UNSTEADY EFFECTS IN 2D MODELLING OF PARTIAL CAVITATION USING BEM

Transcription

1 MÉTODOS COMPUTACIONAIS EM ENGENHARIA Lisboa, 31 de Maio - 2 de Junho, 24 APMTAC, Portugal 24 UNSTEADY EFFECTS IN 2D MODELLING OF PARTIAL CAVITATION USING BEM Guilherme Vaz, Johan Bosschers and J. A. C. Falcão de Campos Instituto Superior Técnico, Lisbon, Portugal (currently at MARIN) g.vaz@marin.nl MARIN, Wageningen, The Netherlands j.bosschers@marin.nl Instituto Superior Técnico, Lisbon, Portugal fcampos@hidro1.ist.utl.pt Keywords: BEM, Foils, Steady, Unsteady, Partial Sheet Cavitation, Reduced Frequency, Pressure Fluctuations Abstract: The authors have developed 2D steady and unsteady models for partial sheet cavitation analysis based on the Boundary Element Method (BEM). Using these models, this paper presents an analysis of the cavitating flow on a foil in a vertical gust similar to a ship wake peak and shows the influence of the reduced frequency on the cavitation characteristics. A quasi-steady analysis based on an equivalent unsteady wetted flow loading is compared to a fully unsteady analysis in order to give additional insight in the unsteady effects on the cavitation behavior. The results show that the quasi-steady analysis done with equivalent loading conditions overpredicts the effects of the reduced frequency. These unsteady effects are correctly taken into account in the full unsteady analysis and revealed to be important for the cavity dynamics. The trends in the variations of the minimum pressure, cavity length, cavity volume, cavity volume speed and cavity volume acceleration with the reduced frequency are evaluated.

2 1 INTRODUCTION Cavitation on ship propellers can be a major source of noise and vibration causing discomfort to crew and passengers. A significant amount of both theoretical and experimental research is devoted to the propeller cavitation behavior and the resulting hull pressure fluctuations during the design phase of ship and propeller. A full three-dimensional analysis of unsteady sheet cavitation on hydrofoils and on propeller blades operating behind a ship s wake has been approached by potential flow theory using Boundary Element Methods (BEM), Kinnas and Fine [1], Achkinadze and Krasilnikov [2], Salvatore et al [3]. A simpler approach for the analysis of ship propellers behind ship wakes consists in using an unsteady wetted flow analysis in combination with 2-D methods for cavitation analysis [4]. The transformation of the propeller blade section to a 2-D equivalent foil is then made on the assumption that the foil will carry the same loading as the blade section in the 3-D unsteady flow. Such an approach neglects the interaction between the cavity and the foil loading and the 3-D unsteady effects on the cavity dynamics. The unsteady effects on the dynamics of partial cavities on a foil, including cavitation inception, are of relevance in understanding wake scale effects on sheet cavitation dynamics. Such unsteady effects on the cavity dynamics can be more easily investigated by an unsteady potential flow analysis of 2-D foils placed in suitable inflow gusts. Therefore, the objective of the present paper is to investigate the influence of the reduced frequency on the intermittent cavitation characteristics for a foil subject to a gust. Further insight into the unsteady effects can be obtained by comparing the full unsteady analysis with a quasi-steady analysis using a steady cavitation analysis at the instantaneous total loading of the unsteady wetted flow analysis. This instantaneous total loading takes into account the reduced frequency effect being therefore analogue to what is done in the classic theory for a flat plate, Kármán and Sears [5]. This analysis is carried out with the previously developed 2D models for partial sheet cavitation based on the Boundary Element Method (BEM). A detailed investigation on the main characteristics of the steady model have been presented in [6]. This investigation included a fully non-linear model with a re-entrant jet cavity closure and a partially non-linear model with and without re-gridding of the cavity surface. From the study of the numerical properties of the different methods regarding convergence, accuracy and robustness, the partially non-linear model with re-gridding appeared to be the best choice for this investigation. The same choices were used in the unsteady model for the present investigation. A full description of the unsteady model can be found in [7]. The paper is organized as follows: Section 2 briefly presents the mathematical formulation for the unsteady potential flow problem of a 2-D foil in a gust. Section 3 describes the major features of the numerical method. Section 4 presents first the results of the models on a foil for a particular condition, and second, the results of the investigation on the influence of reduced frequency in the cavity characteristics for a wake peak type of vertical gust. In section 5 the main conclusions are presented. 2

