Adjoint Linearised Euler solver and Source

Adjoint Linearised Euler solver and Source Modelling for Goldstein acoustic analogy equations for 3D jet flow problems: verification and capability study Vasily A. Semiletov 1 and Sergey A. Karabasov2 Queen Mary University of London, Mile End Rd, London, E1 4NS, UK

Abstract The Adjoint Linearised Euler Solver of Karabasov and Hynes (2006), which includes an efficient iterative solution of linearised Euler equations in the frequency domain, has been extended to a 3D framework. Numerical tests performed for the problem of sound scattering from an axi-symmetric Gaussian jet flow show a very good performance of the method for a range of frequencies and observer angles as compared with the reference semianalytical solution. A new approach has been introduced for evaluating the source terms of the Goldstein acoustic analogy equations based on the second-order statistics provided directly from Large Eddy Simulation (LES). In comparison with the previous works, the new procedure is completely free from modelling assumptions, such as the functional behavior of the auto-covariance stress tensor or the dependence of the correlation scales of frequency. This makes the Goldstein acoustic analogy modelling based on LES practically applicable to general 3D flows and allows not only computing the far-field pressure spectra but also the reconstruction of effective volume noise sources. In comparison with the standard integral surface methods such as the Ffowcs Williams – Hawkings (FW-H) penetrable surface formulation, the new approach is potentially more robust since it has no sensitivity to the acoustic surface location. For the proof-of-concept study, acoustic predictions are made for two static single-stream co-axial jets at acoustic Mach number V /c = 0.875 that correspond to the temperature ratios T /T = 1 and T /T = 2.5. Comparisons with the reference penetrable FW-H solutions and the scaled QinetiQ experiment data are performed. The budgets of acoustic power spectral density for the individual source terms of the acoustic analogy model are also provided.

I. Introduction ONE of the modern acoustic analogies is the one developed by Goldstein and Leib [2,3,4,5] which is based on exact rearranging of the governing Navier-Stokes equations to a nominally linear formulation that includes a linear propagator of the linearised Euler equations-type and nonlinear source terms grouped on the right hand side. By using the reciprocal theorem between the solutions of the direct and adjoint problem, the acoustic solution of these equations is reduced to adjoint formulation and the far-field sound is computed as a convolution integral of the adjoint vector Green’s function with the source. The use of adjoint technique makes the computation of far-field sound very efficient in the situation when the number of far-field microphone locations is much less than the number of points which represent the distributed noise sources in the jet. One particular difference of the acoustic modelling approach of Goldstein from other acoustic models in the literature is that it generically includes full propagation effects in the acoustic analogy formulation thus preserving the same dispersion properties in the propagator as of the original Navier-Stokes equations. As discussed in [6,7,8,9], accounting for realistic mean flow propagation effects is very important for correct capture of distributed sound source / mean flow interference effects.

1

Research Associate, School of Engineering and Materials Science, Mile End Road, AIAA Member. Senior Lecturer in Modelling and Simulation, School of Engineering and Materials Science, AIAA Senior Member. 1 American Institute of Aeronautics and Astronautics 2

On the propagation modelling side, in comparison with the classical acoustic analogy models of Lighthill [10] and Lilley [11], which are based on simplified mean flow representations amenable to analytical Green’s functions, the Goldstein generalised acoustic analogy approach requires a linearised Euler calculation. On the source modelling side, the Goldstein acoustic analogy is considerably simpler in comparison with the classical acoustic analogy approaches. The former requires the evaluation of statistics of fluctuating stresses as well as enthalpy fluctuations in the energy source of the hot jets while the latter include most of the propagation effects in the ‘effective acoustic source’ together with generic turbulent source effects. Both, the propagation and the source evaluation part of the Goldstein analogy acoustic modelling can be quite demanding from the computational viewpoint if a tunableparameter-free “single-button” solution technology is required. On the other hand, the flexibility and generality mentioned would be highly desired in case the acoustic analogy method needs to be competitive with the most advanced integral formulations such as Ffowcs Williams – Hawkings [12]. All of these motivate the present study which objective is to evaluate the feasibility of using linearised Euler solver for 3D sound/non-uniform mean flow propagation problems as well as calculating the effective noise sources from an eddy-resolving simulation in accordance with the Goldstein acoustic analogy framework. In this sense, the current work is a continuation of the studies of [6,9,13] where the Goldstein generalised acoustic analogy was combined with the full solution of adjoint linearised Euler equations and eddy-resolving calculations of jet turbulence.

