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24th Symposium on Naval Hydrodynamics Fukuoka, JAPAN, 8-13 July 2002 Numerical Study on Propulsion by Undulating Motion in :Laminar-Turbulent Flow Zuogang Chen and Yasuaki Doi (Hiroshima University, Japan) ABSTRACT Unsteady viscous flow field around a fish-like advancing body is numerically studied. The main objective is to study the abilities of a fish-like body to produce thrust and achieve higher propulsive efficiency in viscous flow field, and to catch the basic characteristics of fish-like locomotion. The Reynolds number based on the oncoming velocity and the body length ranges from 103 to 3X106. The flow is simulated by solving three-dimensional unsteady Reynolds averaged Navier-Stokes equation in a primitive value formulation, while the eddy-viscosity is described by algebraic model and transitional zone is computed by empirical formula. The numerical scheme is based on the MAC method where the RANS equation is solved by the time marching method on a body fitted coordinate system. The simulations show that pressure is acting as thrusting force while frictional force is acting as resistance during the undulating motion. Strouhal number is the most important governing parameter for the propulsive efficiency. The propulsive efficiency is enhanced with the increase of the Reynolds number or the decrease of body thickness. Two vortices shed from the tail during one undulating period. Vorticity travel shows close connection with pressure distribution. For laminar flow, fish-like locomotion increases frictional resistance, but for turbulence case, fish-like locomotion can reduce frictional resistance by flow relaminarization and achieve a propulsive efficiency more than 1. INTRODUCTION The flow field around a swimming fish or cetacean has been investigated for a long time by the researchers in various field of study such as biology, physical science and engineering. This interest has been inspired not only to understand and simulate an efficient swimming propulsion, but also to utilize the results for engineering application. Fast-starting and 1 maneuvering of flexible hull vehicles can be significantly better than the performance of rigid bodies, because the flow can be controlled over the entire body of the vehicle through the appropriate flexing. The study of live fish swimming can be very instructive in exploring mechanisms of unsteady flow control since fish have evolved over millions of years to optimize their body shapes and locomotive abilities. Triantafyllou et al. (2000) pointed out that the resulting unsteadiness in the flow is exploited by fish and cetaceans to their advantage. The view is based on a series of studies demonstrating that the fish (a) generate large, short-duration forces efficiently, (b) coordinate rhythmic unsteady body and tail motion to minimize the energy required for steady propulsion, and (c) coordinate transient motion of the body and tail to minimize the energy lost in the wake during maneuvering. Fish propel themselves through rhythmic unsteady motions of their body, fins, and tail; they offer a different paradigm of locomotion that conventionally used in man-made vehicles. The most notable proposed mechanisms fall under the categories of laminar boundary layer maintenance, turbulent drag reduction, utilization of shed vorticity and the delay of separation. As described in Wolfgang et al. (1999), a fish benefits from smooth near-body flow patterns and the generation of controlled body-bound vorticity, which is propagated towards the tail, shed prior to the peduncle region and then manipulated by the caudal fin to form large-scale vertical structures with minimum wasted energy. This manipulation of body-generated vorticity and its interaction with the vorticity generated by the oscillating caudal fin are fundamental to the propulsion and maneuvering capabilities of fish. Muller et al. (1997) showed that the fish shed one vortex per half tailbeat when the tail reached its most lateral position. Part of the circulation shed in the vortices had been generated previously on the body by the transverse body wave traveling down the body. The alternating suction and pressure flows form a circulating flow around the inflection points of the
body; this circulating flow is shed when the inflection point reaches the tail. Also validated in Ahlborn et al. (1991), during steady swimming, where vortices may be generated by the fore-body of a fish, the tail could either reverse the rotation immediately, or generate an additional eddy of equal but opposite angular momentum, so that an eddy pair is produced of zero net angular momentum. This eddy pair dissipates quickly, producing a mushroom-shaped flow structure. The momentum of this flow must be equal to the forward momentum of the fish. Recovery of most of the rotational energy and destruction of the eddies allow the fish to swim more efficiently. In addition, the destruction of the vortices, or footprints, also makes the fish less detectable to predators. Azuma (1992) summed up former researchers' results and plotted skin friction drag of fish and cetaceans. In general, the Reynolds number lies in