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Silent Propellers with Unconventional Profile Shapes. Examples Obtained with a New Automatic Optimization Method Stefano Brizzolara and Stefano Gaggero Marine CFD Group, Dept. of Naval Architecture of the University of Genoa, Genoa, Italy ABSTRACT A new CFD based integrated method for design and parametric optimization of unconventional propellers will be presented in the paper. The method is able to optimize the shape of the blades by using unconventional distribution of pitch, skew and rake distributions along the radius and unconventional blade profile shapes, i.e. new hydrofoils with non standard (non-NACA) thickness and camber distribution. The method is based on a largely validated panel method developed by the Marine CFD Group able to solve the three dimensional flow around cavitating propellers working in a (non-uniform) ship wake, an srcinal propeller geometry definition module, based on parametric B-splines modelling of the propellers blade surfaces, and an efficient optimization algorithm. The genetic multi-objective optimization algorithm drives the selection of the main parameters that defines the unconventional propeller geometries. Paper will demonstrate by how far it is possible to reduce cavitation with a globally optimized unconventional propeller design, in two cases: the first regards a cavitating fast patrol boat propeller optimized in uniform flow, while the second addresses the problem of unsteady sheet cavitation, the primary sources of noise for navy propellers. The new design method opens interesting perspectives for the design of silent, efficient propellers and it can be applied for fast open propellers as well as for highly loaded ducted propellers working in nozzles or tunnels. The gains in efficiency and cavitation characteristics achieved by non-conventional optimized designs are discussed in comparison with state of the art conventional reference designs. 1. INTRODUCTION The design procedure for propeller design is still commonly based on the classical vortical theory of Lerbs (1952) with approximated lifting surface corrections, as developed in the 60’s Morgan et al. (1968) or with exact lifting surface methods (Greeley, D., Kerwin, 1982) that represent thes current state of the art of propeller design codes widely nowadays around the world and also in Italy (Grossi, 1980), and adapted recently for the case of counter-rotating propellers (Brizzolara et al., 2007). With these kind of methods the propeller design refers to the mean (uniform) ship wake and includes simple cavitation avoidance and structural strength criteria, that being approximately imposed need ‘a posteriori’ verification procedure that could properly take into account of unsteady cavitation due to non-uniform viscous ship wake. In fact, for navy high speed ships, cavitation and unsteadiness plays the most important role in the design of high performance propellers, to avoid primarily radiated noise and induced vibrations. As demonstrated by several studies (Brennen & Ceccio, 1989; Ceccio & Brennen, 1990; Brennen, 1995) the noise radiated by unsteady cavitation bubbles is well correlated with the variation of their volume. In analogy with an ideal spherical bubble which follows the model of Rayleigh-Plesset, for sheet and for cloud cavitation it is possible to correlate the magnitude of the acoustic pressure p A radiated at a certain distance r from the bubble centre with the variation of the cavity volume over time V(t), according the following formula: (1) Hence it is clear from (1) that in order to reduce the noise in water it is generally important to reduce the cavity volume and its second derivative over time. From this the need of more accurate CDF methods able to adequately predict the cavitation volume, especially in unsteady conditions. A valid answer to this problem can be given by recent potential flow solvers, based on boundary element (panel) methods, developed at different levels by several research groups around the world (e.g. Hsin, 1990; Fine, 1992; Young, 2002; Caponnetto & Brizzolara, 1995, Vaz & Bosschers, 2006) with interesting prediction capabilities. These methods, in fact, can solve the potential flow around the propeller, correctly taking into account for the thickness effects of the blade (neglected or approximately recovered in the lifting surface theories), of the hub and, most important, also of the effect of cavitation. The method recently developed at the University of Genoa (Gaggero & Brizzolara 2007, Gaggero & Brizzolara 2008) can effectively predict the steady and unsteady flow around the propeller with sheet cavitation, including supercavitation cases (Gaggero & Brizzolara, 2009). Nevertheless all such models are conceived for the analysis of known propellers, and so they cannot be directly integrated into an inverse design procedure, differently from the more simpler ones previously cited and, in fact, they are practically used for validation of a given propeller design or for systematic investigation of main parameter influence around a