Computation of Jet Noise using a Hybrid Approach J. Yan∗, K. Tawackolian†, U. Michel‡, and F. Thiele§ Institute of Fluid Mechanics and Technical Acoustics (ISTA), Technical University of Berlin, Straße des 17.Juni 135, D-10623 Berlin, Germany In this work, the subsonic jet flow through a short-cowl coaxial nozzle with both smooth and serrated lips has been investigated. The Mach numbers for the heated primary and cold secondary streams are 0.861 and 0.902, respectively. The Reynolds-averaged NavierStokes (RANS) equations are solved using a finite volume solver with a hybrid scheme employing a blending between upwind-biased high-order and full centred approximations. Detached-Eddy Simulation (DES) is used to treat the turbulence. The three-dimensional form of radiation non-reflective boundary conditions is applied in a modified formulation, which simplifies implementation and was found to suppress reflections successfully from the boundaries of the computational domain. The far-field noise is calculated using an aeroacoustic analogy of Ffowcs-Williams and Hawkings (FWH). The predicted Overall Sound Pressure Levels (OASPL) are generally in good agreement with the experimental data, within a maximum deviation of 3.0 dB for all observer positions. The dominant low frequency component is captured with the relatively coarse grid used. The numerical simulation has indicated the same trend of decreased sound intensity when using serrations as in the experiment, with an observed noise reduction of about 2 dB.
I.
Introduction
Jet noise remains an important source of aircraft noise, especially at take-off. The study of practically relevant coaxial jets requires the consideration of complex nozzle geometries. As noted by Viswanathan & Clark,1 the internal nozzle geometry can influence the radiated sound through modification of the nozzle boundary layer at the exit plane. In the Large Eddy Simulation (LES) of a free jet, the inclusion of nozzle geometry improves the OASPL prediction at all angles and in particular at 90◦ .2 It has further been argued that when the nozzle geometry is included, any specification of forcing parameters is not necessary and the implicit dependence of the jet turbulence and sound field on the forcing is absent.3 Particularly in an engineering context, modification of the nozzle geometry of high bypass ratio turbofan engines is an important design parameter for the control of jet noise, for which the use of serrations is a promising technique. Numerical simulation allows an in-depth analysis of the flow phenomena involved and the noise generation mechanisms. As such, time resolving simulations should ultimately provide an insight into possible methods of reducing jet noise for industrial applications. For RANS the principal difficulty is to find a universal model for the entire unsteady flow field; a static acoustical model applied to a RANS solution has great difficulty in accounting for the large scale motion that is responsible for most low frequencies and has a strong influence on the propagation of sound. In contrast, time resolving methods such as LES can produce reasonable results for low frequencies and large scale motion.4 Although LES allows the direct capturing of noise sources, its resolution requirements in the near-wall area quickly lead to large grids, so that the computer resources needed are enormous, often well beyond those available on present-day machines. It is well known that RANS models are able to give reasonable predictions for attached boundary layers with few grid cells. As a consistently-formulated turbulence approach combining both the RANS and LES, the hybrid DES approach is applied to simulate an industrially relevant short-cowl jet with relatively coarse grid including the upstream nozzle geometry. The far field sound pressure levels are calculated using an ∗ Dr.,
Research Associate. Email:
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aeroacoustic analogy of FWH.5 The feasibility of DES implementation in combination with the numerical methods used in the simulation of both the flow field and sound radiation is the focus of this study.
II.
Flow Configurations
The subsonic jet flow through a short-cowl coaxial nozzle with both smooth and serrated lips has been investigated. Figure 1 depicts the serrated geometry, where 20 equally-spaced serrations are present on the bypass nozzle. The diameter of the primary and secondary streams is Dp =0.1359 m and Ds =0.2734 m, respectively. The static temperature of the heated primary (core) stream is roughly three times that of the secondary (bypass) stream, leading to large temperature gradients in the shear layer between these. The Mach numbers for the primary and secondary streams are 0.861 and 0.902, respectively. The jet is expanded into a stagnant environment and the temperature of the secondary stream is equal to the ambient one. Table 1 gives more details on the flow properties. The investigated Figure 1. Geometry of a shortReynolds number based on the diameter and fully-expanded jet velocity cowl nozzle with serrated lip of the secondary stream is 106 . Table 1. Flow properties
K K
480.7 306.8 0.861 0.902 890.6 335.0
primary (p) and secondary flow (s) fully-expanded jet velocity in the primary stream fully-expanded jet velocity in the secondary stream Mach number in the primary stream Mach number in the secondary stream total temperature in primary stream total temperature in secondary stream
Pa K
101325 288.14
ambient conditions pressure temperature
Up Us Mp Ms Ttp Tts
m/s m/s
P∞ T∞
III.