the range of 103<Re<106 for various fish, and of 106<Re<108 for most cetaceans. It can be seen that fish incur larger friction drag than that of a flat plate of the same surface area, except in rare cases. Anderson et al. (2001) also claimed that swimming fish were found to experience greater friction drag than the same fish stretched straight in the flow. But the confirmation of enhanced friction drag does not exclude the possibility that drag-reducing mechanisms are operating (Fish (1998), Bakenko and Yaremchuk (1998~. Two possible mechanisms observed by Taneda and Tomonari (1974) were suggested in fish boundary layers. They are form drag reduction by delayed separation and friction drag reduction by partial or total laminarization. Their research showed that turbulence in the boundary layer of the plate is suppressed when the speed of propagation of the traveling wave exceeds the free stream velocity. When the wave velocity is larger than the uniform flow, the boundary layer does not separate. The swimming motion shows a strong tendency to laminarize the turbulent boundary layer and reduce the wall shear stress. Wolfgang et al. (1998), Techet et al. (1999a) and Anderson et al. (2001) obtained experimental data of velocity profiles in boundary layer of a swimming fish. They found that undulating motion can convert turbulent flow into laminar flow at some flow domain. Experimental results (Barrett et al. (1999), Techet et al. (1999b)) also demonstrated the ability of flexible-hull vehicle to achieve high maneuverability and turbulence reduction through flow control. Even Barrett et al (1999) pointed out that the power required to propel an actively swimming, streamlined, fish-like body was significantly smaller than the power needed to tow the body straight and rigid at the same speed. A parametric investigation showed sensitivity of drag reduction to the non-dimensional frequency (Strouhal number), amplitude of body oscillation and wavelength, the angle of attack and phase angle of the tail fin. The maximum drag reduction was in excess of 70%. Although experiments with flexible-hull vessels and direct flow measurements on live fish are a reliable way to obtain data on maneuvering flexible bodies and to corroborate analytical predictions, quantifying the locomotor forces experienced by swimming fish represents a significant challenge because direct measurements of force applied to the aquatic medium are not feasible. Numerous studies on two and three dimensional flow around a f~sh-like body have been done since the slender-fish theory of Lighthill (1960) and two dimensional unsteady lifting surface theory of Wu (1961) and have contributed to the present understanding of the hydrodynamic and biological aspects of swimming. The main ingredients of the classical analysis are the two-dimensional potential flow along with linearized boundary conditions, small perturbation velocities, and the assumption of a planar vortex wake. But actually most fish move with amplitudes and frequencies that exceed the limit of linear lifting surface theories. Therefore, it is necessary to develop the capability to compute the flow past surfaces in arbitrary motion (Sandberg and Ramamurti (1999~. The arbitrary motion can include oscillation and deformation of simple and complex surfaces as well as varying separation between the bodies in response to the varying hydrodynamics Interactions. The unsteady viscous flow around an oscillating body has been solved by many researchers. Cheng et al. (1991) developed the three-dimensional waving plate theory by the vortex ring panel method to investigate the swimming performance of fish undulating motion. They found that the undulatory motion can reduce three-dimensional effects. Carting et al. (1998) investigated self-propelled anguilliform swimming by using a computational mode combining the dynamics of both the creature's movement and the two-dimensional fluid flow of the surrounding water. The important role of viscous forces along and around the creature's body and in the growth and dissolution of vortex structures were also illustrated. Kim et al. (1998) investigated the peristaltic propulsion by solving Navier-Stokes equations on unstructured grid in highly viscous fluid, and found that the effective motion of peristaltic propulsion depends on the Reynolds number. Nakaoka and Toda (1994) investigated the flow field around a wing (NACA0010) which moved like a fish by Navier-Stokes solver using moving grid approach. The results showed the fish-like motion was effective to get propulsive force. Three motion models were compared to each other and meaningful conclusion was drawn to produce a larger mean thrust. Doi and Yhomatsu (1999) numerically studied unsteady viscous flow field around a wavy oscillating plate in highly viscous fluid. They investigated the abilities of wavy oscillating plate to produce thrust in highly viscous fluid and the dependence of the wave number and nhase velocity on the thrust and propulsive efficiency. . . .. 2 Liu et al. (1997)
