- Email: [email protected]
Stator Interaction Using CAA
Available online at www.sciencedirect.com
Procedia IUTAM 1 (2010) 203–213
Reprint Prediction of NoiseRotor from Wake/Stator Realistic Rotor Towards of: the Towards Predictionthe of Noise from Realistic Interaction Wake/Stator Interaction Using CAA Using CAA ✩
Ray Hixon, Adrian Sescu, and Vasanth Allampalli Mechanical, Industrial, and Manufacturing Engineering (MIME) Department University of Toledo Toledo, Ohio 43606
Abstract In this work, the NASA Glenn Research Center Broadband Aeroacoustic Stator Simulation (BASS) code is extended for use in the prediction of noise produced by realistic three dimensional rotor wakes impinging on a downstream stator row. In order to accurately simulate such a flow using a nonlinear time-accurate solver, the inflow and outflow boundary conditions must simultaneously maintain the desired mean flow, allow outgoing vortical, entropic, and acoustic waves to cleanly exit the domain, and accurately impose the desired incoming flow disturbances. This work validates a new method for the acoustics-free imposition of three-dimensional vortical disturbances using a benchmark test case. c 2010 Published by Elsevier Ltd. Keywords: Computational Aeroacoustics, Boundary Conditions
1. Introduction There are two main types of noise generated in a jet engine fan: broadband and tone noise. [1] Broadband noise is generated by the interaction of turbulence with the fan rotors and stators. The tone noise is generated by the periodic flow distortions associated with the spinning rotors or by the interaction of flow distortions such as wakes from upstream blades with downstream rotors, stators, or struts. For high-bypass-ratio engines with subsonic fan tip speeds, the interaction of fan wakes with stators is one of the main sources of noise. The computational prediction of wake-stator noise is a demanding problem due to the highly nonuniform swirling flow upstream of the stator vanes and the loaded, complex geometry stator vanes. For tone noise prediction, the frequency domain approach is highly efficient and accurate. In a frequency domain approach, the incoming rotor wakes are decomposed into a number of linear single-frequency perturbations about the mean flow, allowing each frequency to be solved separately. [2, 3, 4, 5] The NASA Glenn Research Center LINFLUX code [6, 7, 8] is an example of a three-dimensional frequency-domain linearized Euler solver designed to provide a comprehensive and efficient unsteady aerodynamic analysis for predicting the aeroacoustic and aeroelastic responses of axial flow turbomachinery. Due to the wide range of length and time scales present in turbulent flows, the prediction of broadband noise requires much more computational effort. In order to accurately and efficiently predict such nonlinear flows, Computational Aeroacoustics (CAA) codes are being developed which combine high-resolution spatial differencing schemes (e.g. Lele [9], Tam and Webb [10], Li [11], Kim and Lee [12], Hixon [13], Sescu, et. al. [14]) with optimized time marching schemes (e.g., Hu, et. al. [15], Stanescu and Habashi [16], and Hixon, et. al. [17]), providing high accuracy while using relatively coarse grids compared to existing CFD approaches. ✩ This article is a reprint of a previously published article. For citation purposes, please use the original publication details: Procedia Engineering 6C (2010) 203–213. DOI of original item: 10.1016/j.proeng.2010.09.022
2210-9838 © 2010 Published by Elsevier Ltd. doi:10.1016/j.piutam.2010.10.022
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The NASA Glenn Research Center (GRC) Broadband Aeroacoustic Stator Simulation (BASS) code is a nonlinear time-domain CAA code, developed for use in the prediction of fan tone and broadband noise. [18] In previous work with the BASS code, the interaction of low-amplitude vortical gusts with realistic 2D [19] and 3D [20] stator geometries was predicted and showed good agreement with linear theory. However, in both cases, the incoming rotor wakes were represented by a combination of three simple harmonic functions which were analytically divergence-free. Current work with the BASS code is focused towards the prediction of broadband noise, supporting the noise reduction goal of the NASA Subsonic Fixed Wing Project. To accomplish this goal, it is necessary to develop computational boundary conditions which allow arbitrary unsteady vortical disturbances to be imposed at the inflow boundary, while also maintaining the desired mean flow conditions and allowing outgoing waves to exit the computational domain without reflection. In earlier work, a framework for accomplishing this task was presented and validated using two dimensional [21] and three dimensional [22] benchmark test cases. Results were also shown for a loaded two dimensional cascade with realistic geometry, using a wake taken from experimental data from the NASA Fan Source Diagnostic Test (SDT) [23]. The wake that was imposed for that test was taken directly from experimental data, and produced a strong acoustic signal when imposed at the inflow boundary. These acoustic signals contaminated the predicted noise from the wakestator interaction. In order to address this problem, a new vortical gust boundary condition (VGBC) has been developed which is formulated to impose an analytically divergence-free vortical gust at the inflow boundary, given an input gust which may or may not be divergence-free. [24] The VGBC has been previously tested in two dimensions, using CAA benchmark test cases with analytic gust specification. This boundary condition is one of the necessary steps towards the prediction of broadband noise, with the generation of realistic divergence-free synthetic turbulence being another major focus of current research. In this work, the VGBC has been extended for use in nonuniform three dimensional mean flows. The threedimensional formulation and validation results using the Third CAA Workshop Category 4 problem [25] are presented. 2. Numerical Approach The NASA Glenn Research Center BASS code [18] is used in this work. The BASS code solves the nonlinear Euler or Navier-Stokes equations in chain-rule curvilinear form: ⎛ ∂ ∂ ∂ ⎜⎜⎜ ξ x ∂ξ {E} + ξy ∂ξ {F} + ξz ∂ξ {G} ∂ ⎜⎜⎜⎜ +η ∂ {E} + η ∂ {F} + η ∂ {G} {Q} + ⎜⎜ x ∂η y ∂η z ∂η ⎜⎝ ∂t +ζ x ∂ζ∂ {E} + ζy ∂ζ∂ {F} + ζz ∂ζ∂ {G}
