Transonic Helicopter Noise

Transonic helicopter noise A. S. Morgans , S. A. Karabasov † , A. P. Dowling ‡ and T. P. Hynes § Department of Engineering, University of Cambridge, CB2 1PZ, UK Abstract Helicopter noise is an increasingly important issue, and at large forward flight speeds transonic rotor noise is a major contributor. In this paper, a method for predicting transonic rotor noise is developed which is more computationally efficient than previous methods, and which furthermore offers physical insight into the noise generation. These benefits combine to make it of potential use to helicopter rotor designers. The permeable surface form of the Ffowcs Williams - Hawkings (FW-H) equation is used to express the sound field in terms of a distribution of monopole and dipole sources over a permeable control surface, and a distribution of quadrupole sources over the volume outside of this surface. By choosing the control surface to enclose the transonic flow regions, the noise from the quadrupole distribution becomes negligible. Only the more straightforward surface sources then need be considered, making the acoustic approach computationally efficient. By locating the control surface close to the blade subject to enclosing the transonic flow regions, efficiency in the CFD approach is also attained. To perform noise predictions, an Euler CFD method to calculate the flow-field was combined with an acoustic method incorporating the retarded time formulation of the FW-H equation. Several rotor blades in hover and steady forward ✁ PhD Student, Student Member AIAA Research Associate, Member AIAA ‡ Professor of Mechanical Engineering and Head of the Division of Energy, Fluid Mechanics and Turbomachinery, Senior Member AIAA § Lecturer, Member AIAA † Post-Doctoral 1 flight were considered, all of which involved transonic flows but for which shock delocalisation did not occur. The predictions showed very good agreement with experimental data and with predictions obtained using more computationally intensive methods. Nomenclature ✁ J ∇y f A ✁✄✂☎✁ ∇η f ✁ c speed of sound in undisturbed flow in ms ✆ f f g 0 defines location of the control surface, S ✂ t ✝ τ ✝ r✞ τ✟ c H Heaviside step function J Jacobian of y ✠ Li η coordinate change pi j n j ✡ ρ ui ✞ un ✝ vn ✟ M Mach number vector M magnitude of M MH magnitude of rotational tip Mach number Mf magnitude of forward flight Mach number Ma magnitude of advancing tip Mach number n unit vector normal to control surface p absolute pressure in Pa pi j compressive stress tensor in Pa r✞ τ ✟ 1 ✁ ✁ x ✞ t ✟☛✝ y ✞ τ ✟ , radiation vector in m 2 r τ ✞ ✁ ✟ r τ , magnitude of radiation vector in m ✁ ✞ ✟ r0 distance from rotor hub to observer in m Rtip radius of blade tip (= blade span) in m S control surface t observer time in s Ti j Lighthill stress tensor = ρ ui u j u fluid velocity in ms ρ ρ0 vi ✞ 1 ✝ pi j 1 ✆ ✂ ✟ ✡ v blade/control surface velocity in ms x observer position in m y source position in m α blade pitch angle in degrees β blade cone angles in degrees η coordinates in which S is stationary δ Dirac delta function ε rotor disc tilt angle in degrees µ helicopter advance ratio ρ density in kgm τ source time in s τ ✁ ✂ retarded time c2 ρ δi j ✝ ρ ρ 0 ui ✂ Ui ✡ M f MH 3 ✆ t ✝ r τ ✞ ✂ ✁ ✟ ψ azimuth angle in degrees () generalised variable c in s 3 1 ✆ ( )0 value in undisturbed fluid () fluctuation about undisturbed level ( )r component in radiation direction ( )n component in surface normal direction 2 wave operator ∇2 ✝ 1 ∂2 c2 ∂ t 2 1 Introduction In recent years, helicopter noise has become an increasingly important issue. This is due to factors such as environmental acceptability of ground noise levels, passenger comfort and, for military helicopters, acoustic detectability. At large forward flight speeds, shock associated rotor noise is a major noise source. Despite this, there is presently no method of predicting it which is sufficiently fast and physically insightful to be useful in the rotor design process. Rotor noise is most efficiently predicted using integral methods which separate the computation of the noise sources and the noise propagation. The aerodynamic field around the blade is evaluated using an unsteady CFD solver, and an integral formulation is used to describe how the sound propagates to the far-field. The two most commonly used integral methods are the Kirchhoff method and Ffowcs Williams - Hawkings (FW-H) equation. The Kirchhoff method involves integration over a surface located in the linear flow region [1, 2, 3]. It has the advantage of not requiring any volume integration, but its drawback for transonic flows is that the linear flow region is typically far from the blade [3, 4] meaning that obtaining a CFD solution which remains accurate and well resolved at the surface is computationally intensive. 4 Hence although the Kirchhoff method has been to accurately predict transonic rotor noise [5, 6, 7, 3], the method is too time consuming for use by designers. Furthermore, it does not provide any physical insight into how the noise is generated. The FW-H equation expresses the noise in terms of a distribution of monopole and dipole sources over a control surface, and a distribution of quadrupole sources over the volume outside the surface [1]. When the control surface is chosen to coincide with the blade surface, these distributions represent the noise due to blade thickness, blade loading and flow non-linearities/entropy variations respectively. It has been shown that the noise generated by the volume quadrupole distribution is negligible for a subsonic volume of fluid, but is significant in regions of transonic flow [8, 9, 10]. Calculating the noise generated by this quadrupole distribution is both time consuming and numerically difficult. Although methods of approximating it exist, there is no satisfactory method of performing