Vol. 46 . No. 1 February 1999 T E C H N I S C H E U N I V E R S I T E I T S cheepsfaydz-omechani ca A r c h i e f
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A S y s t e m for Measuring the Six Degrees of Motions of a Moving B o d y
by Gregory J. Grigoropoulos and Costas G. Politis
Investigation of the Vortex-Wave Wake behind a Hydrofoil by Nikolai Kornev and Andrey Taranov
N u m e r i c a l Simulation of the Unsteady F l o w Past a Hydrofoil by Maurizio Landrini, Claudio Lugni and Volker Bertram
R e - O p t i m i z a t i o n of the H u l l F o r m of a Fast Displacement C a t a m a r a n F e r r y
by Apostolos Papanikolaou and Nikos Dafnias
N u m e r i c a l P r e d i c t i o n of Viscous Propeller Flows by Heinrich Streckwall
C a v i t a t i o n Investigations in T w o Conventional Tunnels and in the H y d r o d y n a m i c s and C a v i t a t i o n T u n n e l H Y K A T by Ernst-August Weitendorf and Harry Tanger
SHIP TE( HilMY MMCH
J o u r n a l f o r Research i n S h i p b u i l d i n g a n d Related Subjects Founded by K . Wendel i n 1952
Editors:
H. Söding, V. Bertram, Technical University of Hamburg-Harburg, Germany Co-editors:
Dr. G. Delhommeau, Ecole Centrale Nantes, Prance
Prof. L . Doctors, University of New South Wales, Australia Prof. J. Jensen, Danish Technical University, Denmark Prof. D. Karr, University of Michigan, USA
Prof K.-Y. Lee, Seoul National University, Korea Prof. H . Miyata, University of Tokyo, Japan
Prof T. Moan, Norw. Inst, of Technology Trondheim, Norway Dr. F. Noblesse, David Taylor Model Basin, USA
Prof A. Papanikolaou, National Techn. Univ. of Athens, Greece
Prof J. Petershagen, Technical University of Hamburg-Harburg, Germany Prof P. Sen, Newcastle University, U K
Dr. G. Thiart, University of Stellenbosch, South Africa Prof T. Yao, Hiroshima University, Japan
Papers and discussions proposed for publication should be sent to Prof. H. Söding, Arbeitsbereiche Schiifbau, Lammersieth 90, D22305 Hamburg, Germany; Fax -1-49 40 2984 3329; e-mail soeding@schiffbau.uni-hamburg.de.
Vol. 46 • No. 1 • February 1999
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This issue of Ship Technology Research is devoted to Prof. D r . - l n g . H a r a l d K e i l who celebrated his 65th birthday on November 18, 1998, and retires at the end of the current lecture term. Page 3
A b s t r a c t s
Gregory J. Grigoropoulos, Costas G. Politis
A System for Measuring the Six Degrees of Motions of a Moving B o d y Ship Technology Research 46/1 (1999), 4-7
A system for measuring the motions of ships or models in a seaway using seven accelerometers mounted on the body is described and experimentally validated. I t is suitable for both linear and nonlinear motions w i t h satisfactory results.
Keywords: seakeeping, motion, measurement, acceleration
Nikolai Kornev, Andrey Taranov
Investigation of the Vortex-Wave Wake behind a Hydrofoil Ship Technology Research 46/1 (1999), 8-13
The coupled vortex sheet and waves behind a hydrofoil are computed by means of a new nonlinear vortex method applicable to steady three-dimensional flows. The method is validated by comparison with measurements. Numerical investigations focus on the necessity to account for nonlinear wake effects in the hydrodynamics of fast ships. The new vortex model appears fully capable of explaining the interaction between a free surface and tip vortices.
Keywords: fast ship, hydrofoil, vortex wake, wave, vortex method
Maurizio Landrini, Claudio Lugni, Volker Bertram
N u m e r i c a l Simulation of the Unsteady F l o w Past a Hydrofoil Ship Technology Research 46/1 (1999), 14-30
The two-dimensional unsteady flow about a hydrofoil moving beneath a free surface is studied by an inviscid model w i t h embedded vorticity, generated through a suitable unsteady K u t t a condition. The mathematical problem is recast in terms of boundary integral equations for which an accurate and efficient solver is developed and coupled w i t h a Runge-Kutta time marching scheme. Favorable com-parisons with both experimental data and computations performed by different numerical techniques are reported for a large variety of test cases, involving also details of breaking waves kinematics. Keywords: hydrofoil, time simulation, free surface, boundary element method
Apostolos Papanikolaou, Nikos Dafnias
R e - O p t i m i z a t i o n of the H u l l F o r m of a Fast Displacement C a t a m a r a n F e r r y Ship Technology Research 46/1 (1999), 31-34
The paper addresses the successful re-design of the hull form of a fast displacement catamaran ferry for increasing its initial displacement and payload capacity. The re-design of the underwater form improved considerably the calm-water performance due to a far better hydrodynamic efficiency. The paper describes the hull form re-design methodology and presents related theoretical and experimental predictions on the ship's calm-water performance and first experience of the converted ship i n service. Keywords: SWATH, catamaran, resistance, ship design
Heinrich Streckwall
N u m e r i c a l Prediction of Viscous Propeller Flows Ship Technology Research 46/1 (1999), 35-42
A commercial Ranse solver was used for propeller open water calculations. Simple concepts were pursued to achieve a quick grid generation procedure, whereby cyclic boundaries helped to keep the number of cells below 200,000. Calculations on two propellers are presented. The theoretical results of thrust, torque, velocities and pressures are in good agreement w i t h experiment. Moreover the Ranse calculations could confirm the outcome of a panel method, which was applied parallel in one case. Keywords: propeller. Ranse, panel method, grid generation
Ernst-August Weitendorf, Harry Tanger
Cavitation Investigations in T w o Conventional Tunnels and in the H y d r o d y n a m i c s and Cavitation T u n n e l H Y K A T
Ship Technology Research 46/1 (1999), 43-56
Cavitation model tests have been carried out in the small, medium size, and in the hydrodynamics and cavitation tunnel H Y K A T of HSVA. The model cavitation in the 3 tunnels for the low-skew propeller of the "Sydney Express" and the skew-back propeller of the "Hongkong Express" is compared with full-scale cavitation. Micro air bubbles were mea.sured w i t h the Phase Doppler Anemometer (PDA), and solid particles w i t h a microscope. Their influence as nuclei on the cavitation process is addressed. The different cavitation characteristics are considered i n view of the different generation of micro air bubbles, flow effects in the test sections, and other scale effects in the 3 tunnels.
Keywords: model, propeller, cavitation, scale effect
Congratulations to Prof. D r . - l n g . H a r a l d K e i l
This issue of Ship Technology Research is devoted to Prof Dr.-lng. Harald Keil who celebrated his 65. birthday on November 18, 1998, and retires at the end o f t h e current lecture term.
Born in Upper Schlesia (now Poland) he grew up in Timmendorfer Strand at the Baltic sea. After practical work at Flender-Werft in Liibeck he studied naval architecture in Han-nover and Hamburg since 1955. His student design project of a research vessel contributed to the ship "Meteor", the first German research vessel built after the 2nd world war. After his diploma examination in 1961 he did research at the Institut für Schiffbau (IPS) in Hamburg. I n 1980 he became professor for computer-aided ship design in Hannover. When the naval ar-chitecture department moved from Hannover to Hamburg during the 80es, Keil returned to the IPS of Hamburg University. He was manag-ing director of this institute from 1985 until the institute was split into four separate working groups belonging now to the Technical Univer-sity of Hamburg-Harburg in 1998.