⎞ ⎟⎟⎟ ⎟⎟⎟ ⎟⎟⎟ = 0 ⎠⎟
(1)
where Q is the vector of conserved properties and E, F, and G are the flux vectors in Cartesian coordinates. For this work, the Euler equations are solved. The BASS code employs a finite-difference time-domain approach for solving the governing equations. Structured multiblock grids are used to provide topological flexibility. For portability and maintainability, BASS is written in standard Fortran 2003, and is parallelized using the MPI-1 [26] standard. BASS has been tested using many compilers and MPI implementations, and has been validated on a range of benchmark aeroacoustics problems. The BASS code has a number of spatial differencing schemes, artificial dissipation methods, and explicit time marching schemes implemented. For this work, the seven-point fourth order accurate Dispersion Relation Preserving (DRP) scheme of Tam and Webb [10]) is employed for spatial differencing. The optimized HALE-RK67 time marching technique (Allampalli et al. [17]) is used to advance the solution in time. A blended second and tenth order shock-capturing dissipation scheme [27], based on the Kennedy and Carpenter explicit filters [28] is used for these cases. The BASS code has been Verified for the nonlinear Euler equations using the EVA-III approach, and showed design spatial and temporal accuracy on 3D curvilinear grids for all schemes. [29] 3. Description of Boundary Condition Implementation In this section, the strategy used for the implementation of boundary conditions in the BASS code is briefly described. In the BASS code, the flow at the boundary is directly calculated using the governing equations; thus, no
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extrapolation is used. [30] The boundary condition is imposed as a ’correction’ to the time derivatives calculated using the governing equations. At the inflow boundary, the boundary condition is decomposed into three components: ∂Q ∂t in f lowBC
∂Q ∂Q ∂Q + + ∂t MFBC ∂t nonre f lecting ∂t imposedDisturbance
=
(2)
Each of these components performs a different task. The mean flow boundary condition (MFBC), described in Reference [31], is used to maintain the correct mean flow conditions at the boundary. A three-dimensional extension of the Giles boundary condition [32] is used to allow outgoing waves to pass through the boundaries with minimal reflection. Finally, the three-dimensional VGBC is used for the imposition of the vortical gust. It is inherently assumed that the flow disturbances at the boundary are of a low enough amplitude that a linear flow assumption can be made. 4. Three Dimensional Vortical Gust Boundary Condition (VGBC) In this section, the formulation of a boundary condition which imposes a three-dimensional divergence-free vortical gust at the inflow boundary of a computational grid is described. The full details of the numerical implementation are given in Reference [24]. In the vortical gust boundary condition (VGBC), the incoming gust is assumed to be linearly convecting with a nonuniform mean flow. The equations for the incoming gust are thus: ut + Uux + Vuy + Wuz + u U x + v U y + w U z vt wt
+
+
Uvx
Uwx
+
+
Vvy
Vwy
+
+
Wvz
Wwz
+ u V x + v Vy + w Vz
+ u W x + v Wy + w Wz ux + vy + wz
=
0
=
0
= =
0 0
(3)
where the overbar quantities refer to the mean flow values and the prime quantities refer to the vortical gust disturbance velocities. Consider the case of a Cartesian grid with an inflow boundary in the x-direction. In this case, the outward-pointing face normal is oriented in the negative x-direction. It is assumed that the velocity components associated with the incoming gust are specified across the boundary face, and thus the tangential (y and z) derivatives of all gust quantities can be computed. It is also assumed that no gust data is available inside the computational domain, and thus that the normal (x) derivatives of the gust quantities cannot be directly computed. In the vortical gust boundary condition, the gust velocity components that are normal to the local flow direction are specified directly using the imposed gust data. The imposed gust velocity component that is aligned with the local flow direction is discarded and replaced with the gust velocity component required for a divergence-free incoming gust. Thus, in the case of a uniform axial mean flow, the equations that are used to compute the unknown gust velocity component are: ut + Uux ux + vy + wz
= =
0 0
(4)
which can be combined to obtain: ut,VGBC
= U vy,speci f ied + wz,speci f ied
vt,VGBC
=
wt,VGBC
=
vt,speci f ied
wt,speci f ied
(5)
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Reference [24] presents the generalized VGBC for use with arbitrary boundary and mean flow orientations. 5. Validation Test Case: Third CAA Workshop Category 4 Problem 5.1. Problem Description The Third CAA Workshop Category 4 problem [25] was chosen as the validation test case for the three dimensional VGBC. In this problem, a stator row consisting of 24 infinitely thin flat plates are mounted in a constant-radius annulus. A uniform axial mean flow convects the wakes from an upstream 16-blade rotor through the stator row, and noise is generated when these wakes impinge on the stators.