analyzed tadpole propulsion using a three-dimensional computational fluid dynamic (CFD) model of undulatory locomotion that simulates unsteady viscous flow around an oscillating body of arbitrary three-dimensional geometry. The simulated results revealed that the shape and kinematics of tadpoles collectively produce a small 'dead water' zone between the head-body and tail during swimming precisely where tadpole can and do grow hind limbs without those limbs obstructing flow. The present paper provides some numerically simulated results which represent the basic characteristic of fish-like locomotion. The objective of present study is to find superiority of propulsion by undulating motion in laminar and turbulent flow. A rectangular plate with aspect ratio 1/3 and NACA wing sections are employed for the simulation. The phenomena of vorticity control and turbulence relaminarization are studied. The relationship between propulsive efficiency and related parameters such as Strouhal number, Reynolds number and body thickness are also investigated. MODELING OF UNDULATION The simulated body undulates actively in unbounded oncoming flow. Related variables are normalized by the body length L and oncoming velocity U. as well as the time is normalized by L/U. The movement of the undulating body in y direction is given as y(`x,t)=axn sint2'zb~x- Sp t)] (1) where a is amplitude, n=1.1, 2 fib is wave number, Sp is phase velocity, t is time, where x and y are the stream-wise and the lateral coordinates whose origin locates at the leading edge of the body. 0<x_ Xend~t) where Xend~t) is x-coordinate of the trailing edge calculated as ~ena,<~dx = 1 0 (2) Equation (2) keeps the body length constant during the undulation. For NACA section, the undulating camber line is computed by Equations (1 ) and (2) while the moving body surface is given by keeping the deformed thickness line length constant. The positive phase velocity propagates waves from the leading edge toward the trailing edge. Typical loci during one period can be seen in Figure 1 at chosen parameters: a=0.051, b=1.0. T is undulating period while 0/8 T corresponds to the time when the tailing edge reaches its positive maximum lateral position and the others denote the configurations after every 1/8 period. The Reynolds number based on the oncoming velocity U and body length L ranges from 103 to 3x 10 . Figure 1: Configurations of an undulating NACAOO10 during one period (a=0.051, b=1.0) NUMERICAL SCHEME The governing equations are Reynolds averaged Navier-Stokes equation (3) and continuity equation (4) u,+uuX+vuy+wuz =-Px+R V2U+Rx v, +uvx +wy +wvz = -PY + R V2v+Ry w, + uwx + vwy + wwz = -Pz + R V2w+ Rz ux + vy + wz O (3) (4) Rx = {V' (`ux +Ux)'~x +lv,fuy +vx)3y + i~v,(~uz +wx)}z Ry = TV, (<vx +uy))x + T,v, ivy + vy)~;y + f,v, fvz + wads Rz = its, (Sax + uz Ax + ~`v, (any + vz))y + TV, (wz + Wz j'Jz Subscripts represent partial differentiations with respect to the referred variables except eddy viscosity v, and Rx, Ry, Rz, which are Reynolds stress components expressed by Equation (54. In Equations (3), (4) and (5), u, v and w are x, y and z components of velocity; p is pressure; Re is Reynolds number. To simulate turbulent flow, Baldwin-Lomax (Baldwin and Lomax 1978) model is applied for eddy viscosity v ,. As the wake field of the undulating body is asymmetric, Paterson and Stern (1993) made a little modification. They applied the updated model for unsteady flow field given by the MIT flapping foil experiment and validated the model as an effective simulation for unsteady flow. Therefore in the present method, the updated model is used to calculate eddy viscosity v, in Equation (5~. 3
The transition onsets when the non-dimensional v, is larger than 14 (Baldwin and Lomax 1978), while the transitional zone ends where v, is larger than 25. The governing equations are solved in a primitive value formulation. A numerical coordinate transformation is introduced into a body fitted curvilinear coordinate system to simplify the computational domain and to facilitate the implementation of boundary conditions. H-type grid system is adopted, but near the leading edge the grid is modified to be adapted for the blunt body head. The numerical scheme is based on the MAC method where the momentum equation is solved by the time marching method on the body fitted coordinate system. The oncoming velocity and the undulating amplitude increase smoothly from zero at ~0 to the steady value at into (to: one specified value). After that the oncoming velocity and undulating amplitude keep constant. The first order difference form of the time derivative is used for an explicit advancement in time. The convection terms are discretized by the third order upwind scheme, while all the other spatial derivatives are discretized by the second order central difference scheme. On the body surface, no-slip condition is applied for the velocity. For the pressure, Neumann type condition is applied to satisfy the momentum equation. A uniform pressure is applied on the inflow boundary, while a zero-gradient extrapolation is used on the outlet boundary. The Poisson equation for pressure is solved by using Successive Over Relaxation method. At Re=103, The computational domain is -3.0Ax_ 5.0, -2.0<y_2.0 and 0.0<z<1.0. The grid consists of t124X73x28] points in x, y and z directions where minimum normal spacing is 5.26X 10-3. For Re=2x 1 o6, the computational domain is -3.0 <x_ 10.0, -2.0 - <y< 2.0 and 0.0 Liz_ 1.0, where the grid consists of t250X73x28] points in x, y and z directions where minimum normal spacing is 6.06X 10-5. SELF-PROPULSION CONDITION AND PROPULSIVE EFFICIENCY The total force acting on the body varies during the undulating motion. Therefore, in the present study, self-propulsion is defined as the condition when the time-averaged total force becomes zero. The amplitude a and wave number 2 ~zb are chosen firstly and the phase velocity Sp is tuned to meet the self-propulsion condition. The related parameters are defined as follows: Cp. , .