Harald Keil's research field is quite diverse. A few of his numerous publications may illustrate this: 1962: The planned research vessel; 1974: Hydrodynamic forces on cylinders at a free surface in shallow water; 1977: Engineers i n ship technology: required skills, education and training; 1998: Energy efiiciency and pollution. He performed and published quite a number of full-scale measurements on board ships, dealing w i t h both vibrations and seakeeping.
Harald Keil placed very much emphasis on teaching students and organizing courses, on research planning and management, and on administrational affairs. As head of a committee on education and training of Schiffbautechnische Gesellschaft STG, he organized a large number of training courses on diverse subjects for engineers in industry, editing many volumes of valuable lecture notes. Since 1990 he edits the 'Handbuch der Werften', a book series summarizing scientific results for practical engineering work. He was vice speaker (acting partly also as a managing director) of the Sonderforschungsbereich 98 "Schiffstechnik und Schiffbau" in which about 40 scientists worked for 15 years on ship economy and safety problems, and was co-editor of the book summarizing the results of this large research project (1989). He is member of the board of the Hamburg Ship Model Basin; he participated i n the reorganization of the East-German universities after the unification; and for many years he was head of the STG and, before that, head of its Technical and Scientific Board. I n all these functions he gained respect and gratitude.
I n teaching. Keil always warned his students of trusting in computer results without understanding them. He not only performed himself all statical calculations for his own house, but he did also much of the timber work during its construction. Performing all his numerous tasks perfectly, he requires the same from his collaborators. Many who did not correspond to his high standards have faced his criticism. On the other hand, many, i f not all, of the researchers of the former IPS profited substantially by being freed from organizational and practical work done by him.
A l l colleagues, former and present students and friends wish him many more active years in good health.
A System for Measuring the Six Degrees of Motions
of a M o v i n g Body
Gregory J . Grigoropoulos, National Technical University of Athens^ Costas G . Politis, Hellenic Register of Shipping S.A.
1. Introduction
A method for measuring the six degrees of freedom ship motions is presented. I t uses seven accelerometers mounted on the body. I t provides a reliable and inexpensive alternative to other systems, e.g. gyro-stabilized platforms or optical tracking devices.
The accelerometers are mounted on the same horizontal plane o f t h e body (Fig. 1). Although six accelerometers suffice to determine the body motions, we use seven accelerometers for simplicity of the equations and for cross-checking of the measured accelerations.
2. Formulation of the problem
We aim to derive the body motion from the measured accelerations. We use two coordinate systems: an inertial system Oxoyo^o moving w i t h the average speed of the body, and a body fixed system Cxyz which coincides w i t h the inertial one i n the absence of body oscillations.
Let Xo{P) be the position vector of a point P i n the inertial system, and X{P) the corresponding vector in the body-fixed system. Then
Xo{P) = Xo{C) + A-'X{P). A is the transformation matrix according to Jeffers (1976):
(1)
A =
cos Ö cos •0 cos Ö sin'0 —sinö - sinV»cos(/? +cos sin(/3sinö c o s c o s Ö + s i n Ö s i n ( / ? s i n c o s ö s i n < / ?
sini/jsin(/5-I-cos V'cos(/?sinö - cos V'sin(/?-|-sinV'sinÖcos</) cosöcos(^
(2)
Using the conventions provided i n standard texts, e.g. SNAME (1952), the Euler angles are
6 = angle of pitch, cp = angle of roll, ip = angle of yaw.
Since A defines an orthogonal transformation, A'^'^ defines also an orthogonal transformation and is the transpose of A.
' N T U A , Dept. N A M E , F O B 64070, G R 15710 Zografou, Greece, gregoryOfluid.mech.ntua.gr
Differentiating eq.(l) twice w i t h respect to time {X{P) is constant) yields
Xo{P)=Xo{C) + A-^X{P). (3)
Applying eq.(3) to the positions of accelerometers 1,2,3,4, and 7, taking the dot product w i t h the unit vector a^ssociated to their sensitive axes, and accounting for gravitational effects, we get (Politis et al.
1995):
Xo{P)^A-'-r + A-'X{P)+g (4)
with
07
{asRi + a4i?3)/(-R3 + Ri) {aiR2 + a2Ri)/{Ri+R2)
(5)
üi {i = 1,...,7) is the output of the ï t h accelerometer, and Ri its distance f r o m the body-fixed
coordinate origin (Fig. 1).
Eq.(4) can be considered as the equation of motion of the body i n terms of the measured acceler-ations (the matrix V) and the transformation matrix A. Thus, i f A can be calculated, eq.(4) can be used to obtain the three translatory velocities and displacements of the arbitrary point P of the body.
Now the calculation of A w i l l be described. Applying eq.(3) successively at each accelerometer position (z = 1,...,6) and subtracting the outputs of each pair of accelerometers, we obtain the following non-linear equation system w i t h respect to the angular velocities w of the body expressed in the body-fixed system (Miles 1986):
LOx + UJyUJz LOy — UJxiOz = Flit) = m ) = F2{t) where Fi{t) = a-b — Q6 Rb + Rn F2{t) = 0-2 ~ 0-1 F,{t) = ^3 ~ Q4 Rz + Ri (6) (7) i ï l +
-^2
These are the fundamental equations of motion of the rigid body defining its angular velocity u i n terms of the measured accelerations. The outputs of the six accelerometers {i = 1,...,6) are not independent; e.g.
ae = {aiR2 + a2Ri)--———~a5 — . (8) Kl +
it2 its
But we use the outputs of 7 accelerometers for the simplicity of the derived equations and to cross-check the measured accelerations.
Once oj is calculated, the Euler angles can be determined by solving the following non-linear equa-tion system (SNAME 1952):
tp = Ux + tan 9{ujy simp + ujz cos (p), 0 = ujy cos (p — u}z sin tp,
lp = {ujyS'mip + u>zCos(p)/cos9.
(9)
The main steps of the procedure to determine the body motions are:
Solve system (6) to obtain the components of the angular velocity cD; solve system (9) to obtain the Euler angles (p, 9, and I/J', use (2) to determine A; and integrate (4) twice to obtain the displacements of any body point P.
3. Numerical solution of the equations
Systein (6) is numerically solved by an iteration of fixed point type: I n step k, knowing the angular velocities ui^\t), J y \ t ) , wi'^^(i), we solve the differential equations
= F2{t)+J^\t)-J^\t), (10)
Initially the angular velocities are assumed to be zero.
Eqs.(10) are integrated using the Fast Fourier Transform (FFT) method, exploiting the following property of Fourier transforms J^:
I f f{t) = h{t) then f{t) = J^-^ (11)
The constants of integration are removed by imposing the condition of zero average angular velocities. A similar procedure is applied in solving system (9) for the Euler angles. Initial estimates for the average roll and pitch angles are
= E{Pi{t)}/g, (/pW = -E{P2{t)}/9. (12)
Here E{Pi{t)} stands for the mean value oi Pi{t). These estimates are readily obtained assuming zero average surge and sway accelerations. These conditions are also used for the final estimation of the integration constants relating to the roll and pitch angles. For the integration constant concerning yaw angle, the condition of zero average value is imposed.
The F F T method uses high-pass and low-pass filters. A n investigation is needed to properly determine the cut-off points of the filters. Furthermore, since the numerical solution uses an F F T integrator, the number of points in each acceleration time history aj{t) must be a power of two, and all signals must be cyclic to prevent errors which otherwise result from Gibb's phenomenon. This is achieved by tapering the signals to zero at both ends of the samphng interval. A n optimum tapering interval length, which greatly reduced these errors, was found to be 5% of the total sampling time T .