II. Mathematical model: the Goldstein acoustic analogy equations with complete mean flow/sound propagation and temperature effects The Goldstein and Leib acoustic analogy equations [4] which will be used as the basis of our work are summarized below. By introducing the fluctuation variables: ′ ≡ − ̅ , ′ ≡ − ̅ , ℎ′ ≡ ℎ − ℎ, ′ ≡ we consider linearised Navier-Stokes equations for modified variables: − 1 ′ ≡ ′+ ′ − ̅ ′ , 2 ≡ ′, which can be written in the following form: ′

+ ′

+ ∇

+

− 1

∇

+∇ ′

+∇ ′

∇

−

− ̅,

= 0, ′

−

∇

∇

= ∇

′′

= ∇ ′′

,

(1)

+ ( − 1) ′′ ∇

where ≡

′

′′

≡−

≡

− 1 ′

′

+

≡ 1 ℎ′ + 2 − 1 2 ′′ ≡

∙+∇

∙ ,

̅ − ′

,

′

′

=

′

+

′

−

+ ′

+

− 1 2

′

,

− 1

′

,

.

From the above equations, the generalised acoustic analogy model consists of two parts: the left-hand-side propagator operator that is of linearised Euler-type with respect to the acoustic variables and the nonlinear source 2 American Institute of Aeronautics and Astronautics

terms on the right-hand-side. The source terms include the fluctuating Reynolds stresses in the momentum equation and also the fluctuating Reynolds stresses source and the fluctuating enthalpy source in the energy equation. Under the commonly made assumption in acoustic modelling, the effect of acoustic feedback on the turbulent sources is negligible in comparison with the effect of turbulence on acoustic waves. Then, the propagation modelling and the source modelling are decoupled and treated separately. The latter one-way coupling scheme allows using most efficient computational methods at each scale: for large-scale acoustic wave propagation and for small-scale turbulence modelling.

III. 3D solver for sound non-uniform mean flow propagation A well-known feature of linearised Euler equations is their shear-layer-type instability. Frequency domain formulation, in principle, allows separating the convective instability from acoustic solution wanted by prescribing the physical radiation boundary condition at the far-field. Direct matrix solution methods are one possibility for this, however, for 3D geometries this is an expensive option. In [1], an efficient iterative solver for Adjoint linearised Euler equations (ALEE) in the frequency domain is suggested, where axi-symmetrical flow equations are solved with a modified Adams-type pseudo-time-stepping scheme with special dual-time-stepping to suppress the numerical shear-layer-type instability. Semiletov and Karabasov [13] extended this framework to solving 3D ALEE with the aid of parallel computing and showed the feasibility of this approach for a test problem of acoustic wave scattering from a parallel Gaussian jet at a single frequency/observer angle. In the current paper, the previous work has been extended to accommodate the full range of frequencies and observer angles by using acoustic-wavelengthadjustable buffer zones close to the boundaries, which is a 3D equivalent of the 2D technique used by Karabasov and Hynes in [1]. Fig.1 shows the computational problem used for verification studies which comprises a parallel axi-symmetric Gaussian jet flow embedded in a 3D Cartesian grid. The incident acoustic wave boundary conditions are specified through the corresponding sources in the equations, similar to the technique used by Tam and Auriault in [14], Karabasov and Hynes in [1], and Karabasov et al in [9]. For best non-reflecting properties a sponge layer zone is used at each open boundary to damp acoustic reflections. The width of the sponge zone is made automatically adjustable to span over at least a few acoustic wavelengths for each frequency (fig.2). For the parallel Gaussian jet flow case, the solution obtained with the new 3D ALEE solver has been verified in comparison with the semianalytical solution obtained from the Lilley-type ordinary-differential equation. Figs. 3,4 and Figs. 5,6 show the results for imaginary and real part of G2 (adjoint variable corresponding to the y-velocity component) at Strouhal number based on the jet diameter St=8 and St=0.2, respectively. Figs. 7 and 8 show detailed comparison of the numerical ALEE solution with the semi-analytical one. The numerical results obtained practically coincide with the semi-analytical solution for 450 and 900 observer angles to the jet flow. Notably, frequency St=0.2 (Fig.8) corresponds to one of the most unstable eigenvalues of the spatial part of the propagator operator. For time-domain linearised Euler equations, the solution at this frequency for a small angle to the jet would have a strong instability wave component. In the frequency domain, it is only the acoustic solution which is retained by applying the corresponding sponge zone boundary conditions and pseudo-time-stepping to control numerical instabilities.