= ~ f ~ pu2S CPX = ~ , CTX = CFX + CPX (6) pu2S 2/ TI F. dt .]o of _ ~ CFX = 1 _pu2S 2 toutputdtl T tTFxpdt _ ~,~ _ _ _ CPX = 1 ~ CTX = CFX + CPX (7) _pu2S 2 U- JDdt P |inputdtlT J[ J(:X +PX) use+ J(ry+ py) vds]dt T T AS iS where Fxf is x-component of frictional force exerted on the body, Fxp is x-component of pressure exerted on the body, T is the period of the undulation, p is the density of the fluid, U is the oncoming velocity, S is the area of the body. The negative symbol of pressure component Fxp (against to x-axis) represents thrust and the positive symbol of frictional force component Fxf (same to x-axis) represents resistance. When the self-propulsion condition is achieved, the time-averaged total force exerted on the body becomes zero. Thus the propulsive efficiency up is defined by Equation (8) in the present study. The output work rate is the product of the oncoming velocity U and the resistance D of a straightly advancing rigid body at the same Reynolds number. The input work rate is the work rate exerted on the body to produce the undulation. In Equation (8), ~ x, Px, ~ y and py are x or y-component of frictional stress or pressure exerted on the body, u and v are x and y-components of the velocity, + S indicates the both sides area of the body. PILOT COMPUTATIONS Flow calculation for a flat plate was performed at first. The computation was taken as the basis for the following simulation. Table 1 shows the dependence of the computed Cal on (/\Y)min for the flat plate at Re=1000. Cal represents the integrated friction drag coefficient along the line z=l.44x10~2 (which is neighboring to the symmetric line), while Cal represents the friction drag coefficient by Blasius solution. ~ i\ Y)min is minimum grid spacing in y-direction. Table 1 shows the results for rectangle plate with aspect ratio of 1.8. It seems that Cal can approach to Blasius solution (CfB) if (AY)min is chosen carefully. Therefore ~ 1\ y~min=0.00526 was adopted for the calculations at Re=1000. The method is also applied to find a minimum spacing for other Reynolds number. Table 1: Dependence of the computed Cal on ~ /\Y)min ( /\Y)min Blasius Solution 0.00847 0.00669 0.00526 4 Cf. (Cf7 - CfB) I COB 0.0840 0.0773 -7.93o/o 0.0809 -3.63% 0.0838 -0.27%
The turbulence flow around a flat plate was numerically simulated at Re=106. Figure 2 shows the computed y+ vs u+, where u+ and y+ are friction velocity and wall coordinate respectively. The flow is laminar at x=0.1091 and transitional at x=0.5933. It can be found that the velocity profiles show good agreement with the theoretical solution when the flow has been fully developed at x=0.8537 and x=0.9494. 30t . 25 .~20 15 10 5 to ~? 000oooooooooooooooooooooooooo 0 ~ , ~ / ,:~^ Ad,,-- o x~.1091 D/ ~ ~63' ~ x=0.5933 /~ v x=0.8537 0 x=0.9494 If+ + - 5.75(logl0Y +5, 5 og,Oy Figure 2: Computed y+ as u+ for a flat plat at Re=1 o6 The computed drag coefficient for NACA0012 at Reynolds number 2.8X106 is 6.7X10-3, which shows good agreement with the experimental result, 6.6X10-3 in Abbott and Doenhoff (1959~. In Figure 3, the computed pressure coefficients also give agreement with experimental data (Gregory and O'Reilly 1970~. -0.5L nit A) 0 5 Computed results Experimental data (Gregory et al. 1970) 02 04 06 08 i x Figure 3: Surface pressure distribution for NACA0012 at Re=2.8x 106 SIMULATION ON AN UNDULATING PLATE Firstly, the flow around a three-dimensional undulating plate with aspect ratio of 1/3 is numerically investigated. Computed results at Re=1000, a=0.1, b=1.5, n=1.1 are shown in Table 2. When the phase velocity Sp=1.56, CTX is greater than zero, while Sp=1.57, CTX is smaller than zero. With the increase of SO, the frictional force component CFX increases slightly while the pressure component ~CPX~ increases sensitively. CTX/CFX is an index to evaluate an error of the self-propulsion condition. From Table 2, it can be seen that alp has an error less than 1%. More accurate alp can be obtained by a linear interpolation from the computed results. For example, when Sp=1.5658, alp is interpolated as 0.56843 while the simulated alp at Sp=1.5658 is 0.56846. In order to find a suitable Sp and corresponding Hip, this kind of linear interpolation is reasonably used for the present study. Table 2: Sensitivity of alp to CTX/CFX at Re=1000, a=O. 1, b=1.5, n=1.1 Sp C Fx 1.56 0.1300 -0.1288 0.12x10~2 0.90% 0.576 1.57 0.1 305 -0. 1 3 1 3 -0.8 x 1 0-3 -0.63% 0.563 1.5658 0.1 303 -0. 