4. E x p e r i m e n t a l verification of the method
The method w£is first evaluated by computer simulation using typical motion spectra. Subsequently a 1:4 model of a high-speed semi-displacement yacht was tested in the towing tank of the laboratory for ship and marine hydrodynamics of N T U A . The model was excited by a Bretschneider wave spectrum with significant wave height 20cm and modal period 1.56s (model scale). The model was attached with large yaw angle to the non-moving carriage via a balance which suppressed surge, sway and yaw while heave, pitch and roll motions relative to the carriage were measured. Fig. 2 compares these measurements w i t h the motions derived from the accelerometers. The agreement is estimated satis-factory, enhancing our confidence i n the reliability of our method. Also two models of passenger/car ferries were tested, w i t h similar success, in the laboratory of harbour works of N T U A . I n one of these models a digital gyro recorded the rolling motion and rolling rate. The proposed data acquisition system could be used also onboard of full-scale ships floating or advancing at any speed in a seaway for the evaluation of their hydrodynamic qualities.
6.00
•6.00
20.0 25.0
Time [sec]
Fig.2: Motion responses of sailing yacht model References
JEFFERS, M.F. (1976), Analytical methods for determining the motion of a rigid body equipped with internal
motion-sensing transducers, DTNSRDC report no. 76-0041
MILES, M.D. (1986), Measurement of six degree of freedom model motions using strap down accelerometers, 21st ATTC, Washington
POLITIS, C; GRIGOROPOULOS, G.J.; LOUKAKIS, T.A. (1995), A measuring system of the six D-O-F
motions of a floating body, NTUA report NAL-138-F-1995
SNAME (1952), Nomenclature for treating the motion of a submerged body through a fluid, SNAME T&R Bulletin 1-5
Investigation of the Vortex-Wave Wake behind a H y d r o f o i l ^
Nikolai K o r n e v and A n d r e y Taranov
Department of Hydromechanics, State Marine Technical University, St.Petersburg^
1 Introduction
The trailing vortices together w i t h the waves behind a hydrofoil, which will be called vortex-wave wake for shortness, play a key role i n predicting the interaction between forward and aft wing systems of fast ships. To avoid a negative influence of the free surface, usually the depth of submergence of modern hydrofoils is chosen sufflciently large. I t is widely accepted that linear potential theory can predict the hydrodynamics of the forward wing w i t h the necessary accuracy. However, in practical calculations of the back wing system, the nonlinearity caused by the vortex wake deformation, and nonlinear terms in the boundary conditions at the free surface must be accounted for, because the aft foil operates in the wake of the forward one.
Most papers on nonlinear hydrofoil hydrodynamics deal with 2-D flows around profiles under a free surface. Various panel methods use vortex, source and dipole distributions (Zaroodny and Greenberg
1973, Zalosh 1976, Ba et al. 1993, Shigunov 1996). Sadovnikov (1994) used a modified panel method
to study afiow w i t h wave breaking using Tulin's model. More sophisticated methods based on finite-volume methods were developed by Kang (1996) and Muzaferija et al. (1996). They may be applied also to investigate wave breaking and the influence of viscosity.
Three-dimensional problems have been investigated by linear methods. Lukashevich(l979) simu-lated the interaction of the hydrofoil t i p vortex w i t h the free surface i n much detail, and he developed practical methods for estimating the interaction between hydrofoils. The vortex-wave interaction at small Froude numbers was computed by Janson(1996), who dealt w i t h the flow around surface-piercing wings and hydrofoils close to a free surface.
Comprehensive experimental studies of hydrofoils were performed by Lukashevich(1979) and
Ep-stein and Blumin (1968). Basin(1993) reported about a vortex-wave resonance, which occurs at small
submergence ( < 0.5 • chord length) and small Froude numbers based on chord length (0.4 < Fn < 0.6). In spite of a considerable body of literature, some practically important questions are still open. Our study is aimed on a numerical method and software for designing fast ships of Auto Jet type
Marinteknik (1998). The key new aspect is the nonlinear analysis of the vortex-wave wake behind
hydrofoils. The analysis shows a strong mutual influence of the free surface and the vortex system behind a hydrofoil. Using a vortex method, we give an intuitive explanation of the wave-induced motion of the t i p vortices.
2 N u m e r i c a l method
We consider a hydrofoil advancing at constant forward speed V in an incompressible, inviscid and irrotational fluid. We use a cartesian coordinate system fixed to the hydrofoil; x points forward, y upward, z i n transverse direction. To simulate nonlinear waves we use an integral equation derived by distributing a vortex sheet of unknown vector density 7 on the free wave surface. 7 can be obtained from the dynamic boundary condition
where v is the velocity on the free surface, g gravity acceleration, and y wave elevation. Using the ' T h e study was made for the St.Petersburg Office of Marine Technology Development L t d .
^Lotsmanskaya 3, 190008 St.Petersburg,Russia, e-mail:kornev@mtu-ic.spb.su
Vv. gy =
0,
(1)well-known properties of the vortex sheet
v = vo + ^ v ^ , Vo = 0 . 5 ( v + + V " ) , v.y = v " - v + = n X 7 , (2) where v + and v ~ denote the limiting values of the velocity above and under the free surface,
respec-tively, and n is the upward normal vector on the free surface, gives
Vv^x = |vop + v^vo + ^ | v ^ p + 2gy - 2Vvox- (3)
Without loss of generality we can decompose 7 at each point on the free surface w i t h i n the computa-tional domain as the sum of two components being tangential to the free surface: 7^ is the component perpendicular to the x axis, 7^ is perpendicular to 7 ^ . By substitution of (2), (3) becomes
1 Y
7c = |voP + (n X 7)vo + + 2 - ^ - 2vox, (4) where 7 = 7 / y , v = v/V, Y = y/C, Fn = V/y/gC are nondimensionalized w i t h respect to V and
wing chord length C. Together with the zero-divergence condition and conditions at infinity,
V 7 = 0, I7I ^ 0 for a; -> 00, I7I ^ 0 for 2 ^ ± 0 0 , (5)
(4) represents a complete system of governing equations for the vector density 7 . The velocity v q can be calculated using Biot-Savart's law.
I n the numerical implementation the free surface is modelled within the rectangle defined hy XQ, xi,
Zo and zi. As usual in the vortex lattice method, the surface vorticity 7 is represented by a number
of ring vortices. Thus the first equation in (5) is satisfied automatically. The kinematic condition on the free surface is used to calculate the shape of the free surface and to adjust the height of the vortex lattice. This works only i f the waves do not break. Both 7 and y are assumed zero at the front and both side boundaries of the modelled free surface region.
Also the hydrofoils are modelled by the vortex lattice method, applying a distribution of horseshoe vortices the centre parts of which are located on the mean line surface of each l i f t i n g element, while the ends continue into the trailing wake.
The strength o f t h e bound vortices is determined from the condition of no flow through the mean foil surface at discrete control points lying in the centers of panels. At the trailing edge a K u t t a condition is satisfied by the special arrangement of control points and vortices. The thickness of airfoil is accounted for by source distribution w i t h strength equal to the thickness gradient. The a priori unknown form of the vortex wake is determined iterativeiy from the streamhne equation according to the Nonlinear Vortex Lattice Method. Details about the method and algorithm are given i n Kornev(1998).