Figure 1: Grid used for the vortical gust boundary condition tests
The azimuthal velocity associated with these wakes is given as:
U
ωx 0.1U cos 16 + θ − ωt + 2πq (2r − 1) U = 0.5
ω
= 0.783
vθ
=
(6)
The parameter q determines the radial variation of the wakes and ranges from a value of 0 to 3. The problem statement as given is not sufficient to allow the imposition of a purely vortical gust. To accomplish this, the axial and radial gust velocities must also be specified to give a divergence-free velocity field:
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vr
=
0
vx
=
−
(7) U v rω θ
(8)
Note the radial variation in the axial gust velocity that is required for a divergence-free gust. 5.2. Numerical Approach The computational grid was generated using the GridPro/az3000 [33] commercial grid generation tool. A single passage grid was generated, and then ’stacked’ in the azimuthal direction. The grid has a minimum of ten grid points per wavelength in all three coordinate directions. Axially, the grid begins 1.5 chord lengths upstream of the stator leading edge and extends 1.5 chord lengths downstream of the stator trailing edge; at this point, the grid is stretched in the axial direction until reaching the outflow boundary at a distance of 4 chord lengths downstream of the trailing edge. The flat plate stators have no thickness in the grid, which gives rise to grid singularities at the leading and trailing edge points of the stators. The grid is clustered about these points to resolve the sharp flow gradients which are generated at these locations. Due to the periodicity of the test problem, the computational domain included only three of the 24 flat plate stators ( 18 of the full annulus). Periodic boundary conditions are specified in the azimuthal direction. A total of 509,000 grid points were used in the computational domain. Figure 1 shows the grid used for all tests. The simulations were run using the BASS code, solving the nonlinear Euler equations. As decribed previously, several boundary conditions are used in combination at the inflow and outflow boundaries. To maintain the correct mean flow conditions, the mean flow boundary condition (MFBC) described in Reference [31] is used. A threedimensional extension of the Giles boundary condition [32] is used to allow outgoing waves to pass through the boundaries with minimal reflection. Finally, at the inflow boundary, the three-dimensional VGBC was used for the imposition of the vortical gust. 5.3. Performance of VGBC The full series of simulations were run using the VGBC, and the results for a q value of 3.0 will be presented here. This value of q was chosen because this test case had the least amount of acoustic response from the wake-stator interaction, and should show the most sensitivity to spurious acoustic waves from the incoming gust. In these simulations, two tests for each q value were run. First, the calculation was run with the stators removed from the computational domain. This test is designed to verify the accuracy of the gust imposition and the vortical gust boundary condition. It should be noted here that only the azimuthal component of the vortical gust is directly specified in this approach; the axial component is generated numerically by the VGBC. The resulting gusts are compared at two locations in the computational domain. The comparison locations are radial lines aligned with the central stator in the domain, located at the inflow boundary and at the stator leading edge. In Figures 2 and 3, the equivalent instantaneous velocity magnitude and pressure at the start of the cycle are shown for the q = 3.0 vortical gust. In Figure 2, the radial ’lean’ of the vortical gust is seen. In this figure, reflections from the outflow boundary condition are generating the largest pressure disturbances in the calculation, and the solution upstream of the stretched grid is free of large pressure waves. Figure 3 shows the effect of the stators on the predicted velocity and pressure. The acoustic response due to the gust is clearly seen; this result compares well with the benchmark solutions [24], as will be shown in the next section. In Figure 4, the computed solutions for q = 3.0 are compared to the analytical solution for the imposed gust. The comparison is shown on a radial line aligned with the leading edge of the center flat plate stator and located at the inflow boundary. In all graphs, the computed solution is denoted by solid lines and the analytical ’gust-only’ solution by circles. In these graphs, the ability of the vortical gust boundary condition to impose the correct axial (vX) velocity is clearly shown. Also, by comparing the computed density and pressure signals between the two test cases, it can be concluded that the amplitude of any acoustic waves generated by the imposition of the vortical gust are much lower than the amplitude of the acoustic waves generated by the gust/stator interaction.
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Figure 2: Velocity magnitude (left) and pressure response (right) for gust-only validation test (q = 3.0)
Figure 3: Velocity magnitude (left) and pressure response (right) for validation test (q = 3.0)
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0.04 0.02 0 -0.02 -0.04
0.04 0.02 0 -0.02 -0.04
vTheta vX
0.0002
0.0002
vR
0.0001
vR
0.0001
0
0
-0.0001
-0.0001
-0.0002
-0.0002 density pressure
0.001 0.0005 0 -0.0005 -0.001 0.5
vTheta vX
0.6
0.7
0.8
density pressure
0.001 0.0005 0 -0.0005 -0.001
0.9
1
0.5
0.6
0.7
0.8
radius
0.9
radius
Figure 4: Comparison of solutions at inflow plane with no stator (left) and with stator (right) for VGBC validation test (q = 3.0)
vTheta vX
0.04 0 -0.04 vR
0.001 0 -0.001
density pressure
0.03 0 -0.03 0.5
0.6
0.7
0.8
0.9
1
radius
Figure 5: Comparison of gust convection to leading edge location for validation test (no stators)
1
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As a final test, Figure 5 compares the analytical ’gust-only’ solution to the calculated ’gust-only’ solution at the leading edge line of the center stator for both validation cases. The comparison between the numerical and the analytical solutions is very good. This illustrates the ability of the numerical scheme to accurately impose the gust and to convect the gust through a complex multiblock grid to the stator leading edge. From the results of these validation cases, it is clear that the VGBC can be used to impose a complex-geometry three-dimensional vortical gust with extremely low levels of spurious acoustic waves. 5.4. Comparison to Benchmark Results The benchmark results given for this problem were calculated by Namba and Schulten using two frequencydomain linearized Euler equation codes. Results for phase and amplitude of the acoustic duct modes were given for the μ = 0, 1, 2 radial modes of the m = −8 and m = +16 azimuthal modes.
Amplitude
0.02 BASS Namba Schulten
0.01 0.005
0.015
0.005 0
0.01
0.01 Re
0
0
0
-0.01
-0.01
0.01
0.01
0 -0.01
BASS Namba Schulten
0.01
Im
Im
Re
Amplitude
0.02 0.015
0 -0.01
0
0.5
1
1.5 q
2
2.5
3
0
0.5
1
1.5 q
2
2.5
3
Figure 6: Comparison of BASS results with benchmark solutions for m = -8 azimuthal mode, μ = 0 radial mode one chord upstream of the leading edge (left) and one chord downstream of the trailing edge (right)
The nondimensionalization of the benchmark results was not given in the workshop proceedings; thus, the BASS results presented here used a scaling and phasing correction factor calculated using the m = −8, μ = 0 upstream results for the q = 0 case. All BASS results were corrected using this factor. Figures 6- 9 compare the BASS results using the VGBC to the benchmark solutions of Namba and Schulten. Each figure shows the overall amplitude of each acoustic mode as well as the amplitude of the real and imaginary components. The acoustic mode amplitudes are measured one chord length upstream of the stator leading edge and one chord length downstream of the stator trailing edge. In general, the figures illustrate the accuracy of the BASS solution compared to the benchmark data. Overall, the BASS code predicts both the upstream and downstream acoustic mode amplitude and phase very well for most of the test cases. In these figures, it can be seen that the acoustic modes are greatly reduced in amplitude when the q parameter is greater than 1.5. This corresponds to the acoustic modes transitioning from cut-on to cut-off. The BASS code predicts this transition well. There is one case where BASS is not predicting as well as expected. The BASS result underpredicts the amplitude for the m = −8, μ = 1 downstream acoustic mode for q = 1, compared to Namba and Schulten. Numerical experiments have shown that this case is very sensitive to the location of the outflow boundary; work is continuing in order to resolve this issue. Another issue of interest is that the phasing of the m = +16 acoustic modes is reversed compared to the benchmark solutions. However, it must be noted that the amplitude of the m = +16 modes are very small compared to the m = −8 modes, and have very little effect on the solution.