1 303 0.2X 1 0-4 0.0 1 8% 0.568 Figure 4 shows the simulated history of hydraulic force coefficients (CFX, CPX, CTX), which are x-components of force coefficient given by Equation (6~. In Figure 4, Yend indicates the position of the trailing edge in y-direction. It is found that the amplitude of CFX is smaller than that of CPX. The motion of plate is sinusoidal, however CFX does not oscillate in the sinusoidal way. The resultant force CTX oscillates around zero. Because of the symmetry of the plate movement, there are two crests and troughs in one undulating period. Figure 5 shows pressure distributions on the symmetric plane (z=0) when the thrust reaches the maximum ACES reaches the maximum at ~2.604) and when the thrust reaches the minimum ACES reaches the minimum at ~2.707~. When the thrust reaches the maximum, there are two zones where the larger pressure difference (red and blue) acting on both sides of the plate generates thrust. When the thrust reaches the minimum, there is only one zone where larger thrust is generated. Cal 0.15 -A 0.05 A ~ ~ ~ / \ :i CTO.O5 - V \\1 \J : V V V V CPX 0O1 ~ ~ 1 ~ ~ ~ ~ /\ ~ ~ -v V V V V V V V -0.2 1 5 2 ' t 2 5 A ~ ~ ~ ~ .W~.~.~. 1.5 2 t 2.5 3 0.1 0.05 to -0.05 Figure 4: Simulated time history of hydraulic force coefficients (the upper) and the corresponding position of the trailing edge in y-direction (the lower) at Re=1000, a=0.1, b=1.5, Sp=1.5658 s
1 P: -0.7 -0.6 -o.s -0.4 -0.3 -0.2 -of o ~ ~ ~ 2 ~ 3 1 n 1 O -0.1 Figure 5: Pressure contours at symmetric plane when thrust reaches the maximum (the upper) and the minimum (the lower) atRe=1000, a=0.1, b=1.5, Sp=1.5658 Figure 6 shows time-averaged distributions of x-components of frictional stress r X' pressure Px and their resultant stress. The frictional stress acts as a resisting force on the whole area of plate, while the pressure acts as a thrusting force. The thrust mainly comes from the aft part of the plate. Figures 7 and 8 show x-components of frictional force and pressure contours at t=2.767 (3/81), 2.821 (4/8~, 2.874 (5/81) and 2.927 (6/81) respectively. It can be seen that there are two larger pressure zones, one is thrust (negative zone) and the other is resistance (positive zone), which move from the middle to the tail periodically. The distribution of frictional stress does not vary so much during the undulation. = === = -0.15 -0.125 -0.1 -0.075 -0.05 -0.025 0 Figure 6: Distributions of time-averaged x-components of fiction stress (the upper), pressure (the middle) and resultant stress (the lower) at Re=1000, a=0.1, b=1.5, Sp=1.5658 0.2 t - ~ /8~ ~ to ^1: ~ ~ ~ ~ C ~ ~ ~ OC A ~ 13 1 0~ _ Figure 7: Distributions of x-components of friction stress at every 0.125T step when Re=1000, a=0.1, b=1.5, Sp=1.5658 -0.3 -0.2 -0.1 0 0.1 0.2 13/8T| u o 0.2 0-4 X 0.6 0.8 ~ Figure 8: Distributions of x-components of pressure at every 0.125T step when Re=1000, a=0.1, b=1.5, Sp=1.5658 Dependence of self-propulsion on wave number was investigated. Table 3 shows the calculated self-propulsion condition at Re=1000' a=0.1' n=1.1. In Table 3' A is the maximum lateral excursion of the trailing edge of the plate and St is Strouhal number, defined by St=f xAlU (9) where f=b xSp is the frequency of undulation and U is 6
the oncoming velocity. As shown in Table 3, Sp decreases when b increases to get self-propulsion. The frictional force depends on the wave number. As b becomes larger, the frictional force increases. On the other hand, when b is very small (b=0.5) Sp is so high that the Fictional force becomes larger. The time-averaged x-component of friction stress for various b is compared in Figure 9. It can be seen that the increase comes from the aft of the plate, where the larger frictional force zone enlarges with the increase of b. Figure 10 shows time-averaged x-directional pressure distribution for various b. It is interesting that the patterns of the x-directional pressure distributions are similar, although their wave numbers are different. The peak of the distribution which contributes to produce thrust locates at 0.55 Cx<0.85 for each wave number, while the end of the plate does not contribute to produce thrusting force. Table 3: Computed results for self-propulsion at Re=1000, a=0.1 b 0.5 1.0 1.5 2.0 2.5 ' Sp 3.10 1.89 - 1.57 1.41 . 1.32 A 0.0988 0.0967 0.0938 0.0906 0.0875 St 0.153 0.183 0.220 0.256 0.289 CFX 0.126 0.125 0.130 0.137 0.143 flat plate 0 0.05 0.1 0.15 0.2 0.25 0.3 LIP 0.369 0.524 0.568 0.558 0.524 l Figure 9: Time-averaged x-components of friction stress for various b atRe=1000, GO.1 Figure 10: Time-averaged x-components of pressure for various b atRe=1000, a=0.1 The calculations for a=0.06, 0.08, 0.12 and 0.14 were carried out. Figure 11 displays the relationship between a, b and Sp when self-propulsion is achieved. As the increase of a or b, Sp decreases but it is greater than 1. Figure 12 shows the dependence of propulsive efficiency on a and b. It can be seen that there exists a zone, in which n p can be beyond 0.55, and that A p decreases gradually when the values of a and b move away Tom the zone. .5 06 0.08 0.1 a 0.12 0.14 Figure 11: Dependence of Sp on a and b 7