3 Results
3.1 Validation tests
A h quantities w i l l be made nondimensional by relating them to wing chord length C and speed of motion V. To calculate the wave elevation a nd forces w i t h necessary accuracy, the forward end of the computational domain should be located at distance > 2C in front of the wing leading edge and the span of the computational domain should be two and a half times greater than the wing span. We used a uniform vortex lattice w i t h mesh size A = 0.25(7 in the region above the wing. A n exponentially stretched lattice was applied outside of this region to reduce computational costs.
First we compute the flow around a submersed rectangular fiat plate of aspect ratio 6.5 at 6° angle of attack. We investigate the influence of the aft end location xi of the computational domain on the wave elevation and the number of iterations N for Fn = Vj^fgC = 4. As an example the surface elevation y at a control point x^jC = - 2 3 , 2: = 0 is used. Fig. 1 indicates the ratio r between y
computed for various xi and N, and the 'best' value computed for xi/C = 4 0 , N = 100. The test indicates convergence and negligible influence of the aft end for |a;i - Xc\ > I O C .
Further tests were performed for a rectangular flat plate of aspect ratio 6.5 at small angles of attack for Fn = 13, submerged under the undisturbed surface by h/C = 0.6 at the leading edge. Results (see Fig. 2 ) are agree well w i t h experimental data by Kosov and Lukashevich (see Lukashevich(1979)).
The wave elevation behind a rectangular, highly loaded wing of aspect ratio 5 at 6 ° angle of attack is shown in Fig. 3. The wing section was circular on the upper and planar on the lower side, w i t h 1 5 % of C maximum thickness, submerged under the undisturbed surface hy h/C = 0 . 3 6 6 7 at the leading edge. Our numerical results for F„ = 5 . 2 1 in the cross-section Xc/C = - 8 compare relatively well with experimental data by Epstein and Blumin (1968). Good agreement was found also for the lift and for results at different F„ and h.
3.2 Nonlinear waves behind a hydrofoil
Figs. 3 and 4 illustrate the wave pattern computed by using semi-linear and nonhnear theories. I n the semi-hnear theory, nonhnear terms in the boundary conditions were omitted, whereas the deformation of the vortex system was included. Taking into account all nonlinear terms results i n the following effects:
- the wave trough becomes deeper, the crest higher;
- the wave crest occurs at shorter distance from the hydrofoil;
- the wave elevation outside of the wave trough is less than in the semi-linear theory; - the trough sides becomes steeper;
- the wave pattern compares better w i t h meeisurements.
The results of the semi-hnear and nonlinear analysis yield a significant difference i n the wetted area of a ship huh (about 1 2 percent for the Auto Jet) and i n hydrodynamic characteristics of the back wing.
3.3 Influence of the free surface on tip vortices of a hydrofoil
Knowing the field of the vortex-induced velocity, it is not difficult to predict the infiuence of the vortex system on the free surface. A presumably new aspect of this problem is how the free surface affects the tip vortices of a hydrofoil. This influence can provoke a connection between a free surface and a tip vortex which, in its turn, leads to the undesirable ventilation of the vortex and a hydrofoil.
Willert and Gharib(1997), dealing w i t h the flow due to a vortex pah under a free surface, confirmed
the connection of a tip vortex w i t h the surface. I n their experiments, spatially modulated vortex pairs were generated below a free surface by two counter-rotating flaps, whose edges approximate a sinusoid. As a result of connection w i t h the free surface, the initially modulated vortex tube was broken into a hne of U-vortices. We show that the presence of the wave trough behind a hydrofoil gives rise to a new mechanism of the vortex-wave interaction in comparison w i t h that described by
Willert and Gharib(1997). Epstein and Blumin (1968) observed the connection between a tip vortex
and the trough side behind a hydrofoil, but the tip vortex trajectory wasn't investigated. Lukashevich
(1979) also reported on the connection between the free surface and tip vortex for smaU submergence
and relatively high l i f t ( Cl > 0 . 1 ) . Here we show that the influence of the free surface results in the inward (convergent) and upward motion of the tip vortices toward the free surface. This is most pronounced for high-lifting hydrofoils at small and moderate submergence. The upward motion leads to the connection of the t i p vortex w i t h the free surface.
For simphcity of explanation, we consider the semi-hnear model. Because in the far wake the term of equation ( 4 ) ïJo^ is negligible compared w i t h 2y/{CF^), the linearized equation for vortex density can be written in the form
7 , « 2y/CFl (6)
Substituting (6) into (5), the equation for the longitudinal vorticity becomes
Thus, the influence of the free surface on the vortex can be easily estimated from the free surface shape. At a short distance {x/C = - 1 0 ) from the hydrofoil, the area of positive surface vorticity 7a; > 0 coincides w i t h the area where § | > 0 (see Figure 5). The maximum positive vorticity jxmax on the free surface is located above the tip vortex or slightly to the left of i t . I n the tip vortex the surface vorticity induces Vy < 0 and v^ < 0. The second longitudinal wave appears at a moderate distance {x/C = - 1 5 ) . I n the region where < 0, the vorticity 7^ is decreased according to (7). The maximum vorticity jxmax is located above the vortex and induces the velocities Vy ^ 0 and
Vz < 0. At a large distance {x/C = - 2 0 ) the maximum vorticity jxTnax is located to the right of the
tip vortex. The induced velocities are Vy > 0 and v^ < 0. Corresponding to the induced velocity, the tip vortex is shifted horizontally toward the plane of symmetry. Vertically, the vortex moves first downward {x/C < 15) and then upward {x/C > 15). As follows from (7), for smaller Fn there is more interaction between tip vortex and a free surface.
Fig. 6 illustrates the horizontal convergence and the vertical trajectory of the tip vortex of a high-lifting hydrofoil at different F „ . The tip vortex and the longitudinal wave are linked to each other, creating a vortex-wave structure (Fig. 7). At large distance from the hydrofoil, the left and right structures collide, producing breaking waves as behind a planing surface. Because the iteration process of the nonhnear theory diverges for this case, the results presented in Fis. 5-7 were obtained using the semi-linear analysis.
The effects described above are very weak and do not appear for large submergence and small CL. In practice, the vortex can also connect w i t h the trough sides because of the free surface instability described by Epstein and Blumin (1968); but this phenomenon is not considered here.
4 References
BA, K.J., KIM, J.W. and HAN,J.H. (1993), Numerical computations of the nonlinear steady waves generated
hy a two-dimensional hydrofoil, Proc. FAST'93, 1003-1014
BASIN, M.A. (1993), Wave formation by the motion of a surface ship hydrodynamic complex near the free
boundary. Classification of nonlinear waves. Wave-vortex resonance, Proc. V I Congress of the IMAM, P.A.
Bogdanov (ed.), Varna
EPSTEIN, L.A. and BLUMIN, V . I . (1968), Some problems of hydrofoils hydrodynamics, Proc. Central Aero-Hydrodynamic Institute, 1103, 152 (in Russian)
MARINTEKNIK, (1998), 32m Superfast Cat, Fast Ferry International, Jan-Feb., pp. 38, 41.
JANSON, CE. (1996), Numerical computation of the flow around surface-piercing wings and hydrofoils close
to a free surface, Proc. 11th Workshop on Water Waves and Floating Bodies
KANG, K.J.(1996), Numerical simulation of nonlinear waves about a submerged hydrofoil, Proc. 11th Workshop on Water Waves and Floating Bodies
KORNEV, N.V. (1998), The computational vortex element method and its application to ship
hydro-aerodynamics. Second Doctor thesis. Marine Technical University St.Petersburg (in Russian)
LUKASHEVICH, A.B. (1979), Three-dimensional flow around a hydrofoil near a free-surface of a heavy liquid, Izvestiya AN USSR, Mechanica zhidkosti i gaza, 2, 54-62 (in Russian; Enghsh translation available)
MUZAFERIJA, S., PERIC, M. and YOO, S.D. (1996), Computation of free-surface flows using moving grids,
Proc. 11th Workshop on Water Waves and Floating Bodies
SADOVNIKOV, D.Y. (1994), Gravity nonlinear and breaking waves in deep water and in shallow, Proc. Int.
