SENSITIVITY ANALYSIS OF THE FACTORS AFFECTING FORCE GENERATION BY WING FLAPPING MOTION

Transcription

1 Proceedings of the ASME 2013 International Mechanical Engineering Congress and Exposition IMECE2013 November 15-21, 2013, San Diego, California, USA IMECE SENSITIVITY ANALYSIS OF THE FACTORS AFFECTING FORCE GENERATION BY WING FLAPPING MOTION Alok A. Rege Doctoral Student, Mechanical & Aerospace Engineering, University of Texas at Arlington Arlington, Texas, Brian H. Dennis Associate Professor, Mechanical & Aerospace Engineering, University of Texas at Arlington Arlington, Texas, Kamesh Subbarao Associate Professor, Mechanical & Aerospace Engineering, University of Texas at Arlington Arlington, Texas, ABSTRACT Insect flight comes with lot of intricacies that cannot be explained by conventional aerodynamics. Insects rely on a peculiar high frequency wing flapping mechanism to produce the aerodynamic forces required for sustainable flight. Broad study of this mechanism for producing forces is imperative to attain a reasonably accurate representation of these forces. In this research, sensitivity analysis is performed on the factors governing the aerodynamic force production due to flapping motion of a two-dimensional wing section of a Micro Air Vehicle (MAV). Published results obtained on a wing section of an MAV model by the authors in their previous work are used for preliminary review. The flapping path parameters are nondimensionalized and the moving mesh problem is solved in a numerical flow solver. A thorough sensitivity analysis is done to realize the effects of the flapping wing Reynolds number, Strouhal number, and the absolute angle of attack on the force generation. INTRODUCTION The aerodynamics involved in the insect flapping mechanism is highly unsteady. The amazing maneuverability obtained by the insects using high frequency flapping motion belies the classical aerodynamics theory and calls for a new approach to study this highly nonlinear aerodynamics. Research is on to find new ways to realize the extraordinary flight capabilities of these insects and engineer a flapping wing Micro Air Vehicle (MAV) with wide range of applications, from detections missions to communication network and area surveillance. Many researchers have focused their research on understanding factors involved in the force production in insect flight. Ellington in [1] talked about the uncertainties involved around defining the lift and induce drag coefficients for unsteady wing motions. Sun et al. in [2] established the lift and power dependencies on the flapping frequencies, stroke amplitude and wing length by solving the Navier-Stokes equations numerically. Jane Wang in [3] found sustainability of insect flight by two dimensional hovering motion by a mechanism involving formation of a jet of counter-rotating vortices. Lee et al. in [4] found that lift is mainly generated during the downstroke of a figure-8 2-D flapping motion whereas thrust is abruptly generated at the end of upstroke. Ramamurti et al. in [5] observed that rotational mechanism in flapping is important and that the combined translational and rotational mechanisms are essential for accurate description of unsteady aerodynamics of flapping wings as was suggested by Dickinson et al. in [6]. Tang et al. in [7] Navier-Stokes equations for fluid flow around a hovering elliptic airfoil to investigate the effects of Reynolds number, reduced frequency and flapping kinematics on the flow structure and aerodynamics. Platzer et al. in [8] found that thrust and lift are strongly dependent on flapping frequency, amplitude and Reynolds number and that thrust is a function of Strouhal number. This paper deals with conducting a sensitivity analysis of the factors that influence force generation in a wing flapping mechanism. Preliminary review is done to identify these factors. Address all correspondence to this author. 1 Copyright 2013 by ASME