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Figure 1. Computational setup for the problem of 3D acoustic wave scattering by a parallel axi-symmetric Gaussian jet flow.

(a) St=8 (b) St=0.2 Figure 2. Adjusting the computational grid for solving the ALEE sound scattering problem depending on the frequency in z=0 symmetry plane of the jet: (a) St=8, (b) St=0.2.

Figure 3. Imaginary part of G2 - Green’s function component for 45 0 observer angle to the jet and frequency St=8 in z=0 symmetry plane of the jet: (a) the numerical ALEE solution, (b) the reference semi-analytical solution.

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(a) (b) Figure 4. Real part of G2 - Green’s function component for 450 observer angle to the jet and frequency St=8 in z=0 symmetry plane of the jet: (a) the numerical ALEE solution, (b) the reference semi-analytical solution.

(a) (b) Figure 5. Imaginary part of G2 - Green’s function component for 45 0 observer angle to the jet and frequency St=0.2 in z=0 symmetry plane of the jet: (a) the numerical ALEE solution, (b) the reference semi-analytical solution.

(a) (b) Figure 6. Real part of G2 - Green’s function component for 450 observer angle to the jet and frequency St=0.2 in z=0 symmetry plane of the jet: (a) the numerical ALEE solution, (b) the reference semi-analytical solution.

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0.04

V, imag

0.02

0

comp semi-analyt

-0.02

-0.04 0.0098

0.0148

x

0.0198

0.0248

(a)

0.04

V, real

0.02

0

comp semi-analyt

-0.02

-0.04 0.0098

0.0148

x

0.0198

0.0248

(b) Figure 7. Comparison of the numerical ALEE solution with the semi-analytical one along line y,z=0 in the jet for G2 -Green’s function component and 450 observer angle to the jet at frequency St=8: (a) imaginary part, (b) real part. 6 American Institute of Aeronautics and Astronautics

1.7

comp

1.65

semi-analyt

V, real

1.6 1.55 1.5 1.45 1.4 1.35 1.3 -0.004

0.001 y, m

Figure 8. Comparison of the numerical ALEE solution with the semi-analytical one along line y,z=0 in the jet for G2 -Green’s function component and 450 observer angle to the jet at frequency St=0.2: real part.

IV. Reconstruction of the non-linear acoustic sources and acoustic modelling results In addition to solving the propagation problem, the other part of the acoustic model requires nonlinear source terms to be specified in accordance with acoustic analogy equations (1). In the previous works, these were evaluated by applying statistical modelling at the source level. For instance, in [4] and [5] there were quasi-normality arguments as well as some calibrations based on the experiment used to define analytical models of the Reynolds stress autocovariance tensor within a RANS approach. On the other hand, [6,9,15,16] used a simplified analytical Gaussian model for the same, where the necessary calibration parameters were obtained from eddy-resolving calculation data (both Large Eddy Simulation and Direct Numerical Simulation). In the latter work, it was realized that the LES/DNS data collection and acquisition for a sufficiently long time in the entire volume of the jet is quite challenging. For instance, the points outside of the jet shear layers which correspond to low correlation amplitudes but large volumes are especially slow to converge in terms of the auto-covariance tensor components (fourth-order correlation velocity functions). In the current work we suggest an alternative approach for computing the nonlinear noise sources required from LES solution (or DNS, if available) without introducing any modelling assumption along the way. In the new approach, the statistical modelling is applied at the observer level, which allows one to reduce source modelling to the calculation of second-order velocity fluctuation statistics which can be done much more computationally efficient and accurate in comparison with computing the fourth-order quantities. We start by recalling that the set of nonlinear source terms of the generalized Goldstein acoustic analogy equations (1) needed for sound predictions are: ∇ ′′ , which is the divergence of turbulent stresses represented in the vector momentum equation and ∇ ′′ and ( − 1) ′′ ∇ , 7 American Institute of Aeronautics and Astronautics

which are the enthalpy fluctuations and turbulent stresses multiplied by tensor velocity gradient in the energy equation. For the current proof-of-concept study, the analysis will be limited to sound predictions for 90 0 observer angle to the jet flow. In this case, the frequency-domain form of the governing equations (1) applies to: w + ∇ ̂ = ∇ ̂ ′′ , (2) w ̂ + c ∇ = ∇ ̂ ′′ + ( − 1) ̂ ′′ ∇ , where hat denotes Fourier transformation, w is sound angular frequency and spectral counterparts of the momentum and pressure solution components.