Shipbuilding Conf. St.Petersburg, Section B: Ship Hydrodynamics, 190-197
SHIGUNOV, V.G. (1996), Vortex method for simulations of unsteady nonlinear wave problems, Ph.D. Thesis, Marine Technical University St.Petersburg (in Russian)
WILLERT, C E . and GHARIB, M. (1997), The interaction of spatially modulated vortex pairs with free surfaces, JFM 345, 227-250
ZALOSH, R.G. (1976), Discretized simulation of vortex sheet evolution with buoyancy and surface tension
effects, AIAA Journal, 14(11), 17
ZAROODNY, S.J. and GREENBERG, M.D. (1973), On a vortex sheet approach to the numerical calculation
of water waves, J. Comput. Phys. 11, 440-446
1.20 1.00 0.80 0.60 0.40 0.20 0.00 -0.20 2.00 r
4^
/
rr
/' /'/ '// /xf-25 /xf-40 /xf-30 —ll ,1 ll '7 !/ — 0,00 -2.00 Yw t f ^ ^ ^ ^ ^ ^ ^ " "'Jl ll/ Q/ii y 1 a 1 12A O 3^ //1 /11 () O 2 1 h ~ A /11 //1 1 f w ^ /11 //1 3 ll ^ 30 40 50 60 70 80 90 100 0.00 0.40 0.80 1.20 1.60Fig. 1. Ratio r = Y{xc,0;xi,N)/
Y{xc, 0; xi = - 4 0 , N = 100) as a function
of the number of iterations N and of xi.
Fig. 2. Wave elevation behind a submerged rectangular flat plate. AR = 6.5, F „ = 13, h = 0.6; 1 : a; = - 8 ; 2 : x = - 1 4 ; 3 :
x = - 2 0 . Sohd line: nonlinear theory; dotted
line: linear theory; circles: measurements
1.00 y/C 0,00 -0.50 A O o 0 < ) © uy / ,
/
—-Q" / O 0 00 2. 00 4. 00 6.Fig. 3. Wave elevation behind a high-lift hydrofoil. AR = 5, 15%-segment, h. = 0.3667, Fn = 5.21, x = - 8 , solid line: non-hnear theory, dotted line: semi-linear theory, circles: measurements
lO.OOi
5.004
o.om -10.00J -30.0 -25.0 -2O0 -15.0 -10.0 -5.0 0.0 -30.0 -25.0 -20.0 -15.0 -10.0 -5.0 0.0 Fig. 4. Comparison of wave patterns beliind Autojet. Fn = 9.7. A: nonlinear analysis, B: linear analysisUndlsturbedfree surface
-25.00 -20.00 -15.00 -10.00 -5.00 0.00
-25.00 -20.00 -15.00 -10.00 -5.00 0.00
Fig. 6. Trajectory of the t i p vortices. Rect-angular wing, 15%-segment, AR = 5, h = 0.3667, a = 6°
0.0 2.0 4.0 6.0 0.0 2.0 4.0 6.0 0,0 2.0 4.0 6.0 Z
Fig. 5. Wave elevation Y, t i p vortex and lon-gitudinal surface vorticity jx behind a rect-angular hydrofoil of aspect ratio 5 at 6° angle of attack (15%-segment, h/C = 0.3667, F „ = 5.21) 1.00 - T - y / C 0.50 0.00 -0.50 -1.00 -1.50 -6.00 -4.00 -2.00 0.00 2.00 4.00 6.00 Z/C 1.00
jy/C
b). 0.50 0.00 -0.50 -1.00 -1.50 -6.00 -4.00 -2.00 0.00 2.00 4.00 6.00 zJcFig. 7. Cross sections of the wave-vortex wake behind a hydrofoil. 15%-segment,
AR = 5, h = 0.3667, F„ = 5.21.
Top: x/C = - 1 0 (solid), -15 (dotted) Bottom: x/C = - 2 0 (solid), -25 (dotted)
Numerical Simulation of the Unsteady Flow Past a H y d r o f o i l
Maurizio L a n d r i n i , Claudio L u g n i , Volker Bertram^, Italian Ship Model Basin^
1. Introduction
Several authors studied numerically the flow around a submerged hydrofoil. We limit ourself in mentioning those papers relevant for the two-dimensional problem dealt w i t h inviscid models. I n this framework, Giesing and Smith (1967) gave a numerical solution of the hnearized steady problem by using boundary integral equations. The steady nonlinear problem is solved in Forbes (1985) by a suitable iterative procedure. I n both cases a (steady) K u t t a condition provides the uniqueness of the (numerical) solution. Less attention has been devoted to the unsteady nonlinear problem which was solved by Coleman (1986) by discretizing the Euler equations and linearizing the wake dynamics. Unsteady free surface flows w i t h embedded concentrated vorticity are discussed by Casciola and Piva
(1990) by using an integral representation for the velocity field (Poincaré representation formula)
coupled with suitable evolution equations.
I n this paper we present a time-domain analysis for the two-dimensional flow around a slightly submerged hydrofoil. Large separation phenomena and cavitation are ruled out and the flow field is described in terms of inviscid-rotational fluid mechanics in which a thin vortical layer mimics the wake hydrofoil and a suitable unsteady K u t t a condition provides the mechanism for vorticity generation. The velocity field is described through an integral representation in terms of the velocity component on the free surface, source and vorticity distribution on the hydrofoil and the wake vorticity. Free surface boundary conditions which hold i n the general rotational case are derived from the Euler equations and the nonlinear free surface dynamics is fully retained.
The model is successfully applied to test cases for which steady regime experimental data are available. Good agreement is generally obtained for wave forms, wave drag and pressure distribution on the foil. The free-surface treatment appears accurate enough to describe some aspects of the kinematics of breaking waves generated by shallowly submerged hydrofoils.