3 2 THEORETICAL FORMULATION 2.1 Governing Equations and Boundary Conditions Consider a 2-D hydrofoil section with an attached cavity, as shown in Fig.(1). The flow is considered to be incompressible, inviscid and irrotational. The hydrofoil is considered to be travelling in space with a translational velocity, V o. In the reference frame fixed to the foil (x,y), we assume the perturbation to the uniform inflow, V w, to take the form of a vertical gust of wave length λ travelling in the positive x direction with velocity V = V o, i.e. V w = (,V wy ) = (,Vwy + V wy f[2π (x/λ) ωt] ), (1) where ω = (2πV )/λ is the frequency and f a specified function. The undisturbed ambient pressure may also vary with the same frequency, p (t) = p + p sin (ωt). (2) The parameters V wy, V wy,p, p and the function f (x,t) define the inflow conditions of the problem. The moving system is constituted by the hydrofoil with boundary S B, the partial cavity surface S C and the wake surface S W. These surfaces bound the domain Ω and constitute the boundary S = Ω of the flow field, Fig.(1). y n s S C n + S W x S B n n V w σ V o Figure 1. Unsteady two-dimensional cavitation problem domain. The non-dimensional frequency of the gust can be defined by the reduced frequency k, k = ωc 2V, (3) where c is the foil chord. In the sequel we introduce non-dimensional quantities using the foil chord c as reference length, the foil speed V as reference velocity, and 1/ω as reference time. The flow velocity is derived from a potential function V = Φ = V w V + φ, where Φ is the total potential and φ the perturbation potential. The flow in the external domain Ω is described by the Laplace equation for potential flow, 2 φ (x,t) =. (4) 3

4 Using Green s identity and the two-dimensional Green s function, the solution for the perturbation potential on a boundary point x for a time t satisfies the equation: [ πφ (x,t) = φ (x,t) ] φ (x,t) [ln r (x, x,t)] + lnr (x, x,t) ds, (5) n n S(t) where / n denotes differentiation along the outward directed normal to the boundary, as depicted in Fig.(1). In the usual interpretation of this equation the potential may be considered as induced by source and dipole distributions, with strengths φ/ n and φ, respectively. The Euler equation for potential unsteady flow, neglecting the effects of gravity, reduces to the Bernoulli equation in the form 2 φ t + V r o + φ 2 V ro 2 = C p, (6) where V ro = V w V is the undisturbed velocity in the reference frame fixed to the foil, C p = (p p )/(1/2ρV ) 2 the pressure coefficient, p the pressure, p the undisturbed pressure and ρ the density of the fluid. For the wetted part of the body surface, S B, the impermeability condition must be satisfied by imposing the boundary condition, φ n = V r o n, (7) where n is the unit normal to the foil surface. On the body surface the potential is unknown. Consider a cavity surface defined by F (s,n,t) = or n = η (s,t), η being the cavity thickness in a local tangential-normal coordinate system (s,n) attached to the cavity surface at each time step t. Since the position of the cavity surface is not known, we need two boundary conditions: a kinematic and a dynamic boundary condition. The kinematic boundary condition states that the flow must be tangent to the cavity surface, η t = Φ n Φ η s s. (8) The dynamic boundary condition ensures that the pressure on the cavity is the vapour pressure p v, σ 2 φ t + V r o 2 Φ Φ =, (9) where σ = (p p v )/(1/2ρV ) 2 is the cavitation number. The total tangential velocity U and the total normal velocity V to the surface are given by U = Φ s = ( V ro s + φ s ), V = Φ n = ( V ro n + φ ), (1) n where s and n are the unit vectors of local coordinate system (s, n). The dynamic boundary condition, Eq.(9), is used to prescribe the potential variation on the cavity surface and the strength of the source distribution has to be determined. The application of the exact form of the dynamic boundary condition requires the knowledge of the location of the cavity surface, which is not known from the outset. 4