2 Nondimensional parameters are defined that incorporated these factors and a full factorial design of experiments is conducted and response surface methodology is used to approximate the expressions for average force coefficients. These expressions are evaluated over the region bounded by the parameter limits to observe the relationships of parameters and the time averaged force coefficients. This study is a subsequent step of the continuing effort of the authors to determine a new set of aerodynamic expressions to be used in a nonlinear flight dynamics model of a flapping wing MAV [9 11]. METHODOLOGY The following sections give a step by step procedure adopted to perform the sensitivity analysis. Preliminary Review Previous work done by the authors [10, 11] dealt with determining the variables that affect the force generation by a flapping wing motion. The research considered the unsteady motion twodimensional wing section of an MAV model through an incompressible viscous fluid. In their study on straight, elliptical and figure-8 flapping trajectories the authors detected the presence of two counter-rotating vortices created by continuous flapping at very low free stream velocities. These vortices created a downwash on the wing thereby reducing the net lift force. Moreover, the location of these vortices changed for each trajectory and so did the values of lift coefficients time averaged over a flap cycle (C Lavg ). A parametric study was performed on straight and figure-8 flapping paths using computational techniques at different forward flight conditions. It was found that the C Lavg values were higher at the start of flap, but gradually converged to a lower value after a period of time. Increasing the free stream inlet velocity yielded greater lift coefficient values. It was also noted that the time history of flapping, especially at low velocities, is an important factor that affects the force production and needs to be considered while developing the new aerodynamic model for force representation. Parameters for Sensitivity Analysis Sensitivity analysis is performed on the nondimensional parameters that include the factors influencing the force production. Observations from the previous study suggested that the average force coefficient values depend on the free stream velocity as well as the velocity generated due to wing flapping. These velocities are incorporated by defining two non dimensional parameters, the free stream Reynolds number and the local Reynolds number. The free stream Reynolds number is given by where ρ is the density of air, V is the free stream air velocity, c is the MAV wing chord which is taken as the reference length, and µ is the dynamic viscosity of air. The local Reynolds number is given by Re l = ρv lc µ, (2) where V l is the local velocity due to wing flapping and is given by V l = ωl (3) where ω is the wing flapping frequency and L is the wing semi-span. The Strouhal number is defined as the ratio of local and free stream Reynolds numbers. It is given by St = Re l Re f = ωl V (4) This parameter gives the direct relation between the local and the free stream velocities. The absolute angle of attack α is defined as the angle of the free stream velocity vector with respect to the body x-axis. Straight path is chosen as the flapping trajectory for the sensitivity analysis. In Fig. 1(a), the circle and the diamond show the start and the direction of the wing stroke respectively. Figure 1(b) shows the instantaneous wing positions during the two half strokes;the downstroke or the forward flapping motion of the wing, and the upstroke or the backward flapping motion. The circular head on the wing depicts the wing leading edge. The orientation of the trajectory is fixed parallel to the body x-axis. As shown in Fig. 2, each wing chord along the wing span will traverse a certain maximum path distance based on the geometry of the wing. To accommodate this geometric constraint of the wing, a geometric function G is defined as one of the parameters and is given by G = c L, (5) The parameters St, Re l, G, along with α, represent the independent non-dimensional parameters that influence force production for a given airfoil geometry and trajectory. The dependent horizontal and vertical forces are also non-dimensionalized as shown below. Re f = ρv c µ, (1) C x = F x.5ρ(ωl) 2 c, C F y y =.5ρ(ωL) 2 c (6) 2 Copyright 2013 by ASME

3 (a) Flapping direction FIGURE 2. WING VARIATION OF PATH DISTANCE FOR AN ELLIPTIC (b) Instantaneous wing flapping positions FIGURE 1. Straight flapping trajectory The independent parameters defined above are used in the design of experiments (DOE) to perform the force coefficient sensitivity analysis. Design of Experiments It is understood that the force generation by wing flapping depends on a number of factors. To sense the combined or individual effect of these factors a design of experiments is carried out. A full factorial design method with two intervals per factor is adopted to conduct response surface methodology. The local Reynolds number, Strouhal number and the absolute angle of attack are chosen as the parameters for the design analysis, while the geometric function G is fixed at a constant value of 0.4 for the entire set of experiments. Since the number of parameters involved is 3, the full factorial design would be 3 k = = 27. So a total 27 simulation runs are needed to conduct a full factorial design for the given parameters. For the simulations, a pressure-based finite volume method is used to solve the unsteady incompressible Navier-Stokes equations on an unstruc- tured moving mesh using Ansys-Fluent as described in [11]. Pressure-velocity coupled scheme along with standard 2nd-order upwinding differencing method is chosen to run all the simulations. A moving mesh scheme is used to maintain a boundary conforming grid. The appropriate smoothing and remeshing parameters are employed. User-defined functions are used to initialize the simulation runs and to control the motion of the airfoil as a function of time. The grid and the boundary conditions are parametrized in terms of the chord length c, the flapping frequency ω, the wing semispan L, the free stream velocity V and the absolute angle of attack α. The wing section is set at a no-slip wall condition. The simulations are run for 5 flapping cycles with 1000 time steps per cycle. The unsteady forces produced by the wing section are computed by integrating the time-variant static pressure over the wing section to produce instantaneous force coefficients. The reference values used for calculating the corresponding horizontal and vertical force coefficients C X and C Y respectively are parametrized similar to the boundary conditions. These force coefficents are then further integrated over the cycle to produce the cycle averaged force coefficients. The values of some of the parameters used for the simulation are given in Tab.1. For the full factorial design, the maximum and minimum values of the local Reynolds number and Strouhal number are calculated based on the chosen values. The values used for all the three design of experiment parameters are given in Tab.2. A quadratic polynomial graduating function is used as a re- 3 Copyright 2013 by ASME

