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Journal of Experimental Biology partnership with Dryad

On the role of form and kinematics on the hydrodynamics of self-propelled body/caudal fin swimming
I. Borazjani, F. Sotiropoulos

SUMMARY

We carry out fluid–structure interaction simulations of self-propelled virtual swimmers to investigate the effects of body shape (form) and kinematics on the hydrodynamics of undulatory swimming. To separate the effects of form and kinematics, we employ four different virtual swimmers: a carangiform swimmer (i.e. a mackerel swimming like mackerel do in nature); an anguilliform swimmer (i.e. a lamprey swimming like lampreys do in nature); a hybrid swimmer with anguilliform kinematics but carangiform body shape (a mackerel swimming like a lamprey); and another hybrid swimmer with carangiform kinematics but anguilliform body shape (a lamprey swimming like a mackerel). By comparing the performance of swimmers with different kinematics but similar body shapes we study the effects of kinematics whereas by comparing swimmers with similar kinematics but different body shapes we study the effects of form. We show that the anguilliform kinematics not only reaches higher velocities but is also more efficient in the viscous (Re∼102) and transitional (Re∼103) regimes. However, in the inertial regime (Re=∞) carangiform kinematics achieves higher velocities and is also more efficient than the anguilliform kinematics. The mackerel body achieves higher swimming speeds in all cases but is more efficient in the inertial regime only whereas the lamprey body is more efficient in the transitional regime. We also show that form and kinematics have little overall effect on the 3-D structure of the wake (i.e. single vs double row vortex streets), which mainly depends on the Strouhal number. Nevertheless, body shape is found to somewhat affect the small-scale features and complexity of the vortex rings shed by the various swimmers.

FOOTNOTES

  • This work was supported by NSF Grants 0625976 and EAR-0120914 (as part of the National Center for Earth Surface Dynamics) and the Minnesota Supercomputing Institute. We are grateful to Professor Smits at Princeton University for providing the lamprey morphology data from the CT data of Professor Fish.

  • LIST OF SYMBOLS AND ABBREVIATIONS

    amax
    tail-beat amplitude
    a(z)
    amplitude envelope
    A
    width of the wake
    Am
    oscillation amplitude
    BCF
    body/caudal fin
    c
    damping factor
    ccr
    critical damping factor
    CF
    mean force coefficient
    CFo
    mean non-dimensional force in the axial direction
    CP
    mean power coefficient
    CPo
    mean non-dimensional power
    CT
    mean thrust coefficient
    CTo
    mean non-dimensional thrust
    CT
    computed tomography
    D
    cylinder diameter
    D(t)
    drag
    EBT
    elongated body theory
    f
    tail-beat frequency
    fc
    oscillation frequency
    fn
    natural frequency
    fo
    non-dimensional tail-beat frequency
    F
    force
    F
    force vector exerted on the virtual swimmer's body by the fluid
    FSI
    fluid–structure interaction
    h
    lateral displacement of fish body
    Embedded Image
    lateral velocity of fish body
    hmax
    maximum displacement of the tail
    h(z,t)
    lateral excursion of the body at time t and location z
    HCIB
    hybrid Cartesian immersed-boundary
    IB
    immersed boundary
    k
    wave number
    KC
    Kuelegan–Carpenter number
    L
    total body length
    LC–FSI
    loose coupling fluid–structure interaction
    LL
    lamprey swimming like a lamprey
    LM
    lamprey swimming like mackerel
    m
    mass of the virtual swimmer
    ML
    mackerel swimming like a lamprey
    MM
    mackerel swimming like a mackerel
    MPG
    miles per gallon
    Mred
    reduced mass
    n
    normal vector the surface
    nj
    jth component of the normal vector
    p
    non-dimensional pressure
    PIV
    particle image velocimetry
    Pside
    power losss due to lateral undulations of the fish body
    QT
    rotation matrix of the non-inertial frame relative to the inertial frame
    ra
    non-dimensional position in the inertial frame
    rr
    non-dimensional position relative to the non-inertial frame
    r.m.s.
    root mean squared
    Re*
    mean Reynolds number based on swimming speed U*
    Reo
    non-dimensional viscosity or Reynolds number based on characteristic velocity Uo
    S
    symmetrical parts of the velocity gradient
    SC–FSI
    strong coupling fluid–structure interaction
    St*
    mean Strouhal number based on swimming speed U*
    Sto
    Strouhal number based on the characteristic velocity Uo
    t
    time
    T
    mean thrust
    T
    period of a tail-beat cycle
    ua
    non-dimensional Cartesian absolute velocity vector of the fluid in the inertial reference frame
    uc
    velocity of the non-inertial frame or the center of mass
    um
    maximum oscillation velocity
    un
    fluid velocity normal to the body
    ur
    non-dimensional Cartesian relative velocity vector of the fluid in the non-inertial reference frame
    ut
    fluid velocity vector tangential to the body
    U
    mean swimming speed
    U*
    mean non-dimensional swimming speed
    Uo
    characteristic (tether) velocity
    Ured
    reduced velocity
    VIV
    vortex-induced vibration
    xc
    position vector of center of mass non-dimensionalized by L
    xc
    location of the center of the cylinder
    z
    axial direction measured along the fish axis from the tip of the fish's head
    α
    under-relaxation coefficient
    Γ(t)
    a dynamically evolving surface
    η*
    mean efficiency (velocity over power)
    ηf
    Froude efficiency
    λ
    wavelength
    ν
    kinematic viscosity
    ρ
    fluid density
    τij
    viscous stress tensor
    ω
    angular frequency
    ξ
    damping coefficient
    Ω
    angular velocity
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