Governing Equations
The equations solved are the Reynolds-averaged continuity, momentum, energy and state equations: ∂ρ ∂(ρui ) + =0, ∂t ∂xi ∂(ρui ) ∂(ρui uj ) ∂p ∂σij ∂τij + =− + + , ∂t ∂xj ∂xi ∂xj ∂xj ∂(ρT ) ∂(ρui T ) ∂p ∂p ∂ q˙i ∂uj + + ui Cp = − + τij , ∂t ∂xi ∂t ∂xi ∂xi ∂ui p = ρRT ,
(1) (2) (3) (4)
where σij and τij are the viscous stress tensor and the modeled Reynolds stress tensor, respectively. These are defined as 1 σij = 2µ(Sij − Skk δij ) , 3 1 2 τij = 2µt (Sij − Skk δij ) − ρk δij , 3 3 2 of 13 American Institute of Aeronautics and Astronautics
(5) (6)
whereby Sij is the strain rate tensor Sij =
1 2
∂ui ∂uj + ∂xj ∂xi
,
(7)
and µt is the turbulent dynamic viscosity. The heat flux appearing in the energy equation is modeled using a temperature gradient approach: qj = Cp
IV.
µt ∂T P rt ∂xj
,
Prt = 0.9 .
(8)
Turbulence Modeling
The turbulent dynamic viscosity µt is modeled by the Wilcox k − ω model, which is formulated as ∂ µt ∂k ∂ρk ∂(ρuj k) (9) + = µ+ + ρPk − Cµ ρkω , ∂t ∂xj ∂xj σk ∂xj ∂ρω ∂(ρuj ω) ∂ µt ∂ω ω + = µ+ + Cω1 ρPk − Cω2 ρω 2 , (10) ∂t ∂xj ∂xj σω ∂xj k k µt = ρ , (11) ω p with Pk = νt S ∗ 2 , S ∗ = 2Sij Sij and the model constants are determined as Cµ = 0.09,
cω1 = 5/9,
cω2 = 3/40,
σk = σω = 2.0 .
In order to simplify the expression, Eqs. (9) and (11) are rewritten as follows ∂ ρk 3/2 ∂ρk ∂(ρuj k) µt ∂k + = , µ+ + ρPk − ∂t ∂xj ∂xj σk ∂xj L µt
= ρCµ Lk 1/2 ,
(12) (13)
where L denotes the modeled length scale. For the RANS region, it is defined as L = LRAN S =
k 1/2 , Cµ ω
(14)
while for the LES-mode region it is estimated from the local grid space as will be described. Within the present DES framework, the so-called DES length scale LDES is introduced, viz: LDES = min(LRANS , CDES ∆)
with
∆ = (∆1 ∆2 ∆3 )1/3 .
(15)
Hereby ∆ represents the local spatial dimension, the subscripts 1, 2 and 3 denote the three coordinate directions defined by a curvilinear grid. CDES is a model constant. The definition of LDES is a continuous function given by the minimum of the two length scales. Hence the DES behaves in one of two modes – RANS or LES. In addition to the standard substitution of the DES length scale only in the destruction term of Eq. (12), the present DES formulation includes substitution in the turbulence eddy viscosity expression as µt = ρcµ LDES k 1/2 .