the volume integration exactly. To predict shock-associated noise while avoiding the need to compute the problematic quadrupole term, the FW-H equation can be applied to a permeable control surface which encloses the blade but is not coincident with it. If the control surface is also chosen to enclose all transonic regions of flow, the flow volume outside of the control surface is fully subsonic and the noise generated by the quadrupole distribution outside of the surface is negligible. Thus by moving the control surface outwards, the effect of the quadrupoles within it can be accounted for by the surface source terms. Furthermore, the transonic region is always well defined and the surface source terms continue to have physical meaning; the monopole distribution is related to mass flux through the surface and the dipole distribution to momentum flux. Also, by comparing the prediction obtained using the permeable control surface to that obtained using the blade surface, the thickness, loading and 5 shock-associated contributions to the overall noise can be identified. The permeable surface form of the FW-H equation has recently been used to successfully predict transonic rotor noise [11, 7]. However, in these cases the permeable control surfaces enclosing the transonic region were generally Kirchhoff-type control surfaces which were far from the blade, resulting in more computational effort than was necessary. To take full advantage of the permeable surface form, the surface should be as small as possible while enclosing all transonic flow regions. This ensures that the quadrupoles responsible for significant noise generation are accounted for, while minimising the computational effort needed to obtain the surface information. 2 Acoustic methodology 2.1 Background: the permeable surface form of the FW-H equation The permeable surface form of the Ffowcs Williams-Hawkings equation follows from the fluid conservation laws in the same way as the more familiar impermeable surface form [1]. A permeable control surface, S, is considered which is defined by the equation f x t ✞ encloses all solid boundaries and moves with velocity v. permeable surface, S f<0 H( f ) = 0 enclosed body f>0 H( f ) = 1 velocity, v Figure 1: Permeable control surface 6 ✟ 0. S Generalised flow variables are defined (denoted by an overbar) which hold over infinite space. Outside the surface, S, the generalised variables are equal to the real flow variables, while inside the surface they have the value zero. The continuity and momentum equations valid over all space are, H f ✞ ∂ρ ∂t ✟ ∂ ρ ui ∂t ✞ H f ✞ ✟ ∂ ρ ui ∂ xi ✞ ✡ ∂ p ∂xj ij ✟ ✁ ✟ ρ ui u j ✞ ✡ 0 ✡ ✟ (1) ✁ 0 (2) which can be rearranged to give, ∂ρ ∂t ∂ ρ ui ∂t ✞ ∂ ρ ui ∂ xi ✞ ✡ ∂ p ∂xj ij ✟ ✞ ✡ ✡ ρ 0 un ✟ ✞ ρ ui u j ✟ ρ un pi j n j ✞ vn δ f ∇x f ✁ ✝ ✞ ✡ ✟ ✟ ρ ui un ✞ (3) vn δ f ∇x f ✝ ✁ ✞ ✡ ✁ ✟ ✟ ✟ ✞ ✁ (4) ✟ Following the notation of di Francescantonio [11], new variables, Ui and Li are introduced to simplify the algebra. These represent mass-like and momentum fluxes through S. Ui ✂☎✄ 1 ✆ ρ ✝ ρ ✞ v ✟ ρ ✝ ρ u ✠ 0 Li 0 i i ✂ pi j n j ✄ ✆ v✞ ✟ ρ ui un ✄L ✂ n m Li Mi ✞ (5) By subtracting the divergence of (4) from the time derivative of (3), an inhomogeneous wave equation is obtained. 2 c 2 ρ xt ✞ ✟ ✝ ∂ L δ f ∇x f ∂ xi i ✡ ✞ ✁ ☞☛ ✁ ✟ ✡ 7 ∂ ρ Un δ f ∇x f ∂t 0 ✡ ✞ ✁ ☞☛ ∂ 2 Ti j ✁ ✟ ✡ ∂ xi ∂ x j (6) This is the equation governing the generation and propagation of sound, and is the differential form of the FW-H equation. On the right hand side, surface monopole, surface dipole and volume quadrupole distributions act as acoustic sources, while on the left the wave operator describes the propagation of sound from the sources to the observer. Ti j is the generalised Lighthill stress tensor, which has value Ti j ρ ui u j pi j ✡ c2 ρ δi j outside of the surface S. Since S is chosen to enclose the ✝ blade and all transonic flow regions, Ti j is negligible outside of S and so the last term on the right can be neglected. This is consistent with the observation that for blades around which the flow is entirely subsonic (up to tip Mach numbers of approximately 0 8), the total noise contribution from Ti j is negligible [13, 14, 15]. The equation is valid in all of three dimensional space, owing to the fact that generalised variables have been used. The integral form can therefore be obtained by convolving with the free space, 3-D Green’s function for the wave equation, which has the well known form δ t ✞ ✁ ✁ ✂ ✂ x c ✟ 4π x [12]. The substitution p ✁ ✞ ✝ c2 ρ can be made on the left, requiring linearity and no ✁ ✟ entropy variation at the observer location (although no such restriction is placed on the flow at the control surface). ∞ ✁ p xt ✞ ✟ ✝ ∂ ∂ xi ✂ ✁ Li δ f δ g ∇y f 3 d yd τ 4π r ✁ ✞ ∞ ✟ ✞ ∂ ∂t ✁ ✟ ✡ ✆ ∞ ρ0Un δ f δ g ∇y f 3 d y dτ 4π r ✁ ✞ ∞ ✟ ✞ ✁ ✟ (7) ✆ ✁ where r ✂ x ✝ yτ ✞ ✁ ✟ and g t ✝ τ ✂ ✝ r c A coordinate change from fixed coordinates, y, to coordinates which move with the control surface, η , allows the source strengths to be considered in a frame moving with the surface. The Jacobian for the coordinate change is J, and represents the ratio of volume elements in the η and y 8 spaces. ✁ p xt ✞ ✟ ∂ ∂ xi ✝ ✂ ∞ Li δ f δ g ∇y f J 3 d η dτ 4π r ✞ ∞ ∞ ✁ ✁ ✟ ✞ ✁ ✟ ✆ ✡ ∂ ∂t ✂ ρ0Un δ f δ g ∇y f J 3 d η dτ 4π r ✁ ✞ ∞ ✟ ✞ ✁ ✟ (8) ✆ Equation (8) is the most general integral form of the permeable FW-H equation. To implement it numerically, it is necessary to integrate the two delta functions. The method of performing these integrations determines which formulation of the FW-H equation is used. 2.2 Numerical implementation of the FW-H equation The various possible formulations have previously been discussed in some detail by Brentner [16]. The retarded time formulation is conceptually the most physical and is computationally relatively straightforward to implement. Although it has