Aerodynamics of Insect Flight Effects of wind gusts on a rigid flapping NACA 0012 airfoil at Re = 3000 Marcus Lundberg [email protected] Degree Project in Engineering Physics, First Cycle, SA104X at KTH Mechanics Supervisors: Luca Brandt & Walter Fornari Examiner: M˚artenOlsson Stockholm, Sweden 2015 Abstract Insects and small flyers operate at Reynolds numbers ranging from ap- proximately 10 − 105, where viscous forces are important. Due to their small size and weight, they are sensitive to small changes in the free stream during flight, such as wind gusts. First, the aerodynamics of flapping flight is briefly explained. Then the lift, drag and power us- age for a flapping NACA 0012 airfoil is simulated in ANSYS Fluent for different oncoming wind directions. The aim of the report is to un- derstand how the pitching amplitude, the flapping frequency and the plunging amplitude can be adjusted to compensate for oncoming wind gusts. The simulation is modelled as quasi-static since the time-scale of the flapping wings of the insect is much shorter than the time-scale of the wind gusts. 2 Contents 1 Introduction 4 1.1 Objectives . 4 2 Theory of flapping wing aerodynamics 5 2.1 Definitions . 5 2.1.1 Wing nomenclature . 5 2.1.2 Reynolds number, Re ...................... 5 2.1.3 Reduced frequency, k ...................... 5 2.1.4 Strouhal number, St ....................... 6 2.2 Aerodynamic forces . 6 2.2.1 Lift . 6 2.2.2 Drag . 7 2.2.3 Thrust . 7 2.2.4 Leading edge vortices (LEVs) . 8 2.2.5 Dynamic stall . 8 2.3 Wing model . 8 2.3.1 Equations for wing motions in 2D . 8 2.4 Power considerations . 9 3 Numerical simulations 10 3.1 Governing equations and turbulence model . 10 3.2 Geometry and mesh parameters . 10 3.2.1 Grid characteristics . 11 3.2.2 Validation . 12 3.3 Simulation model . 13 3.3.1 Boundary conditions . 15 3.3.2 Calculation of power usage . 16 3.3.3 Moving airfoil setup . 16 3.4 Results . 17 3.4.1 Reference simulation . 17 3.4.2 Varying the reduced frequency, k . 19 3.4.3 Varying the plunging amplitude, ha . 21 3.4.4 Varying the pitching amplitude, αa . 23 3.5 Conclusion . 25 3.5.1 Wind from below, β > 0◦ .................... 25 3.5.2 Wind from above, β < 0◦ .................... 25 3.5.3 Final remarks . 25 4 Appendix A 27 4.1 User Defined Functions . 27 3 1 Introduction The flight of birds, bats and insects has intrigued humans for many centuries. There exists almost a million species of flying insects and over 13000 warm-blooded vertebrate species have the ability to roam the sky. Although great progress has been made in aeronautical technology over the past 100 years, birds', bats' and insects' ability to manoeuvre a body efficiently through air is still very impressive. Studying their flight mechanisms is important in the development of Micro Air Vehicles with flapping wings. By studying the flapping wing flight in animals, we can improve the designs and build smaller, more efficient and more manoeuvrable MAVs. However, a consequence of the small size of insects is that they are very sensitive to disturbances in the environment, such as wind gusts. 1.1 Objectives It is the aim of this report to first explain the basics of flapping wing flight and then study the effects of wind gusts on lift, drag and thrust on insect-sized bodies, and to give a recommendation for how the wings should be aligned to counteract the gust and have optimal performance. The frequencies of typical wind gusts, around O(1) Hz, are low compared to the flapping frequencies of the insects, which are O(101) Hz to O(102) Hz. Therefore, in the time-scale of the flapping wings, the wind gusts can often be modelled as quasi-steady [1]. 4 2 Theory of flapping wing aerodynamics In this section a short theoretical background on flapping wing aerodynamics at low Reynolds numbers is introduced. 2.1 Definitions 2.1.1 Wing nomenclature Figure (1) shows the different parts of an asymmetric airfoil. Figure 1: The terminology for lift producing airfoils. The chord line is the imaginary straight line between the leading edge and the trailing edge of the airfoil. The camber line is defined as the curve that lies halfway between the upper and lower surfaces of the airfoil. The chord line and camber line coincide for symmetric airfoils, such as the NACA 0012 airfoil, which will be used in the simulations. The maximum distance between the camber line and the chord line is the maximum camber, which is a measure of the curvature of the airfoil. 