Figure 1. Schematic diagram of the self-propelled airfoil with pitching motion (a) Near the flat ground (b) Near the wavy ground.

Figure 1. Schematic diagram of the self-propelled airfoil with pitching motion (a) Near the flat ground (b) Near the wavy ground.

Figure 2. Schematic of computational domain with an immersed boundary.

Figure 2. Schematic of computational domain with an immersed boundary.

Figure 3. Lift and drag coefficients for different St as h0=c.

Figure 3. Lift and drag coefficients for different St as h0=c.

Figure 4. Instantaneous vorticity contours at h0=c, St=0.1 (a) pitching downward (b) pitching upward.

Figure 4. Instantaneous vorticity contours at h0=c, St=0.1 (a) pitching downward (b) pitching upward.

Figure 5. Computational meshes (a) full computational domain (b) zoomed view of the meshes around the airfoil.

Figure 5. Computational meshes (a) full computational domain (b) zoomed view of the meshes around the airfoil.

Figure 6. Motion displacements of the airfoil pitching with different h0. (a) T=1 s  (b) T=2 s (c) T=3 s (d) T=4 s (e) T=5 s.

Figure 6. Motion displacements of the airfoil pitching with different h0. (a) T=1 s  (b) T=2 s (c) T=3 s (d) T=4 s (e) T=5 s.

Figure 7. Comparison of mean drag coefficient, mean lift coefficient and lift-to-drag ratio for different pitching periods (a) C¯l (b) C¯d (c) C¯l/C¯d.

Figure 7. Comparison of mean drag coefficient, mean lift coefficient and lift-to-drag ratio for different pitching periods (a) C¯l (b) C¯d (c) C¯l/C¯d.

Figure 8. Displacement of the airfoil over flat ground and wavy ground with wavelength parameter λ=2c and λ=c under different pitch periods (a) flat ground (b) λ=2c  (c) λ=c.

Figure 8. Displacement of the airfoil over flat ground and wavy ground with wavelength parameter λ=2c and λ=c under different pitch periods (a) flat ground (b) λ=2c  (c) λ=c.

Figure 9. Average lift and drag coefficients and average lift drag ratio of airfoil under different pitching periods.(a) C¯l (b) C¯d (c) C¯l/C¯d.

Figure 9. Average lift and drag coefficients and average lift drag ratio of airfoil under different pitching periods.(a) C¯l (b) C¯d (c) C¯l/C¯d.

Figure 10. Time histories of Cl, Cd corresponding to different initial heights (a) Cl at T=1 s (b) Cd at T=1 s (c) Cl at T=2 s (d) Cd at T=2 s  (e) Cl at T=3 s (f) Cd at T=3 s (g) Cl at T=4 s (h) Cd at T=4 s  (i) Cl at T=5 s (j) Cd at T=5 s.

Figure 10. Time histories of Cl, Cd corresponding to different initial heights (a) Cl at T=1 s (b) Cd at T=1 s (c) Cl at T=2 s (d) Cd at T=2 s  (e) Cl at T=3 s (f) Cd at T=3 s (g) Cl at T=4 s (h) Cd at T=4 s  (i) Cl at T=5 s (j) Cd at T=5 s.

Figure 11. Vorticity contours as pitching period T=1 s (a) h0=0.5c (b) h0=1.0c  (c) h0=2.0c (d) Without ground.

Figure 11. Vorticity contours as pitching period T=1 s (a) h0=0.5c (b) h0=1.0c  (c) h0=2.0c (d) Without ground.

Figure 12. Pressure distribution around the airfoil as the pitching period T=1 s (a) h0=0.5c (b) h0=1.0c.

Figure 12. Pressure distribution around the airfoil as the pitching period T=1 s (a) h0=0.5c (b) h0=1.0c.

Figure 13. Vorticity contour as the pitching period T=5s (a) h0=0.5c (b) h0=1.0c.

Figure 13. Vorticity contour as the pitching period T=5s (a) h0=0.5c (b) h0=1.0c.

Figure 14. Time histories of lift and drag coefficients of the airfoil in one period under different pitching periods (a) Cl as λ=2c (b)  Cd as λ=2c (c) Cl as λ=c (d) Cd as λ=c.

Figure 14. Time histories of lift and drag coefficients of the airfoil in one period under different pitching periods (a) Cl as λ=2c (b)  Cd as λ=2c (c) Cl as λ=c (d) Cd as λ=c.

Figure 15. Vorticity contours in a pitching period (a) T=1 s, λ=2c (b) T=2 s, λ=2c (c) T=1 s, λ=c (d) T=2 s, λ=c.

Figure 15. Vorticity contours in a pitching period (a) T=1 s, λ=2c (b) T=2 s, λ=2c (c) T=1 s, λ=c (d) T=2 s, λ=c.

Figure 16. Pressure distribution (a) T=1 s, λ=2c  (b) T=2 s, λ=2c (c) T=1 s, λ=c (d) T=2 s, λ=c.

Figure 16. Pressure distribution (a) T=1 s, λ=2c  (b) T=2 s, λ=2c (c) T=1 s, λ=c (d) T=2 s, λ=c.

Table 1. Mesh and time step sensitivity tests.

Table 1. Mesh and time step sensitivity tests.

Number of GridsΔx/cΔt(s)C¯dC¯l50,0000.020.010.29230.6925150,0000.010.0050.22240.3580250,0000.0050.0050.22870.3473250,0000.0050.00250.22750.3466