Drag-thrust decomposition and optimality in swimming N. A. Patankar*, A. A. Shirgaonkar, O. M....
1
Drag-thrust decomposition and optimality in swimming N. A. Patankar *, A. A. Shirgaonkar, O. M. Curet, R. Bale, M. Hao, A. Bhalla, M. A. MacIver, Department of Mechanical Engineering, Northwestern University Overview This poster highlights the following key results: 1. A novel constraint-based formulation to simulate self- propulsion has been developed. The numerical approach is used to obtain the results below. 2. A drag-thrust decomposition that is different from that proposed by Lighthill is found in the propulsion by undulatory ribbon fins of gymnotiform and balistiform swimmers. 3. The height of the ribbon fin of a gymnotiform swimmer seems optimized with respect to the mechanical Cost of Transport (COT). This can be explained based on the drag-thrust decomposition in 2, above. 4. Contrary to prior theoretical results by Lighthill, we find that there is no momentum enhancement in balistiform and gymnotiform swimming. 5. Pectoral fin movements of larval zebrafish seem optimal with respect to COT. 1.A novel constraint-based formulation for self- propulsion. 2. Drag-thrust decomposition in undulatory propulsion of ribbon fins Correspondence Neelesh A. Patankar: n- [email protected] 3. Optimal height of the ribbon-fin for a given body Black ghost knifefish (Aptenorotus albifrons) : Undulatory motion of the ventral ribbon fin. Left photo: Neil Hepworth. Right photo: M. A. MacIver Drag-thrust decomposition (right): U is the forward translational velocity of fin and V w is the lateral velocity due to the wave motion at a point. V is the resultant velocity. The force generated by the undulatory ribbon was found to be approximately equal to forces from two separate sets of simulations identified as “Drag” and “Thrust” mechanisms. The decomposition is kinematically consistent and the only decomposition we found that leads to force decomposition. Why can drag-thrust be decomposed? Left figures show velocity contours while the right figures show pressure contours. Top figures are for the drag mode while the bottom figures for the thrust mode. It is seen that the velocities due to each mechanism are dominant in physically distinct regions. The drag mechanism is dominated by boundary layer-type flow while the thrust mechanism is dominated by separation-type flow. There is weak coupling of the two modes by way of the non-linear convection term. Consequently, drag-thrust separation is observed in spite of large Reynolds numbers (~ 10 3 ). This result has been experimentally verified. U swim H = 2cm h = ? L = 10 cm 2 waves 3 Hz Water Fin thrust Fin drag Body drag The h scaling of the swimming velocity U s is a consequence of the scalings of the fin drag and fin thrust with respect to h. The mechanical Cost of Transport (COT) shows optimality. COT ~ Mean power = h 3 Mean vel U s (h) Power scaling based on lateral flapping work. U s (h) which depends on drag/thrust scaling of the fin causes the optimum in COT scaling. 4. There is no “momentum enhancement” Lighthill’s theory Our simulations What is “momentum enhancement”? If the thrust force generated by a ribbon is greater in the presence of a rigid body attached to it then it’s termed “momentum enhancement.” If true, then the presence of a rigid body would help in propulsion. Why is there no momentum enhancement? Our calculations indicate that the thrust force is not enhanced when the ribbon fin is attached to a rigid body. We find that the axial flow at the bottom edge of the fin is the dominant thrust generating flow field. This flow is not affected by the presence of a rigid body attached at the top edge of the fin. This is unlike the approach by Lighthill where cross-sectional flow (see sketch above) was considered the dominant thrust producing flow field. Sketches: Triantafyllou et al. 5. Another example of COT based optimality “Normal case” (expt. kinematics) Breast stroke 180 o out of phase wrt normal case Effect of pectoral fins on COT in larval zebrafish: Different pectoral fin movements relative to body undulations were tested. Phase indicates the phase difference with respect to body undulations. A no-fin case was also considered. It is seen that the naturally observed case with “normal” pectoral fin movements had the lowest COT. Scaling of power and COT: It is found that the dominant power expenditure comes from lateral oscillations of the body. Thus, the power scales as the lateral force times the lateral velocity and not force times velocity in the swimming direction. This leads to new insights into the scaling of COT. ~ 6.5%
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Transcript of Drag-thrust decomposition and optimality in swimming N. A. Patankar*, A. A. Shirgaonkar, O. M....
