Investigation of Methods for Including Real Vane Geometry

Investigation of methods for including real vane geometry in predictions of fan broadband noise I. Introduction Many low-order methods for simulating broadband interaction noise downstream of a fan stage in a turbofan engine have emerged recently 1–4. These methods all rely on a simplified model of the fan exit guide vane (FEGV) that replaces the real vane with a flat vane. These models are effective for trend prediction and do reasonably well in predicting the actual noise spectrum for a given case. However, the individual case results are dependent upon the selection of vane stagger angle. Figure 1 depicts this issue. The Source Diagnostic Test (SDT) baseline vane case at approach and take-off speeds was simulated. 5 The stagger angles from hub to tip selected with a leading edge weighting, a mid chord weighting, and a trailing edge weighting are shown in Figure 1(a). Figures (b) and (c ) then show the results obtained for the exhaust power level spectrum using the exact same wake turbulence input with the different stagger angle specification. The predicted spectrum clearly depends on the stagger angle specifications. Therefore it is of interest to explore whether increasing the fidelity of the model can be increased by utilizing more realistic vane geometry. Recently, a method for including the effect of camber in a low-order cascade model for tonal noise was described by Laborderie et al. 6 The cascade response to an unsteady disturbance that serves as the core of their model utilizes a method that separates the influence of the leading and trailing edges on the response. Therefore they are able to solve two flat plate problems one that matches the leading edge stagger and the other that matches the trailing edge stagger. The full response of the actual vane is then pieced together based on the appropriate trailing and leading edge contributions. Such an approach should be applicable for extending the broadband model of Posson et. al. 2 which is based on the same cascade response methodology. For low-order methods based on cascade response solvers that treat the entire vane section simultaneously such a splitting of leading-edge and trailing-edge stagger is not possible. Also, it is of interest to consider inflow angle and thickness as well as camber. Therefore alternate models are being explored. 40 35 Stagger angle 30 25 20 15 10 5 0 0.5 0.6 0.7 0.8 0.9 Nondimensional radial location 1 (a) Stagger angle selection. 105 115 Measured 90/10 50/50 10/90 100 Measured 90/10 50/50 10/90 110 95 PWL (dB) PWL (dB) 105 90 100 95 90 85 85 80 3 10 4 10 frequency (Hz) (b) Approach speed. 80 3 10 4 10 frequency (Hz) (c) Take off speed. Figure 1: Exhaust power levels for SDT baseline case. Simulation performed with three different choices of stagger angle. Mean flow and turbulence values taken from CFD results. Previously, an attempt to include real vane geometry in the broadband model focused on utilizing the asymptotic airfoil response method of Ayton and Peake. 7,8 While the method was shown to be computationally efficient when used together with the low-order broadband method of Grace 5 it was unable to fully capture the effect of real vane geometries such as those found in the SDT because of thin-airfoil assumptions on which the theory is based. 8,9 This paper describes two alternatives that have been considered and the progress made in developing these alternatives. The first method is motivated by methods that have been developed to model propeller response to turbulence ingestion in underwater applications. In this method, the unsteady response of the vane is computed using a blade vortex interaction simulation. The response could then be utilized in the broadband model of Grace 5 with some minor alterations. Details concerning this method, progress in the development of the method, and additional thoughts regarding such an implementation are given in Section II. The second method is motivated by a higher fidelity modeling method for fan-stage broadband noise described by Logue 10,11 . Here a linearized Euler approach is used to determine the acoustic modes in the duct downstream of a 3D blade or vane row due to unsteady fluid-structure interactions. These modes are then used together with the upstream turbulence information to build the broadband response. This method is not a low-order method like those mentioned above; however, it is low order when compared to a full direct simulation. 