4 TABLE 1. Parameter PARAMETER VALUES USED FOR SIMULATION. Wing chord length, c Wing semi-span, L Wing flapping frequency, ω Value m m 100 Hz Density, ρ kg/m 3 Viscosity, µ TABLE E-5 kg/m-s PARAMETER VALUES FOR DOE. Parameter Re l S t α (deg) maximum mean mimimum sponse function to approximate both the average horizontal and vertical force coefficient values. The method of least squares minimization is used to estimate the coefficients of the approximation function that gives the best fit through the data. The expression for the vertical force coefficient based on the DOE is given by C Yavg = Re l St α Re 2 l 0.654St α Re l α 0.119Stα. (7) The expression for the horizontal force coefficient based on the DOE is given by C Xavg = St + 179St 2. (8) We can see from the above expressions that while the average vertical force coefficient is dependent on all the three parameters, the average horizontal force coefficient is a function of Strouhal number only. It is to be noted that these expressions are valid only for the first 5 flapping cycles. and surface plots of these coefficients are observed to study the relationships between the parameters and the coefficients. Figures 3-4 shows that C Xavg value entirely depends on the Strouhal number for a fixed α. Similarly, Fig. 6 shows C Yavg value is not affected by Reynolds number but only for lower values of Strouhal number. As the Strouhal number approaches the largest value tested, the Reynolds number starts to influence C Yavg. Figure 3 and Fig. 5 give the surface plots showing highest values of the average force coefficients. Figure 7 and Fig. 8 shows C Xavg and C Yavg values depend more on the Reynolds number than the angle of attack. Figure 9 and Fig. 10 show a strong quadratic response behavior of the C Xavg and C Yavg values with respect to the Strouhal number, albeit the C Xavg -St curve has a positive slope while the C Yavg -St curve has a negative slope. A positive value of C Yavg would indicate a net lifting force. So according to Fig. fig:stresurfcy and Fig. 10, a lifting force will only be produced if St > 14. Once this critical value of St is reached, the lift force increases slightly with increased Re l. Note that one may increase St by either increasing the frequency or the length scale or by decreasing the free stream speed. Therefore, a MAV based on the flapping motion given here would require an increasing in flapping frequency to maintain level flight at increased forward speeds. Since the lift force, as defined in this paper, is a function of frequency squared, the required change in frequency may be slight, but still necessary to maintain steady level flight conditions. For C Xavg, a positive value would indicate a thrust force. From Fig. 9, the response surface indicates a thrust force would be produced with St > 14 and is not a function of Re l. Again, maintaining thrust force with increased freestream speed would require an increase in flapping frequency to hold the required value of St. Figure 11 shows that C Xavg -α trend is same for different Strouhal numbers but the magnitude is different. However one can see from Fig. 12 that different Strouhal numbers yield entirely different slopes of the C Yavg -α curve. We can see from Fig. 13 that C Xavg value decreases at lower values of Reynolds number but increases again after Re l value crosses 14. Figure 14 shows that C Xavg increases steeply as Reynolds number increases. The response surface predicts optimal values of the nondimensional parameters to achieve maximum C Yavg and minimum C Xavg. Future work would involve running these points with the flow solver to gage the accuracy of the response surface and then update it, if necessary. RESULTS The vertical and horizontal force coefficients are calculated using the approximate expressions given by eqn. 7 and eqn. 8 respectively for the parameter limit range given by Tab.2. Contour CONCLUSION A sensitivity analysis of the factors affecting the aerodynamic force production due to flapping motion of a 2-D wing section MAV model. The dimensional variables influencing the 4 Copyright 2013 by ASME

