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First published online December 1, 2006
Journal of Experimental Biology 209, 4841-4857 (2006)
Published by The Company of Biologists 2006
doi: 10.1242/jeb.02526

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Simulations of optimized anguilliform swimming

Stefan Kern and Petros Koumoutsakos^*

Institute of Computational Science, ETH Zurich, CH-8092, Switzerland

View larger version (51K): [in this window] [in a new window]   Fig. 1. Illustration of the analytical description of the three-dimensional body. (A) Length of the two half axes w(s) and h(s) of the ellipsoid cross sections according to Eqn 1 and Eqn 2; (B) slice through the deforming grid used to solve the fluid dynamics part of the fluid-body interaction. The computational domain consists of circular cylinder of radius R and length L+Ldstr capped with a hemisphere of radius R.

View larger version (26K): [in this window] [in a new window]   Fig. 2. Development of the longitudinal velocity U|| and the lateral velocity U as the undulating body accelerates from rest for the four investigated swimming cases. (A) U|| (solid lines) and U (broken lines) of the 2D case with the reference motion pattern (cyan) and of the 3D case with reference motion pattern (black). (B) U|| (solid lines) and U (broken lines) of the 3D case with motion pattern optimized for efficiency (blue) and the 3D case with motion pattern optimized for maximal swimming velocity (red).

View larger version (42K): [in this window] [in a new window]   Fig. 3. Non-dimensional net force and moment coefficients for the 2D (cyan) and 3D (black) reference case. (A) Longitudinal net force coefficient C||, (B) lateral net force coefficient C, (C) net yaw moment coefficient CM.

View larger version (62K): [in this window] [in a new window]   Fig. 4. Velocity field and contours of vorticity normal to the image plane (z) for (A) the 2D case and (B) the 3D case with the reference motion pattern (Eqn 5) for an entire swimming cycle after the body has reached its asymptotic mean swimming speed.

View larger version (25K): [in this window] [in a new window]   Fig. 5. Development of the fitness function values f and fU and the parameters K1, K2, K3, K4 and tail defining a realization of the motion pattern s(t)=K(s)·sin(2[t/T-(s)]) in the course of optimizing swimming efficiency (A,B) and optimizing swimming velocity (C,D).

View larger version (22K): [in this window] [in a new window]   Fig. 6. Amplitude envelopes of the body mid-line including the feedback from the fluid forces of the reference motion pattern (black), the efficient swimming pattern (blue), and the fast swimming pattern (red).

View larger version (56K): [in this window] [in a new window]   Fig. 7. Non-dimensional net force and moment coefficients for the 3D cases with optimized swimming efficiency (blue) and maximized swimming velocity (red). (A) Longitudinal net force coefficient C||, (B) lateral net force coefficient C , (C) net yaw moment coefficient CM.

View larger version (53K): [in this window] [in a new window]   Fig. 8. Velocity field and vorticity normal to the image plane (z) for (A) the 3D case with optimized swimming efficiency and (B) the 3D case with maximized swimming velocity for an entire swimming cycle.

View larger version (35K): [in this window] [in a new window]   Fig. 9. Snapshots of isosurfaces of vorticity magnitude ||||2 for (A) the efficient swimming motion, and (B) the fast swimming motion.

View larger version (50K): [in this window] [in a new window]   Fig. 10. Formation of the secondary flow structures visualized by plotting isosurfaces of axial vorticity |x|1 for (A) the efficient swimming motion, and (B) the fast swimming motion.

View larger version (82K): [in this window] [in a new window]   Fig. 11. Flow structure at the tail visualized by isosurfaces of vorticity magnitude ||||2 colored by contours of vorticity in lateral direction y for (A) the efficient swimming motion, and (B) the fast swimming motion.

View larger version (41K): [in this window] [in a new window]   Fig. 12. Unsteady fluid forces acting on the fish surface for the five segments (1-5) from head to tail. The forces on the left half are colored in blue, the ones on the right half in red, and the black line plots the cumulated force of left and right part. The total force in swimming direction C|| is plotted in green as reference. Positive values relate to thrust, negative to drag. Results for the efficient swimming case are plotted in (A) and for the fast swimming motion in (B).

View larger version (10K): [in this window] [in a new window]   Fig. A1. Coordinate systems used to describe the undulating centreline. The description by prescribed curvature s(s,t) is transformed to a description as xs'(s,t) and ys'(s,t) in the system (0',x',y') where the center of mass of the deforming body remains in (x',y')=0 and the total rotational impulse of the deforming body is 0. The fluid-body interaction is computed in the inertial system (0,x,y,z) reducing the forces and torque of the fluid to the center of mass and moving the local coordinate system (0',x',y') in order to satisfy the dynamic equations.

View larger version (14K): [in this window] [in a new window]   Fig. A2. Falling sphere benchmark. (A) Falling velocity of the sphere reaching an asymptotic value of U=1.006. (B) Contours of vorticity magnitude of the flow at time t=20.

View larger version (28K): [in this window] [in a new window]   Fig. A3. Convergence study in space and time for the reference motion pattern defined in Eqn 5. (A) Relative error in F|| vs grid size; (B) relative error in F|| vs time step size; (C) longitudinal force F|| and (D) swimming velocity U for the grids in Table 1 with 70x103 (blue), 200x103 (green), 300x103 (red) and 420x103 cells (black).