12 This second approach and initial findings are discussed in Section III. II. Method 1 The first investigation utilizes the blade vortex interaction (BVI) method that is often applied in underwater propeller analysis. 13,14 Because the spacing is large between the propellers and the Mach number is low for underwater applications, analyses for the response of a propeller to ingested turbulence can be based on the single-airfoil response to flow disturbances. This is very different than in the turbofan application where the spacing on both blade and vane rows is usually less than one and the flow Mach number is subsonic. In order to develop an approach for the turbofan modeling that is analogous to the propeller work, a BVI-like interaction method for a cascade must be developed. In order to understand how the BVI method integrates into a broadband model, the next section will provide a brief overview of the low-order model that has been described in detail elsewhere. 5 A. Fan Broadband Noise Model The previously tested low-order model for predicting the noise produced by rotor wake turbulence interacting with a downstream exit guide vane has these essential features: 1. Vanes are treated as flat plates 2. Strip theory is used to account for spanwise vane geometry 3. Single Fourier components of the turbulence are modeled as gusts and a flat-plate cascade gust response model based on the work of Ventres 15 provides the unsteady surface pressure 4. The cascade response to a 3D gust is modeled using the Graham similarity method 16 to extend the 2D Ventres results 5. The Liepmann spectrum is used to model the rotor-wake turbulence. An average passage value of the wake turbulence intensity and length scale are used in the Liepmann function 6. The downstream duct is modeled as a cylindrical annulus and the Green’s function for an infinite annulus is used to determine the acoustic duct pressure from the vane unsteady pressure. 7. The acoustic power level per frequency at the duct exit is calculated based on the acoustic intensity which can be related to the acoustic pressure in the duct. The final mathematical form for the downstream duct power per frequency is  4 ω‡ −βann c  Z rT Z Z P ower/f req U (r)Lr (r) b b 2 V   = R (r, ω)eiµ(r, ω)(z1 −z2 )  2  3 4 ‡ ρann c20 Rtip c R ‡ ‡ 0 8π 2 (1 − σ 2 )kmn r −b −b tip H ωc + Mann kmn     Z X E2,2 (k1 , A2 , k3 , r) k3 Lr dk3 dz2 dz1 dr F0 F0∗ sinc 2 c20 c1 (1) Here the duct modes and axial wave numbers from the Green’s function equation are contained in R and µ respectively. kmn is related to the roots of the Green’s Function modes, k = ( Uω )2 + (A2 )2 + k32 , A2 = c1 V −m . σ is the hub-to-tip ratio, c0 the speed of sound, ωc‡ = ωRtip /c0 , σ = Rhub /Rtip , and βann = pr 2 . F0 is the cascade response function to a 3D, unit amplitude gust. The ann subscript indicates 1 − Mann flow values in the annulus downstream of the vane. E2,2 is the Liepmann function describing the transform of the expected value of the turbulent upwash velocity correlation 2 E2,2 (k, t) = (2π)3 2u02 L5s (k12 + k32 ) π 2 (1 + (kLs )2 )3 (2) with u02 and Ls being the turbulence intensity and streamlength scale respectively. B. Cascade BVI If a compressible, cascade-BVI method can be developed, the results from the BVI would replace the third feature in the broadband method given above. The cascade response would incorporate vane section shape including thickness and camber. Potentially, flow incidence could be included a lso. In this section, a BVI method for single airfoils will be described and then the extension for the cascade will be given. The simulation of a vortex passing an airfoil is carried out in the time domain. The vortex can either move along a streamline or be moved by the local mean flow. If the vortex strength is small enough and the distance at which it will pass the airfoil is great enough, then it will remain on a streamline when being convected by its local flow. When these conditions are not met, the vortex will cross streamlines if it moves with its local velocity. A wake is generated due to the interaction of the vortex with the airfoil. The wake can be modeled using point vortices or constant doublet panels or another suitable elementary flow. The wake path can follow a fixed trajectory or be defined by the path that the panels or point vortices trace out when convected by the local mean flow. I t h as b een s hown p reviously t hat t he F ourier t ransform o f the time dependent lift function that arises from a BVI simulation for a flat-plate airfoil at low Mach number reproduces the Sears