Numerical simulation of the effect of li

NUMERICAL SIMULATION OF THE EFFECT OF LIFTING APPENDAGES ON BOAT SEAKEEPING Manh Hung Nguyen, Université de Poitiers, France Malick Ba, ENSMA, Poitiers, France Serge Huberson, Université de Poitiers, France Michel Guilbaud, Université de Poitiers, France Laboratoire d’Etudes Aérodynamiques (UMR CNRS n°6609) SUMMARY A linear diffraction radiation hydrodynamic solver for flows around ship was extended to take into account for the lifting effects on fins, rudders or stabilizers (code Poseidon: panel method in the frequency domain using the diffractionradiation Green function) This modelisation is based on a doublet distribution set on the symmetry plane of the body, and on semi infinite strips extending from the trailing edge, simulating the wake assumed to be plane A linearized unsteady Kutta-Joukowsky condition enforcing the continuity of the unsteady pressure at the trailing edge is satisfied The distribution law for the doublet intensity on the body symmetry-plan was selected in order to improve the resulting matrix conditioning Validation was obtained through comparisons of our results with those available for the cases of a deeply submerged body and weakly submerged and surface piercing 2D and 3D bodies Application to flows with lifting effects is presented on the wake, due to the Kelvin theorem, the doublet strength is oscillating The results were validated by means of comparison with available numerical or test results in aerodynamics or hydrodynamics (submerged D profile or 3D surface-piercing body) This code was also used to study the influence of moving appendices (rudder or stabilizing fins) on the seakeeping abilities of boats In particular the effect of the phase lag between the waves and the appendices motion was investigated The effect of the propulsive force of paddles periodicity on the unsteady motion of a canoe or a kayak was also investigated INTRODUCTION It is well known that boat seakeeping can be widely affected by appendages either fixed as sailboat keel, bilge keel or active as rudder and stabilizing foils Although accounting for flap or wing oscillations has been a common practice in unsteady aerodynamics for a long time, Ashley et al (1), Geissler (2) and others, the effects of oscillating foils are seldom accounted for in ship hydrodynamics, even when the unsteady motion of the ship itself due to waves is the objective of the calculation Among some few exceptions, calculations around submerged two-dimensional bodies, Kyozuka et al (3), Kyozuka (4), or yet in three-dimensional flows, Bertram (5), are worth mentioning The case of transverse waves with ship with rounded stern was also briefly addressed in Zou (6) and the lifting surface method was also used for surface-piercing bodies by Nontakaew (7), Nontakaew et al (8) These effects, not essential for hydrodynamic flows, can be important in the case of a sailing boats, a maneuvering ship or for studying stabilization by mean of foils The present work deals with the introduction of lifting effects in the code Poseidon (panel method in the frequency domain using the forward speed diffractionradiation Green function) developed by Boin (9), Guilbaud et al (10), Boin et al (11) It has been validated in the case of non lifting flows, Maury et al (12) Green doublets on the symmetry plane of the body and on semi-infinite strips extending from prescribed trailing edges were added to the source distribution on the body This is, actually, a linear modelisation of the wake which was assumed to be plane The strength of each strip was determined by a linearized unsteady Kutta-Joukowsky condition enforcing the unsteady pressure continuity at the trailing edges; an arbitrary law for the doublet intensity on the plane of symmetry was also used in order to avoid numerical difficulties, while Session A PROBLEM FORMULATION JOUKOWSKY CONDITION AND KUTTA- Problem to solve and equations The unsteady flow around a ship travelling at constant forward speed U, is considered in the frequency domain for