Pig.1: Problem definition and adopted nomenclature
2. P r o b l e m formulation
We consider the flow field around a hydrofoil H moving beneath a free surface Fig. 1. The main motion of the foil is i n the horizontal direction w i t h velocity U{t) i n an otherwise quiescent fluid which therefore has asymptotically zero velocity at large distances from H i n the inertial frame of reference TZf, ( s i , ^2). A second frame of reference TZ, {x, y), moving w i t h velocity —U{t) — —U(t)ei with respect to TZf, is also considered. I n the general case, arbitrary oscillations in heave, surge and pitch are allowed. The velocity V relative to TZf and the acceleration A of a point P fixed on V. are ' T h e research was supported by Ministero dei Trasporti e della Navigazione through I N S E A N Research Program 1997-99
^Visiting Scientist f r o m T U Hamburg-Harburg ^ I N S E A N , V i a d i Vallerano 139, 00128 Roma, I t a l y
then respectively
ViP,t) = -U{t)ei + V{0,t) +n{t)x{P-0)
heave-surge pitch
= - ^ e i + i ( Ö ) + ^ x ( P - Ö ) + j l W x [ j 1 x ( P - Ö ) ]
Ö is the origin of moving w i t h velocity V{0, t), and Ü is the angular velocity. In the moving frame TZ, the flow is described by the Euler equations
V - . - ^ O £ | = - l v p + , ^ + f (2)
Dt p dt
q = U + u is the fluid velocity, D/Dt = d/dt + g • V is the substantial derivative, g is the gravity
acceleration and p is the pressure of the fluid w i t h density p. The term dU/dt accounts for the non-inertial character of TZ. Observing that V • U = 0 and that
Dq _ Du dU 'm ~ ^ ^ ~dt '
the field equations are recast i n terms of the 'perturbation' velocity ü
V - Ü = 0 ^ = - ^ V p + g (3)
In the frame of reference TZ, the standard impermeability boundary condition holds:
{Ü - V{Ö,t)-Ü{t)x{P-Ö))-u = -U-i^ onn (4)
The unit normal vector u points into the body. 2.1. Free-surface equations
The free-surface boundary equations follow from the kinematic condition that the fluid does not cross it and from the dynamic constraint that the pressure is atmospheric on !F. These requirements are fulfllled i f the motion of points P{(, t) on T is described by
f dP{^,t)
— = w = q^iy + WrT
(5)
dUr{^,t) ^ 1 ^ 1 , ^ dTi^,t) = r-^{w-q)-Vu + g--Vp.^ + u - ^ ^
V, T are the unit normal and tangent vectors to T. The first equation states that P{i,t) moves w i t h
velocity w: the normal component w • u is fixed by the corresponding component q,^ = q • V oi the fiuid below, while the tangential component Wr can be chosen arbitrarily. Anyway, regardless the actual value of WT, the tangential velocity component Uj of the fluid evolves according to the second equation which follows f r o m the tangential projection of the Euler equation. Unless otherwise stated, the Lagrangian description of the free surface is adopted i n the following by choosing w = q in the above equations. This choice allows for large wave motions, up to breaking events. I t is therefore preferred to the Eulerian description which can be recovered for w = {0,8?]/dt) where only pure vertical displacement of the free surface is allowed.
2.2. T h i n wake model
In the following i t is assumed that vorticity, created and convected within the body boundary layer, leaves continuously the hydrofoil at the trailing edge to enter a zero-thickness wake W downstream the hydrofoil. Such a t h i n wake is modeled by a vortex layer w i t h local strength
7w = iwk = {q+-q-) xu = [qr]k . (6) In (6), [qr] = [ur] is the j u m p of the tangential component of the fluid velocity across W while the
normal component q,y is continuous due to mass conservation, e.g. Friedrichs (1966). A Lagrangian description of the wake is given by
accounting for the kinematics and the dynamics o f t h e wake, respectively. A wake point P ( x , i ) , marked by the curvilinear coordinate x , moves w i t h velocity deflned as w{x,t) = + ^-)- Consistently, the derivative d/dt keeping x constant is the rate of change of a quantity seen by an observer moving with the wake point. The second equation, where J(X)*) = d P / d x , describes the time evolution of the wake vorticity jyviXit) according to the Kelvin theorem.
A thin wake cannot sustain a pressure j u m p and, for consistency, the density -JTE of the 'nascent' wake element is determined by enforcing a zero trailing edge loading. Namely, upon considering a body fixed frame of reference TZ-u, such unsteady Kutta-like condition follows from the Euler equation in the form
dV 4 2n
a is the total fiuid velocity i n TZ-H- Eq. (8) relates the rate of change of the circulation about the
hydrofoil, dV/dt, to the fiux —w • JTE of vorticity JTE = {^TE ~ ^TE) injected i n W w i t h velocity
2.3. H y d r o d y n a m i c forces The evaluation of the loads
F = j p{s)ud£ ^ ^ j ^^^^^ ^
requires the pressure distribution p(s, t) on the foil. To the purpose, the tangential projection of the Euler equation i n a body-fixed frame TZu is considered,
da Ida^ I dp _ _ _
+ 0 1 ^ = — ^ + fa-T + g-T 10
dt 2 dr pdr
where the apparent forces
fa = - ( ^ - ^ 4 + A{0) + ^ X ( P - Ö ) + Ü{t) x [ Ü x { P - Ö ) ] J (11)
per unit of mass are included and a = ar. Integrating (10) along the body surface gives the pressure
p { s , t ) - p { s o , t ) = -p
f
adt-]-pa^' +P r{fap + 9)-Tdi (12)Jso ^ *o Jso
The reference pressure p(so,^), say the pressure at the trailing edge, is undetermined. The loads (9) are not affected by its actual value, while the pressure distribution on the foil at different time instants are comparable except for a constant time dependent shift.
3. N u m e r i c a l solution
We assume that the motion of the hydrofoil and the flow field are completely known at some instant <0. Then, a subset of the boundary data can be prolonged i n time by observing that
- the body motion fixes in advance the normal velocity u^l^^ through (4).
- the free-surface geometry T can be updated by the first equation of (5); the tangential velocity
Ur\j: can be prolonged by the second one.
- the knowledge of the velocity field on the hydrofoil allows computing the newly generated vor-ticity 7r£;, while W changes according to (7).
The normal component u^l^- and the tangential component Ur\-}i are still to be determined and are evaluated, following Casciola and Piva (1990), through a boundary integral equation approach to solve the kinematic problem the unknown
V - ? / = 0 V x u = C (13) where the vorticity C = V x u reduces to the generalized wake vorticity 7. The more general case w i t h
vorticity spread over a finite area is discussed, e.g. by Landrini et al. (1998) in the case of the viscous fiow around a circular cylinder beneath incoming waves.
The solution of the kinematic problem w i t h boundary data
Uj on T onH
7w
on W (14)provides ah the information requested to step the solution forward i n time according to Fig. 2. Both the free surface evolution equations and the wake evolution equations are integrated i n time by a fourth-order Runge-Kutta scheme.
S O L U T I O N O F T H E B O U N D A R Y G E O M E T R Y A N D B O U N D A R Y D A T A ( 1 4 ) K I N E T I C P R O B L E M (13) B O U N D A R Y G E O M E T R Y A N D B O U N D A R Y D A T A ( 1 4 ) S T E P F O R W A R D I N T I M E O F F R E E - S U R F A C E E V O L U T I O N E Q U A T I O N S (5)
J
W A K E E V O L U T I O N E Q U A T I O N S (7-8)J
H Y D R O F O I L M O T I O NFig.2: Schematic description of the numerical procedure 3.1. Integral representation of the velocity field
In the moving frame of reference U, the velocity ^ -> as |x| ^ 00 for finite time. Then the perturbation velocity field u is asymptotically zero far from the hydrofoil and, for a generic point Q in the fluid domain, can be represented by
u{Q) = VQ j^u-vGdSp + VQX j uxPGdSp free-surface infiuence
+ V Q / aGdSp + ^VqX [ kGdSp hydrofoil influence (15)
JH 1- Jn
G = l n | P - Q|/27r is the two dimensional free space Green function. The first two terms in (15) account for the presence of the free surface 'directly' in terms of the normal u • v and tangential
ÜX u components of the fiuid velocity. The body effects are taken into account by the gradient of a
simple layer potential f a G dS and the curl of a uniform vortex sheet w i t h density T/C related to the instantaneous circulation
Ü • Td£ (16)
around the foil of perimeter C The Biot-Savart integral mimics the effects of the wake W . 3.2. Discretized integral equations
By introducing Nj:- points on the free surface and N-H (second order) panels on the hydrofoil, an algebraic system of the form
n
-<
^ n n - \ n
i - 1
= r (17)
A
is obtained for the unknown normal velocity component and the source strength. I n (17) the matrix coefficient A follows by assembling the self-inffuence and the cross-inffuence of the free surface and the hydrofoil, while the right hand side
r =
{ - }
r' n - ' w
-+
n - w
-• 7w }
(18)accounts for the known free surface contribution due to the tangential velocity component Ur, the influence o f t h e body circulation F and the wake vorticity 7w> respectively. For not deforming bodies, the self-influence matrix [H —>• H] and the self-induced velocities can be computed once and for all. The quadrature formula adopted to discretize the free-surface integrals is described i n Graziani and
Landrini (1998), while the analytical formulas for the source and vorticity panels are reported e.g. in Katz and Plotkin (1991). A fast summation algorithm based on multipoles expansion is employed to
compute the influence of wake vortices. The solution of (17) is flnally evaluated by an iterative solver.