been demonstrated that it suffers from limitations at tip speeds approaching or exceeding the speed of sound [17], such conditions are associated with the noisy phenomenon of shock delocalisation and in practice helicopters do not operate in this regime. These high speed limitations are therefore less important for design applications and the retarded time formulation is a suitable formulation on which to be basing a realistic transonic noise prediction method. To obtain the retarded time formulation from equation (8), integration over τ is firstly performed: since the surface S is fixed in η coordinates, the δ f term is unaffected by this. Noting ✞ that ∂ g ∂ τ ✁ ✂ ✁ ✁ ✟ ✁ 1 ✝ Mr and integrating over one space dimension using the remaining delta func- 9 tion gives, p ✞ x t✟ ✝ ∂ ∂ xi Li A ✁ ✁✂✁ 4π r 1 ✝ Mr ✂ S dS ✞ η ✟ τ ✡ ∂ ∂t ρ0Un A ✄✁ 4π r 1 Mr ✂ ✁ ✁ τ ✝ S dS ✞ η ✟ (9) Equation (9) is the permeable surface form of the retarded time formulation with the quadrupole term neglected. A ✁ J ∇y f ✁✄✂☎✁ ✁ ∇η f represents the ratio of area elements in the η and y spaces. If the surface is undistorted in motion then A is equal to unity. τ is the retarded time, given implicitly ✁ by the relationship, τ t✝ r✞ τ c ✁ ✁ ✁ ✟ t✝ x ✞ t ✟☛✝ y ✞ τ c ✁ ✟ ✁ (10) Sound emitted by the source at retarded time τ will reach the observer at the time of interest, ✁ t. For a fixed observer position and time and for subsonic surface motion, τ can only have one ✁ value. To express equation (9) in a form suitable for computation, further manipulation is required. Numerical differentiation of the integrals can be avoided and the speed and accuracy of the compu✂ tation is improved if the derivatives are taken inside the integrals [14]. The ∂ ∂ x i can be replaced using (11) and the integration surface S ✞ η ✟ is independent of t so time derivatives inside the inte✂ grals can be replaced with ∂ ∂ τ terms using (12). ∂ ∂ xi ✂ Qi dS S ∂ ☎ Q τ ✝✆ τ ∂t ✞ ✟ ∂ ∂t Qi r i dS ✡ cr ✂ S Qi r i dS r2 ✂ (11) S 1 ∂Q✞ τ ✟ ✞ 1 ✝ Mr ✟ ∂ τ 10 ✁ τ (12) Differentiating and gathering terms together gives, 4π p x t ✁ ✂ρ ✄ ✠ U̇n Uṅ r 1 Mr 2 ✂ ✞ S ✁ ✂ ✟ ✆ 1 c ✞ ✞ ✄ ☎ L̇r r 1 Mr ✁ ✂L S ✟ ✄ ✄ 1 c ✟ 0 τ 2 ✆ ✄ r ✞ ✁ ✂ρU 0 n dS η ✄ ✁ ✂ S ✄ ☎ dS η τ ✄ ✟ ✞ r ∂ M ∂ τ r c Mr r2 1 Mr 3 ✄ ✟ ✄ ✞ Lr r2 1 ✝✆ M ✆ Lm Mr ✆ ✆ ✄ 2 ✆ ✞ ✞ ✞ ✄ ✞ 2 ✞ ✟ c Mr ✄ Mr ✆ 3 τ ✝✆ M ✆ 2 ✆ ✞ ✞ ✄ ☎ dS η τ ✞ ✄ ☎ dS η ✄ ☎ dS η τ ✆ S r ✝ ✄ ✄ S ✝ ✄ r ∂M ∂τ r2 1 ✄ ✟ ✞ ✄ ✞ ✄ ✞ (13) ✄ ✞ Equation (13) is the form of the impermeable retarded time formulation that has been used for numerical computation. It is valid for subsonic surface motion and it assumes that the integration 1 and ∂ A ∂ τ ✂ surface is undistorted in motion, so that A 0. In the computer program used to numerically implement the retarded time formulation, it is assumed that the blade is a rigid body which can rotate but not deform. In reality, blade loading may lead to significant blade twist and bending; it is assumed that these can be reasonably well accounted for using modified pitch and cone angles Rotation matrices and an overall angular velocity vector are used to describe the effect of the blade pitch, cone, rotation and rotor disc tilt angles. Positions, velocities and accelerations in the blade-fixed frame can then be related to values in a stationary frame by combining the matrices/vector with rigid body relations for rotating reference frames. A knowledge of the helicopter forward flight behaviour is also required. 11 3 CFD methodology The primary interest is in shock-associated noise, and typically the shock is present over the outer part of the span for azimuthal angles corresponding to the blade advancing. For this phase of the cycle in level flight, the interaction with other blade wakes and the disturbance due to the presence of the fuselage and tail rotor are likely to be small. The CFD calculations used to generate the data for the acoustic calculations were therefore performed for a single, oscillating and advancing blade in an otherwise undisturbed flow. Calculations were performed using a grid which was fixed relative to the blade, with the problem formulated in terms of the relative velocity as in Zhong and Qin [18]. This made it relatively easy to form smooth acoustic surfaces and to ensure good grid and solution quality in their vicinity. In addition, it removed errors associated with the pitching grid technique [19] which arise when re-interpolating the solution at each time step onto a new grid position Although the blade is treated as a rigid body, this approach has been used successfully to account for a wide range of blade motions, including collective and cyclic pitch variations, tilt and coning. Non-inertial terms due to the highly convoluted motion of the blade frame appear as volumetric sources in this formulation. It is important to handle these terms in a manner which is compatible with the conservative nature of the discretisation scheme for the equations of motion and with the non-reflective nature of the boundary conditions. Provided this is done, however, no extra complication appears to arise from using this accelerating frame approach [20]. The governing equations were discretised in a control volume fashion by assembling numerical fluxes across interfaces using well-proven shock-capturing methods. The explicit character- 12 istic Roe scheme was implemented together with Van Leer’s variable extrapolation (MUSCL) of 2nd/3rd order in characteristic variables in each grid direction. Unwanted numerical oscillations were minimised using the Total Variation Diminishing approach, which was applied to limit the slopes of the characteristic variables using the MinMod limiter. Slight modifications were made to the standard MUSCL TVD variable extrapolation formulae to account for non-uniform grid spacing. The solid-wall rotor blade boundary conditions were applied by assembling the