both cases of a ship oscillating on an initially undisturbed free surface or ship motion in regular waves A Galilean framework attached to the ship that advances along a straight path was used together with Cartesian coordinates (x,y,z) The x axis lies along the ship path and points toward the bow and the z axis is vertical and points upward with the mean free surface taken as the plane z=0 The fluid is assumed to be inviscid and incompressible For an irrotational motion, an unsteady velocity potential can be used It is defined in the frequency domain as ϕ(x,y,z)exp(iωt), where ω is the encounter frequency The uniform flow is assumed to be the mean steady flow This potential satisfies the Laplace equation in the fluid domain, the body condition on the ship averaged position and a linearised free-surface condition as well as conditions at infinity In the case, of lifting appendices, the pressure must be continuous at the trailing edges of the lifting parts and across the semi- 189 ∂ ∂ −U  ϕ ( M ∈Σ+ ) eiωt − ϕ ( M ∈Σ− ) eiωt  = (3) ∂x   ∂t infinite wake extending from these parts We use the diffraction-radiation with forward speed Green function, G(M,M'), which satisfies the Laplace equation, the free-surface boundary and the radiation conditions The 3rd Green formula leads to an integral equation to compute the unsteady potential, cf Boin et al (9,10) The unsteady linearised pressure is derived from the Bernoulli equation: ρ By writing : µ%( M ∈Σ;t ) = µ ( M ∈Σ) eiωt = ϕ ( M ∈Σ+ ) − ϕ ( M ∈Σ− )  eiωt (4) p% ( M , t ) = p ( M ) eiωt , with p=ρ(iωϕ-U∂ϕ/∂x) and using Equation (3), the value of the doublet intensity on a point of a wake is related to the value of this strength on the trailing edge (subscript te) on a parallel to the x axis, by: When lifting effects need to be accounted for, a Kutta-Joukowsky condition has to be applied; it is written indirectly as a pressure continuity condition written on the trailing edge of the acute parts of the lifting parts of a body in the form: iω / U µ x ∫ xte dx = ∫ dµ / µ (5) µ te which can be integrated as:  ∂ϕ ( + ) ∂ϕ ( − )  iω ϕ ( + ) − ϕ ( − )  −  − =0 U∞  ∂xM + ∂xM −  µ ( M ∈ Σ ) = µ ( M te ) eiω ( x − x (1) te (6) where for a point M on the wake with abscissa x, the point Mte with abscissa xte is the intersection of a parallel to Ox axis passing par M with the lifting part trailing edge where the superscripts (+) and (-) stand for the suction and pressure sides at the sharp trailing edge Furthermore, to the source distribution σ(M) on the wetted areas of the ship used in classical seakeeping calculations, a Green doublet distribution µ(M) is added on the ship mid surface and on a plane extending from the trailing edges So equation (1) becomes equation (2), r with nM ' , the unit outward normal at M’ located on the Body discretisation The body surface is divided into NB strips and NC columns in order to define a set of panels NTC=NB*NC The centerplane is divided into the same number of strips, but in order to have the same panels boundaries, the total number of panels is half the number of panels on the body, NT=NC/2*NB=NTC/2 The wake is also divided into NB strips For the longitudinal space step ∆x, we have chosen a distribution along the chord with a cosine law in order to have panels with small chord lengths close to the bow and stern, while the transversal step ∆y has been kept constant The number of equations to be solved is NTC body conditions and NB Kutta-Joukowsky conditions, so we have only NB degrees of freedom for the unknown doublet intensities while we have NTC degrees for the sources The source intensities were assumed to be constant on each body panel The doublet intensities were linked to the value µtej at the trailing edge, for each strip The body condition are written on the centroid of the hull panels while the Kutta condition is written at the centroid of the panels of the last column, adjacent to the sharp trailing edges In order to avoid numerical difficulties, discontinuities have to be removed at both leading and trailing edges