studied by Lighthill [1], Wu [2] and other workers. The main focus of these studies has been ... Some of them report very high efficiency of propulsion. But for mere ...
In 5th Inter. Symposium on Computational Fluid Dynamics, volume 1, pages 13-18, Sendai, Japan, 9 1993.
Finite-Volume Simulation of a Flow about a Moving Body with Deformation Hiromichi Akimoto* and Hideaki Miyata*
*Department of Naval Architecture and Ocean Engineering, University of Tokyo,7-3-l Hongo, Bunkyo-ku,Tokyo,Japan
In a large number of fluid flow problems the.geometry of the body may change in time. The motion of fishes, the flow about oscillating cylinders of drilling platforms, the motion of flaps of airfoils and hydrofoils and the vibration of structure in fluid are examples. In order to cope with these problems of practical importance a new method is developed being based on the finite-volume method. The moving coordinate system is employed and the grid system is regenerated at each time step by an algebraic method. On the body boundary the moving boundary condition is implemented so that the conservation laws are fulfilled.
Here a swimming fish is treated as a first example. The original shape data is given and the deformation of the backbone is given as a function of time and the x-coordinate in the longitudinal direction.
1
Introduction
The hydrodynamics of swimming fish have been
studied by Lighthill [1], Wu [2] and other workers. The main focus of these studies has been to un
methods. Some of them report very high efficiency of propulsion. But for mere precise understanding of fish propulsion, the viscous flow with boundary layer and separation must be considered. In this paper, a fish-like body passing a wave down its body is numerically studied. The simu
derstand the high efficient propulsion mechanism of fish and marine mammals, and to develop a high ef ficiency oscillatory propulsor. This is attractive for ship propulsion where the oscilla'ting propulsor system can use larger blades compared to rotating propeller system, to overcome
ing the Navier-Stokes equations, in the moving grid system fitted to the deforming body boundary. The resulting flow patterns around this fish-like body moving with deformation is presented, and propul
free-surface and draft restrictions of the ship.
sion mechanism of fish is visualized.
There have been several models of the swimming motion of fish. One is based on the pitching and heaving oscillatory wing which is rigid or flexible and corresponds to the fin part of a fish. This is suitable for the fishes with a high aspect ratio fin. The second models the deforming wing by creating an advancing wave from its leading edge to trail ing edge. This model demonstrates the motion of all length of a fish with low aspect ratio fin, which uses after part of the body for propulsion. The first
of the advancing wave, phase velocity c and wave length A, are varied to examine their relation with the propulsive performance.
model is also used in the dynamic stall or flatter problems and has been widely studied. However, there are few investigations of the sec ond model, and most of them are restricted by
the assumption of inviscid fluid or small amplitude movement. Many of previous works uses discrete vortex method or other singularities distribution
lation code is based on a Finite Volume Method us
2
Parameters
Numerical Method
The simulation code is based on the 3D flow solver
WISDAM-V[6]. Basic equations are A.L.E. (Arbitraly Lagrangean and Eulerian) description of the Navier-Stokes equations and the mass conservation law as follows.
du
~dt
+ V • (u —v)u =
-VP+^V-[Vu +(Vu)T], V •u = 0 .
(1) (2)
I
Fig.l: Grid system (all region),t —9.0, A
0.1, c =
Fig.2: Grid system (around the body),2 = 9.0, A
1.5, A = 2/3
0.1,c= 1.5, A = 2/3
Here Re is the Reynolds number (Re = Uooc/v),
the grid velocity v.
u is the dimensionless velocity, v is the velocity of
The body boundary conditions are set for the
moving grid points, c is the chord length and P is the pressure divided by density p. These equations are non-dimensionalized with re spect to the charactaristic length c and velocity
convective and diffusiv.e flux of momentum and
pressure on the body boundary. The momen tum flux resulting from convection across the body boundary is zero, because convective velocity u —r> is zero on the body boundary. The diffusive mo
^co in the moving grid system, the convective term in mentum flux is evaluated from the body surface ve eq.(l) contains two components. One is the usual locity v and u near the body. The deforming body momentum flux caused by convective velocity u. boundary is treated as the fluid with velocity v, and The second is the convection of momentum caused the velocity gradient V-u on the body is calculated by the deformation of control volumes. Mass flux entering into a deforming control vol
from v and two u points by Taylor series expansion. The body boundary condition for pressure is in
ume through one of its area elements is treated as the volume swept by this area element. And the
troduced from the Navier-Stokes equations. Using
momentum contained in this volume is then the mo
mentum flux acrossing the area element caused by
grid motion. This treatment is a modification of the method by Rosenfled et al.[5] A H-type grid system is generated on every time steps,'fitted to the time dependent body boundary using an arithmetic method as shown in Fig.2 and 1. The 2nd order QUICK scheme is employed for the convective term. Pressure and velocity convergence
is reached by the MAC like solution procedure.