function and the effect of airfoil thickness, camber, and angle of attack on the unsteady lift of a single airfoil at low Mach number have been studied using the BVI method in the past 13,14 . A 2D linear vortex panel method, as described in Kuethe and Chow 17 , has been extended to perform a BVI analysis. For a vortex of strength 0.01, starting 4 chords upstream and 0.12 chords above NACA 0012 and NACA 2412 airfoils, the frequency domain lift results are shown in Figure 2. The normalized unsteady c lift is plotted against normalized frequency, k1 = πf U , in the figure. The magnitude of the Sears function S(k1 ) and the magnitude of the lift with an airfoil thickness correction described by Moriarty et. al. via 2π ln(10) |S(f )|2 e−β(τ )f c/U with β(τ ) = [1.123D(τ ) + 5.317D(τ )2 ] and D(τ ) = (τ /c)1% + (τ /c)10% 10 where τ /c is the airfoil thickness ratio 18 . The scallops in the result for the vortex passing the NACA 0012 are most likely a function of the time discretization used in the BVI calculation and will be further investigated in the future. Even with the scallops it is clear that the BVI method captures the thickness trend. And that camber also effects the trend. Normalized Lift 10 0 Sears BVI Thickness Analytical Thickness BVI Thickness & Camber 10 -2 10 -4 10 -6 10 -8 0 5 10 15 20 Normalized Frequency Figure 2: Normalized lift results for a vortex passing a real geometry airfoil. NACA 0012 and NACA 2412. Incompressible. Normalized Lift The same airfoil geometry effects can be considered at subsonic Mach numbers also. Again the NACA 0006 airfoil is considered, now at a Mach number of 0.3. The results from the BVI simulation are compared to the results obtained with the semi10 -1 analytical integral approach. See Figure 3. Possio BVI These airfoil response results for a simple 2D -2 gust can be extended to account for a three10 dimensional disturbance using Graham’s similarity rules. 16 10 -3 The BVI results for the single airfoil demonstrate that if the method can be extended to the cascade case, it could provide the un10 -4 steady response noted as the third essential feature of the broadband method. Therefore a -5 10 cascade-BVI simulation is considered here. It 0 5 10 15 20 Normalized Frequency is based upon a mapping given by Howell 19 and Thwaites 20 that transforms an infinite cascade into a single object. In a paper by Bhi- Figure 3: Normalized lift for a vortex passing a NACA marasetty and Govardhan 21 this mapping was 0006 with freestream Mach number 0.3. used together with the Kuethe and Chow vortex panel method to compute the mean flow around a cascade of airfoils. Figure 4 shows a cascade of NACA 6412 airfoils, the transformed shape utilized in the method, and the computed meanflow streamlines in the cascade frame of reference. After transformation Cascade of Airfoils 3.5 1 3 2.5 0.5 2 1.5 1 0 0.5 0 −0.5 −0.5 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 −1 −0.5 (a) Cascade. 0 0.5 1 (b) Transformed shape 0.4 0.3 0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −1 −0.5 0 0.5 (c) Mean flow streamlines. Figure 4: Original cascade, shape in transformed plane, and computed streamlines of mean flow around the cascade. In the transformed plane, the freestream flow is represented by a source and vortex located at (−1, 0) and a sink and vortex located at (1, 0). Points far upstream of the cascade transform into basically the single point (−1, 0) and points far downstream of the vane transform into the single point (1, 0). For large vane-to-vane spacing, the single vane in the transformed plane is relatively far from these points. So the imposition of a vortex “far” upstream and the continuation of a wake “far” downstream are straight forward as the “far” locations do not immediately coincide with these points. Given a very large spacing-to-chord, h/c = 30, the results from the cascade and the single airfoil calculations coincide as shown in Figure 5. 10 0 When the spacing-to-chord is 1.0 or less, the points between a cascade airfoil leading 10 -2 edge and ”far” upstream are compressed into a very small region in the transformed plane -4 10 highlighted by the blue box in Figure 6. This poses some difficulty because if the imposed -6 10 vortex or wake vortices are too close to the source/sink/vortices that model the freestream 10 -8 in the transformed plane, then their trajecto0 5 10 15 20 Normalized Frequency ries will be unnaturally altered producing inaccurate unsteady lift results. Careful control Figure 5: Comparison between results for cascade of of the time step and regularization of the inNACA 0006 airfoils with spacing-to-chord 30 and a single fluence of the source/sink/vortex on the imNACA 0006 airfoil. posed vortex and the wake vortices is necessary. For a spacing-to-chord of 1.0, the response of a cascade of NACA 0006 airfoils was computed using the BVI and compared to the flat-plate response values given by the Ventres method. 