(16)
The aim of this work is to study the performance of DES for the prediction of jet noise. A commonly acknowledged problem in DES is the so-called “grey area” problem, which refers to the region between the attached (URANS-mode) boundary layer and the fully developed LES region. The term grey area is associated with the undefined modeling which occurs when the DES length scale switch becomes activated, thereby switching the background RANS model to LES mode, however the flow field in this region does not contain resolved turbulence (“LES content”). Furthermore, fairly high levels of eddy viscosity are convected from the RANS region into the grey area, which acts to further damp the development of LES content. 3 of 13 American Institute of Aeronautics and Astronautics
The grey area phenomenon is the reason why DES is ideally suited to massively separated flows behind bluff bodies, where the instability in the separated shear layer is so strong that the extent of the grey area is minimized, and its effect becomes negligible. This is further enhanced by the effect of wake-flow recirculation, whereby high levels of resolved turbulence from the wake are directed back towards the shear layer. For jet flows, the situation is rather different: the strength of the instability in the free shear layer is much weaker, and no recirculation of the turbulent jet occurs. Therefore, the grey area phenomenon would be expected to cause a strongly-delayed development of resolved turbulence in the shear layer compared to LES, compounded by the high levels of eddy viscosity convected from the RANS boundary layers. In order to alleviate the grey area problem, a variation of DES studied by Yan, Mockett and Thiele6 has been used. The standard DES substitution, with the DES length scale only in the destruction term of the ktransport equation acts to reduce the level of eddy viscosity through an increased destruction term when the LES-mode region is entered. The present DES variant with the DES substitution also in the viscosity term additionally causes a reduction in the production and diffusion terms of the k equation, thereby damping the levels of eddy viscosity more strongly, i.e. not just through increased destruction, but reduced production + diffusion. This is intended to reduce one of the mechanisms by which the grey area problem is enhanced. It has been found that ∆ = max(∆1 , ∆2 , ∆3 ) as the grid filter width in the DES length scale delays the development of resolved turbulence noticeably, as the grid cells in the shear layer are strongly stretched. ∆ = (∆1 ∆2 ∆3 )1/3 has been found to reduce the modeled turbulent dissipation and hence allow improved resolution of turbulence. However, this filter definition worsens the well-documented DES problem of “modeled stress depletion”,7 leading to corruption of the attached boundary layers caused by the undesired encroachment of the LES-mode. To remedy this, a simple shield function is introduced as f = [0, 1] (if d > CDES ∆ then f =1 else f =0) or a function of f = max(0, 1 − (CDES ∆/d)4 ). Therefore, the DES length scale is rewritten as LDES = LRAN S − max(0, LRAN S − CDES ∆) ∗ f . (17) According to our experience, this function assures the RANS-mode in the near-wall region. The use of the cubic root formulation for ∆ is often viewed as controversial, as ∆ should represent the size of structure that the grid can resolve. Along this line of thinking, the “maximum” formulation is clearly more valid in highly-stretched cells. However, as the turbulent structures in the early shear layer are very small, resolving these with an isotropic grid would result in an unaffordable computational expense. Using the cubic root formulation is therefore seen as a pragmatic remedy for this problem. An alternative approach, explored by Shur et al.8 with considerable success, is to eliminate the sub-grid model entirely by using an implicit LES (ILES) approach.
V.
Non-Reflective Boundary Conditions
Numerical results can be seriously disturbed by reflections from the boundaries of the computational domain. To avoid such reflections, the non-reflective radiation boundary conditions in three-dimensional ∂ ∂ form of Bogey and Bailly9 are applied. They are modified by replacing ( ∂r + 1r ) through ∂n where r is the distance from the acoustic source location to the boundary and n is the normal vector on the boundary. This modified form is implemented in the flow solver. At the inflow and at the lateral sides of the computational domain, the modified radiation boundary conditions read as ρ ρ−ρ ∂ ∂ (18) u i + Vg · ui − ui = 0 , ∂t ∂n p p−p where Vg is the acoustic group velocity and defined as Vg = c + Vn , c is the speed of sound and ρ, ui and p are the mean density, velocity components and pressure, respectively. The outflow conditions are written as ∂ρ 1 ∂p + u · ∇(ρ − ρ) = 2 + u · ∇(p − p) , (19) ∂t ∂t c ∂ui 1 ∂(p − p) + u · ∇(ui − ui ) = − , (20) ∂t ρ ∂xi ∂p ∂ + Vg · (p − p) = 0 . (21) ∂t ∂n 4 of 13 American Institute of Aeronautics and Astronautics
Unlike the original formulations of Bogey and Bailly, the apparent location of the sound sources is not needed in the present modification. This simplifies the implementation of non-reflective boundary conditions in CFD-codes and has been validated with different test cases at ISTA. It had been found to be capable of suppressing acoustic reflections from the boundaries.