211
Amplitude
0.03 0.025 0.02 0.015 0.01 0.005 0 0.02 0.01 0 -0.01 -0.02
Re
BASS Namba Schulten
0.02 0.01 0 -0.01 -0.02
Im
Im
Re
Amplitude
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0
0.5
1
1.5 q
2
2.5
0.03 0.025 0.02 0.015 0.01 0.005 0 0.02 0.01 0 -0.01 -0.02
BASS Namba Schulten
0.02 0.01 0 -0.01 -0.02
3
0
0.5
1
1.5 q
2
2.5
3
Amplitude
0.003 0.0025 0.002 0.0015 0.001 0.0005 0 0.002 0.001 0 -0.001 -0.002
Re
BASS Namba Schulten
0.002 0.001 0 -0.001 -0.002
Im
Im
Re
Amplitude
Figure 7: Comparison of BASS results with benchmark solutions for m = -8 azimuthal mode, μ = 1 radial mode one chord upstream of the leading edge (left) and one chord downstream of the trailing edge (right)
0
0.5
1
1.5 q
2
2.5
3
0.003 0.0025 0.002 0.0015 0.001 0.0005 0 0.002 0.001 0 -0.001 -0.002
BASS Namba Schulten
0.002 0.001 0 -0.001 -0.002 0
0.5
1
1.5 q
2
2.5
3
Figure 8: Comparison of BASS results with benchmark solutions for m = -8 azimuthal mode, μ = 2 radial mode one chord upstream of the leading edge (left) and one chord downstream of the trailing edge (right)
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Amplitude
0.001 BASS Namba Schulten
0.0005
0 0.0005 Re
0
0
0
-0.0005
-0.0005
0.0005
0.0005
0 -0.0005
BASS Namba Schulten
0.0005
0.0005
Im
Im
Re
Amplitude
0.001
0 -0.0005
0
0.5
1
1.5 q
2
2.5
3
0
0.5
1
1.5 q
2
2.5
3
Figure 9: Comparison of BASS results with benchmark solutions for m = +16 azimuthal mode, μ = 0 radial mode one chord upstream of the leading edge (left) and one chord downstream of the trailing edge (right)
6. Conclusions and Future Directions In this work, a Vortical Gust Boundary Condition (VGBC) was presented, which is designed to impose a numerically acoustics-free incoming gust at the inflow boundary of a computational domain. The VGBC was tested using a three-dimensional CAA benchmark problem. The results showed the accuracy of the VGBC and the ability of the NASA BASS code to calculate wake/stator interaction problems. The VGBC is currently being validated for nonuniform swirling flows and is a necessary step towards applying the BASS code to the prediction of broadband noise from turbulent rotor wake/stator interaction. Acknowledgments This work is supported by the Subsonic Fixed Wing Project of the NASA Fundamental Aeronautics Program. The technical monitor for this work is Dr. Edmane Envia of the NASA Glenn Research Center. [1] D. L. Huff, Fan noise prediction: Status and needs, in: AIAA Paper 98-0177, 1998. [2] S. N. Smith, Discrete frequency sound generation in axial flow turbomachines, Tech. Rep. 3709, Aeronautical Research Council (1973). [3] D. S. Whitehead, A finite element solution of unsteady two dimensional flow in cascades, International Journal of Numerical Methods in Fluids 10 (1990) 13–24. [4] K. C. Hall, J. M. Verdon, Gust response analysis for cascades operating in nonuniform mean flows, AIAA Journal 29 (1991) 1463–71. [5] K. C. Hall, W. Clark, Linearized Euler predictions of unsteady aerodynamic loads in cascades, AIAA Journal 31 (1993) 540–50. [6] J. M. Verdon, M. D. Montgomery, A. H. Chuang, Development of a linearized unsteady Euler analysis with application to wake/blade-row interactions, Tech. Rep. NASA CR-1999-208879, National Aeronautics and Space Administration (1999). [7] J. M. Verdon, Linearized unsteady aerodynamic analysis of the acoustic response to wake/blade row interaction, Tech. Rep. NASA CR-2001210713, National Aeronautics and Space Administration (2001). [8] D. Prasad, J. M. Verdon, A three-dimensional linearized Euler analysis of classical wake/stator interactions: Validation and unsteady response predictions, International Journal of Aeroacoustics 1 (2002) 137–163. [9] S. K. Lele, Compact finite difference schemes with spectral-like resolution, Journal of Computational Physics 103 (1992) 16–42. [10] C. K. W. Tam, J. C. Webb, Dispersion-relation-preserving finite difference schemes for computational aeroacoustics, Journal of Computational Physics 107 (1993) 262–281. [11] Y. Li, Wave-number extended high-order upwind-biased finite-difference schemes for convective scalar transport equations, Journal of Computational Physics 133 (1997) 235–55. [12] J. W. Kim, D. J. Lee, Optimized compact finite difference schemes with maximum resolution, AIAA Journal 34 (1996) 887–93. [13] R. Hixon, Prefactored small-stencil compact schemes, Journal of Computational Physics 165 (2000) 522–41. [14] A. Sescu, R. Hixon, A. A. Afjeh, Multidimensional optimization of finite difference schemes for computational aeroacoustics, Journal of Computational Physics 227 (2008) 4563–88. [15] F. Q. Hu, M. Y. Hussaini, J. Manthey, Low dissipation and dispersion Runge-Kutta schemes for computational acoustics, Journal of Computational Physics 124 (1996) 177–91.