15 Figure 7 shows the BVI method provides reasonable results. Previous BVI analyses for single airfoils showed a deviation from the gust response results under k1 = 0.5. Here the deviation continues until k1 = 5.0. The source of this error will be identified in the near future. Normalized Lift Single Airfoil Cascade h=30 The initial results indicate that the BVI model can be use to get the unsteady response of a cascade of real airfoils. And the Graham Upstream Vortex & Sink 0.8 Downstream Vortex & Sink similarity rules allow the results to be extended 0.6 to include oblique gust effects (i.e. 3D distur0.4 bance effects). However, there is an important 0.2 caveat that must be addressed before adopting this as the third element in the broadband model 0 for the fan-stage interaction noise. The trans-0.2 formation method for obtaining the cascade-0.4 vortex interaction inherently models the case -0.6 when the number of rotor wakes matches the -0.8 number of FEGVs. This is true because the transform from the cascade plane to the single -1 -0.5 0 0.5 1 object plane assumes perfect periodicity in the cascade and thus in the position of the upstream Figure 6: Transformation of cascade of NACA 0006 airvortex. Therefore, the vortex is passing through foils to single body. Flow singularities that account for the cascade with the exact same gap position vane- mean flow in the transformed plane are shown as orange to-vane. The broadband noise calculation re- circles. The region into which the entire space upstream quires the assessment of the cascade response of the airfoil is transformed is shown as the blue box. at multiple k2 values. In practice this corresponds to multiple inter-blade phase angles. So the flexibility t o i nclude t he i nter-blade p hase a ngle i n t he r esponse i s e ssential f or t he b roadband calculations. Methods to incorporate real inter-blade phase angles are being pursued. One potential method artificially amplifies or reduces the induced velocity from the imposed vortex on the lower surface of the vane based on a desired inter-blade phase angle. Normalized Lift 10 -1 Ventres BVI Cascade 10 -2 10 -3 5 10 15 20 Normalized Frequency Figure 7: Unsteady lift response of a cascade of NACA 0006 airfoil via BVI and cascade of flat plates via the Ventres method. Spacing-to-chord of 1.0. III. Method 2 The methods that have been considered thus far to extend the low-order model to include real vane shapes have some insufficiencies. P reviously, the insufficiencies in the asymptotic method for computing the response of airfoils and cascade with real geometries were shown to be inherent in the method and thus insurmountable. Here, the BVI method has been shown to have some insufficiencies w hich m ay be overcome. Given this uncertainty, and the fact that computational resources are more abundant now, a completely different higher fidelity model has been c onsidered. The method was introduced by Atassi and Logue 10,11 and uses a 3D linearized Euler solution method to obtain the cascade-gust response. In particular, the duct acoustic modes are computed and used in the acoustic simulation, not the vane unsteady surface pressure. The calculation of the vane response via linearized Euler is far more time-intensive than the semi analytical gust-response evaluation for a flat-plate cascade and the BVI method described above for real geometry cascades. The method would however provide response values for all rotor-FEGV counts (varying interblade phase angles) unlike the BVI method while including vane geometry. The introduction of linearized Euler analysis can actually take several forms. One form would utilize a 2D linearized Euler calculation to provide the unsteady surface pressure on a strip as is currently done with the semi-analytic flat-plate r esponse a nd i s p lanned f or t he r eal v ane, B VI c omputed, r esponse. A second form would utilize a 2D linearized Euler calculation to obtain the 2D cascade acoustic modes and incorporate these into a broadband model as was done with the semi-analytic flat-plate c ascade acoustic modes previously. 