and ̂ are the corresponding

As discussed previously, the right-hand-side of (2) does not contain the acoustic variable and the far-field pressure signal at the specified observer location, can be computed with a Green’s function method, which (after a few integrations by parts) gives the following equation for the complex far-field pressure: ̂ w, where

=

̂ ′′ ∇ ∇

−

w

̂ ′′ ∇

−

− 1 ̂ ′′

∇

,

(3)

is the corresponding pressure free-stream Green’s function: w

w, | − | =

|

|

4 | − |

.

Once the complex far-field pressure signal is computed the corresponding acoustic power can be obtained from it by multiplying the pressure signal by its complex conjugate: ̂ ∗ w, w, = ̂ w, (4) In accordance with equation (3), the source data required is the spectrum of the fluctuating Reynolds stress tensor and the enthalpy fluctuation terms. In the present work, these data are provided from the LES solutions, which we conducted for two static singlestream co-axial jets at acoustics Mach number / = 0.875 that correspond to the temperature ratios / = 1 (cold jet) and / = 2.5 (hot jet). The LES calculations are based on the Monotonically Integrated LES approach. The calculations included an axi-symmetric nozzle geometry as well as the jet flow. The grid was circa 20 million cells and was of implicit-multi-block hexagonal type. For the calculations, the high-resolution CABARET method was used which details can be found in [18,19,20]. The cases computed correspond to a recent QinetiQ experiment which data are not yet publically available. Hence, the present acoustic modelling results will be validated with respect to the (scaled) data of the previous QinetiQ experiment which already is in the public domain [21]. In addition to the acoustic analogy solution, a reference permeable Ffowcs Williams – Hawkings (FW-H) calculation has been performed based on the same LES data. The FW-H solution corresponded to a large closed permeable control surface with multiple closing disks at the outlet side in accordance with the best practice [22]. Detailed description of the present LES calculations and the FW-H modelling will be the subject of a separate publication. For the sake of the present study, the results for 900 observer angle will be considered. The LES data used in both acoustic modelling methods correspond to about 260 convective time units. In comparison with the multiple-disk FW-H method, the current Goldstein acoustic analogy implementation computes the volume sources rather than surface ones. It is at least as fast in terms of the CPU but requires about 20 times more disk space since all the post-processing is done offline at the moment. In future, this should be reduced my making most of the operations online. It is also worth emphasizing that the current acoustic analogy modelling is completely free from the modelling assumptions such as the functional behavior of the auto-covariance tensor or the correlation scale frequency behavior. In addition, our investigations have shown that integral (3) quickly converges with the jet volume and is insensitive to the areas outside the jet (farther than approximately 1.5D from the jet axis in the radial direction and 20D from the nozzle exit in the axial direction). Figs.9 and 10 show real and imaginary parts and magnitude of the spectra of the Reynolds stresses source components at St=0.2 and St=2 for the cold jet. Figs. 11 and 12 show real and imaginary parts and magnitude of the spectra of the fluctuating enthalpy source components at St=0.2 and St=2 for the hot jet. 8 American Institute of Aeronautics and Astronautics

There is one common feature of all sources: the area of significant effective noise sources at St=0.2 appears to extends up to 20D while the same at St=2 extends only about upto 10D. Further work will be directed to including the Green’s function effect as well as the source volume to provide a quantitative answer to the question which part of the jet provides which part of the acoustic energy budget at which frequency.

a)

b)

c)

d) Figure 9. Magnitude, real and imaginary parts of Reynolds stresses components at St=0.2 for the cold jet: a) ̂ ′′ ; b) ̂ ′′ ; c) ̂ ′′ ; d) ̂ ′′ .

a)

b)

c)

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d) Figure 10. Magnitude, real and imaginary parts of Reynolds stresses components at St=2 for the cold jet (ycoordinate has been scaled by factor 4 for better visibility): a) ̂ ′′ ; b) ̂ ′′ ; c) ̂ ′′ ; d) ̂ ′′ .

a)

b) Figure 11. Magnitude, real and imaginary parts of enthalpy fluctuation components at St=0.2 for the hot jet: a) ̂ ′′ ; b) ̂ ′′ .

a)

b) Figure 12. Magnitude, real and imaginary parts of enthalpy fluctuation components at St=2 for the hot jet (ycoordinate has been scaled by factor 4 for better visibility): a) ̂ ′′ ; b) ̂ ′′ .