outer fluxes and eliminating the velocity component normal to the wall. In order to reduce the entropy generation near the solid boundary more accurate estimates for fluxes adjacent to the boundary were made by the introduction of symmetric fictitious points lying inside the blade surface. At the outer boundary, a range of non-reflecting boundary conditions were used to judge the domain size necessary to ensure that the solution at the acoustic surfaces was unaffected by the boundary presence. The computational grid consisted of a series of chord-wise O-type grids stacked in the blade span-wise direction. At the blade tip the grid was wrapped around forming a hemispherical blade cap to provide a uniform mesh distribution away from the blade. A modification of the flux assembling procedure was needed for where the confluence of the blade cap radial lines and the blade grid met to ensure that the algorithm remained uniform and conservative. This grid strategy proved to be quite robust in application to a number of blade geometries. The inviscid solver was tested in a number of one and two-dimensional initial value problems [21, 22]. A simplified but effective way of applying non-reflecting boundary conditions was also developed and tested on a number of problems involving a 2-D transonic pitching blade sec- 13 Figure 2: Computational grid and pressure field around a hovering UH-1H rotor blade at MH 0 88 tion accelerating and decelerating in the free stream [23]. The three-dimensional Euler solver has been validated against several hover and forward flight benchmarks and has been shown to give good agreement of the calculated blade surface/near-field pressure variations with experiments and other calculations. For example, figure 3 shows the nearfield comparison for a two-blade rotor at MH 0 7634 and µ ✂ 0 25 at the r0 Rtip 0 88 blade station in forward flight [24]. Figure 4 shows the CFD grid used for the more complex rotor blade geometries of the HELISHAPE test cases, in which the blade is tapered, twisted and drooped. The calculated blade surface pressure distributions at two outer blade sections are compared to experimental data for two azimuth angles in Figure5. In general the agreement is very good, despite the fact that the numerical model ignores viscous effects, the influence of the other blades and variations in the blade pitch angle due to elasticity. 14 Figure 3: Simulation of UH-1H-type rectangular unloaded blade in forward flight Figure 4: Blade surface and half-domain CFD grid around the tapered blade for 3-D Euler code validation against HELISHAPE data 4 Noise prediction results Noise predictions were performed for various rotor blades in three-dimensional motion. In all cases, the flow in the vicinity of the blade was transonic for at least part of the rotational cycle, although tip Mach numbers were sufficiently low to avoid shock delocalisation. The CFD method was used to calculate the aerodynamic field around the blade; the permeable surface form of the FW-H equation was then used to generate noise predictions. 15 Figure 5: Computed and experimental pressure coefficient for two outer blade sections and azimuth positions during advancing blade motion: tapered HELISHAPE blade 4.1 Hover For comparison with available experimental results [25, 26, 27, 28], an isolated UH-1H blade with an aspect ratio of 13 7 was considered in non-lifting hover. Although non-lifting hover does not 16 represent a realistic flight condition, these test cases have become benchmark tests for transonic rotor noise prediction methods and therefore provide a first means of validating the noise prediction approach for three-dimensional motion. They also allow parameters such as CFD grid resolution and control surface location to be investigated. Since the UH-1H blade is symmetric and non-lifting test cases were being considered, the blade pitch and cone angles were zero. Noise calculations were performed at two different tip Mach numbers, 0 85 and 0 88; for both a supersonic flow pocket formed on the outer part of the blade and was present throughout the rotational cycle. Previous work [10, 26, 25] suggested that the tip Mach number marking the onset of shock delocalisation for a non-lifting UH-1H blade was between 0 88 and 0 90. The tip Mach numbers of 0 85 and 0 88 should therefore have nondelocalised shocks, although in the MH 0 88 case the shock will be very close to the onset of delocalisation and hence this is a challenging test case. Calculations for both tip Mach numbers were performed using two different CFD grid resolutions. The first was a 99 30 36 grid, meaning that there were 99 grid points in the wrap- around direction, 30 in the outwards direction and 36 along the blade span. The second was a finer 139 50 36 grid. The contours of the Mach number relative to the blade and the pressure as cal- culated by the CFD solver are shown for both tip Mach numbers and both CFD grids in figures 6 to 9. For the tip Mach number of 0 85, it can be seen from the Mach number contours that the supersonic flow region and shock wave are confined to being above/below the blade and do not extend significantly beyond the blade tip. The pressure contours show subtle changes on increasing the grid resolution, but both essentially reveal the same directionality of the aerodynamic pressure 17 Figure 6: Contours of relative Mach number (left) and pressure (right) for MH 30 36 CFD grid 0 85 and a 99 Figure 7: Contours of relative Mach number (left) and pressure (right) for MH 50 36 CFD grid 0 85 and a 139 field. For the tip Mach number of 0 88, it can be seen that the supersonic region and hence shock wave do now extend beyond the blade tip. However, they do not intersect the region of the flow which is supersonic in the frame in which the blade is stationary, meaning that the shock is not delocalised. The pressure contours show minor differences with grid resolution, but both reveal an aerodynamic pressure field with a directionality which is slightly more pronounced than for the tip Mach number of 0 85. 