VI.
Damping Algorithm
In addition to the non-reflective boundary conditions, two damping (or absorbing) zones are placed adjacent to the nozzle inlets as well as the outlet of the computational domain, as depicted in Figure 3. The damping function is smooth, so that no reflections occur inside the damping zone. It consists of a smooth matching of an evolving and a “target” flow field within the damping zone by use of the relation φ = (1 − g)φ∗ + gφ ≈ φ∗ − g(φn−1 − φ) .
(22)
Here g denotes the damping function, φ∗ the “calculated” flow quantity on the current outer iteration n at one time step and φ is the time-averaged value. The damping function is set as g = s(l/L)3 . Here s denotes the damping strength (s < 1), L the width of the layer and l the distance from the internal boundary of the layer to the considered point (0 < l < L). If the damping function drops out, i.e. g = 0, it reduces to φ = φ∗ . The relation (22) is implemented in the discretized approximation of the momentum equations. In finite-volume methods, the discretized form of the momentum equations can be written as X aP φ∗ = anb φ∗nb + Sφ . (23) With the substitution of (22), the discretized approximation can be written as X X n−1 aP φP = anb φnb + Sφ + anb gnb (φn−1 − φP ) . nb − φnb ) − aP gP (φP
(24)
Under consideration of damping, the pressure p is defined as p = p∗ + p0∗ + g(p − pn ) ,
(25)
p0 = p0∗ + g(p − pn ) .
(26)
and the pressure correction as 0∗
With this substitution into the discretized approximation of p X aP p0∗ anb p0∗ P = nb + Sp0 ,
(27)
the discretized approximation for p0 reads as X X ap p0P = anb p0nb + Sp0 + anb gnb (pnb − pnb ) − ap gP (pnp − pP ) .
(28)
VII.
Numerical Features
The three-dimensional unsteady Navier-Stokes equations are discretized using a cell-centered finitevolume method based on block-structured grids. For the discretization of the convective fluxes, both second order central and higher order upwind-biased limited schemes are available. The diffusive fluxes are approximated by a second order central difference scheme (CDS), and the pressure field is solved by the SIMPLE algorithm. A close coupling between pressure and convective velocity is achieved by the Rhie and Chow interpolation method.10 Computational grid The computational domain extends axially to 50 Ds downstream of the nozzle exit and radially to 18 Ds . It is discretized with a structured grid of 3.8 million cells for the baseline configuration and 11 million cells for the serrated case. The baseline geometry is resolved with 60 cells in the azimuthal direction, while 160 5 of 13 American Institute of Aeronautics and Astronautics
cells have been used for the serrated case, with a resolution of 8 cells per serration. The grid distribution is concentrated in the wall regions and shear layers, as seen in Figure 2a, where every second grid line is plotted. Figure 2b shows the grid distribution around the lip and Figure 2c the block structure on the cone tip, with a square cross section used at the center to avoid a singularity. In order to better resolve the development of turbulence in the early, relatively thin and annular shear layers, 53 grid points are used in the radial direction for the primary and secondary streams. There are at least 8 grid points in the boundary layers and a dimensionless wall distance of Y + < 2 is assured at all walls. Both grids differ only with repect to the cell number in the circumferential direction; they have identical axial-radial cell distribution, except in the region near the serrated lip.
(b)
(c)
(a)
Figure 2. (a) Mesh overview, (b) grid distribution around the lip and (c) block structure on the cone tip
Boundary conditions
Absorbing layer
Steady outflow (ambient P fixed) Non−reflective acoustic outflow cond.