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Reprint Prediction of NoiseRotor from Wake/Stator Realistic Rotor Towards of: the Towards Predictionthe of Noise from Realistic Interaction Wake/Stator Interaction Using CAA Using CAA ✩
Ray Hixon, Adrian Sescu, and Vasanth Allampalli Mechanical, Industrial, and Manufacturing Engineering (MIME) Department University of Toledo Toledo, Ohio 43606
Abstract In this work, the NASA Glenn Research Center Broadband Aeroacoustic Stator Simulation (BASS) code is extended for use in the prediction of noise produced by realistic three dimensional rotor wakes impinging on a downstream stator row. In order to accurately simulate such a flow using a nonlinear time-accurate solver, the inflow and outflow boundary conditions must simultaneously maintain the desired mean flow, allow outgoing vortical, entropic, and acoustic waves to cleanly exit the domain, and accurately impose the desired incoming flow disturbances. This work validates a new method for the acoustics-free imposition of three-dimensional vortical disturbances using a benchmark test case. c 2010 Published by Elsevier Ltd. Keywords: Computational Aeroacoustics, Boundary Conditions
1. Introduction There are two main types of noise generated in a jet engine fan: broadband and tone noise. [1] Broadband noise is generated by the interaction of turbulence with the fan rotors and stators. The tone noise is generated by the periodic flow distortions associated with the spinning rotors or by the interaction of flow distortions such as wakes from upstream blades with downstream rotors, stators, or struts. For high-bypass-ratio engines with subsonic fan tip speeds, the interaction of fan wakes with stators is one of the main sources of noise. The computational prediction of wake-stator noise is a demanding problem due to the highly nonuniform swirling flow upstream of the stator vanes and the loaded, complex geometry stator vanes. For tone noise prediction, the frequency domain approach is highly efficient and accurate. In a frequency domain approach, the incoming rotor wakes are decomposed into a number of linear single-frequency perturbations about the mean flow, allowing each frequency to be solved separately. [2, 3, 4, 5] The NASA Glenn Research Center LINFLUX code [6, 7, 8] is an example of a three-dimensional frequency-domain linearized Euler solver designed to provide a comprehensive and efficient unsteady aerodynamic analysis for predicting the aeroacoustic and aeroelastic responses of axial flow turbomachinery. Due to the wide range of length and time scales present in turbulent flows, the prediction of broadband noise requires much more computational effort. In order to accurately and efficiently predict such nonlinear flows, Computational Aeroacoustics (CAA) codes are being developed which combine high-resolution spatial differencing schemes (e.g. Lele [9], Tam and Webb [10], Li [11], Kim and Lee [12], Hixon [13], Sescu, et. al. [14]) with optimized time marching schemes (e.g., Hu, et. al. [15], Stanescu and Habashi [16], and Hixon, et. al. [17]), providing high accuracy while using relatively coarse grids compared to existing CFD approaches. ✩ This article is a reprint of a previously published article. For citation purposes, please use the original publication details: Procedia Engineering 6C (2010) 203–213. DOI of original item: 10.1016/j.proeng.2010.09.022
2210-9838 © 2010 Published by Elsevier Ltd. doi:10.1016/j.piutam.2010.10.022
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The NASA Glenn Research Center (GRC) Broadband Aeroacoustic Stator Simulation (BASS) code is a nonlinear time-domain CAA code, developed for use in the prediction of fan tone and broadband noise. [18] In previous work with the BASS code, the interaction of low-amplitude vortical gusts with realistic 2D [19] and 3D [20] stator geometries was predicted and showed good agreement with linear theory. However, in both cases, the incoming rotor wakes were represented by a combination of three simple harmonic functions which were analytically divergence-free. Current work with the BASS code is focused towards the prediction of broadband noise, supporting the noise reduction goal of the NASA Subsonic Fixed Wing Project. To accomplish this goal, it is necessary to develop computational boundary conditions which allow arbitrary unsteady vortical disturbances to be imposed at the inflow boundary, while also maintaining the desired mean flow conditions and allowing outgoing waves to exit the computational domain without reflection. In earlier work, a framework for accomplishing this task was presented and validated using two dimensional [21] and three dimensional [22] benchmark test cases. Results were also shown for a loaded two dimensional cascade with realistic geometry, using a wake taken from experimental data from the NASA Fan Source Diagnostic Test (SDT) [23]. The wake that was imposed for that test was taken directly from experimental data, and produced a strong acoustic signal when imposed at the inflow boundary. These acoustic signals contaminated the predicted noise from the wakestator interaction. In order to address this problem, a new vortical gust boundary condition (VGBC) has been developed which is formulated to impose an analytically divergence-free vortical gust at the inflow boundary, given an input gust which may or may not be divergence-free. [24] The VGBC has been previously tested in two dimensions, using CAA benchmark test cases with analytic gust specification. This boundary condition is one of the necessary steps towards the prediction of broadband noise, with the generation of realistic divergence-free synthetic turbulence being another major focus of current research. In this work, the VGBC has been extended for use in nonuniform three dimensional mean flows. The threedimensional formulation and validation results using the Third CAA Workshop Category 4 problem [25] are presented. 2. Numerical Approach The NASA Glenn Research Center BASS code [18] is used in this work. The BASS code solves the nonlinear Euler or Navier-Stokes equations in chain-rule curvilinear form: ⎛ ∂ ∂ ∂ ⎜⎜⎜ ξ x ∂ξ {E} + ξy ∂ξ {F} + ξz ∂ξ {G} ∂ ⎜⎜⎜⎜ +η ∂ {E} + η ∂ {F} + η ∂ {G} {Q} + ⎜⎜ x ∂η y ∂η z ∂η ⎜⎝ ∂t +ζ x ∂ζ∂ {E} + ζy ∂ζ∂ {F} + ζz ∂ζ∂ {G}