18 Figure 8: Contours of relative Mach number (left) and pressure (right) for MH 30 36 CFD grid 0 88 and a 99 Figure 9: Contours of relative Mach number (left) and pressure (right) for MH 50 36 CFD grid 0 88 and a 139 In hover, the inner part of the blade moves at substantially smaller speeds than the outer part. Control surfaces could therefore be used which enclosed only the outer part of the blade and the supersonic flow regions. Examples of the control surfaces used, in this case for the finer grid MH 0 88 calculations, are shown in figure 10. Beyond the blade tip, all pass between the edge of the supersonic region and the radius at which their motion would become sonic. They thus enclose the transonic flow region while undergoing subsonic motion, allowing noise prediction to 19 be carried out via the retarded time formulation of the FW-H equation. Figure 10: The four permeable control surfaces and the supersonic flow pocket at MH the 139 50 36 CFD grid 0 88 for The noise at an observer lying in the rotor plane a distance of 3 09 rotor radii from the axis of rotation was calculated by applying the permeable surface form of the FW-H equation to four different control surfaces for each test case. Since the blade position with respect to the stationary observer was periodic at the blade rotation frequency, the observer sound signature was also periodic. One period of the signature for each of the tip Mach numbers is shown in figures 11 and 12. For both rotor tip Mach numbers, the observer pressure signal has the form of a large negative peak preceded and followed by much smaller positive peaks, the characteristic signature of rotor plane shock-associated noise [26, 25, 3, 29, 11]. All four control surfaces are seen to predict very similar results. The outermost control surface (surface 4) has the largest maximum speed, and some slight errors due to the need for a finer acoustic grid are observed in the predictions for the hover tip Mach number of 0 88. 20 M = 0.85 M = 0.85 H 40 20 20 0 0 pressure fluctuation in Pa pressure fluctuation in Pa H 40 −20 blade surface surface 1 surface 2 surface 3 surface 4 −40 −60 −80 −20 −40 −60 −100 −100 −120 −120 −140 0 0.05 0.1 −140 0 0.15 blade surface surface 1 surface 2 surface 3 surface 4 −80 0.05 time in s 0.1 0.15 time in s Figure 11: Non-lifting hover at MH 0 85: one period of the predicted sound for an in-plane observer, r0 Rtip 3 09, CFD grids of 99 30 36 (left) and 139 50 36 (right) ✂ MH = 0.88 100 50 50 0 0 pressure fluctuation in Pa pressure fluctuation in Pa MH = 0.88 100 −50 −100 blade surface surface 1 surface 2 surface 3 surface 4 −150 −50 −100 −200 −200 −250 −250 −300 0 0.05 0.1 −300 0 0.15 time in s blade surface surface 1 surface 2 surface 3 surface 4 −150 0.05 0.1 0.15 time in s Figure 12: Non-lifting hover at MH 0 88: one period of the predicted sound for an in-plane observer, r0 Rtip 3 09, CFD grids of 99 30 36 (left) and 139 50 36 (right) ✂ Experimental noise measurements in the vicinity of the negative pressure peaks exist [27, 25, 26, 28], and by enlarging the scale of the graphs around the negative pressure peak, as in figures 13 to 16, the experimental and predicted results can be compared. The experimental results were for ✂ a 1 7th scale blade model and so the time axis has been scaled to correspond to a full-size blade. The contributions from the loading and thickness noise are also included, as well as their combined total in the form of a prediction based on the blade surface. The difference between the total noise 21 M = 0.85 H 40 20 pressure fluctuation in Pa 0 −20 −40 −60 −80 −100 −120 −140 blade surface thickness noise loading noise surface 1 surface 2 surface 3 surface 4 experiment (scaled Purcell) 0.022 0.024 0.026 Figure 13: Non-lifting hover at MH of 99 30 36 0.028 0.03 0.032 time in s 0.034 0.036 0.038 0.04 0.042 0 85: comparison of predictions and experiment, CFD grid M = 0.85 H 40 20 pressure fluctuation in Pa 0 −20 −40 −60 −80 −100 −120 −140 blade surface thickness noise loading noise surface 1 surface 2 surface 3 surface 4 experiment (scaled Purcell) 0.022 0.024 0.026 Figure 14: Non-lifting hover at MH of 139 50 36 0.028 0.03 0.032 time in s 0.034 0.036 0.038 0.04 0.042 0 85: comparison of predictions and experiment, CFD grid 22 M = 0.88 H 50 pressure fluctuation in Pa 0 −50 −100 −150 −200 −250 −300 −350 blade surface thickness noise loading noise surface 1 surface 2 surface 3 surface 4 experiment (scaled Purcell) 0.022 0.024 0.026 Figure 15: Non-lifting hover at MH of 99 30 36 0.028 0.03 0.032 time in s 0.034 0.036 0.038 0.04 0.042 0 88: comparison of predictions and experiment, CFD grid M = 0.88 H 50 pressure fluctuation in Pa 0 −50 −100 −150 −200 −250 −300 −350 blade surface thickness noise loading noise surface 1 surface 2 surface 3 surface 4 experiment (scaled Purcell) 0.022 0.024 0.026 Figure 16: Non-lifting hover at MH of 139 50 36 0.028 0.03 0.032 time in s 0.034 0.036 0.038 0.04 0.042 0 88: comparison of predictions and experiment, CFD grid 23 prediction and the blade surface prediction is the shock-associated noise. It can be seen that for both tip Mach numbers, the predicted and experimental results are in good agreement, although the size of the negative pressure peak is under-predicted by approximately 8% for the tip Mach number of 0 85 and 14% for the tip Mach number of 0 88 . In both cases, the largest contribution is from thickness noise, with a smaller but substantial contribution from shockassociated noise. The loading noise contribution is small; the blade is non-lifting and so all of the loading noise is associated with non-compactness across the blade section and numerical drag. The