Figs 13-15 show the comparison between the current implementation of the generalised acoustic analogy with the scaled QinetiQ experiment data and the reference FW-H calculations for the cold jet case. Figs 16-18 show the same for the hot jet. The frequency is normalized by the jet exit velocity and jet diameter as usual, and the sound spectral density is computed with the standard reference pressure 20mPa and reference frequency of 1 Hz. To investigate jet noise directivity, the same predictions are obtained for two observer locations corresponding to R=120 nozzle diameters, D, from the nozzle exit: x=(0,-R,0) and x=(0,0, -R), where the first coordinate component corresponds to the jet stream direction and the other two are in the normal plane of the Cartesian coordinate system. In addition to the total spectra, various contributions corresponding to the source terms in the Goldstein acoustic analogy model has been evaluated for the observer locations x=(0,-R,0) and x=(0,0, -R) which correspond to the dame 900 angle to the jet flow. The analysis includes (i) the divergence of turbulent stresses term, ̂ ′′ (ii) turbulent stresses multiplied by tensor velocity gradient term, ′′ ∇ and (iii) the enthalpy fluctuations term ̂ ′′ . The following can be observed: 10 American Institute of Aeronautics and Astronautics

· ·

· ·

For the cold jet, the predictions of the Goldstein acoustic analogy model are within 2-3dB from the scaled experiment data and the reference FW-H solution (which is within 1-2dB from the scaled experiment); the acoustic analogy predictions for the two observer locations are in an excellent agreement with each other; For the hot jet, the predictions of the Goldstein acoustic analogy model are within 3-5dB from the scaled experiment data and the reference FW-H solution (the latter captures the scaled experiment within 2-3 dB); the acoustic analogy predictions for the two observer locations are also in an excellent agreement with each other; For the cold jet, it is the divergence of turbulent stresses term, ′′ which is dominant, but the enthalpy fluctuations term ̂ ′′ is dominant for the hot jet; For both cold and hot jet, it is the radial-in-plane-observer component of the divergence of turbulent stresses which are most significant for this source term: ′′ and ′′ for the observer locations at x=(0,R,0) and x=(0,0, -R), respectively; the contributions of stress sources corresponding to out-of-plane observer directions are least significant. All this is in accordance with the statistically axi-symmetric nature of the source expected.

90 degrees

60

60

55

55

50

50

SPL, dB

SPL, dB

90 degrees

45 40

exp., QinetiQ, 1983

35

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30 0.01

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exp., QinetiQ, 1983

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30

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0.01

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(a)

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SPL, dB

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60

40

St (b)

10

65 60 55 50 45 40 35 30

SPL,y=-12 SPL,z=-12

0.01

St

0.1

1 St (d)

(c)

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10

Figure 13. Comparison of far-field sound spectra results for the cold jet: a) the FW-H solution vs experiment; b) the Goldstein acoustic analogy vs experiment; c) the Goldstein acoustic analogy results for two observer locations.

90 degrees

80

50 SPL, dB

60 SPL, dB

90 degrees

70

40 SPL,y=-12 S_Tij S_Tij_grad_j_v_i S_ent

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10

(b)

90 degrees

SPL, dB

90

40

-10 SPL,y=-12 S_T42

S_T41 S_T43

-60 0.01

0.1

St

1

10

(c) Figure 14. Comparison of far-field sound spectra results for the cold jet with the observer location placed in ydirection: a) turbulent stresses term, enthalpy fluctuations term and turbulent stresses multiplied by tensor of gradient velocity term; b) various components of the turbulent stress term; c) enthalpy fluctuations term

90 degrees

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70

70

60

50 SPL, dB

SPL, dB

50 40 30

SPL,z=-12 S_Tij S_ent S_Tij_grad_j_v_i

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-10 -30

0 0.01

0.1

St

1

10

0.01

(a)

0.1

S_T11 S_T33 S_T13

St (b)

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1

10

90 degrees 80 60 SPL, dB

40 20 0

-20

SPL,y=-12 S_T41 S_T42 S_T43

-40 -60 0.01

0.1

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1

10

(c) Figure 15. Comparison of far-field sound spectra results for the cold jet with the observer location placed in zdirection: a) turbulent stresses term, enthalpy fluctuations term and turbulent stresses multiplied by tensor of gradient velocity term; b) various components of the turbulent stress term; c) enthalpy fluctuations term

90 degrees

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55

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exp., QinetiQ, 1983

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exp., QinetiQ, 1983

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0.01

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St

St (d)

(c)

Figure 16. Comparison of far-field sound spectra results for the hot jet: a) the FW-H solution vs experiment; b) the Goldstein acoustic analogy vs experiment; c) the Goldstein acoustic analogy results for two observer locations.