⎞ ⎟⎟⎟ ⎟⎟⎟ ⎟⎟⎟ = 0 ⎠⎟
(1)
where Q is the vector of conserved properties and E, F, and G are the flux vectors in Cartesian coordinates. For this work, the Euler equations are solved. The BASS code employs a finite-difference time-domain approach for solving the governing equations. Structured multiblock grids are used to provide topological flexibility. For portability and maintainability, BASS is written in standard Fortran 2003, and is parallelized using the MPI-1 [26] standard. BASS has been tested using many compilers and MPI implementations, and has been validated on a range of benchmark aeroacoustics problems. The BASS code has a number of spatial differencing schemes, artificial dissipation methods, and explicit time marching schemes implemented. For this work, the seven-point fourth order accurate Dispersion Relation Preserving (DRP) scheme of Tam and Webb [10]) is employed for spatial differencing. The optimized HALE-RK67 time marching technique (Allampalli et al. [17]) is used to advance the solution in time. A blended second and tenth order shock-capturing dissipation scheme [27], based on the Kennedy and Carpenter explicit filters [28] is used for these cases. The BASS code has been Verified for the nonlinear Euler equations using the EVA-III approach, and showed design spatial and temporal accuracy on 3D curvilinear grids for all schemes. [29] 3. Description of Boundary Condition Implementation In this section, the strategy used for the implementation of boundary conditions in the BASS code is briefly described. In the BASS code, the flow at the boundary is directly calculated using the governing equations; thus, no
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extrapolation is used. [30] The boundary condition is imposed as a ’correction’ to the time derivatives calculated using the governing equations. At the inflow boundary, the boundary condition is decomposed into three components: ∂Q ∂t in f lowBC
∂Q ∂Q ∂Q + + ∂t MFBC ∂t nonre f lecting ∂t imposedDisturbance
=
(2)
Each of these components performs a different task. The mean flow boundary condition (MFBC), described in Reference [31], is used to maintain the correct mean flow conditions at the boundary. A three-dimensional extension of the Giles boundary condition [32] is used to allow outgoing waves to pass through the boundaries with minimal reflection. Finally, the three-dimensional VGBC is used for the imposition of the vortical gust. It is inherently assumed that the flow disturbances at the boundary are of a low enough amplitude that a linear flow assumption can be made. 4. Three Dimensional Vortical Gust Boundary Condition (VGBC) In this section, the formulation of a boundary condition which imposes a three-dimensional divergence-free vortical gust at the inflow boundary of a computational grid is described. The full details of the numerical implementation are given in Reference [24]. In the vortical gust boundary condition (VGBC), the incoming gust is assumed to be linearly convecting with a nonuniform mean flow. The equations for the incoming gust are thus: ut + Uux + Vuy + Wuz + u U x + v U y + w U z vt wt
+
+
Uvx
Uwx
+
+
Vvy
Vwy
+
+
Wvz
Wwz
+ u V x + v Vy + w Vz
+ u W x + v Wy + w Wz ux + vy + wz
=
0
=
0
= =
0 0
(3)
where the overbar quantities refer to the mean flow values and the prime quantities refer to the vortical gust disturbance velocities. Consider the case of a Cartesian grid with an inflow boundary in the x-direction. In this case, the outward-pointing face normal is oriented in the negative x-direction. It is assumed that the velocity components associated with the incoming gust are specified across the boundary face, and thus the tangential (y and z) derivatives of all gust quantities can be computed. It is also assumed that no gust data is available inside the computational domain, and thus that the normal (x) derivatives of the gust quantities cannot be directly computed. In the vortical gust boundary condition, the gust velocity components that are normal to the local flow direction are specified directly using the imposed gust data. The imposed gust velocity component that is aligned with the local flow direction is discarded and replaced with the gust velocity component required for a divergence-free incoming gust. Thus, in the case of a uniform axial mean flow, the equations that are used to compute the unknown gust velocity component are: ut + Uux ux + vy + wz
= =
0 0
(4)
which can be combined to obtain: ut,VGBC
= U vy,speci f ied + wz,speci f ied
vt,VGBC
=
wt,VGBC
=
vt,speci f ied
wt,speci f ied
(5)
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Reference [24] presents the generalized VGBC for use with arbitrary boundary and mean flow orientations. 5. Validation Test Case: Third CAA Workshop Category 4 Problem 5.1. Problem Description The Third CAA Workshop Category 4 problem [25] was chosen as the validation test case for the three dimensional VGBC. In this problem, a stator row consisting of 24 infinitely thin flat plates are mounted in a constant-radius annulus. A uniform axial mean flow convects the wakes from an upstream 16-blade rotor through the stator row, and noise is generated when these wakes impinge on the stators.