predicted relative sizes of the thickness, loading and shock-associated noise contributions are in agreement with previous work [26, 30, 3, 29]. Since the blade drag is likely to be significantly smaller for the Euler-based predictions than the experiments, the corresponding deficit in loading noise may go some way towards explaining the under-prediction of the negative pressure peaks. The larger difference for the tip Mach number of 0 88 may be caused by the extreme sensitivity to tip Mach number that occurs close to the onset of shock delocalisation; it was commented in reference [26] that the negative pressure peak for the MH 0 89 case is almost 25% larger than for the MH 0 88 case. The accuracy to which the experimental Mach number could be measured is likely to result in a large possible range for the size of the negative pressure peak. Other transonic noise prediction methods have invariably also under-predicted the size of the 0 88 tip Mach number negative pressure peak, adding weight to this argument [26, 25, 31, 30, 10]. The predictions from the four control surfaces are converged everywhere except in the immediate vicinity of the negative pressure peak for the high Mach number case, where small discrepancies exist. For the tip Mach number of 0 85 these discrepancies are almost insignificant, but for 24 the tip Mach number of 0 88 they are larger, showing that the predicted noise is more sensitive to control surface location at this larger tip Mach number. The predictions obtained using the two different CFD grid resolutions are almost indistinguishable for the tip Mach number of 0 85, suggesting that convergence with grid resolution has been obtained. However, for the tip Mach number of 0 88, the predictions from the two grids are seen to differ in the region of the negative pressure peak. The finer grid results in the capture of an increased amount of shock-associated noise, which results in the size of the negative pressure peak being approximately 5% larger. The combined contribution of thickness and loading noise is seen to be virtually unaffected, although the loading noise is slightly reduced when using the finer grid, probably because the increased spatial resolution results in reduced numerical drag. These results confirm that when the flow around the blade is transonic, shock-associated noise contributes significantly to the in-plane negative pressure peak. The contribution increases with blade tip Mach number. Accurate capture of the noise requires use of a sufficiently fine spatial CFD grid and integration over a sufficiently large permeable control surface. The required number of CFD grid points and control surface size both increase with blade tip Mach number. The results obtained using the permeable surface form of the FW-H equation are compared in figures 17 and 18 to recent results obtained using direct Euler CFD calculations [26], the Kirchhoff method [5, 3] and the impermeable surface form of the FW-H equation with the quadrupole term approximated [3]. The results have been scaled, where necessary, so that they apply to a full-sized UH-1H blade. The predictions obtained using the permeable surface form of the FW-H equation compare favourably with those from the other prediction methods. Furthermore, since they only require 25 M = 0.85 H 40 20 pressure fluctuation in Pa 0 −20 −40 −60 experimental permeable FW−H CFD Euler Kirchhoff −80 −100 −120 −140 0.022 0.024 0.026 0.028 0.03 0.032 time in s 0.034 0.036 0.038 0.04 0.042 Figure 17: Comparison of permeable FW-H noise predictions with those from other methods for a non-lifting UH-1H blade in hover, MH 0 85, r0 Rtip 3 09 ✂ MH = 0.88 50 pressure fluctuation in Pa 0 −50 −100 experimental permeable FW−H CFD Euler Kirchhoff impermeable FW−H −150 −200 −250 −300 −350 0.022 0.024 0.026 0.028 0.03 0.032 time in s 0.034 0.036 0.038 0.04 0.042 Figure 18: Comparison of permeable FW-H noise predictions with those from other methods for a non-lifting UH-1H blade in hover, MH 0 88, r0 Rtip 3 09 ✂ 26 near-field CFD data and surface integration, the approach is the most computationally efficient. 4.2 Non-lifting forward flight The noise prediction method is now applied to the more challenging problem of forward flight. For comparison with available experimental results [25, 32, 33], a rectangular OLS blade moving at a constant forward flight speed with a rotational tip Mach number of 0 664, an advance ratio of 0 2605 and an advancing tip Mach number of 0 837 was considered. The blade aspect ratio was 9 22 and the blade was once again non-lifting to avoid BVI noise. The coning and rotor disc tilt angles were zero. In forward flight, the blade tip speed varies throughout the rotational cycle. In practice a supersonic flow pocket exists only during the advancing part of the cycle. Using the azimuth convention of ψ increasing in the direction of rotation and being equal to 90 at the advancing position and 270 at the retreating position, a supersonic flow region existed between azimuth angles of approximately ψ at ψ 139 70 and ψ 135 for the test case being considered. The strongest shock occurred 105 and the Mach contours for this position are shown in figure 19. A CFD grid of 30 36 was used. For a hovering blade, rotating the location of an observer within a horizontal plane while maintaining distance from the blade hub results in a pressure variation which is identical except for a time shift. This is not the case for forward flight; the observer sound signature depends on orientation within the horizontal plane as well as distance from the blade hub. To gain an insight into the rotor plane noise for this forward flight test case, the sound at three observer locations within the rotor plane was considered. All three observers translated with the same forward flight 27 r0 /R tip = 3.44 −30o ω o 0 rotor disc o +30 forward flight