225t 2 1.75 1.5 1.25 1 0.75 ~ _ 0.06 0.08 0.1 0.12 0.14 a Figure 12: Contours of propulsive efficiency From Figures 11 and 12, it can be observed that alp seems to depend on a, b and Sp. These three parameters are combined in one parameter St (Strouhal number) as shown in Equation (9~. Figure 13 shows the dependence of the propulsive efficiency alp on Strouhal number St. The curves of alp concentrate within a narrow range so that, for an undulating plate, the governing parameter for the propulsive efficiency is not the amplitude, wave number or phase velocity, but Strouhal number. The highest alp is about 0.58 when St is around 0.23. The higher alp, which is greater than 0.55 for example, is achieved at 0.2< St< - 0.26 when Reynolds number is 1000. 0.6 _ 0.55 so 0.5 .O ~ 0.45 c' <,, 0.4 0.35 0.3 . a=0.06 · a=0.08 a=0.1 0 · a=0.1 2 1 · a=0.14 0.15 0.2 0.25 0.3 0.35 Strouhal number Figure 13: Dependence of propulsive efficiency on Strouhal number SIMULATION ON A TWO-DIMENSIONAL UNDULATING NACA WING Because Strouhal number was verified as the important parameter to achieve high propulsive efficiency at one chosen Reynolds number, the emphasis of this section is given on the dependence of propulsive efficiency on the body thickness and Reynolds number, as well as basic characteristics in fish-like locomotion. Like the simulation in the last section, the parameters are still chosen as a=0. 1, b=1.5, n=1. 1 for Re=1000. The basis for enhancing performance through unsteady flow control is the formation of large-scale vortices through body motion, the sensing and manipulation of these vortices as they move down the body, and the eventual repositioning through tail motion. These concepts constitute the essence of vorticity control. Figure 14 displays pressure distributions and streamlines around NACA0010 at Re=1000, a=0. 1, b=1.5, Sp=1.78. The obvious vortices have been generated at concave parts with the undulation of the body and travel down the body before they shed into the wake. The propagating undulation improves the pressure exerted on both body sides. The high pressure zone (indicated by red) travels simultaneously with the vortices down the body. Meanwhile from the figure, it is obvious that high pressure zone and low pressure zone always exert on both sides of the body and act as thrust, while in the wake the pressure is lower in every center of vortex. It shows the body sheds one vortex per half period when the tail is near its most lateral position (0/81~. The vortices shed from the tail and fade in the wake. They look like reverse Karman street. Thus the presence of vorticity in the wake of an undulating body in viscous fluid is a consequence of the need for a propulsive jet to counter the body drag. The relationship between the propulsive efficiency and body thickness was investigated. Figure 15 shows the computed Sp and alp for NACA wing with different relative thickness. The increase of body thickness leads to the increase of phase velocity and the decrease of propulsive efficiency, which implies some advantages for slender swimmers. Laminar and turbulent flow around an undulating NACA0010 was investigated. Figure 16 shows the variations of propulsive efficiency and frictional force coefficient. CFO is the frictional force coefficient for rigid body while CFX is that for undulating body at self-propulsion state. At low Reynolds number, CFX is over CFO by more than 70%, but at enough higher Reynolds number, CFX is smaller than CFO Table 4 lists the values of drag reduction and propulsive efficiency for NACA00 10 at a=0.05 1, b=1.0. Frictional resistance reduction is about 22% at Re=2xlo6 and about 32% at Re=3xlo6. It is found that the propulsive efficiency defined by Equation (8) is greater than 1 at Re=2xlo6 and Re=3xlo6. It numerically shows that the power required to propel an actively swimming fish-like body is smaller than the power needed to tow the body straight and rigidly at the same speed. 8
'.9L ~.8 -0.7 ~.6 ~.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 no in, -0.5 n ~ n is 1.8 1.7 1.6' 0.6 0.58 0.56 ~ c, 0 54 Ha : 0.52 6 ~ 10 12 14 1 60.5 Relative percentage thickness Propulsive efficiency / ~ / --^ ~ ~ - -^ -A Figure 15: Dependence of Sp and Alp on body thickness at Re=1000, a=0.1, b=l.S O O CFO for rigid body O O CFX for unduladog body _ G · Propulsive efficiency ~ ° 0 ~ . .~1.2 Figure 14: Pressure distributions and streamlines for NACAOO10 atRe=103, a=0.1, b=l.S, Sp=1.78 9 0.11 0.06 x ,~ A Lo 0.01 . . ' . . . 103 104 105 1o6 Reynolds number 1 . 1 c, 1 a) .O 0.9= a) 0.8 .> _ 0.7 Q 0 0.6 Q - c _ 0.5 Figure 16: Variations of fictional force coefficient and propulsive efficiency with Reynolds number for NACAOO 10 Table 4: Drag reduction and propulsive efficiency for NACAOO10 at a=0.05 1, b=1.0 Figure 17 shows the distributions of unsteady local friction stress coefficients on the upside body surface during one undulating period at Re=2x 106. The thicker line plots the friction stress for a rigid body. In the figure, x=0 and x=1 represent the leading and trailing edge respectively. As shown in Figure 17, the friction stress on undulating body surface obviously diminishes on most part of the body (0.3_x_0.9~. Figure 18 plots the calculated transition zone. During a period, the laminar flow domain is enlarged by the undulating motion, which implies that the fish-like undulation delays the turbulence transition. Figure 19 compares the velocity profiles at x=0.1416, 0.3510 and 0.7617. The thicker lines plot the velocity profiles for a rigid body, while the marks represent the unsteady profiles during one undulating period. The dashed line plots the velocity of logarithm law. It is found, at x=0.1416, the undulation makes the profiles deviate laminar form in a small degree, which increases local frictional stress as shown in Figure 17. While at