22 A third form would use a full 3D linearized Euler calculation to obtain the duct acoustic modes and incorporate these into a broadband model as was done by Logue. 11 In Logue’s work, a 3D simulation was used where the vanes were modeled as flat, c ambered, p lates. T he l inearized E uler code used by Logue is not generally available. So, in order to explore this option further, an alternative linearized Euler solver is needed. LINFLUX is such a solver that is distributed by NASA. It has has been previously utilized by the authors for tonal studies. LINFLUX is a three-dimensional, finite-volume, linearized Euler solver 23 developed to simulate the aerodynamic and acoustic response of a turbomachinery blade row to prescribed structural and/or aerodynamic excitations. It is available via the NASA computational code server. LINFLUX provides both the unsteady surface response as well as the normalized duct pressure. For example, the unsteady surface pressure and the duct pressure for the 2BPF tone related to the SDT baseline-vane, approach-speed case are shown in Figure 8. (a) Magnitude of surface pressure. (b) Phase of surface pressure (c) Real part unsteady duct pressure. Figure 8: Unsteady, normalized FEGV suction-side surface pressure and duct pressure. SDT baseline vane geometry. Approach speed. 2BPF. Because a linearized Euler based computation provides both the unsteady surface response and the duct acoustic pressure and velocity, one can utilize the duct information directly without working through the Green’s method. In fact, it is more accurate to utilize the duct acoustic information unless one goes through the effort to compute the correct Green’s function for the nonuniform mean flow that exists near the vanes. If one uses the Green’s function for the constant, infinite annulus, one has assumed that the mean flow is constant. This can lead to the propagation of modes that would actually be cut-off and vice versa. Therefore the emphasis has been on understanding the method required to utilize the duct acoustic pressure and velocity and to consider the accuracy of LINFLUX for computing the required wave-number-frequency combinations for the broadband calculation. A. Broadband noise based on duct modes The acoustic power at an axial location along the annulus downstream of a vane row is obtained by integrating the axial component of intensity over the cross section of the annulus. The intensity is given by   pac I · x̂axial = + U · uac (ρo uac1 + U1 ρac ) (3) ρo  Z ZZZ Z ZZZ  1 p̃ac = + U · ũac e−iωt+ik·x 8 (2π) ρo ω,k × (ρo ũac1 + U1 ρ̃ac )∗ν,K eiνt−iK·x dω dk dν dK (4) (5) To compute this using LINFLUX individual gust frequency-wave-number combinations are considered. A gust disturbance upstream is modeled in general as X ~u = ~amg (r)ei(−ωt+mg θ+αx) When LINFLUX is used for rotor-stator interaction tonal noise analysis, the upstream input disturbance is determined from experimental or computational wake data. The time-dependent, streamwise component of the wake velocity at several radial positions is Fourier transformed to obtain the disturbances amplitude at the frequency of interest. The k2 component denoted as the nodal diameter is known from the disturbances periodicity in the circumferential direction (i.e. sB where B is the number of rotor blades and s is the harmonic of interest). For broadband noise, a complete set of nodal-diameters must be considered with no connection to the number of rotor blades, and canonical, physically supported, mode shapes for the disturbance in the spanwise direction can be considered. The representative form of the streamwise disturbance then becomes   πng r ug = amg ,ng ei(−ωt+mg θ+αx) cos rt − rh For any incident disturbance, LINFLUX computes the acoustic pressure and acoustic intensity factors in the duct based on a modal representation. In the annulus downstream of the vane, the pressure can be represented as a series of modes with azimuthal and radial mode numbers m and n respectively such that the pressure takes the form XX cmn Pmn ei(kmn x+mθ) p(~x) = m n with P normalized such that its maximum value is unity. The intensity factors are similarly represented in the LINFLUX output as  I1 = pac + U · uac ρo  = cmn I1mn ei(kmn x+mθ) ∗ ∗ I2 = (ρo uac1 + U1 ρac ) = c∗mn I2mn ei(kmn x+mθ) (6) (7) with I representing the modal shapes. LINFLUX then computes the axial mean value of the area-averaged acoustic power via ) ( Z rt X π ∗ I1mn I2mn rdr (8) Re |cmn |2 P ower = AD r h m,n In order to utilize the modal content in the duct for the broadband prediction, Logue suggests that first the