base design geometry. The time required to solve a steady cavitating propeller problem is in the order of few minutes on a modern PC and the type of data and accuracy of the results obtained is of the same level of accuracy of the RANS solvers wide range of off-design working conditions, as demonstrated for instance by Brizzolara et al. (2008). Hence the straightforward idea is to include the panel method into a parametric optimization procedures for the automatic optimization of an initial propeller geometry, allowing for a more congruent and effective search of the best geometry subjected to more stringent constraints: the optimization procedure is driven by a multi-objective optimization algorithm which drives the selection of the main geometric parameters of propeller blades to converge on the optimum solution: i.e. the one that can minimize the cavitation volume, keeping the same thrust and possibly increasing or not decreasing the efficiency. To obtain significant and feasible geometries, first of all a robust parametric description of the blade geometry is needed. The propeller geometry is represented by a series a B-Splines curves describing the distributions (along radius or chord) of all the significant geometrical characteristics of the propeller (such as pitch, chord, maximum camber and thickness, skew and rake). The vertexes of the control polygon of these B-Spline curves become the free parameters of the optimization algorithm. Finally, the solution is driven to convergence using the developed panel method coupled with some evolutionary algorithms into the ModeFRONTIER environment (Esteco, 2008). The proposed procedure have been initially applied with success (Brizzolara et al, 2009) in case of open and ducted propellers in uniform flow. The applications presented in this paper, instead, are focused, first, on the optimization of a modern highly skewed propeller designed for medium speed patrol boat (24 kn) via a traditional lifting line/lifting surface code and subjected to a mean equivalent uniform inflow deriving from the actual ship wake. The second example, instead, features the optimization of a CP propeller designed for high speed frigates (35 knots), extensively tested in the cavitation tunnel of the Department, and optimized as operating in a non uniform inflow, by means of an srcinal method in order to reproduce the unsteady conditions behind an hull, by means of a restricted set of stationary conditions. 2. FLOW SOLVER A potential flow solver, developed by the Marine CFD Group of the University of Genoa, has been adopted to compute the hydrodynamic characteristics of non cavitating and cavitating marine propellers in steady and unsteady conditions. The code is based on an inviscid and irrotational approach. With these assumptions, the flow around an arbitrary body (in a simple connected domain) is known as the solution of a differential Laplacian equation, that can be further simplified applying Green’s second identity. Consider a right handed propeller rotating with constant angular velocity ω ωω ω in a axisymmetric incoming flow field. In the (,,) ppp xyz coordinate system that rotates with the propeller, the total velocity vector V can be written as the sum of the relative undisturbed inflow rel V (known in the propeller reference system) and the perturbation potential velocity ind q , due to the velocity influence of the propeller itself on the velocity field: relind = + V V q (2) where the relative velocity rel V , in the propeller reference system, can be written as: rel ∞ = − × V V r ω ωω ω (3) With the assumption of an inviscid, irrotational and incompressible fluid, the perturbation velocity can be written in terms of a scalar function, the perturbation potential, that satisfies the Laplace equation: 2 0 ind and φ φ ∇ = =∇ q (4) By applying Green’s second identity for the perturbation potential, the differential problem (4) can be written in integral form with respect to the potential p φ at every point p laying onto the geometry boundaries. The perturbation potential i φ represents the internal perturbation potential, that must be set equal to zero in order to simulate fluid at rest inside the boundaries of all the bodies subject to the external inflow (blades, hub). 1112 BBW CBCB qqi pqqiqqpqqqpqqpqSSSSS dSdSdS rrr φ φ πφ φ φ φ + + ∂ ∂∂ ∂= − − − + ∆∂ ∂ ∂ ∂ ∫ ∫ ∫ n n n n (5) The subscript q corresponds to the variable point in the integration, n is the unit normal to the boundary surfaces and r pq is the distance between points p and q . Equation (5) expresses the potential on the propeller blade as a superposition of the potential induced by a continuous distribution of sources ( σ ) on the blade and hub surfaces and a continuous distribution of dipoles ( φ or φ ∆ ) on the blade, on the hub and on the wake surfaces: a certain number of boundary conditions, depending on the type of problem (non cavitating or cavitating, steady or unsteady), are needed to invert and solve equation (5). In non cavitating steady condition, classical kinematic (6) and Kutta (7) boundary conditions hold, while in the cavitating case some other conditions must be forced on the cavity surface. qqq n φ ∂= ⋅∂ − V n (6) ...... ULTETETE φ φ φ ∆ = − (7) Equation 6, as proposed by Morino (1974), is rigorously valid only for two dimensional flows and can be applied with success to the three dimensional cases in which the cross flow effects are negligible (lightly loaded propellers or propellers without excessive skew). For a more accurate solution of three dimensional flow, a more rigorous pressure Kutta condition has to be applied: driven by an Newton-Raphson iterative algorithm, the dipoles intensity on the blade wake is adjusted until the pressure jump at the blade trailing edge (and, consequently, on the wake) is equal to zero: ( ) ........ ( ) ULTETETETE f p φ φ φ φ + ∆ = − ∆ (8) In cavitating conditions, appropriate kinematic and dynamic boundary conditions hold on the cavity surface. Representing the total velocity vector V with its covariant and controvariant components on a local ( l,m,n ) non orthogonal reference system (see, for instance, Caponnetto & Brizzolara, 1995, Gaggero & Brizzolara, 2008 and Fine, 1992), the dynamic condition (i.e. the pressure on the cavity surface must be equal to the water vapour tension) yields to an expression (equation 8) for the perturbation potential on the cavity surface: ..22.. 0 cossin (,)() Bub Bub TE lmVapm LE UUUdlmm mlm φ φ θ θ φ φ ∂ ∂ − + + + − + ∂ ∂ = + ∫ V (9) while the kinematic condition (10) yields to a partial differential equation for the cavity thickness, whose iterative solution, mixed with a cavity closure condition at the bubble trailing edge, allow to obtain the right cavity planform: 2 coscossin0 mmnll tt UUUUU lmlmlmn φ φ φ φ φ θ θ θ ∂ ∂ ∂ ∂ ∂ ∂ ∂+ − + + + − + + + =∂ ∂ ∂ ∂ ∂ ∂ ∂ (10) For the unsteady solution (i.e. a propeller operating in a spatial non uniform inflow), the integral equation (5) and the kinematic and dynamic boundary conditions, previously expressed only as functions of relative position between field points p and q , have to be adjusted to account for the time dependency of the inflow velocity field and the Bernoulli law, applied to calculate pressure from velocities, has to be applied in its unsteady formulation. Finally, in order to achieve the right circulation around the blade, also the Kelvin theorem (equation 10) has to be enforced for the dipoles shed on the propeller wake: ( ) 0 t φ φ = ∂∆+ ⋅∇ ∆∂ V (11) The numerical solution of equation (5) is achieved substituting integrals over the surfaces with discrete sums performed over the surfaces discretized (panelized) with a planar and non planar quadrilateral mesh (Morino, 1974, Lee, 1987, Gaggero & Brizzolara, 2007 & Gaggero & Brizzolara, 2008) and solving the resulting linear system subjected to boundary conditions of equations (6), (8), (9), (10) and (11). The pressure forces are obtained from Bernoulli’s theorem by differentiating the potential on the blade with respect to a local reference system. The viscous forces on the wetted part of the blade, instead, are recovered by calculation of a friction coefficient as a function of the local Reynolds number, as in Brizzolara et al. (2008). 2.1 Validation Before proceeding in the optimization, some validation test are carried out to assess the accuracy and the robustness of the devised panel method. The custom propeller named E033, available at the cavitation tunnel of the University of Genoa, that has been adopted for the unsteady optimization procedure, has been also adopted for the validation of the code, together with the well known DTMB 4119 propeller (Jessup, 1989), for which pressure coefficient distributions along the chord are available. The E033 four bladed CP propeller was designed for high speed (35 knots) displacement vessels with a traditional lifting surface code, allowing for partial cavitation at lowest cavitation indexes. It has a pitch ratio P/D=1.5 at r/R = 0.7 and an expanded area ratio of AE/A0=0.685. The DTMB 4119, instead, is a more traditional propeller, designed to operate in non cavitating condition via a lifting line/lifting surface approach, with a pitch over diameter equal to 1.08 at r/R = 0.7. The pressure distributions available for propeller DTM 4119 are useful to test the ability of the panel code to predict pressure forces. As presented in figure 1, the prediction of the pressure, at r/R = 0.7 and r/R = 0.9 is good, as well as good is the pressure closure at tip achieved via the iterative Kutta condition. Fig. 1. Predicted and measured pressure coefficient - Propeller DTMB 4119, r/R = 0.7 (left) and r/R = 0.9 (right). For the cavitating condition, a wide series of experiments carried out on the E033 propeller is available, composed of thrust and torque measures and cavity sketches and photos. 0 1 2 3 4 5 6 700.10.20.30.40.50.60.70.80.9 σ N K T 1 0 K Q K T Exp.10K Q Exp.K T Num.10K Q Num. 0 1 2 3 4 5 600.10.20.30.40.50.60.70.8 σ N K T 1 0 K Q K T Exp.10K Q Exp.K T Num.10K Q Num. Fig.2. Predicted and measured thrust and torque coefficients as a function of cavitation index σ N - Propeller E033, J = 0.8 (left) and J = 0.9 (right).
