confirms the validity of the proposed unsteady Kutta condition. References [1] S.H. Chen and C. M. Ho. Near wake of an unsteady symmetric airfoil. Journal of Fluids and Structures, 1:151–164, 1987. [2] C.P. Ellington. The aerodynamics of hovering insect flight. iv. aerodynamic mechanisms. Phil. Trans. R. Soc. Lond. B, 305:79–113, 1984. [3] M-K. Huang and C-Y. Chow. Trapping of a free vortex by joukowski airfoils. AIAA, 20(3), 1981. [4] D.R. Poling and D.P. Telionis. The trailing edge of a pitching airfoil at high reduced frequency. Journal of Fluid Engineering, 109:410–414, 1987. [5] P.G. Saffman and J.S. Sheffield. Flow over a wing with an attached free vortex. Studies in Applied Mathematics, 57:107–117, 1977. [6] L. Schouveiler, F. S. Hover, and M. S. Triantafyllou. Performance of flapping foil propulsion. Journal of Fluids and Structures, 20(7):949959, 2005. [7] X. Xia and K. Mohseni. Lift evaluation of a two-dimensional pitching flat plate. Physics of Fluids, 25(9):091901, 2013. [8] X. Xia and K. Mohseni. A flat plate with unsteady motion: Effect of angle of attack on vortex shedding. AIAA paper 2014-0746, National Harbor, MD, Jan 13-17 2014..
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