linearized Euler calculation be done with a unit amplitude gust disturbance, amg ,ng = 1 and then the radial integration in Eq. (8) can be weighted by the appropriate magnitude of the inflow turbulence spectrum for the frequency wave-number combination of interest at each radial location. This is less intuitive when a 3D solver has been utilized to find the vane response as when the strip theory is used. The 3D solution depends on the spanwise distribution of the disturbance. If a k3 wave number has been used to model the disturbance and then the disturbance weightings from hub-to-tip based on the local turbulence parameters alter this spanwise distribution it is not clear that the solution is still valid. However, Logue showed that for the SDT low-count vane case at approach, this method provided a good prediction. Hence, it is considered here. The derivation of this formulation is a generalization of that reported previously 22 in which a strip theory, 2D cascade response was considered but the acoustic modes were incorporated directly into the broadband calculation and the Green’s method was not used. The important pieces are repeated here. First, the acoustic power in the annulus downstream of the vanes is given by Z 2π Z Rtip (I · x̂axial )rdrdθ P ower = 0 (9) Rhub The intensity in the axial direction can be rewritten noting that the modal amplitudes are understood to be the solution to a unit input disturbance 1 (2π)8 I · x̂axial = × Z ZZZ Z ZZZ X X m n
∗ ∗ ∗ ∗ iνt−iK·x cmn ug I2mn ν,K e dω (cmn ug I1mn )ω,k e−iωt+ik·x dk dν dK (10) The turbulence spectrum given in Eq. (2) can be used to describe the disturbance correlation via ~ = (2π)5 δ(ω − ~k · U ~ )δ(ν − K ~ ·U ~ )E22 (~k, r)δ(~k − K) ~ ug (ω, ~k)u∗g (ν, K) (11) ~ gives Integration of Eq. (10) with ν and K I · x̂axial 1 = (2π)3 Z ZZZ X X m ~ )E22 (~k, r)I1mn I ∗ dωd~k |cmn |2 δ(ω − ~k · U 2mn (12) n Integration with respect to k1 and the substitution of dω = 2πdf gives I · x̂axial = 1 (2π)2 Z ZZ X X |cmn |2 m Ux n ∗ E22 (~k, r)I1mn I2mn dk2 dk3 df (13) The relations between k2 , k3 in this formulation and the coordinates used for the initial gust description are mg ωUθ 1 (U − ) Ux r U   πng = rt − rh k2 = (14) k3 (15) such that U dmg rUx   πdng = rt − rh dk2 = (16) dk3 (17) but mg and ng take on integer values so one finds I · x̂axial = Z π X X X X U |cmn |2 ~ ∗ (k, r)E22 (~k, r)I1mn I2mn df 2 r(r − r ) (2π)2 m n U t h x m n g (18) g One can then obtain the differential power per frequency by integrating over the area of the annulus to find P ower X X X X 1 = f req 2(rt − rh ) m n m n g g Z rt rh  U Ux2  ∗ |cmn |2 E22 (~k, r)I1mn I2mn dr (19) where the ~k values in the Liepmann spectrum function is understood to utilize the connections that k1 = ω/U and Eqs. (14)-(15). B. Applying the method to an SDT case LINFLUX has been used to analyze the tonal interaction noise produced by the SDT rotor-FEGV combinations. 24,25 Figure 8 above shows results obtained for the 2nd blade pass frequency of the baseline vane case at approach speed. Because the mesh and mean flow are already available for this case, it was used for initial consideration of utilizing LINFLUX for broadband noise calculations. At a given frequency, the number of circumferential and radial gust modes that should be included in the broadband calculation is unknown a priori. For lower frequencies, fewer gust modes lead to cut-on propagating acoustic waves, so it is clear that more modes are necessary as the frequency increases. A preliminary computation was made at frequencies of 1000 and 5000 Hz. The integrand of Eq. (19) was to be plotted vs. mg and ng separately to ascertain reasonable limits for mg and ng . The test failed at the higher frequency because there were multiple modes for which LINFLUX would not run properly. To investigate the problem, the (mg , ng ) = (1, 0) gust mode was run for frequencies between 1000 Hz and 6000 Hz in 25 Hz increments. The amplitude of the cut-on, propagating modes for this gust mode are plotted for all frequencies in Figure 9. The relatively horizontal lines in the figures depict the amplitudes of the different acoustic (m, n) modes. The figure shows that the code has difficulty providing accurate results near the cut on of a mode. There are also other issues that do not seem to be related to the cut on of a mode where the solver is failing to give accurate results. The same type of result is found for all (mg , ng ) gust modes and as mg increases, the number of inaccurate solutions not near a cut-on mode increases. While it seems