of the lifting parts So an auxiliary oscillating function f(M,ω) was defined on the body symmetry plane by: body surface; S is the body surface, S0 is the center plane, C is the waterline and Σ is the wake:    S  iω  − ∫∫σ (M ') ( G(M + , M ') − G(M − , M ')) dSM '  U∞ S  + − r U∞ ∂G(M , M ') ∂G(M , M ') r )(nM ' x)dyM '  − ∫σ (M ')( − (2) ∂xM+ ∂xM− g C  =0 iωU∞ r r + − σ (M ') ( G(M , M ') − G(M , M ')) (nM '.x)dyM '  +  g C∫   ∂ G(M+ , M ') ∂2G(M − , M ')   − + ∫∫ µ(M ')   dSM '   ∂x + ∂nM ' ∂ ∂ x n M' S0 +Σ   M− M   − ∂ G M M ( , ')   ∂G(M , M ') iω + − − M dS ( ') µ   M '  ∂ ∂ U∞ S∫∫ n n   M' M' +Σ   ∂G(M + , M ') ∂G(M − , M ')  −  dSM '  ∂x + ∂xM− M   ∫∫σ (M ') This equation involves second order derivatives of the Green function, leading to numerical difficulties as described in (10) On the wake Σ, assumed to be plane without thickness, with both sides named Σ+ and Σ-, the pressure is also continuous, so we can write: Session A )/U   ( x −x)  (iω(x −x)/U ) f (M;ω) =1/21+cosπ te  e te  ( xte −xle )     190 This function is zero at the leading edge, its real part is equal to and its imaginary part equal to at the trailing edge; so on each panel j, the actual doublet strength is µ(M)=f(M,ω)µtej, M being the panel center, in order to have a constant value on the symmetry plane So, the only doublet unknowns are the values of the doublet intensities at the trailing edges The set of discretised equations reads: strength, we use a technique proposed by Geissler (2), for the Rankine terms First the infinite integration domain is bounded into a finite number of wavelengths on the wake, Λ=2π/k’, k’=ω/U being the wave number The integral defined by: [S → G]ij σj +[D → G]ik µk = Bil ; i, j = 1, NTC with i = 1, 2,3 and j = 2,3 I = ∫∫ e iω ( x ' − xte ) / U ∂xi ∂x ' j ΣB [S → KJ ]nj σk + [D → KJ ]nk µk = 0; k, n = 1, NB where Φ I = A B f ( x ') = ∫ ω0 ik [ x cos β + y sin β ] ∂xi ∂x ' j (10) dl ' The integral defined by Equation (9) can be eventually calculated as: (8a) I = e − ik ' xte xte ∫ xte −Λ / N eik ' x ' ∑ ( −1) f ( x '− nΛ / ) dx ' n (11) n =0 The study of the influence of the value of the number N shows that the convergence is reached for N>10 The remaining part of the Green function which does not depends on the Froude number, appearing also in the Rankine methods, was computed as usual Once the set of Equation (7) has been solved and the unknown source and doublet strengths are computed, pressure distribution, the added-mass and damping, the forces and the moments on a ship hull or its motion in regular waves are easily obtained is the potential of the incoming waves where β denotes the angle between course heading and wave direction (β=180°, head waves), A denotes the wave amplitude ω0 is the regular wave frequency and is related to the encounter frequency ω by the dispersion relation in infinite depth ω=ω0Ukcosβ, with the wave number given by k=ω02/g The new terms, with respect to calculations without lifting effects Boin et al (9,10), are those given by the influence of the sources (S) and doublets (D) on the Kutta-Joukowsky condition (KJ) and those of the influence of the doublets on the body condition (G) Details can be found in Nguyen et al (13) Difficulties appear from the integrations of the second derivatives of the Froude dependant part of the Green function, (11) The numerical difficulties clearly increase with the order of derivation (order being for the function itself) and are even more difficult than that of the wave resistance Green function For integration on quadrilateral panels, the technique involves interchange of the integrations, the boundary integration on the panel being performed before the Fourier integration required in the definition of the Green function The area integration is performed analytically using the Stokes theorem to transform the surface integral into a contour integral, leading to a smoothing of the integrand, which is less singular than the original Green function For the integrals on the semiinfinite strips of the wake with an oscillating doublet