x=0.35 10, the undulation makes the turbulent flow relaminarize at some instants. At 318T, 4/8 T and 518T, the profiles deviate from logarithm law form, so the Fictional force is reduced greatly as shown in Figure 17. Meanwhile, the profiles at x=0.7617 are close to the logarithm law, but the undulation decreases the magnitude of the friction stress. O.On3 c o 0.2 0.4 0.6 0.8 1 X 4n 30 20 10 o rigid 0 0/8T 1/8T 2/8T D 3/8T ~ 4i8T ~20 a A/RT 30 10 an Figure 17: Distributions of local friction stress coefficient for NACAOO10 at R~2X 1 o6 1n o.s 0.8 0.7 0.6 ~0.5 <0.4 0.3 1 0.2 0.1 O 0/8 The area between two dashed lines: transitional zone of rigid body The area between two solid curves: t~anstional zone of undulating body , 1/8 2/8 3/8 4/8 5/8 6/8 7/8 0.5 T Figure 18: Transition zone variations caused by undulation at Re=2x 1 o6 for NACAOO10 Figure 20 plots the variations of pressure gradient in x-direction at the mesh points most near the upside body surface. The undulation reduces the mean pressure gradient at almost whole zone, which may be the main reason for turbulence suppression. The corresponding flow phenomena can be observed in Figures 21 and 22, which respectively plot the distributions of vorticity ~ co ~ and relative velocity uref at the mesh points most near the upside body surface. The relative velocity uref denotes x-component velocity subtracted by velocity component on the body surface. From Figure 21, globally the vorticity becomes smaller under undulation, which implies that turbulence strength is also reduced because the eddy viscosity is bound to diminish by mixing length formula (Baldwin and Lomax 1978~. From Figure 22, when x ranges between 0.3 and 0.9, the relative velocities Uref are smaller than those for rigid case. The decrease of relative velocity leads to the reduction of friction stress. inner ,0~ o o ~ a o o 0/8T X=0.4446 / t/\ ~ i' 2/3T / ~ 4/8T /°~ ----a 5/8T it ----------- 7/8T_ - f v D O O * 1 IO910{ 2 G <O ~ ~ O O O O X=~. . . . , . . . . . 1 log,Oy. 2 ,~ s - ;~-~ ~ ) 1 log,Oy+2 3 Figure 19: Effects on velocity profiles by undulation for NACAOO10 at Re=2x 106 .... nglc~ 0 0/8T v 2/8T ~ 4/8T - o 6/8T · mean value ~ ~-1---~- -u.oO 0.2 04 X 0 6 0 8 1 Figure 20: Pressure gradient in x-direction at mesh points most near the upside body surface Inn 2000 1 ~ 11500 Anon 500,~ rigid _ ° O/8T t · 1/8T 2/8T 3/8T ' 4/8T ° 5/8T Ol y ;~' 0 0.2 0~4 X 0.6 0.8 1 Figure 21: Comparisons of vorticity at mesh points most near the upside body surface 0
040 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 ngid · 0/8T t · 1/8T 2/8T - ~ 3/8T 4/8T it. r 0 5/8T 'id 0 0.2 0~4 X 0.6 0.8 1 Figure 22: Comparisons of relative velocity at mesh points most near the upside body surface The computations for different phase velocity were performed. Figure 23 shows the variations of hydraulic coefficients with phase velocity for NACA0010 at Re=2xl06, a=0.051, b=1.0. When Sp=1.2, c TX becomes zero so self-propulsion is achieved. When C TX iS larger than 1.2, the net thrust can be obtained. The present study evaluates the propeller efficiency when the flexible-hull body is taken as a propulsor. The related propeller efficiency ~7 is defined as J outputdt / T U |TXdI q= r = T (10) |inpu~dt/T J.[| (a +Px) ups+ ~ (ry +py) vds]~ T IS IS where the input work rate is defined as same as that in Equation (8) for up, the output work rate is the product of net thrusting force Tx and oncoming velocity U. Certainly ~ is zero for self-propulsion state, just like case 1 in the Table 5. Cases 2, 3 and 4 represent the computed results for NACA0010 at different phase velocity. With the increase of SP, ~ and the thrust coefficient CTX increase monotonously when SP ranges from 1.2 to 1.6. The difference between case 4 and case 5 is the body thickness. The thinner flexible-hull can achieve higher efficiency and generate larger thrust. The comparison between cases 5 and 6 explores that ~7 increases with Reynolds number. The high efficiency of ~ shown in Table 5 indicates that fish-like undulating body can be used as a propulsor because of its satisfied propulsive performance at higher Reynolds number. 0.0z , 1~ 0.01 ~ . 1C) 1C) -0 01 ~ Ado_ A_ _ ______ _1____~ _ ~~ __ _ L _ _ I \ \ I _ _ _ _ _ _ _ A_ _ _ _ _ _ _ _ _ _~ _ _~ _ ~ 1 -9 _ _ _ _ _ _ _ _ ~ _ _ _ _ _ _ _ _ _ _ _ _ _ >~< _ I 1~ . . , . . . . . . . . . . , · ~ 0 05 Sp 1 1.5 Figure 23: Hydraulic coefficients with phase velocity for NACAOO10 atR~2xlo6'a=0.051, b=1.0 Table 5: Investigation on propeller efficiency Case _ 2 -3 4 5 6 Re 2xlo6 2xlo6 2xlo6 2xlo6 2X106 3X106 .