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90 degrees

70

90 degree

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50 SPL, dB

SPL, dB

50 40 30

SPL,y=-12 S_Tij S_Tij_grad_j_v_i S_ent

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(c) Figure 17. Comparison of far-field sound spectra results for the hot jet with the observer location placed in ydirection: a) turbulent stresses term, enthalpy fluctuations term and turbulent stresses multiplied by tensor of gradient velocity term; b) various components of the turbulent stress term; c) enthalpy fluctuations term

90 degree

70 60

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(a)

S_T11 S_T33 S_T13

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1 (b)

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10

90 degree

80 60 SPL, dB

40 20 0 SPL,y=-12 S_T41 S_T42 S_T43

-20 -40 -60 0.01

0.1

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(c) Figure 18. Comparison of far-field sound spectra results for the hot jet with the observer location placed in zdirection: a) turbulent stresses term, enthalpy fluctuations term and turbulent stresses multiplied by tensor of gradient velocity term; b) various components of the turbulent stress term; c) enthalpy fluctuations term

V. Conclusion The Adjoint Linearised Euler Solver of Karabasov and Hynes [1], which includes an efficient iterative solution of linearised Euler equations in the frequency domain, has been extended to a 3D framework. Numerical tests performed for the problem of sound scattering from an axi-symmetric Gaussian jet flow show a very good performance of the method for a range of frequencies and observer angles as compared with the reference semianalytical solution. A new approach has been introduced for evaluating the source terms of the Goldstein acoustic analogy equations based on computing the second-order statistics from an eddy-resolving solution (LES). In comparison with the previous works, this procedure is completely free from modelling assumptions such as the functional behavior of the auto-covariance stress tensor or the dependence of the correlation scales of frequency. For the proof of concept study, acoustic predictions are made for two static single-stream co-axial jets at acoustic Mach number V /c = 0.875 that correspond to the temperature ratios T /T = 1 and T /T = 2.5. Comparisons with the reference penetrable Ffowcs Williams – Hawkings solutions and the scaled QinetiQ experiment data are performed. The results can be summarized as the following: · For the cold jet, the predictions of the Goldstein acoustic analogy model are within 2-3dB from the scaled experiment data and the reference FW-H solution (which is within 1-2dB from the scaled experiment); the acoustic analogy predictions for the two observer locations are in an excellent agreement with each other; · For the hot jet, the predictions of the Goldstein acoustic analogy model are within 3-5dB from the scaled experiment data and the reference FW-H solution (the latter captures the scaled experiment within 2-3 dB); the acoustic analogy predictions for the two observer locations are also in an excellent agreement with each other; · For the cold jet, it is the divergence of turbulent stresses term, ̂ ′′ which is dominant, but the enthalpy fluctuations term ̂ ′′ that is dominant for the hot jet; · For both cold and hot jet, it is the radial-in-plane-observer component of the divergence of turbulent stresses which are most significant for this source term: ̂ ′′ and ̂ ′′ for the observer locations at x=(0,R,0) and x=(0,0, -R), respectively; the contributions of stress sources corresponding to out-of-plane observer directions are least significant. All this is in accordance with the statistically axi-symmetric nature of the source expected. Further work will be devoted to comparing the current predictions with the actual (full-scale) QinetiQ data as soon as the latter are released as well as making predictions for other observer angles with the use of the 3D ALEE solver we have developed.

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Acknowledgment The work of VS and SK has been supported by the Engineering and Physical Sciences Research Council (EP/I017747/1). Partly, this work has also been supported in the framework of “Evaluation of an alternative CFD/CAA method for jet noise” project with Aerospace Technology Institute (ATI) and “Understanding and Predicting Jet Noise” project with Ohio Airspace Institute. The use of computing time on HECToR and Lindgren Supercomputing Facilities is gratefully acknowledged. The second author acknowledges the continuing support of the Royal Society of London.

References 1

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16 American Institute of Aeronautics and Astronautics