direction Figure 19: Contours of Mach number relative to the blade for the upper blade surface at ψ 105 Figure 20: The observer locations in the frame of the rotation axis speed as the blade hub, such that the distance between the rotation axis and the observers was fixed ✂ at r0 Rtip 3 44. The angle between the observer and the downstream direction was varied as shown in figure 20. For the noise predictions, four control surfaces were used which enclosed the supersonic region when at its largest. Noise prediction results for one period of the motion are shown in figure 21 and are compared to experimental measurements [25] in figure 22 (it is not clear how flight test results for a non-lifting blade were obtained). The azimuth angle of the blade when sound is received at the observer, ψ t , ✞ ✟ is plotted on the x-axis. This is equivalent to plotting scaled observer time; it should be noted that ψ t is greater than the azimuth angle corresponding to sound emission, ψ τ . Noise predictions ✞ ✟ ✞ ✁ ✟ based on the blade surface are also included; since the blade is non-lifting, these are essentially equal to the contribution from blade thickness noise. The general shape of the noise prediction is the same for all three observers; a negative pressure peak is immediately preceded and followed by a smaller positive peak. For each of the observers, 28 Non−lifting OLS rotor blade in forward flight, MH = 0.664, µ = 0.258 40 20 pressure fluctuation in Pa 0 −20 −40 −60 −80 −100 0 −30, blade surface −30, surface 1 −30, surface 2 −30, surface 3 −30, surface 4 0, blade surface 0, surface 1 0, surface 2 0, surface 3 0, surface 4 +30, blade surface +30, surface 1 +30, surface 2 +30, surface 3 +30, surface 4 50 100 150 200 250 blade azimuth angle in deg 300 350 ✂ Figure 21: Forward flight noise predictions for 3 rotor plane observers at r 0 Rtip Non−lifting OLS rotor blade in forward flight, MH = 0.664, µ = 0.258 40 20 pressure fluctuation in Pa 0 −20 −40 −60 −80 −100 180 −30, blade surface −30, surface 1 −30, surface 2 −30, surface 3 −30, surface 4 0, blade surface 0, surface 1 0, surface 2 0, surface 3 0, surface 4 +30, blade surface +30, surface 1 +30, surface 2 +30, surface 3 +30, surface 4 wind tunnel flight 200 220 240 260 280 300 blade azimuth angle in deg Figure 22: Comparison of the predicted noise with experimental data 29 3 44 the predictions from the four control surfaces are in good agreement, with the predictions from the outer two surfaces being almost indistinguishable. This suggests that little extra shock-associated noise could be captured by moving the control surface out further. Once again, thickness noise is the main in-plane component, with a significant additional amount caused by shock-associated noise. The predicted signatures are in very good agreement with those from experiment, although there is a slight shift in the azimuth response. The reason for this is unclear; although there is some ambiguity in the Baeder paper [25] as to the advance ratio / advancing tip Mach number combination that has been used, noise predictions have been performed for all possible combinations and confirm that the changes are too subtle to explain the azimuth shift. One possibility is that in-plane bending, known as lead-lag motion, occurs in the experiments and is responsible [34, 35]. The sizes of the negative pressure peaks are under-predicted by approximately 10%, although again this may be partially due to the loading noise deficit associated with using an Euler CFD code. For both the predicted and experimental results, the peak response occurs somewhere between the ✝ 30 and 0 observer stations. This corresponds to a location upstream of the advancing rotor ✂☎✁ ✁ blade. At this location, the Doppler factor, 1 1 ✝ Mr , is large and the radiation vector magnitude, r, small when the blade speed and shock strength are large; these effects amplify the noise. 4.3 Lifting forward flight Noise predictions were now carried out for lifting rotors in forward flight. Two rotors were considered, each comprised of four identical blades of radius of 2 1 m and aspect ratio 15. The first rotor was comprised of rectangular blades consisting of a span-wise blend of OA213 and OA219 30 Boeing aerofoils; the second was comprised of tapered blades with the same span-wise blend of aerofoils but with a different geometrical twist, such that the blade tips were slightly swept back and drooped. For each of the rotors, a different combination of rotational tip Mach number, MH , advance ratio, µ , effective rotor disc tilt angle, εe f f , and pitch variation, α t , was considered, as sum✞ ✟ marised in table 1. The forward flight and advancing tip Mach numbers are denoted by M f and Ma ✂ respectively. The pitch angle of the blades varied cyclically about the 1 4 chord axis as a function of blade azimuth angle, and was always negative to give upwards lift. The blade cone angle was taken to be zero, as indicated in the experimental data, and second order harmonic flap angles were neglected. Blade shape rectangular tapered MH 0.662 0.660 µ 0.331 0.328 Mf 0.219 0.216 Ma 0.881 0.877 εe f f in deg 4.81 5.10 αav in deg -7.61 -7.75 Table 1: The parameters for the two HELISHAPE test cases It is clear that the flight parameters for the two cases are very similar, with the advancing tip Mach number differing by less than 0 5%. In both cases, the advancing tip Mach number is sufficiently high that a shock wave forms above the outer part of the blade during the advancing part of the cycle. The CFD calculations were performed on a 127 30 36 grid; the pressure contours for the advancing positions are shown in figure 23. For each rotor, the sound at two observers was considered. Both observers translated at the rotor forward flight speed and were thus fixed with respect to the rotor hub. Such observers are equivalent to stationary