Graziani and Landrini (1998) developed a more efficient algorithm to solve the integral equations
arising from (15) based on fast summation of multipoles expansion coupled to an iterative solver. A deeper discussion of the numerical method can be found there.
3.3. N u m e r i c a l treatment of the free surface
A truncated computational domain has to be used and, in general, both the velocity components
Ur and Ul, and the wave height 77 do not vanish at the edges, at least for long enough evolution time.
Therefore, unphysical refiections and numerical errors due to unbalanced edge singularities may occur. This drawback is avoided by artificial damping layers which force the tangential velocity component and the wave disturbances to vanish when approaching the outer portions of T. I n particular, modified free-surface evolution equations can be written by introducing a damping function /i(a;), zero every-where except within the damping regions, which multiplies Ur and r] in the dynamic and kinematic free-surface equations, respectively. The length o f t h e damping layers and the maximum damping coef-ficient have been heuristicaUy determined. The accuracy of the computation is eventually maintained under control by comparing different solutions for increasing domain lengths.
To handle the mean horizontal motion U{t) in the frame of reference TL of the free-surface points, a fixed computational window is considered and those points dropping out downstream are thrown away and new markers are inserted upstream. The sawtooth instability, that usually sets in for long enough evolutions, is removed by high-order filtering procedures.
3.4. N u m e r i c a l treatment of the wake
The wake W is continuously emanated from the trailing edge of the hydrofoil. I n actual compu-tations, W is discretized by a set of point vortices w i t h circulation Fk fixed at the time of shedding according to (8). Hence, the Kelvin theorem (7) implies that stays constant during the follow-ing motions. Consistently, the Biot-Savart integral in (15) is replaced by the sum of the velocities
QG{Q,Pk) X k induced by the point vortices embedded in the fiow field.
For long time evolution, to prevent irregular motions and instabilities discussed by Krasny (1991), the kernel is desingularized by convolution w i t h a cut-off function which implies a finite velocity induced on near vortices and a zero self-induced velocity. Additional difficulties can appear when dealing with large deformations of the discretized wake. When the distance between the vortices increases/decreases too much, vortices are split/merged enforcing conservation of total circulation and of the center of vorticity.
For large number of vortices, the computational effort w£is significantly reduced using multipoles expansion with fast summation techniques for computing the self-induced velocity and the infiuence of the vorticity on the other fiuid boundaries.
4. Discussion of the results 4.1. Introductory results
Fig. 3 shows the generation of the wave field by a NACA 0012 towed w i t h an angle of attack a = 5°. The velocity U{t) of the foil smoothly increases from zero up to the final value corresponding to a Froude number F „ = U/^/gc = 0.5669 in a time Tr ~ 1200At according to the ramp-function ( - 2 0t ^ + 70t2 - 8 4t -I- 35)t'', ( t : = t/Tr) smooth enough to avoid tilting behaviors of the loads related
to discontinuities of the body acceleration, x is made non-dimensional by the chord length c of the foil, the wave height T] by the steady state value \r]t\ of the trough above H. A t the beginning a smooth dip over the hydrofoil gradually increases w i t h an horizontal extent initially larger than the steady state trough (about two times the wavelength A ~ = 2.019). Later on (sixth frame), as a signature of this local depression, the first wave crest w i h be located lower than the more downstream ones. After about 1200 time steps the velocity reaches the final value (third frame) but stih the wave pattern is changing and only after more than 1600 time steps (fourth frame) the free surface appears stationary close to the foil. Eventually (sixth frame) a regular steady wave train downstream the foil is obtained and the five different wave profiles, although made by distinct Lagrangian markers, are well superimposed and the only sign of unsteadiness is the lengthening further downstream of the wave train w i t h the group velocity Cg = U/2. A vigorous vortex shedding JTE is observed during the initial stage, Fig. 4, and the circulation F about the body grows together w i t h the l i f t force component d -Quasi-steady values are reached and only small oscillations of JTE are detectable.
Computations are performed in double precision and all the results shown have been tested by doubhng the number of elements on the fluid boundaries and halving the time step At until convergence was obtained. For a given number Nj^ of markers on J^, the effect of doubling the number (N-u = 30, 60,120,240) of the body-panels was studied. Because of the invariance of the numerical wave elevation up to the flrst 8 digits for N-^ - 120 and 240, N-^ was selected in the considered example. Similar findings are obtained for MUjc = 1/25,1/50,1/100.
Fig. 5 shows the convergence analysis of the wave pattern. The solutions achieved by using
N/X — 5,9,17 markers per wavelength (denoted by • ) are contrasted w i t h the most accurate solution
(sohd hne) obtained by 34 markers each wavelength. The coarsest discretization, N/X = 5, captures the local disturbance above the foil and the first peak past the body but eventually fails in following the more downstream wave motion. By doubling the total number of markers, N/X = 9, the qualitative agreement of the solution over the entire free surface is apparent and further increasing
Np nicely reproduce the most refined solution. The high degree of convergence is probably due to
and Landrini (1998), which for smooth solutions would guarantee a spectral convergence, Sidi and Israeli (1988). .04 .02 -.02 -.04 .8 .4 -.4 .4 7 7 / 7 7 ^ -.4 .4 -.4 .4 -.4 .4 t/At= 100-500 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ~ l/At=600-1000 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 t/At=1100-1500 . 1 1 1 1 1 1 1 1 1 R / 1 1 1 1 1 l l l l l l l l l i i i l ~ t/At= 1600-2000 A A 1 1 1 1 1 1 1 1 1 Vy 1 1 1 1 1 l l l l l l l l l i i i l t/At=2100-2500 A A 1 1 1 1 1 1 1
""i^^^^i^V/M
l l l l l l l l l l l l l i i i l t/At=2600-3000 A A 1 1 1 1 1 1 1 \ l l l l l l l l l l l l l i i i l -8 -4 4 8 12 16 x / cFig.3: Time evolution of free surface past a NACA 0012 hydrofoil placed w i t h the mid-chord point at (x=0,y=-d/c=-1.28), angle of attack a = 5°, final non-dimensional speed F„ = U/^/gc = 0.5669. The wave height 77 is normalized by the amplitude \r]t\ of the trough above the foil.
Another point, relevant to discuss the unsteady behavior of the solution, is related to the transient adopted to start the flow field. We used a smooth increase of the foil velocity. A sudden start would
.7 .1 .2 .4 .3 .5 .6
r /
c U c 10 20 30 40 50 t U / c Fig4: Time evolution of lift coefficient C/,, body circulation F and vortex shedding •^TE for a NACA0012 starting f r o m rest, see Fig. 3-4 4 8 12 16 x / c Fig.5: Effect of increasing the number of Lagrangian markers per wavelength. From top down the
most refined computation w i t h 34 points per wavelength (solid line) is compared w i t h coarser and coarser discretizations (symbols). Flow conditions: The mid-chord point of the NACA0012 profile is located at (x=0,y = - d / c = —1.28) w i t h an angle of attack a = 5", F„ = 0.5669. Other discrete parameters: At — ^c/U, Nf, = 120.
imply an oscillatory behaviour of the solution about the steady one w i t h oscihation period Tc = B-KU/Q
and decaying amplitudes, Liu and Yue (1996). This behaviour is described in Fig. 6, where the time history of drag and lift is reported for several length Tr of the smooth ramp function adopted to start the flow. For Tr = 0, the very first time steps are deleted for representation purposes.