Non−reflective acoustic inflow cond. Steady inflow (Pt , T t ) Nozzle inlet
Steady inflow (u, T) For the steady simulation, the total pressure and total temperature are Non−reflective acoustic inflow cond. 0000 1111 prescribed at the nozzle inlets for the primary and secondary streams 0000 1111 0000 1111 0000 1111 respectively. They are calculated from the given Mach number, am0000 1111 0000 1111 bient temperature and pressure. A turbulence intensity of 10% and 0000 1111 10 0000 1111 10 0000 1111 10 a turbulent viscosity ratio (µt /µ) of 10 are specified. Adiabatic no 0000 1111 10 0000 1111 10 0000 1111 10 slip wall boundary conditions are used for the solid surface of the noz0000 1111 10 0000 1111 0000 1111 zle. For numerical stability a velocity of 10 m/s is specified for the 0000 1111 0000 1111 Absorbing layer 0000 1111 coflow boundaries, with a turbulence intensity of 0.1%, a turbulent (inside nozzle) 0000 1111 0000 1111 viscosity ratio of 1 and a temperature of 288.14 K, corresponding to 0000 1111 Non−reflective acoustic inflow cond. Steady inflow (u, T) a Mach number of 0.03. As boundary condition at the outflow plane, the ambient static pressure is fixed. For the unsteady simulation, the converged RANS solution is used as a starting condition. The non- Figure 3. Overview of damping reflective boundary conditions described above are used for all sides. zones and boundary conditions
Convective scheme In the LES-mode region of DES the upwind-biased scheme TVD (Total Variation Diminishing) is not appropriate, as it exhibits excessive numerical dissipation.11 In order to decrease the numerical dissipation, a hybrid scheme between CDS and TVD using a blending function σ is used, F = σFT V D + (1 − σ)FCDS ,
(29)
where FT V D and FCDS are the convective fluxes corresponding to the upwind-biased limited scheme TVD and CDS respectively. The blending function σ is defined as σ = χ3 , 6 of 13 American Institute of Aeronautics and Astronautics
in which the parameter χ is taken as χ = min(1, CDES ∆/d) ,
(30)
with χ = 1 in the near wall region, whereas χ < 1 and close to zero in the region far away from the wall. For numerical stability, the blending factor is limited to σ ≥ 0.05 ∼ 0.1. This means that in the near-wall region TVD is used, while in the far region almost complete CDS is used. This hybrid scheme has been applied to all unsteady simulations of this study. Time step As the time discretization of the flow solver is implicit, the CFL-criterion is not as restrictive as for explicit methods. Nonetheless, the physical restraints that define a certain time scale provide an upper limit for the time step size. One is the convective time scale Tc which is defined by Tc = Ds /Us ' 9 · 10−4 s, the other is the lowest acoustic time scale Ta which is defined to match the speed of acoustic waves propagating downstream with a velocity of Us + c so that Ta = Ds /(Us + c) ' 4 · 10−4 s. In this work, the time step ∆t is taken as ∆t = 2.5 · 10−6 s. It corresponds to approximately 360 time steps per convective time Tc and to a Strouhal number, St = f Ds /Us = Ds /(∆tUs ), of 359.
VIII.
Results
In this section, results for the simulations conducted for the aforementioned configurations are presented. As a preliminary investigation the difference between the present DES formulation and the standard one has been investigated, the details of which are not described here. Only the density isosurfaces in the shear layer between the secondary stream and the surrounding flow are shown in Figure 4, demonstrating model performance. It can be seen that the present DES formulation produces much finer flow structures behind the nozzle exit than the standard one. It can be further expected that the low frequencies are over-predicted and the high frequencies under-predicted using the standard DES formulation.
(a)
(b)
Figure 4. Isosurfaces of density showing flow structures in the shear layer between the secondary stream and the surrounding flow. (a) present DES formulation used in this work, (b) standard DES formulation
In order to save storage space, three velocity components and the static pressure on two predefined surfaces are stored every fourth time step during the unsteady simulation. The first surface encloses the jet flow and is used for the aeroacoustic analogy of FWH. The other is an xy-cut through the computational domain along the jet axis, on which 100 ms of data has been sampled for each calculation. They are used to evaluate the time-averaged values and to define the aerodynamic near-field plotted in the following. The experimental data is taken from LDV measurements supplied by the CoJeN project partners (see Acknowledgment). A.