5 In a partially non-linear formulation, [6], we assume that the cavity thickness is small but the cavity induced velocities are not. This approach does not lead to a fully linearized method, where both cavity thickness and velocities are considered small. Expanding the velocity components around the foil surface n =, we may translate the boundary conditions from the cavity to the foil as follows: U = U + U n=η n= n η + O ( η 2), (11) n= V = V + V n=η n= n η + O ( η 2). (12) n= Since the flow field is irrotational and divergence free, we may write for the normal derivatives of the velocity components, U n = κu, V n = U s, (13) where κ is the body curvature. Neglecting the terms involving the normal velocity V in the dynamic boundary condition we obtain Φ s = σ + Vr 2 o 2U 2 κη 2 φ t, (14) and for the kinematic boundary condition η s = V η U U + ηκu, (15) where the velocities are evaluated at n =. Eq.(14) is used to prescribe the perturbation potential on the body surface s [ ] φ = φ + σ + V ro 2 2U 2 κη 2 φ t V r o s ds, (16) s s η t where φ is the potential at the cavity detachment point s = s. For the application of the boundary condition (16), we need to determine the cavity length from the position of the cavity surface. This is done iteratively. The cavity update may be obtained from the kinematic boundary condition Eq.(15) in the form s U V η η = ds. (17) s U + ηκu At a given time step, for a given cavity length, the cavity thickness can be computed from Eq.(17). Convergence of the spatial iteration is obtained when the cavity thickness at the last point of the cavity η (s c ) = δ is zero. For that a non-linear equation, s η t δ (l c ) = (18) has to be solved, being l c the cavity length at each time step. When the length of the cavity estimate is lower than the converged one, the thickness distribution η does not cross the foil 5

6 surface at the end of the cavity (δ(l c ) > ). However, when the initial length estimate is higher than the converged value this thickness is negative (δ(l c ) < ) at the end of the cavity. Using a simple non-linear numerical scheme, a good estimate of the correct length can be found, where δ should be zero at the last point of the cavity. 2.2 Cavity Detachment and Cavity Closure The cavity detachment point location is known to be very important in the prediction of the correct cavity lengths, especially for thick foils. Different criteria investigated in the literature comprise the location of minimum pressure, the location where the pressure becomes equal to vapour pressure, the smooth separation condition by Brioullin and Villat, and the laminar separation position (see Franc and Michel [8]). All these options were implemented in the current models, but in the present paper the influence of the cavity detachment location is not considered, and, therefore the detachment point location has been fixed at the leading edge. For the cavity closure, a pressure recovery model has been used for the steady model [6]. For the unsteady model the dynamic boundary condition Eq.(14) has been used without any pressure recovery at the cavity closure. 2.3 Wake Surface In an unsteady lifting flow, as the circulation around the foil varies in time, vorticity is shed into the wake at the trailing edge. We assume the vorticity to be convected along the x axis in the positive direction with speed V. It is easily seen that the potential jump ( φ) is also convected along the wake with the speed V, φ (x,t) = φ (x V t). (19) The initial condition for Eq.(19) is given by the Morino Kutta condition ( φ) t.e = φ + φ, (2) relating the potential jump in the wake at the trailing edge ( φ) t.e to the surface potentials on the upper and lower sides of the foil at the trailing edge φ + and φ. 3 NUMERICAL METHOD 3.1 BEM For the solution of the integral equation Eq.(5) a low order BEM is used, i.e. linear panels and piecewise constant singularities distributed on the foil. Influence coefficients are computed analytically. For the solver, a simple LU decomposition is used. For post-processing, all the integrations and differentiations are performed using cubic splines. 3.2 Spatial and Time Discretization For all models, tight stretching is needed near the leading edge of the foil, detachment of the cavity and re-attachment of the cavity. For the wetted flow, the usual cosine stretching is ap- 6