Figure 1: Grid used for the vortical gust boundary condition tests
The azimuthal velocity associated with these wakes is given as:
U
ωx 0.1U cos 16 + θ − ωt + 2πq (2r − 1) U = 0.5
ω
= 0.783
vθ
=
(6)
The parameter q determines the radial variation of the wakes and ranges from a value of 0 to 3. The problem statement as given is not sufficient to allow the imposition of a purely vortical gust. To accomplish this, the axial and radial gust velocities must also be specified to give a divergence-free velocity field:
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vr
=
0
vx
=
−
(7) U v rω θ
(8)
Note the radial variation in the axial gust velocity that is required for a divergence-free gust. 5.2. Numerical Approach The computational grid was generated using the GridPro/az3000 [33] commercial grid generation tool. A single passage grid was generated, and then ’stacked’ in the azimuthal direction. The grid has a minimum of ten grid points per wavelength in all three coordinate directions. Axially, the grid begins 1.5 chord lengths upstream of the stator leading edge and extends 1.5 chord lengths downstream of the stator trailing edge; at this point, the grid is stretched in the axial direction until reaching the outflow boundary at a distance of 4 chord lengths downstream of the trailing edge. The flat plate stators have no thickness in the grid, which gives rise to grid singularities at the leading and trailing edge points of the stators. The grid is clustered about these points to resolve the sharp flow gradients which are generated at these locations. Due to the periodicity of the test problem, the computational domain included only three of the 24 flat plate stators ( 18 of the full annulus). Periodic boundary conditions are specified in the azimuthal direction. A total of 509,000 grid points were used in the computational domain. Figure 1 shows the grid used for all tests. The simulations were run using the BASS code, solving the nonlinear Euler equations. As decribed previously, several boundary conditions are used in combination at the inflow and outflow boundaries. To maintain the correct mean flow conditions, the mean flow boundary condition (MFBC) described in Reference [31] is used. A threedimensional extension of the Giles boundary condition [32] is used to allow outgoing waves to pass through the boundaries with minimal reflection. Finally, at the inflow boundary, the three-dimensional VGBC was used for the imposition of the vortical gust. 5.3. Performance of VGBC The full series of simulations were run using the VGBC, and the results for a q value of 3.0 will be presented here. This value of q was chosen because this test case had the least amount of acoustic response from the wake-stator interaction, and should show the most sensitivity to spurious acoustic waves from the incoming gust. In these simulations, two tests for each q value were run. First, the calculation was run with the stators removed from the computational domain. This test is designed to verify the accuracy of the gust imposition and the vortical gust boundary condition. It should be noted here that only the azimuthal component of the vortical gust is directly specified in this approach; the axial component is generated numerically by the VGBC. The resulting gusts are compared at two locations in the computational domain. The comparison locations are radial lines aligned with the central stator in the domain, located at the inflow boundary and at the stator leading edge. In Figures 2 and 3, the equivalent instantaneous velocity magnitude and pressure at the start of the cycle are shown for the q = 3.0 vortical gust. In Figure 2, the radial ’lean’ of the vortical gust is seen. In this figure, reflections from the outflow boundary condition are generating the largest pressure disturbances in the calculation, and the solution upstream of the stretched grid is free of large pressure waves. Figure 3 shows the effect of the stators on the predicted velocity and pressure. The acoustic response due to the gust is clearly seen; this result compares well with the benchmark solutions [24], as will be shown in the next section. In Figure 4, the computed solutions for q = 3.0 are compared to the analytical solution for the imposed gust. The comparison is shown on a radial line aligned with the leading edge of the center flat plate stator and located at the inflow boundary. In all graphs, the computed solution is denoted by solid lines and the analytical ’gust-only’ solution by circles. In these graphs, the ability of the vortical gust boundary condition to impose the correct axial (vX) velocity is clearly shown. Also, by comparing the computed density and pressure signals between the two test cases, it can be concluded that the amplitude of any acoustic waves generated by the imposition of the vortical gust are much lower than the amplitude of the acoustic waves generated by the gust/stator interaction.
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Figure 2: Velocity magnitude (left) and pressure response (right) for gust-only validation test (q = 3.0)
Figure 3: Velocity magnitude (left) and pressure response (right) for validation test (q = 3.0)
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0.04 0.02 0 -0.02 -0.04
0.04 0.02 0 -0.02 -0.04
vTheta vX
0.0002
0.0002
vR
0.0001
vR
0.0001
0
0
-0.0001
-0.0001
-0.0002
-0.0002 density pressure
0.001 0.0005 0 -0.0005 -0.001 0.5
vTheta vX
0.6
0.7
0.8
density pressure
0.001 0.0005 0 -0.0005 -0.001
0.9
1
0.5
0.6
0.7
0.8
radius
0.9
radius
Figure 4: Comparison of solutions at inflow plane with no stator (left) and with stator (right) for VGBC validation test (q = 3.0)
vTheta vX
0.04 0 -0.04 vR
0.001 0 -0.001
density pressure
0.03 0 -0.03 0.5
0.6
0.7
0.8
0.9
1
radius
Figure 5: Comparison of gust convection to leading edge location for validation test (no stators)
1
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As a final test, Figure 5 compares the analytical ’gust-only’ solution to the calculated ’gust-only’ solution at the leading edge line of the center stator for both validation cases. The comparison between the numerical and the analytical solutions is very good. This illustrates the ability of the numerical scheme to accurately impose the gust and to convect the gust through a complex multiblock grid to the stator leading edge. From the results of these validation cases, it is clear that the VGBC can be used to impose a complex-geometry three-dimensional vortical gust with extremely low levels of spurious acoustic waves. 5.4. Comparison to Benchmark Results The benchmark results given for this problem were calculated by Namba and Schulten using two frequencydomain linearized Euler equation codes. Results for phase and amplitude of the acoustic duct modes were given for the μ = 0, 1, 2 radial modes of the m = −8 and m = +16 azimuthal modes.
Amplitude
0.02 BASS Namba Schulten
0.01 0.005
0.015
0.005 0
0.01
0.01 Re
0
0
0
-0.01
-0.01
0.01
0.01
0 -0.01
BASS Namba Schulten
0.01
Im
Im
Re
Amplitude
0.02 0.015
0 -0.01
0
0.5
1
1.5 q
2
2.5
3
0
0.5
1
1.5 q
2
2.5
3
Figure 6: Comparison of BASS results with benchmark solutions for m = -8 azimuthal mode, μ = 0 radial mode one chord upstream of the leading edge (left) and one chord downstream of the trailing edge (right)
The nondimensionalization of the benchmark results was not given in the workshop proceedings; thus, the BASS results presented here used a scaling and phasing correction factor calculated using the m = −8, μ = 0 upstream results for the q = 0 case. All BASS results were corrected using this factor. Figures 6- 9 compare the BASS results using the VGBC to the benchmark solutions of Namba and Schulten. Each figure shows the overall amplitude of each acoustic mode as well as the amplitude of the real and imaginary components. The acoustic mode amplitudes are measured one chord length upstream of the stator leading edge and one chord length downstream of the stator trailing edge. In general, the figures illustrate the accuracy of the BASS solution compared to the benchmark data. Overall, the BASS code predicts both the upstream and downstream acoustic mode amplitude and phase very well for most of the test cases. In these figures, it can be seen that the acoustic modes are greatly reduced in amplitude when the q parameter is greater than 1.5. This corresponds to the acoustic modes transitioning from cut-on to cut-off. The BASS code predicts this transition well. There is one case where BASS is not predicting as well as expected. The BASS result underpredicts the amplitude for the m = −8, μ = 1 downstream acoustic mode for q = 1, compared to Namba and Schulten. Numerical experiments have shown that this case is very sensitive to the location of the outflow boundary; work is continuing in order to resolve this issue. Another issue of interest is that the phasing of the m = +16 acoustic modes is reversed compared to the benchmark solutions. However, it must be noted that the amplitude of the m = +16 modes are very small compared to the m = −8 modes, and have very little effect on the solution.