2.1.2 Reynolds number, Re The Reynolds number is defined as the ratio between the inertial and viscous forces. It is defined as U L Re = ref ref (1) ν where Uref is a reference velocity, Lref is a reference length and ν is the kinematic viscosity of the fluid. In flapping wing flight, the reference length is generally taken to be either a mean chord length or the wing length R. The mean chord length is defined as the average chord length in the spanwise direction. In forward flight, the reference velocity Uref is the forward velocity [1]. 2.1.3 Reduced frequency, k The ratio between the forward velocity and the flapping velocity is another impor- tant dimensionless parameter used to analyse the aerodynamic performance of a natural flyer [2]. It is expressed using the reduced frequency k, 5 πfc k = ; (2) U where f is the flapping frequency, c is the mean chord length and U is the forward velocity U. 2.1.4 Strouhal number, St The Strouhal number St is a dimensionless parameter that describes the wing kine- matics for flying animals [2]. It is well known to govern the vortex dynamics and shedding behaviour for airfoils undergoing pitching and plunging motions. It is defined as the stroke frequency f times a reference length Lref = 2ha, where ha is the plunging amplitude, divided by the forward speed U. 2fh 2kh St = a = a ; (3) U πc where St is expressed in the reduced frequency using equation (2). The Strouhal number for natural flyers and swimmers in cruising conditions has been found to be in the range 0:2 < St < 0:4 [2]. 2.2 Aerodynamic forces 2.2.1 Lift When an airfoil moves forward through air, the air is deflected and exerts an aero- dynamic force on the airfoil. Lift is defined as the component of this force perpen- dicular to the direction of the oncoming flow, as shown in Figure (2). The airfoil exerts a downward force on the air when it flows past. According to Newton's third law, the air exerts an equal but opposite force on the airfoil, the lift. The direction of the air flow changes as it passes over the airfoil and curves downward. This change of direction in the air results in a reaction force opposite to the directional change. Reynolds numbers for insects ranges between approximately 10 to 105 [3]. Elling- ton [4] observed, in a comprehensive investigation, that classical, steady state the- ories predicted insufficient forces in flapping flight, to explain the characteristics of insect and bird flight [5]. It has been shown in experiments that the reduced frequency increases as the mass and size of the flying animal decreases, which in- dicates that unsteady effects are important in lift and thrust generation for small flyers [5]. Figure 2: Directions of forces. Lift L, drag D, total aerodynamic force R and angle of attack α 6 2.2.2 Drag Drag is a mechanical force generated by the contact and interaction of a solid body and a fluid. It is a non-conservative force which depends on the speed of the body relative to the speed of the fluid. If the relative speed is zero, there is no drag. The drag force is the component of the aerodynamic force that is opposed to the motion of the object, as shown in Figure (2). It is a type of friction and refers to forces acting in the opposite direction of a body's motion through a fluid. A drag force can exist between a body and a fluid or between two fluid layers. It acts to reduce the relative speed between the body and the fluid [6]. 2.2.3 Thrust Generally, bird and insect flight can be separated into two types; unpowered (gliding and soaring) and powered (flapping wings). The thrust is the component of the total aerodynamic force parallel to the direction of the flying animal and depends on the power output of its flight muscles. Consider an airfoil with a sinusoidal plunging motion, with no pitching, flying forward through a stationary fluid. During the motion of the airfoil, the effective angle of attack changes, as can be seen in Figure (3). Positive lift is generated during the downstroke since the airfoil is exposed to a flow with positive angle of attack, and vice versa on the upstroke. Figure 3: The effective angle of attack varies during the plunging cycle [1]. Cross-sections of the wings further from the body will move faster than parts closer to the body during upstrokes and downstrokes. Consequently, the flow di- rection hitting the wing during forward flight will vary along the wing span. Bird wings are flexible and can twist to make sure the angle of attack is correct along the entire wing span. Since the vertical speed of the wings is highest at the tips, they are angled more forward during down-stroke than the inner parts of the wing.