7 plied. The spatial discretization is characterized by: N w -number of panels for the wetted flow calculations N l -number of panels on the lower side of the foil; N c -number of panels on the cavity; N d -number of panels on the part of the foil downstream of the cavity; N-total number of panels. The spatial discretization of the wake surface in inherently related with the temporal discretization of the model. Two parameters have to be specified: the number of wave lengths N λ, which determines the wake length, and the number of panels within a wave length N Nλ, which determines the time step. 3.3 Wake Modelling and Vorticity Shedding The degree of approximation of the wake dipole strength is an important numerical choice to be made. The previous work done by Hsin, [9], showed that if constant dipole distribution on the first wake panel is considered, the results become dependent on the ratio of the length of the first wake panel to the length of the trailing edge panel. To overcome this dependency, linear dipole distributions for the whole wake are used, being the influence coefficients of these singularities calculated analytically using our previous higher-order studies [1]. The dipoles strengths are located not on the center of the panels (as in the body and cavity) but rather on the boundary points of the wake panels. 3.4 Algorithm For performing the unsteady calculations, an initial solution is needed. Therefore, for the startup of our calculations, a steady wetted calculation is done for the average loading conditions, being the resultant wake strengths taken into account for the first unsteady calculation. For each time step, the occurrence of cavitation inception is tested. If cavitation inception does not occur, a wetted flow calculation is performed. If there is inception, the spatial iterative scheme for the cavity is started. This involves an initial guess for the cavity length. Geurst s fully linearized model, [11] is used for that purpose. The spatial iterative scheme consists of solving the non-linear equation for the zero thickness at the end of the cavity. For each spatial iteration, when using the correction terms Eqs.(11), (12) and (13), inner iterations are needed for solving the boundary conditions coupling. This scheme is computationally cheap, since no influence coefficients are re-computed and therefore only the right hand side of the linear system is changed. This process is stopped when the differences between the correction terms of two consecutive iteration steps are smaller than the specified tolerance of Also, for each length estimate, the part of the foil beneath the cavity estimate is re-panelled and the influence coefficients re-calculated. When convergence is achieved, the thickness is added to the foil coordinates to obtain the cavity surface. This process is repeated until spatial convergence is achieved for the cavity length l c, i.e. ( ) lc n lc n 1 lc n δ ǫ. (21) A value of is used for ǫ, following our previous studies [6]. 7

8 When spatial convergence is achieved, the unsteady terms are calculated, the vorticity shed and cyclic updated, and the time stepping iterative procedure continues. In this implementation of the unsteady models the process is stopped when the simulation time T s is reached. This time must be high enough to let the numerical transients to fade away and then obtain physically correct values. 4 RESULTS 4.1 Tests Conditions Having available the steady and unsteady models for cavitation analysis, denoted by and PNL UnStd, respectively, an interesting practical application consists of a propeller typical design section under a 2-D gust qualitatively similar to a real 3-D ship wake. In this work only periodic vertical gusts will be studied, being in practice the most important case. A case where V wy =, Vwy V =.1 and the wake length is 5% of a cycle, is here considered. The wake peak (maximum) is at θ = 18 position, being θ = ωt. A cavitation number of σ = 1. is considered. Fig.(2) shows the vertical velocity variation along one cycle (or revolution) of the unsteady motion. A typical NACA16-6 foil widely used for numerical and experimental tests on cavitation, and also previously used for the assessment of the steady model, [6] is chosen as test geometry V wy / V Local α [º] at L.E Figure 2. Inflow conditions. 2-D wake peak gust. The angle of attack α corresponds to the local instantaneous angle of attack calculated at the leading edge of the foil. The same picture also shows a local angle of attack at the leading edge of the foil for each 8