A l l solutions oscillate around the steady state value, although the oscillation amplitudes decrease faster for greater Tr/Tc- This is more pronounced for the drag, probably because the l i f t is dominated
-I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I
Fig.6: Effect of the body acceleration of the drag and l i f t force components. Tc = SnU/g is the oscillation period of forces. Flow conditions: NACA0012 profile, a = 5°, d/c = 1.2857, F„ = 0.5669. Discrete parameters: N/X = 17, At = ^c/U, Nb = 120.
.1
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d / c = 1.2857 7 7 / c 0 / -.1 _ ^^^^ 3^ ^—rro ^ ~ r 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 .1r
d / c = 1 . 1 6 2 6 \ A \ ° r ] / c 0 / \ / \ ° / \ ° \ /o \ / \ a. ^ V / \ r ] / c A , / \ o / 0 \ -.1 O c o o ° 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I l l l l l l .1 d / c = 1,0345 ^ 0 / 0 / 0 / A tf^ A. /W ° \ B I \ / „° \ S 7 ] / c=-0 / 0 / - f l f l - O © . ^ 0 / ^ ^ ^ o S \ ° / o\o \ c/ il 9 s / o \ o / ° \ 9 .
n / \ A ^
9> \ V \ 1° \ -.1 0 / . , I l l l i l l l W ° i l l l l l 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I I I I I I -1 x / c 1 2 3 4 5 6 7Fig.7: Wave height r]/c past a NACA 0012 at an incidence {a = 5°, F„ = 0.5669). The depth of submergence d/c is measured from the mid-chord point and decreases from top down i n the figure. Solid lines: present unsteady results, o: measurements from Duncan (1988).
by the vortex shedding which is affected by the free surface to a lesser extent, at least for this relatively large depth of submergence. The solutions in the next chapter are obtained by using a starting period Tr = 3Tc which minimizes the time needed to reach a steady state.
4.2. Quasi-steady solutions
After the transient stage, the wave profile is constant i n a body-fixed frame of reference and loads reach constant values. I n these circumstances it is easier to compare the numerical results with experimental data available in literature. I n particular, the predicted wave patterns are compared to experiments of Duncan (1983) for a NACA 0012, Fig. 7. The depth of submergence d/c is measured from the mid-chord profile to the undisturbed water level. For d/c = 1.2857, the first trough just above the profile is well predicted as weh as the downstream waves. For d/c = 1.1626 some differences appear. The depression above the suction side is shghtly under predicted and this feature is repeated for hollows more downstream. The first and second experimental wave humps are captured well, while the third crest is underestimated. Finally, a 'shift' of the wave forms is detected and would require an analysis o f t h e discrete dispersion relation, Dold (1990). This behaviour is even stronger for d/c = 1.0345. The free-surface deformation above the hydrofoil is more pronounced in the experiments. Computations were originally performed without including the effects of the bottom of the basin. We eventually included the finite depth effects by the method of images but the results obtained were practically identical.
-1 ^ 1 2 3 4 5 6 7
x / c
Fig.8: Wave height T]/C past a NACA 0012 (see bottom plot Fig. 7) Top plot: — present unsteady solution, o experiments Duncan (1983), steady B E M Thiart and Bertram (1998). Bottom plot: — present unsteady solution, o experiments Duncan (1983), unsteady Euler solution Muzaferija
(1998), • unsteady Euler solution Yoo (1998).
To speculate on this marked difference, i n Fig. 8 our results for d/c = 1.0345 are compared with those achieved by different methods. I n the top plot, the steady solver of Thiart and Bertram (1998) based on a boundary integral formulation for potential flows w i t h lifting effects (dashed line) provides a wave profile in qualitative agreement w i t h the our unsteady computations, although the former underestimates the wave amplitude. I n the bottom plot, solutions obtained by solving the unsteady Euler equations through the same finite volume technique but different free-surface treatments are
reported. Close to the hydrofoil, the solution from Yoo (1998), •, is in good agreement w i t h our result. Further downstream the two solutions slightly disagree, probably due to the truncation of the computational domain. Muzaferija's results (private communication) agree much better w i t h the experimental results. We are at present not able to explain quantitatively the differences between the two solvers. A last point to mention is the experimental uncertainty: no error bar is reported in Duncan paper and he generally claimed that 'distances were measured to an accuracy of about ± 0 . 3 cm', which can partly explain the differences between the experiments and the reported numerical results.
- 1 ^ / ^ 1 2 3 4 5 6
Fig.9: Wave profile 7?/c past a NACA 0012 (F„ = 0.5669, a = 5°, d/c = 0.9507). Sohd line: present unsteady solution at tU/c = 10.9; o: measured 'non-breaking' wave height Duncan (1983); •, *: steady Navier-Stokes solutions Tzabiras (1997) for fully turbulent and transitional body boundary layer, respectively.
For d/c = 0.9507 the waves are close to breaking: Duncan reported a steady non-breaking profile which, upon perturbing the free surface ahead the foil, became characterized by a steady breaker located at the first wave crest past the foil. I n the numerical computation the breaking is triggered by simulating the sudden start of the foil: after tU/c = 10.9 a tiny jet develops at the first crest past the body. I t is shown i n natural scale i n the top left frame of Fig. 9. Because of the breaking, the computation stops and, due to the limited duration of the simulation, a regular wave train cannot develop, nor are we able to model the post-breaking evolution. I n spite of this, close to the hydrofoil, the non-breaking measurements, o, agree well w i t h our unsteady solution. I n the same plot, two different numerical solutions of the steady Navier-Stokes equations are reported, Tzabiras (1997), which both agree well w i t h the experiments and our solution. I n particular, when transition to turbulence in the body boundary layer is modeled, • , a larger peak past the foil is obtained which strikingly resembles the one predicted by our inviscid-rotational unsteady model. Results w i t h a fully developed turbulent boundary layer perform slightly better than the other computations at least close to the hydrofoil. Anyway, further downstream, all the methods suffer phase lag and under-prediction of the wave height w i t h respect to the experimental profile. For viscous computations this could be due to the finite extent of the computational domain. Upon observing the non-uniformity of the measured wave train, i t is possible that the experimental data are still largely affected by the transient and that on a suitable time scale the breaking would anyway appear.
Salvesen (1966,1969) gave an attempt to analyze the nonlinearities in wavy fiows generated by a
submerged body. He performed a wide set of experiments, Salvesen (1966), by using a mathematically generated profile. Salvesen (1969) developed a second-order theory in which i) the thickness and the chord are comparable and ii) the free surface is far enough to assume the disturbance on T one order lower in the perturbation expansion scheme. Here we reproduce some of the cases tested by Salvensen. We start from Fig. 10 where, for the larger submergence d/c = 1.37 shown on the left-plots, the wave forms are reported for increasing forward speed (from top down F„ = 0.422,0.590,0.759,0.928, respectively). The foil is placed at zero incidence w i t h the traihng edge at x/c = 0, y/c = -1.37. The