5 FIGURE 3. Re l -St SURFACE PLOT FOR HORIZONTAL FORCE COEFFICIENT C Xavg AT α = 0 deg FIGURE 5. Re l -St SURFACE PLOT FOR VERTICAL FORCE CO- EFFICIENT C Yavg AT α = 0 deg FIGURE 4. Re l -St CONTOUR PLOT FOR HORIZONTAL FORCE COEFFICIENT C Xavg AT α = 0 deg FIGURE 6. Re l -St CONTOUR PLOT FOR VERTICAL FORCE CO- EFFICIENT C Yavg AT α = 0 deg force generation were identified. Nondimensionalized parameters like the local and free stream Reynolds numbers, Strouhal number and Geometric Function were defined that incorporated the influencing variables. All these parameters along with the absolute angle of attack were used in design of experiments to perform the sensitivity analysis. Geometric function was set at a constant value for the entire experiment. A three-level full factorial design method was adopted with three chosen parameters to conduct response surface methodology. Straight flapping trajectory was used for the experiments. Unsteady incompressible Navier-Stokes equations were solved on an unstructured dynamic mesh using CFD solver with user defined functions. The mesh, boundary conditions, and the reference values required to calculated the force coefficients were parameterized in terms of the chord length, semi-span, free stream velocity and the angle of attack. The simulations were run for 5 5 Copyright 2013 by ASME

6 FIGURE 7. Re l -α CONTOUR PLOT FOR HORIZONTAL FORCE COEFFICIENT C Yavg AT St = FIGURE 9. PLOT FOR C Xavg VERSUS St FIGURE 8. Re l -α CONTOUR PLOT FOR VERTICAL FORCE CO- EFFICIENT C Yavg AT St = FIGURE 10. PLOT FOR C Yavg VERSUS St flapping cycles and the average values of force coefficients C Xavg and C Yavg were calculated. The average force values for all the simulation runs were noted and a quadratic polynomial graduating function was used to approximate an expression for C Yavg and C Xavg in terms of the three parameters given by eqn.7 and eqn.8 respectively. It was found that while the average vertical force coefficient is a function of all the three parameters, the average horizontal force coefficient is a strong function of the Strouhal number only. This finding was corroborated by the contour plots of the force coefficients and the three parameters. These are preliminary results and are valid only for the fifth flapping cycle and specific to the chosen Geometric Function value of 0.4. The future work will aimed at including all the parameters in the design of experiments to add more data points and come up with a function based on new techniques like the neural networks that would fit the data more accurately. 6 Copyright 2013 by ASME

7 FIGURE 11. PLOT FOR C Xavg VERSUS α FIGURE 13. PLOT FOR C Xavg VERSUS Re l FIGURE 12. PLOT FOR C Yavg VERSUS α FIGURE 14. PLOT FOR C Yavg VERSUS Re l REFERENCES [1] Ellington, C., The aerodynamics of hovering insect flight. V. A vortex theory. Philosophical Transactions of the Royal Society of London. Series B: Biological Sciences, 305(1122), pp [2] Sun, M., and Gang, D., Lift and power requirements of hovering insect flight. Acta Mechanica Sinica, 19(5). [3] Jane Wang, Z., Two dimensional mechanism for insect hovering.. Physical review letters, 85(10), Sept., pp [4] Lee, J.-S., Kim, J.-H., and Kim, C., Numerical Study on the Unsteady-Force-Generation Mechanism of Insect Flapping Motion. AIAA Journal, 46(7), July, pp [5] Ramamurti, R., and Sandberg, W. C., A three-dimensional computational study of the aerodynamic mechanisms of insect flight.. The Journal of experimental biology, 205(Pt 10), May, pp [6] Dickinson, M. H., F.-O., L., and Sane, S., Wing Rotation and the Aerodynamic Basis of Insect Flight. Sci- 7 Copyright 2013 by ASME

8 ence, 284(5422), June, pp [7] Tang, J., Viieru, D., and Shyy, W., Effects of Reynolds Number and Flapping Kinematics on Hovering Aerodynamics. AIAA Journal, 46(4), Apr., pp [8] Platzer, M. F., Jones, K. D., Young, J., and S. Lai, J. C., Flapping Wing Aerodynamics: Progress and Challenges. AIAA Journal, 46(9), Sept., pp [9] Rege, A. A., Subbarao, K., and Dennis, B., Open Loop Simulation of the Nonlinear Dynamics of a Flapping Wing Micro Air Vehicle. In AIAA Guidance, Navigation, and Control Conference, pp [10] Rege, A., CFD Based Aerodynamic Modeling to Study Flight Dynamics of a Flapping Wing Micro Air Vehicle. Master s thesis, University of Texas at Arlington, May. [11] Rege, A. A., Dennis, B. H., and Subbarao, K., Parametric Study on Wing Flapping Path of a Micro Air Vehicle Using Computational Techniques. In 51st AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, pp Copyright 2013 by ASME