9 time step. This angle of attack will be the inflow condition for the steady model at each time step, of the quasi steady analysis, valid in the limit of zero reduced frequency. Part of the unsteady effects can be taken into account in the quasi-steady analysis by reducing the angle of attack to match the unsteady total load. For the unsteady analysis, first the results will be shown for a fixed moderate reduced frequency of k =.5 and afterwards the influence of the reduced frequency will be studied. The typical results of the models studied in this work will be the cavity length l c, cavity volume V c and the minimum pressure coefficient C pmin. Also of interest are the time derivatives of the cavity volume. The second derivative has the largest contribution to the source term of the pressure fluctuations at a point far from the hydrofoil. A harmonic analysis of the cavity volume second time derivative is therefore considered with respect to the gust frequency. For the BEM models, the numerical parameters were set based on our previous exhaustive tests on the influence of the discretization levels (space and time) on the results. For the steady model,, [6] and for the unsteady model PNL Unstd [7]. The numerical parameters chosen are: Detachment point at the leading edge. Spatial discretization: N w = 1,N c = 5,N d = 5. Time discretization: N λ = 16, N Nλ = 6, with a total of N λt = 96 time steps. 4.2 Typical Results As an example of what our models can produce we show for the chosen inflow, typical results obtained by the steady model and the unsteady model PNL UnStd. For the steady model an angle of attack of 4 was chosen, being this in the range of angles for the particular inflow. Fig.(3) shows the steady cavity layout and the pressure distribution on the foil. For the unsteady model PNL Unstd the obtained cavity layouts for each time step, Fig.(4) are shown. This picture together with Fig.(5), show that for the chosen inflow we have intermittent cavitation. Fig.(5) also shows that for the particular numerical conditions we have time accurate converged results. 4.3 Vertical Gust Analysis Reduced Angle of Attack In order to perform a fair comparison between the steady and the unsteady models under a spatial gust variation, an average equivalent steady loading has to be calculated from the fully unsteady 2-D wake gust. The lift coefficient is chosen to be matched. For that purpose, we consider a so called reduced angle of attack or corrected angle of attack, α cor, input for the steady model. The process to determine this quantity involves three steps: Calculate C lα, slope of C l vs α curve, and α, angle of zero lift coefficient, for the selected geometry, using. 9

10 y/c y/c.4.2 Foil x/c -C p C p Wetted Flow x/c x/c x/c Figure 3. Cavity layout (left) and pressure distribution (right). α = 4. results. Calculate for the selected inflow the lift coefficient C l for a wetted flow calculation, using PNL UnStd. From the values calculated above, for each time step of the gust cycle, calculate a reduced angle of attack, α cor = C l C lα α. (22) Fig.(6) illustrates the results for the α cor. The reduced frequency influence on the lift has the effect of smoothing the gust by decreasing the amplitude of α and by introducing a phase lag. In the same picture it is also visible that the application of this reduced angle of attack induces also a decrease of the minimum pressure and shifts its position. This is important for the inception point of cavitation Results for a Fixed Reduced Frequency The results of both models for the inflow of Fig.(2) are shown for k =.5 in Fig.(8). The results show that for this reduced frequency using the reduced angle of attack leads to small differences in the cavity length and cavity volume amplitude between both models. However, there is still a clear phase difference. The values for the cavity dynamic analysis and the amplitude of the power spectrum of the FFT analysis, for the unsteady model PNL UnStd are higher than the ones obtained by the steady model. With respect to the second time derivative of the cavity volume, in spite of the phase and amplitude differences, the qualitatively behavior is very similar, as seen in the amplitude spec- 1

11 y/c y/c x/c Foil ωt= 15x36 o Foil ωt=3x36 15x36 o + o o y/c x/c Foil ωt= 15x36 o + 21 o y/c x/c Foil ωt= 15x36 o + 24 o x/c Figure 4. Cavity layouts for some time steps of the last cycle. PNL UnStd results. trum picture. For both models the 4 th harmonic with respect with the gust frequency is the dominating one. For the model the higher order components are due to the numerical oscillations and have no physical meaning Reduced Frequency Influence The effects of the variation of reduced frequency are investigated in the range (steady) to 1.. Fig.(9) shows the influence of the reduced frequency on the angle of attack correction. With increasing k the reduced angle of attack shifts to the end of the cycle and its amplitude is also decreased. Fig.(1) shows the effects on the minimum pressure coefficient. For the quasi-steady analysis the effect of the reduced angle of attack is to decrease the minimum pressure amplitude and to shift its position to the end of the cycle. The qualitative behavior of both quantities α cor and C pmin for the model results is therefore very similar. However, for the PNL UnStd model, the results show that the shift is not so pronounced and that the amplitude does not decrease at the same rate. There is, however, a phase lag of the minimum pressure in relation with the maximum of the gust at 18 position. Fig.(11) shows the trends of the cavity length and cavity volume with the reduced frequency for both models. While the amplitudes of the cavity length and cavity volume look similar, even if there are increasing differences when the reduced frequency is increasing, the phase lag 11