no-slip velocity condition (u = v : on the body surface) eq.(l) is rewritten as :
^ =-VP+^V-[V« +(W)T] (on the body surface) .
On the assumption that the diffusion term is neg ligible compared to the body surface acceleration du/dt , eq.(3) is rewritten as : dv
VP - —— 3
Moving body boundary conditon
A no-slip velocity conditon is imposed on the body boundary with time dependent deformation. The fluid velocity u on the body boundary is equal to
(3)
(on the body surface) .
(4)
Eq.(4) shows the relation between pressure gradi ent and body boundary acceleration on the body surface. Pressure on the body boundary is extrap
olated from the pressure points near the body and
7
./
1. (0 < t < 0.5) Increase inflow velocity from 0 to Uoo- Oscillation amplitude A = 0.
2. (0.5 < t < 1.0) Relax flow field to obtain steady state solution. -4 = 0.
3. (1.0 < t < 2.0) Increase amplitude of oscilla
0.4
tory deformation A from 0 to AmaxTime
4. {t > 2.0) Steady oscillatory deformation.
where t is the non-dimensionalized time {c/Uoq unit). The initial body profile is two-dimensinal
Fig.3: Offset h{x,t) from the initial center line, A — NACA0012 foil.
The Reynolds number Re =
Uooc/v is 5.0 x 103, time increment dt —1.0 x 10~3,
0.1, c= 1.5, A = 2/3
and the number of grid points is 80 x 60 x *1. The computational domain is (—5 < z < 5, —3 < y < ary. In the case that the body motion is static or
3), and the leading edge of the body is located at (0,0). The wave length is set at A = 1,2/3, phase
non-accelerative, eq.(4) gives the usual zero gradi
velocity c = 1.2 ~ 1.7, and the maximum amplitude
ent pressure condition along the normal direction of
of deformation is fixed at A = 0.1.
pressure gradient(from eq.(4) ) on the body bound
the body surface. 5
4
Deformation of the swimming body
Numerical results
The time history of the hydrodynamic forces is shown in Fig.4 and 5. These forces are non-
The deformation of the 2D fish is treated as a simple example. The initial form of the fish is a NACA0012
dimensionalized by ^pUoo2c. The positive value of
wing section, which deforms around its center line. Body motion from the initial center line h(x,t) is given by the following equations :
produces drag, and the negative one thrust produc tion. In Fig.5, the time averaged value of Fx is almost zero, and this means that the fish can con tinue swimming at this cruising speed Uoo by this
h(x,t)
--= f(x) g(x,t) ,
(5) (6)
the streamwise component means that the motion
tial center line. The motion of the body is changed from steady state to oscillatory deformation in fol
deformation. Propulsive force occurs twice in a pe riod of trailing edge oscillation, when the trailing edge makes downstroke and upstroke. Fig.6 shows the time average Fx. In the cases when Fx > 0, this swimming motion cannot sustain cruising speed Uoq. On the other hand, in the cases Fx < 0, the body will be accelerated, which measns that this is a overaction for this cruising speed. The optimum motion at the cruising speed U^ is attained when Fx — 0. Fig.6 shows that the opti mum point is when c = 1.60 for the case of A = 1 and c — 1.47 for A = 2/3. Fig.7 shows streamwise velocity u\ contour map. The parameters of the motion are c — 1.5 and A = 2/3, and this has
lowing four stages,
the ability to cruise at Uoo {Fx —0)- The contours
/(*)
9(x,t)
== A{-(x-iy + i}, 2x.