- Relative thickness 10% 10% 10% 10% _ O Sp .2 .3 1.5 1.6 1.6 _ 1.6 CTX x 1 0 . -0.001 1 -0.321 1 -0.9986 -1.353 -1.935 -1.981 o 0.2992 0.5019 0.5389 0.6264 0.6438 SIMULATION ON THE THREE-DIMENSIONAL UNDULATING BODY The two-dimensional results were expended to three-dimensional cases by setting the body width is 0.24. At the plane of z=0, the body section is NACA wing, while the body thickness is zero on the fringes (z=+0.12~. The body thickness varies smoothly at every x=const. plane. The body is called as "flat fish" in the present study. The simulated results at Re=103 are listed in Table 6. When the computed result of 3D (case 8) is compared with 2D (case 7), both the phase velocity and frictional drag increase becomes smaller. That the propulsive efficiency becomes small shows that the 3D effects weaken the propulsion performance. The prediction is same to the conclusion by Liu et al. (1997), whose simulated object was a swimming tadpole. Figure 24 shows three transverse cuts (parallel to yz axes), demonstrating that the flow varies along the longitudinal direction. The longitudinal vortices exist both near the body and in the wake. Probably the strong rotating flow has significant effects on propulsive efficiency for 3D case. Also, like the conclusion drawn in two-dimensional case, the increase of body thickness leads to the increase of phase velocity and the decrease of propulsive efficiency (see cases 8 and 9~. Table 6: Investigations on self-propulsion for the flat fish at Re=103, a=0. 1, b=1.5 . Case 7 8 9 Maximum relative thickness 10% (2D) 10% (3D) 14% (3D) T: 1.761 1.610 [ 1.694, I ( C Fr / COO) - 1 1 75.7% l 1 65.0% T 76.4% 1 UP 1 1 1 1 0.547 T 0.4581 1 0.408 1 The simulated results for a flat fish with maximum body thickness of 14% at Re=2x105 are listed in Table 7. Turbulent flow simulation shows smaller frictional drag increase and higher propulsive efficiency, which implies again that the undulation can suppress turbulence. From cases 10 and 9, the propulsive efficiency enhances with the increase of Reynolds 11
number. One phenomenon is shown in Figure 25, in which x-component of pressure stress is compared for the two cases. Pressure acts as thrusting force, but for lower Reynolds, the time-averaged pressure on a small area near the trailing edge is not thrust but resistance. Table 7: Comparison on flow state for a flat fish with maximum body thickness of 14% at Re=2x 105, a=0.067, b=1.087 Case Flow state SP (Cry/ CFO)-1 ~ P 10 Laminar 1.278 55.9% 0.568 1 1 Turbulent 1.287 17.5% 0.662 ~ i , .~! :~.: ~~ ~~.~; :: ~ :~: ~1 Pressure: ~.7 -0.6 -0.5 -0.4 ~.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 `~ it- ~~.~ Figure 24: Pressure and velocity distributions along planes perpendicular to the advancing direction of a flat fish with maximum thickness of 10% at Re=103, a=O. 1, b=1.5, Sp=1.61 n 1 0.2 n 1 n- O ~ ~ . . i . , . . _ 0 0.2 0.4 0.6 0.8 1 X ~ () 0.2 0.4 X 0.6 0.8 1 Figure 25: Time-averaged x-component of pressure distributions atRe=103 and 2XlOs for a flat fish CONCLUSIONS The MAC method with a body fitted coordinate system is applied to simulate the laminar-turbulent flow around an undulating advancing body. A modified turbulence model is adopted for Reynolds averaged Navier-Stokes equation. The simulation shows that pressure is acting as thrusting force while frictional force is acting as resistance during the undulating motion. Thrust goes up with the increase of undulating wave propagation velocity. Strouhal number is the most important governing parameter for the propulsive efficiency. The propulsive efficiency is enhanced with the increase of Reynolds number or the decrease of body thickness. For laminar flow, frictional force enlarges with the increase of phase velocity. But for turbulent flow, fish-like locomotion can reduce frictional resistance by turbulence relaminarization and achieve a propulsive efficiency more than 1. Two vortices shed from the tail during one undulating period. Vorticity travel shows close relation to pressure distribution. Fish-like undulating body shows the potential as a propulsor. REFERENCES Abbott, I. H. and Doenhoff, A. E. V., "Theory of Wing Sections----Including a Summary of Airfoil Data", Dover Publications, INC. New York, 1959. Ahlborn, B., Harper, D. G. Blake, R. W., Ahlborn, D. and Cam, M., "Fish without Footprints", Journal of Theoretical Biology, Vol. 148, pp. 521-533, 1991. Anderson, E. J., McGillis, W. R. and Grosenbaugh, M. A., "The Boundary Layer of Swimming Fish", The Journal of Experimental Biology, Vol. 204, pp. 81-102, 2001. Azuma, A., "The Biokinetics of Flying and Swimming", Springer-Verlag, Tokyo, 1992. Babenko, V. and Yaremchuk, A. A., "On Biological Foundations of Dolphin's Control of Hydrodynamic Resistance Reduction", Proceeding of International Symposium on Seawater Drag Reduction, pp. 451~52, Newport, RI, July, 1998. Baldwin, B. S. and Lomax, H., Approximation and Algebraic Model "Thin Layer for Separated Turbulence Flows", AIAA 16th Aerospace Sciences Meeting, pp. 1-8, Huntsville, Alabama, January 16-18, Barrett, D. S., Triantafyllou, M. S., Yue, D. K. P., Grosenbaugh, M. A. and Wolfgang, M. J., "Drag Reduction in Fish-Like Locomotion", Journal of Fluid 12
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