Aerodynamic Near Field
Time-averaged velocity profiles Figures 5a and 5b show the predicted axial velocity profiles directly behind the nozzle outlets in comparison with the experimental data, from which it can be seen that the predicted velocities are too large. In the simulation, a velocity of 10 m/s is specified for the surrounding flow and the ambient pressure is fixed at the outlet plane. This leads to a lower pressure at the nozzle exits than in the experiment with a stagnant surrounding, resulting in a higher prediction of axial velocity compared to the experimental data. 7 of 13 American Institute of Aeronautics and Astronautics
Figure 5c shows the predicted axial velocity profiles at different axial positions in comparison with the experimental data. The agreement is generally satisfactory, except in the region near the jet axis. The main reason for this is the grid distribution in this region.
(a)
(b)
(c)
Figure 5. Comparison of the axial velocity with the experimental data behind the nozzle exits (a),(b) and at the different axial positions (c)
Velocity correlations Figure 6 shows the comparison of time-averaged velocity correlations < u0 u0 > and < u0 v 0 > with the experimental data for the baseline configuration. The profiles have been staggered according to their axial positions. It can be seen that the velocity fluctuations directly behind the nozzle (the first two profiles from the left) are not well resolved. As noted by Wu,12 the very small scales generated in this region cannot be captured by LES methods on affordable grids, while the larger scales further downstream are more easily resolved. The fluctuations downstream are captured with a small over-prediction between 2Ds and 4Ds . It is believed that the resulting enhanced mixing is responsible for the under-prediction of the potential core length. A highly satisfactory agreement is found in the comparison of < u0 v 0 >.
(a)
(b)
Figure 6. Comparison of the velocity correlations < u0 u0 > and < u0 v 0 > with the experimental data at different axial positions for the baseline configuration
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Turbulent structures Figure 7 shows instantaneous snapshots of density isosurfaces to demonstrate the turbulent structures for both the baseline and serrated configurations. It can be observed that although no artificial unsteady perturbations are prescribed at the inlet, all the typical features of the initial region of high-Reynolds number jets such as shear-layer roll-up, three-dimensionalization and well defined turbulence structures in the developed jet region are reproduced by this method. The shear layer between the core and bypass streams is relatively thin and stable, its degree of mixing is very low. This is consistent with the findings of Andersson et al.13 For the baseline configuration the ring-shape structure is deformed, transported and broken up downstream, while for the serrated case an increasing in finer structures is seen close to the serrated lip, indicating the high frequency content. The serrations help to precipitate the transition to turbulence and formation of small eddies, leading to an enhanced mixing.
baseline
serrated
Figure 7. Isosurfaces of density showing flow structures in the shear layer. Above: between the primary and secondary stream (ρ = 0.90kg/m3 ); below: between the secondary stream and surrounding flow (ρ = 1.19kg/m3 )
Pressure fluctuations in the irrotational part of the flow The aerodynamic near field has been investigated by Tinney et al.14 using a linear microphone array located outside the rotational region of the flow. The line array is positioned 6.5 cm outside of the secondary nozzle lip and expanded at 8.5◦ relative to the jet axis, as shown in Figure 2a. In this region the grid is relatively coarse, ∆r >1 cm, hence the high frequencies cannot be resolved.