some interpolation across frequencies could be used to deal with the anomalous solutions, this only seems feasible at the lower frequencies where the anomalies are well separated. It has been determined that if the LINFLUX solution was stable for all of the gust modes, the broadband simulation could be completed in a day or two if multiple processors are used. Gust frequency-mode combinations that have acoustic waves cut-on take approximately 20 minutes. Each computation is fully decoupled from the others so the use of multiprocessors to do the calculations of the different frequencymode combinations is perfectly efficient in terms of speed-up of the c alculations. This makes the approach potentially reasonable once the grid and mean flow solutions are d eveloped. Therefore, it warrants further investigation. In particular, it must be determined if LINFLUX can be used in any way to obtain the required information for the broadband duct power prediction, Eq. (19). In addition, it is still possible to pursue the use of a 2D linearized Euler solver that could be used together with the strip approach. Both of these options will be explored in the future. IV. Conclusions Low-order broadband noise models for fan-stage, rotor-stator interaction noise exist and have been shown to provide good trend predictions. However, because the models employ flat-plate vanes, the vane stagger becomes a non-exact input parameter. Predictions performed using the leading edge stagger, an averaged stagger, and the trailing edge stagger captured trends properly but differed in the exact spectral shape. As such, methods for incorporating real vane geometry in a broadband noise model have been considered. First, methods that would simply provide an alternative to the 2D flat-plate cascade gust response model that serves as the core of the low-order broadband calculation are considered. An asymptotic method for including thickness, camber and angle of attack to the response model was shown to be too limited by thin-airfoil constraints to be applicable to real geometries of interest. In this paper, a method that uses the blade-vortex interaction (BVI) technique to determine the response of a 2D cascade of real geometry vanes is described. The ability of the BVI technique to reproduce the single-airfoil compressible-flow gust response is demonstrated. Additionally, preliminary results from the BVI technique applied to cascades shows that the flat-plate cascade response is well reproduced. The next step will be to test the method for the SDT 0.01 0.008 |cmn| (0,0) 0.006 0.004 0.002 0 1000 (0,1) (0,2) (0,3) 3000 4000 Frequency (Hz) 2000 (0,4) 5000 6000 Figure 9: SDT baseline vase geometry at approach speed. mg = 1, ng = 0. Amplitude of cut-on, propagating modes. (m, n) modes vane geometry and validate the cascade response. Second, the standard BVI will have to be modified in order to handle adjustable inter-blade phase angles. Finally, the response method must be incorporated into the broadband model which will require validation that the Green’s method can provide reasonable duct acoustic values based on the unsteady vane surface pressure. In addition to the consideration of methods for adding real geometry directly into the existing low-order model, preliminary exploration of a a higher fidelity method for studying fan-stage broadband interaction noise has been undertaken. The motivation to consider a higher fidelity method comes both from the uncertainty that a technique to extend the low-order method will be fully realized and from previously reported research. The higher fidelity model is based on the linearized Euler equation solution to the fluid-structure interaction problem for a cascade. Both 2D cascade and 3D vane linearized Euler simulations are of interest with the former being more of an extension of the acoustic mode method that was described previously. 22 In this paper, the potential to use the 3D solver LINFLUX was explored. The results show that in its current operating condition, LINFLUX is not stable enough to be used directly to predict broadband fan-stage interaction noise. Further investigations may reveal ways in which LINFLUX can be utilized to this end. V. Acknowledgments The authors acknowledge the financial support of the Aeroacoustics Research Consortium and collaboration of its members. The authors also acknowledge that the computational work reported on in this paper was performed on the Shared Computing Cluster which is administered by Boston Universitys Research Computing Services.