Session A ∂ 2G ( M , M ') A (8b) e kz e (9) is thus transformed into a finite sum thanks to the definition of the auxiliary function defined as: where (n1,n2,n3)= n , (n4,n5,n6)= r ∧ n and (m1,m2,m3) =(0,0,0); (m4,m5,m6)=(0,n3,-n2), corresponding to the choice of a uniform flow as the mean steady flow and enabling to uncouple the steady and unsteady problems For the diffraction problem l=7, we have: Bi7 = −∂Φ I / ∂n dS ', (7) depending on the diffraction-radiation problem l=1 to to be solved The right hand term of the set of linear equations to be solved for the six radiation problems reads: Bil = [iω nl ( M i ) + ml ( M i ) ] ; l = 1, ∂ 2G ( M , M ' ) NUMERICAL RESULTS VALIDATION Convergence study was performed on a rectangular NACA 0024 wing with aspect ratio λ = in forced heaving motion, at a reduced frequency ν=ωc/U=0.4 Results show that convergence is about reached as soon as a grid with 20 strips columns is used; results are independent of the total number of panels when NTC is about 400 for the added-mass and 500 for the damping For the grid, it had been take care of avoided panels with too small or large aspect ratio To compare with available 3D aerodynamic results, computations have been first performed for a wing with an aspect ratio λ=0.5 at deep submergence (f/C=10) Further computations were performed using 10 strips and 35 columns by strip leading to 350 panels on the body symmetry plane and 700 on the hull On figure 1, the lift coefficient amplitude ( C = C eiΦ , top) and the phase lag L L L (bottom) versus the reduced frequency ν=ωC/U the agreement with the results of (1) or those of Nontakaew et al (7,8), both using a lifting surface method, is satisfactory Some differences can be attributed to the 191 fact that the present method takes into account the body thickness, whereas the other methods not To check the accuracy of the method, 3D calculations on a test case given in Geissler (2) were performed with the focus put on the pressure distribution A 25° swept wing with angle, with a NACA0012 profile, aspect ratio =2.94, equipped with an oscillating flap extending on 25% of the chord was considered The mean incidence is zero and the reduced frequency ν=0.744 The pressure jump coefficient distribution along the chord, ∆Cp, is plotted on figure for a section located at y=1.02C (real part on the top and imaginary part on the bottom) The agreement with non linear (panel method) and linear (lifting surface) results of Geissler (2) and with the tests of Försching et al (15) shows a good agreement except close to the hinge, may be due to a lack of grid refinement close to the hinge in our calculations It can be also observed that the KuttaJoukowsky condition is very well satisfied Re(∆Cp) 0 0.2 1.5 2.8 2.7 0.4 ν=ωC/U 0.6 0.8 Tests Forsching et al (14) Non linear comp Geissler (2) Linear comp Geissler (2) Present method |CL/(iνξ/C)| 0.5 Im(∆Cp) 2.6 2.5 2.4 -0.5 y/C=1.02 2.3 -1 2.2 -1.5 2.1 -2 1.9 1.8 0.1 210 0.2 0.3 ν=ωC/U 0.4 ΦL 195 190 185 Present method Ashley et al (1) Nontakaew et al (7,8) 175 0.2 0.3 ν=ωC/U 0.4 0.5 Figure 1: Lift amplitude coefficient and phase lag versus the reduced frequency for an aerodynamic wing with aspect ratio λ=0.5 with a NACA0012 profile Session A 0.4 ν=ωC/U 0.6 0.8 The effect of the Kutta-Joukowsky condition on the pressure jump coefficient ∆CP along the chord was also investigated Figure plots the real (top) and imaginary (bottom) parts of the pressure coefficient jump along the chord for an aerodynamic computations on a 2D NACA0024 profile, deeply immerged (f/C=10), at the reduced frequency ν=0.64 The present results with the Kutta-Joukowsky condition (wi) are represented by full lines They are compared with results for the same case with the same code, but without any Kutta-Joukowsky conditions and doublet distribution (wo), Boin (8), given by a dashed line Large discrepancies close to the trailing edge for x/C>0.6~0.7 can be observed There are also important differences particularly for the real part of the pressure coefficient jump, where for x/C