12 Last Cycle l c l c Iter θ [ o ] Figure 5. Cavity length for the complete simulation (left) and for the last cycle (right). PNL UnStd results. 6 α α cor α [º] 3 C pmin C pmin α C pmin α cor PNL UnStd C pmin Figure 6. Reduced angle of attack and its influence on the minimum pressure. is considerably different for the two models. For the cavity volume and length the trends are the same in this respect. 12

13 .7.6 Cl α Cl α cor PNL UnStd Cl.5.4 C l Figure 7. Loading equivalence for the wetted flow PNL UnStd 3.5 x PNL UnStd l c.1 V c d 2 V c /dθ PNL UnStd Amp. Spectrum of d 2 V c /dθ Figure 8. Results for k =.5. and PNL UnStd. PNL UnStd nω 13

14 6 5 4 α α cor k=.5 α cor k=.1 α cor k=.25 α cor k=.5 α cor k=.75 α cor k=1 α [º] 3 k increasing Figure 9. Reduced frequency influence on the α cor. 5 5 C pmin k increasing C pmin k increasing 1 1 α k=.5 k=.1 k=.25 k=.5 k=.75 k= α PNL UnStd k=.5 PNL UnStd k=.1 PNL UnStd k=.25 PNL UnStd k=.5 PNL UnStd k=.75 PNL UnStd k= Figure 1. Reduced frequency influence on the C pmin. (left) and PNL UnStd (right). Another interesting fact is related with the growing and collapsing phase of the cavity. While for the steady model these two phases seem to have the same velocity since in Fig.(11)(left) the curves for cavity length and volume seem symmetric around the maximum point, for the unsteady model the results show a different speed, which is physically consistent. The growing phase appears to be slower than the collapsing phase. This is very clear for the cavity length 14

15 k=.5 k=.1 k=.25 k=.5 k=.75 k= PNL UnStd k=.5 PNL UnStd k=.1 PNL UnStd k=.25 PNL UnStd k=.5 PNL UnStd k=.75 PNL UnStd k=1 l c.4 l c k=.5 k=.1 k=.25 k=.5 k=.75 k= PNL UnStd k=.5 PNL UnStd k=.1 PNL UnStd k=.25 PNL UnStd k=.5 PNL UnStd k=.75 PNL UnStd k=1 V c.2 V c Figure 11. Reduced frequency influence on the cavity length l c and volume V c. (left) and PNL UnStd (right). curve. Being the cavity volume more important from a dynamic point of view, it is worthwhile to investigate this different speed of the growing and shrinking phase of the cavity for the cavity volume. Fig.(12) shows that for the steady model there is almost no change between these two phases while for the unsteady analysis the velocities of the collapsing phase have higher absolute values than the ones for the growing phase of the cavity. The reduced frequency has also influence on the cavity aspect ratio. For both models it is seen in Fig.(13) that the cavity maximum thickness to cavity length ratio is decreasing. The cavity maximum thickness decreases from 8.% of the cavity length to around 6.25%, for both models. However, for the unsteady model the cavity aspect ratio is in the growing phase different from the collapsing phase. Finally the variation of the inception point and cavity length with the reduced frequency will be compared for both models. Fig.(14) shows the curves of maximum absolutes values of the minimum pressure coefficient and the cavity length with the reduced frequency. For the maximum amplitude it is shown that both models present curves with different slopes with the results of the steady model being more sensitive to variations in the reduced frequency. For the phase lag the differences of the two models are considerably high. For the steady model the phase is always increasing with the reduced frequency, while for the unsteady model the phase 15