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Amplitude
0.03 0.025 0.02 0.015 0.01 0.005 0 0.02 0.01 0 -0.01 -0.02
Re
BASS Namba Schulten
0.02 0.01 0 -0.01 -0.02
Im
Im
Re
Amplitude
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0
0.5
1
1.5 q
2
2.5
0.03 0.025 0.02 0.015 0.01 0.005 0 0.02 0.01 0 -0.01 -0.02
BASS Namba Schulten
0.02 0.01 0 -0.01 -0.02
3
0
0.5
1
1.5 q
2
2.5
3
Amplitude
0.003 0.0025 0.002 0.0015 0.001 0.0005 0 0.002 0.001 0 -0.001 -0.002
Re
BASS Namba Schulten
0.002 0.001 0 -0.001 -0.002
Im
Im
Re
Amplitude
Figure 7: Comparison of BASS results with benchmark solutions for m = -8 azimuthal mode, μ = 1 radial mode one chord upstream of the leading edge (left) and one chord downstream of the trailing edge (right)
0
0.5
1
1.5 q
2
2.5
3
0.003 0.0025 0.002 0.0015 0.001 0.0005 0 0.002 0.001 0 -0.001 -0.002
BASS Namba Schulten
0.002 0.001 0 -0.001 -0.002 0
0.5
1
1.5 q
2
2.5
3
Figure 8: Comparison of BASS results with benchmark solutions for m = -8 azimuthal mode, μ = 2 radial mode one chord upstream of the leading edge (left) and one chord downstream of the trailing edge (right)
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Amplitude
0.001 BASS Namba Schulten
0.0005
0 0.0005 Re
0
0
0
-0.0005
-0.0005
0.0005
0.0005
0 -0.0005
BASS Namba Schulten
0.0005
0.0005
Im
Im
Re
Amplitude
0.001
0 -0.0005
0
0.5
1
1.5 q
2
2.5
3
0
0.5
1
1.5 q
2
2.5
3
Figure 9: Comparison of BASS results with benchmark solutions for m = +16 azimuthal mode, μ = 0 radial mode one chord upstream of the leading edge (left) and one chord downstream of the trailing edge (right)
6. Conclusions and Future Directions In this work, a Vortical Gust Boundary Condition (VGBC) was presented, which is designed to impose a numerically acoustics-free incoming gust at the inflow boundary of a computational domain. The VGBC was tested using a three-dimensional CAA benchmark problem. The results showed the accuracy of the VGBC and the ability of the NASA BASS code to calculate wake/stator interaction problems. The VGBC is currently being validated for nonuniform swirling flows and is a necessary step towards applying the BASS code to the prediction of broadband noise from turbulent rotor wake/stator interaction. Acknowledgments This work is supported by the Subsonic Fixed Wing Project of the NASA Fundamental Aeronautics Program. The technical monitor for this work is Dr. Edmane Envia of the NASA Glenn Research Center. [1] D. L. Huff, Fan noise prediction: Status and needs, in: AIAA Paper 98-0177, 1998. [2] S. N. Smith, Discrete frequency sound generation in axial flow turbomachines, Tech. Rep. 3709, Aeronautical Research Council (1973). [3] D. S. Whitehead, A finite element solution of unsteady two dimensional flow in cascades, International Journal of Numerical Methods in Fluids 10 (1990) 13–24. [4] K. C. Hall, J. M. Verdon, Gust response analysis for cascades operating in nonuniform mean flows, AIAA Journal 29 (1991) 1463–71. [5] K. C. Hall, W. Clark, Linearized Euler predictions of unsteady aerodynamic loads in cascades, AIAA Journal 31 (1993) 540–50. [6] J. M. Verdon, M. D. Montgomery, A. H. Chuang, Development of a linearized unsteady Euler analysis with application to wake/blade-row interactions, Tech. Rep. NASA CR-1999-208879, National Aeronautics and Space Administration (1999). [7] J. M. Verdon, Linearized unsteady aerodynamic analysis of the acoustic response to wake/blade row interaction, Tech. Rep. NASA CR-2001210713, National Aeronautics and Space Administration (2001). [8] D. Prasad, J. M. Verdon, A three-dimensional linearized Euler analysis of classical wake/stator interactions: Validation and unsteady response predictions, International Journal of Aeroacoustics 1 (2002) 137–163. [9] S. K. Lele, Compact finite difference schemes with spectral-like resolution, Journal of Computational Physics 103 (1992) 16–42. [10] C. K. W. Tam, J. C. Webb, Dispersion-relation-preserving finite difference schemes for computational aeroacoustics, Journal of Computational Physics 107 (1993) 262–281. [11] Y. Li, Wave-number extended high-order upwind-biased finite-difference schemes for convective scalar transport equations, Journal of Computational Physics 133 (1997) 235–55. [12] J. W. Kim, D. J. Lee, Optimized compact finite difference schemes with maximum resolution, AIAA Journal 34 (1996) 887–93. [13] R. Hixon, Prefactored small-stencil compact schemes, Journal of Computational Physics 165 (2000) 522–41. [14] A. Sescu, R. Hixon, A. A. Afjeh, Multidimensional optimization of finite difference schemes for computational aeroacoustics, Journal of Computational Physics 227 (2008) 4563–88. [15] F. Q. Hu, M. Y. Hussaini, J. Manthey, Low dissipation and dispersion Runge-Kutta schemes for computational acoustics, Journal of Computational Physics 124 (1996) 177–91.
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