(a)
(b)
(c)
Figure 8. rms of pressure, (a) along the microphone array with fixed averaging time of 0.01s; (b) at x/D = 4 on the line array with different averaging times; (c) along the microphone array with an averaging time of 0.08s
The comparison of predicted rms of pressure with the measurement is depicted in Figure 8. Since there 9 of 13 American Institute of Aeronautics and Astronautics
is only 0.1 s of physical time data available, special attention has been paid to the averaging process. With a fixed averaging time of 0.01 s, the rms levels vary strongly in the region of x/Ds > 2 when a different sample of computed physical time is selected, as seen in Figure 8a. Considering the averaging process as a kind of low-pass filter, the filtered data shows that this region contains long wavelength fluctuations. To assess the averaging time required for meaningful statistics, the time dependence at the position of x/Ds = 4 is depicted in Figure 8b, indicating that the fluctuations of the average values are reduced monotonically when the averaging time is increased. It can be expected that an averaging time of 0.08s will give representable results within +/- 0.5dB. The rms of pressure on the line array with an averaging time of 0.08s are presented in Figure 8c. There is a clear under-prediction in the region of x/Ds < 1.5, due to insufficient grid resolution. The grid damping is especially strong for high frequencies which are dominant in this region. An associated reason is the delay of resolved turbulence onset due to the coarse grid size in the azimuthal direction, so the initial shear layer behind the outer nozzle lip is unresolved in this direction. The rms levels match more closely with a slight over-prediction in the region of 1.5 – 5 Ds , which is supposed to be caused by the under-predicted potential core length. Thereafter, the predicted fluctuation levels slowly decrease because of numerical damping. An improvement in prediction can be expected if the grid is refined. Flow regimes The flow field in a jet and in its vicinity can be separated into four regions. The non-linear vortical region inside the shear layer. The pressure fluctuations depend on the fluctuating flow field. The irrotational near field. It is dominated by the velocity field that is induced by the vorticity distribution inside the shear layer. This field was described by Michel.15 He showed that the decay of the fluctuating velocities outside the shear layer is more rapid than near a plane shear layer,16 where the decay is u2 = (r − rn )−4 . Later, Arndt et al.17 found that the decay could be characterized by (r − rn )−6.7 . The pressure fluctuations are proportional to the velocity fluctuations. The radial decay outside the shear layer depends on the decomposition of the flow field into azimuthal orders. The acoustic near field. The decay of the pressure fluctuations is in the order of (r/rn )−1 . The geometric far field. The decay of the pressure fluctuations is proportional to (r/rn )−2 . The onset of the decay moves outwards with increasing x/Ds in Figure 9a. One reason is that the shear layer thickness increases. A second reason is that the limits between the three outer regions depend very much on the wave length of the sound and the wave length of the dominating part of the pressure field increases in the downstream direction.
In the last region, a normalization of the sound pressure level Lp that compensates for the linear decay of the sound pressure level (quadratic decay of acoustic intensity) can be used to compare the sound pressure levels for different observer locations: Lpn = Lp + 20 log
r rn
(31)
Fourier analysis of the flow field Figure 9b shows the Fourier decomposition (the real part of the complex amplitude) of the pressure field for the dominant frequency of 488Hz. This is in the frequency range where large amplitudes are noticed for downstream observers in the acoustic far field, as will be seen. The filtered field shows a strong directivity of the radiated low frequency noise towards the downstream direction.
B.
Acoustic Far Field
This section gives a presentation of the predicted far field sound pressure levels. The noise is calculated through surface integrals using the FWH aeroacoustic analogy. For comparison, the axisymmetric FWHsurface has the same radial position for both the baseline and serrated configurations. It encloses the jet flow as a tube, as depicted by the solid line in Figure 10a. It starts from the position of x/Ds ≈ −0.25 10 of 13 American Institute of Aeronautics and Astronautics
(a)
(b)
Figure 9. (a) Decay of pressure fluctuation levels perpendicular to the jet axis, (b) Fourier analysis of the pressure field for the dominant frequency 488 Hz
and y/Ds ≈ 1.5 and extends streamwise to x/Ds =20, widening at an angle of about 6.2◦ following the expansion of the jet flow, and the downstream side is open. The surface is located well outside of any stronger fluctuation of vorticity, because strong vortical fluctuations passing a surface are known to cause strong spurious sound in the far-field prediction. As shown in Figure 9a, the interface lies inside the linear hydrodynamic region, meaning that non-acoustic fluctuations exist when pressure and velocity are not in phase. These fluctuations should not be transported in the acoustic far field and therefore should be filtered out by use of the FWH-equations. Overall sound pressure levels The OASPL in the far field is of interest to noise prediction. The sound pressure levels are presented for 10 observers equally spaced between 30◦ − 120◦ as seen in Figure 10a. These observers are located on an arc of 30 Ds with its origin on the center of the outer nozzle exit plane. Figure 10b compares the predicted OASPL with those of Andersson et al.13 and the experimental data that was conducted at the QinetiQ Noise Test Facility in the UK and scaled using Eq. (31). The agreement with the measurements is quite satisfactory, with the maximum deviation of 3.0 dB occurring at the observer locations of 40◦ − 50◦ for both configurations. This is attributed to the gradual coarsening of grid cells downstream of the potential core in the axial direction, meaning that noise sources in this region are not resolved. Visual evidence of this can be found in Figure 10a, showing instantaneous contours of the z-vorticity in the xy-plane. Downstream of x/Ds = 5, the structures show a fast decay in magnitude until they almost disappear altogether at larger axial positions.