16 d V c /d θ d V c /d θ.2.4 k=.5 k=.1.6 k=.25 k=.5 k=.75 k= PNL UnStd k=.5 PNL UnStd k=.1 PNL UnStd k=.25.8 PNL UnStd k=.5 PNL UnStd k=.75 PNL UnStd k= Figure 12. Reduced frequency influence on the cavity volume velocity dvc dθ. (left) and PNL UnStd (right) t cmax / l c.5.4 t cmax / l c k=.5 k=.1 k=.25.1 k=.5 k=.75 k= PNL UnStd k=.5 PNL UnStd k=.1 PNL UnStd k=.25.1 PNL UnStd k=.5 PNL UnStd k=.75 PNL UnStd k= Figure 13. Reduced frequency influence on the cavity maximum thickness to chord ratio tcmax l c. (left) and PNL UnStd (right). lag seems to approach a constant value. 16

17 PNL UnStd max C pmin 5 PNL UnStd C pmin max l c k 1 PNL UnStd max l c.2.4 k PNL UnStd k k Figure 14. Reduced frequency influence on the amplitude and phase of minimum pressure and cavity length. 5 CONCLUSIONS This work presents results of a steady and unsteady analysis of a hydrofoil under a periodic wake peak type of vertical gust. Two previously implemented steady and unsteady BEM models are used. The steady analysis is done with same loading conditions as predicted by the unsteady model for the wetted flow. This procedure produces comparable results to the unsteady analysis for low reduced frequencies. From this study the following conclusions can be drawn: The equivalence of loading for high reduced frequencies is questionable since in that situation the non-linear effects of the gust on the loading and cavity are higher. The unsteady analysis shows that the growing phase of the cavity is faster than the shrinking phase, a fact well known from experiments. The quasi-steady computations do not show this behavior. The cavity aspect ratio changes with the reduced frequency. The results show that the cavity length and minimum pressure phase lags are over predicted by the quasi-steady analysis: the differences to the fully unsteady analysis increase with the reduced frequency. The cavity length and the minimum pressure phase lags tend to a constant value. Further 17

18 investigations should be addressed on this topic and the analogy with the classical analytical theory of Sears should be further investigated. Future research could concentrate on the influence of changes in the gust amplitude and length on the cavity dynamics, due, for instance, scale effects on the ship wake. More representative gusts derived from section operational conditions for propellers in ship wakes may be also considered. REFERENCES [1] Kinnas, S. A. and Fine, N. E. A Numerical Nonlinear Analysis of the Flow Around Two and Threedimensional Partially Cavitating Hydrofoils. Journal of Fluid Mechanics, 254: , September [2] Achkinadze, A. S. and Krasilnikov, V. I. A Velocity Based Boundary Element Method with Modified trailing Edge for Prediction of the Partial Cavities on the Wings and Propeller Blades. Proceedings of Cav21, Pasadena, California USA, July 21. [3] Salvatore, F. and Esposito, P. G. Numerical Analysis of Cavitating Propellers Including Viscous Effects. Proceedings of 8th Int. Symp. on Practical Design of Ships and Mobile Units (PRADS 21), 21. [4] Van Gent, W. Pressure Field Analysis of a Propeller with Unsteady Loading and Sheet Cavitation. Proceedings of 2 th ONR, pages , [5] Kármán, Th. Von and Sears, William R. Airfoil Theory for Non-Uniform Motion. Journal of the Aeronautical Sciences, 5(1), August [6] Vaz, G., Bosschers, J. and Falcão de Campos, J. Two-dimensional Modelling of Partial Cavitation with BEM. Analysis of Different Models. Cav23 Proceedings, Osaka, Japan, November 23. [7] Vaz, G. 2D Unsteady BEM Cavitation Model. Technical Report RP SP, MARIN, December 22. [8] Franc, J. P. and Michel, J. M. Attached Cavitation and the Boundary Layer: Experimental Investigation and Numerical Treatment. Journal of Fluid Mechanics, 154:63 9, [9] Hsin, C. Development and Analysis of Panel Methods for Propellers in Unsteady Flow. PhD thesis, Massachusetts Institute of Technology - MIT, September 199. [1] Vaz, G., Falcão de Campos, J. and Eça, L. A Numerical Study on Low and Higher-Order Potential Based BEM for 2D Inviscid Flows. Computational Mechanics, December 23. [11] Geurst, J. A. Linearized Theory of Two-Dimensional Cavity Flows. PhD thesis, University of Delft, May