microphones within a wind tunnel, allowing the noise predictions and wind 31 Figure 23: Pressure contours for the advancing position of the rectangular blade (left) and the tapered blade (right) tunnel measurements to be directly compared. It was assumed that the wind tunnel boundaries were sufficiently far from the rotor for their effect on the measurements to be negligible. The observer coordinates were ✞ ✝ 5 27 ✝ 2 28 2 69 for observer 1 and ✟ ✞ ✝ 5 27 ✝ 2 28 2 15 ✟ for observer 2, where a coordinate system which translates with the rotor hub and uses the convention shown in figure 24 was used. rotation forward flight y x z Figure 24: The coordinate system for forward flight Noise prediction was again carried out using several control surfaces. Since the observers translated at the rotor forward flight speed, the behaviour of each blade relative to the observer 32 was identical except for phase shift or time delay. Noise prediction was therefore carried out for a single rotating blade and the results were time delayed and summed to obtain the noise prediction for all four blades. The predictions were compared to measurements from equivalent wind tunnel test cases ¶ . Since the rotor blades were lifting, the comparisons were complicated by the presence of vortex induced loading noise in the measurements, which could not be accounted for in the predictions due to the use of an Euler CFD solver. Some discrepancy between the predicted and measured noise was therefore expected. The results are shown in figures 25 and 26. The observer pressure fluctuations are plotted against non-dimensionalised time for one period of the rotor motion. The starting time for the experimental results is arbitrary and hence a phase difference between the experimental and measured results is expected. Both the experimental and predicted pressure variations consist of four equally spaced negative pressure peaks, corresponding to the advancing phase of the cycle for each of the four blades. The peaks are separated by regions of positive pressure at low levels, in which a proportionally larger amount of the measured noise is likely to be contaminated by effects such as vortex induced loading noise. For both cases, the magnitude of the negative pressure peaks are slightly larger for observer 2 than for observer 1. This is because the z-coordinates of the observer positions are 2 69 m and 2 15 m respectively, with the blade radius being 2 1 m. Observer 2 is closer to being directly upstream of the advancing position, meaning that the amplification effects of the Doppler factor ¶ HELISHAPE noise measurements, permission to publish from Dr V. Kloeppel, Eurocopter Deutschland 33 Rectangular Blades, Observer 1 Rectangular Blades, Observer 2 40 40 20 20 0 pressure fluctuation in Pa pressure fluctuation in Pa 0 −20 helishape 162 data −40 −60 −80 −100 −120 0 −20 helishape 162 data −40 −60 −80 −100 −120 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 −140 0 1 50 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.9 1 50 0 0 blade surface surface 1 surface 2 surface 3 blade surface surface 1 surface 2 surface 3 −50 −100 0 0.8 −50 0.1 0.2 0.3 0.4 0.5 t/T 0.6 0.7 0.8 0.9 −100 0 1 0.1 0.2 0.3 0.4 0.5 t/T 0.6 0.7 0.8 0.9 1 Figure 25: Observer noise for 1 period of rotor motion, rectangular blades Tapered Blades, Observer 2 Tapered Blades, Observer 1 30 20 20 pressure fluctuation in Pa pressure fluctuation in Pa 10 0 −10 −20 helishape 142 data −30 −40 −50 0 −20 helishape 142 data −40 −60 −60 −70 −80 0 −80 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 40 40 20 20 0 −40 −60 −60 0.1 0.2 0.3 0.4 0.5 t/T 0.6 0.7 0.8 0.9 0.3 0.4 0.5 0.6 0.7 1 −80 0 0.8 0.9 0.1 0.2 0.3 0.4 0.5 t/T 0.6 0.7 0.8 0.9 Figure 26: Observer noise for 1 period of rotor motion, tapered blades 34 1 blade surface surface 1 surface 2 surface 3 surface 4 −20 −40 −80 0 0.2 0 blade surface surface 1 surface 2 surface 3 surface 4 −20 0.1 1 and radiation vector are larger. Despite the fact that the flight condition parameters are very similar for both rotors, the noise levels differ significantly; they are approximately 35% smaller for the rotor with tapered blades. This is primarily due to differences in the shock-associated noise, since the thickness and loading contributions are similar. The difference is likely to be associated with the blade shape. Whereas the rectangular blade is unswept, the tapered blade is slightly swept back and drooped at the tip. Sweep serves to reduce the flow Mach number relative to the blade. It therefore follows that the strength of the shock and the shock-associated noise are reduced, an effect that has been observed in the past [36]. The extent of the supersonic region is clearly seen to be smaller for the tapered blade in figure 23. The predicted magnitudes of the negative pressure peaks are approximately 20% less than the measured. However, the predictions from the different control surfaces are converged, suggesting that little extra real quadrupole noise could be captured by further enlarging the control surface. The differences are therefore likely to be caused by physical effects, such as vortex induced loading noise, which are not accounted for in the noise predictions, the drag deficit associated with the Euler CFD code and CFD grid coarseness. 5 Conclusions The permeable form of the FW-H equation has been used to predict transonic rotor noise with much less computational effort than has previously been possible. The methodology involved applying the FW-H equation to control surfaces which were very small while enclosing both the blade and 35 all transonic flow regions. Such a choice of control surface ensured that the quadrupole term in the FW-H equation was negligible and meant that an accurate CFD solution was required only in the vicinity of the blade. 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