Drag-thrust decomposition and optimality in swimming N. A. Patankar*, A. A. Shirgaonkar, O. M. Curet, R. Bale, M. Hao, A. Bhalla, M. A. MacIver,
Department of Mechanical Engineering, Northwestern University
Overview
This poster highlights the following key results:1.A novel constraint-based formulation to simulate self-propulsion
has been developed. The numerical approach is used to obtain the results below.
2.A drag-thrust decomposition that is different from that proposed by Lighthill is found in the propulsion by undulatory ribbon fins of gymnotiform and balistiform swimmers.
3.The height of the ribbon fin of a gymnotiform swimmer seems optimized with respect to the mechanical Cost of Transport (COT). This can be explained based on the drag-thrust decomposition in 2, above.
4.Contrary to prior theoretical results by Lighthill, we find that there is no momentum enhancement in balistiform and gymnotiform swimming.
5.Pectoral fin movements of larval zebrafish seem optimal with respect to COT.
1.A novel constraint-based formulation for self-propulsion.
2. Drag-thrust decomposition in undulatory propulsion of ribbon fins
CorrespondenceNeelesh A. Patankar: [email protected]
3. Optimal height of the ribbon-fin for a given body
Black ghost knifefish (Aptenorotus albifrons):Undulatory motion of the ventral ribbon fin. Left photo: Neil Hepworth. Right photo: M. A. MacIver
Drag-thrust decomposition (right):U is the forward translational velocity of fin and Vw is the lateral velocity due to the wave motion at a point. V is the resultant velocity. The force generated by the undulatory ribbon was found to be approximately equal to forces from two separate sets of simulations identified as “Drag” and “Thrust” mechanisms. The decomposition is kinematically consistent and the only decomposition we found that leads to force decomposition.
Why can drag-thrust be decomposed? Left figures show velocity contours while the right figures show pressure contours. Top figures are for the drag mode while the bottom figures for the thrust mode. It is seen that the velocities due to each mechanism are dominant in physically distinct regions. The drag mechanism is dominated by boundary layer-type flow while the thrust mechanism is dominated by separation-type flow. There is weak coupling of the two modes by way of the non-linear convection term. Consequently, drag-thrust separation is observed in spite of large Reynolds numbers (~ 103). This result has been experimentally verified.
Uswim
H = 2cm
h = ?
L = 10 cm
2 waves 3 HzWater
Fin thrust Fin drag Body drag
The h scaling of the swimming velocity Us is a consequence of the scalings of the fin drag and fin thrust with respect to h.
The mechanical Cost of Transport (COT) shows optimality.
COT ~ Mean power = h3
Mean vel Us(h)
Power scaling based on lateral flapping work. Us(h) which depends on drag/thrust scaling of the fin causes the optimum in COT scaling.
4. There is no “momentum enhancement”
Lighthill’s theory Our simulations
What is “momentum enhancement”? If the thrust force generated by a ribbon is greater in the presence of a rigid body attached to it then it’s termed “momentum enhancement.” If true, then the presence of a rigid body would help in propulsion. Why is there no momentum enhancement? Our calculations indicate that the thrust force is not enhanced when the ribbon fin is attached to a rigid body. We find that the axial flow at the bottom edge of the fin is the dominant thrust generating flow field. This flow is not affected by the presence of a rigid body attached at the top edge of the fin. This is unlike the approach by Lighthill where cross-sectional flow (see sketch above) was considered the dominant thrust producing flow field.
Sketches: Triantafyllou et al.
5. Another example of COT based optimality
“Normal case”(expt.
kinematics)
Breast stroke
180o out of phase wrt normal case
Effect of pectoral fins on COT in larval zebrafish: Different pectoral fin movements relative to body undulations were tested. Phase indicates the phase difference with respect to body undulations. A no-fin case was also considered. It is seen that the naturally observed case with “normal” pectoral fin movements had the lowest COT.Scaling of power and COT: It is found that the dominant power expenditure comes from lateral oscillations of the body. Thus, the power scales as the lateral force times the lateral velocity and not force times velocity in the swimming direction. This leads to new insights into the scaling of COT.
~ 6.5%