--= cos -r-{x —ct) ,
(?)
where, g{xyt) is the motion of body wave advanc ing from head to tail with the phase velocity c, the wave length A and maximum amplitude A. f(x) is the weight function to prevent the head of the body from moving, and increase oscillation amplitude to ward the trailing edge accelerating fluid near the body. Fig.3 shows the offset h{x,t) from the ini
f
••/
/
drawn in bold lines {u\ > Uoo) indicate the fluid accelrated and they are tranported backward by this wave motion, of which reaction force is the thrust.
The contour map of Cp is shown in Fig.&. With the wave motion, the pressure on the body surface facing backward is higher than that of forward fac ing surface. Integration of the pressure difference between backward and forward facing surfaces gives thrust that is equal to the viscous drag.
7.4
6
7.6
Conclusion
Time
This paper has presented the results of a twoFig.4: Time history of hydrodynamic forces and the offset at the trailing edge, c — 1.6, A = 1
0.2
0.15 0.1
.
.
...--.,
y""\
•
w"
•
dimensional flow
simulation around
a, fish
like
body performing large amplitude transverse swim ming motion. The solution of the Navier-Stokes equations is accomplished using the finite volume method with a deforming coordinate system. The influence of the phase velocity and wave length on the hydrodynamic thrust and drag forces has been numerically investigated. These results show:
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1. The swimming motion of the body can create enough thrust to overcome the body's viscous drag.
Fx
0.1*Fy
-0.15
T.E. motion
.
-0.2 7.4
7.2
1
7.6
7.8
I
Time
2. The swimming body produces adequate thrust at cruising speed Uoo when Fx = 0 which oc curs at c =
1.60 for A =
1.0 and c =
1.47 at
A = 2/3. Fig.5: Time history of hydrodynamic forces and the offset at the trailing edge, c —1.5, A = 2/3
3. The numerical results show the propulsive force occurs twice in a period of the tail edge motion during the upstroke and downstroke.
0.03
X wave length 1
0.02
\v o
wave length 2/3
However, still we have many differences between calculated motions and those of real fishes.
Amplitude of the side force Fy is so large that,
0.01
if the body is not restrained, it can not swim in a straight course. This large side force comes partially -0.01
from 2D condition (which gives large added mass) and the mode of swimming (not optimized as real fishes), and this causes power loss. The oscillation
-
-0.02
1.25 1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65 1.7 1.75
Phase velocity c
Fig.6: Time average of drag, Fx
of thrust may also gives lower efficiency, but this may not be significant compared to the side force. For better understanding of the fish propulsion, carefull evaluation of power used in this motion must be made at a higher Reynolds number.
../
§s§
Fig.7: Streamewise velocity u\ contour map, c = 1.5, A = 2/3 (bold line : i*i > Uoot thin line : u\ < Uoo, contour interval is 0.1)
Fig.8: Cp contour map, c = 1.5, A= 2/3 (bold line is (+), thin line (-), contour interval is 0.1)
/
References
[1] Lighthill, M. J. "Noteon the swimming ofslender fish" J. Fluid Mechanics vol.9 pp.305-317(1960) [2] Wu, T. Y. "Hydrodynamic of swimming propul sion. Part 1. Swimming of a two-dimensional flexible plate at variable forward speeds in an inviscid fluid" J. Fluid Mechanics vol.40 pp.337355(1971) [3] Todor A. Videv, Yasuyuki Doi "Numerical Study of the Flow and Thrust Produced by a Pitching 2D Hydrofoil" J. of The Society of Naval Archi. of Japan, Vol 172, pp. 165174(1992)
[4] Isshiki,S. Morikawa,H "Study on Dolfin-Style Fin Ship" Bulletin of the Society of Naval
archi. of Japan, vol.642, Dec. 1982, pp.2-9 (In Japanese)
[5] Moche Rosenfeld and Dochan Kwak "Time dipendent solution of viscus incompressible flows in moving co-ordinates" Int.Journal for Numerical Methods in Fluids VOL.13,1311-
1328(1991) [6] M. Zhu, H. Miyata, and H. Kajitani, "A FiniteVolume Method for the Unsteady Flow about a Ship in Generalized Coordinate Systems", J.
Soc. Naval Archit. Japan, Vol. 167,(1990)