(a)
(b)
Figure 10. (a) FWH-surface relative to flow field of the vorticity ωz , (b) comparison of predicted OASPL with those of Andersson et al. and the experimental data
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The results for the baseline configuration with a grid of 3.8 million cells are very similar to those of Andersson et al.13 with 20 million points. It can be concluded from these results that the grid effects are not so important for the far field as for the near field. In terms of predicted OASPL, the use of serration leads to a noise reduction of 2dB. The trend of sound intensity reduction compares favorably with the experimental observations. Spectra Exemplary far-field spectra for observers at angles of 30◦ , 60◦ , 90◦ and 120◦ are displayed in Figure 11 for both configurations. The simulated physical time is too short to allow a proper Fourier analysis of frequencies below 100Hz. The dominant low-frequency range of the simulation matches closely with the experimental data which explains the good agreement of OASPL. As expected, the numerical simulation shows a stronger descent of the spectrum in the high frequency range above 3 kHz. This is because of the under-resolved initial shear layer close to the outer nozzle lip which should be the main source for the high frequency noise. This is most noticeable for upstream observers. There is no clear difference between the spectral distributions of both configurations, even with a much better resolution in the azimuthal direction for the serrated case, which is attributed to the grid used as well. Due to the same axial-radial cell distribution, the high frequencies in the serrated case cannot be better resolved than those in the baseline case.
Figure 11. Comparison of spectra obtained from predicted and measured far-field data
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IX.
Conclusion
The subsonic jet flow through a short-cowl coaxial nozzle with both smooth and serrated lips has been investigated. The computational domain including the upstream nozzle geometry is discretized with a structured grid of 3.8 million cells for the baseline configuration and 11 million cells for the serrated case. Without any artificial forcing, the present method reproduce all the typical features of the initial region of high-Reynolds number jets such as shear-layer roll-up, three-dimensionalization and well-defined turbulence structures in the developed jet region. The far field sound pressure levels are calculated by using the aeroacoustic analogy of FWH. The predicted overall sound pressure levels are generally in good agreement with the experimental data, within a maximum deviation of 3.0 dB for all observer positions. The high-frequency noise, which is mainly associated with the region very close to the nozzle is under-predicted, while the dominant low frequencies are captured with the relatively coarse grid. The use of serrations generates finer, more numerous structures, making the transition to turbulence faster and leading to an enhanced mixing. The numerical simulation has indicated the same trend of decreased sound intensity when using serrations as in the experiment, with an observed noise reduction of about 2 dB. Possible future work to be considered would include a simulation of the baseline case using the same azimuthal grid resolution as the serrated case, thereby enhancing the comparability between these.
Acknowledgment The authors would like to thank C. Mockett, D. Eschricht and L. Panek for the informative discussions. The financial support of the EU 6th Framework Project CoJeN (Computation of Coaxial Jet Noise), contract number: AST3-CT-2003-502790, and the use of the SGI Altix 4700 computer at the Bavarian Supercomputing Facility (H¨ ochstleistungsrechner in Bayern - HLRB) are gratefully acknowledged.
References 1 Viswanathan,
K. & Clark, L.T. Effect of nozzle internal contour on jet aeroacoustics. Int. J. Aeroacoustics. 3:103-135,
2004. 2 Wu, X. Tristanto, I.H., Page, G.J. & McGuirk, J. Influence of nozzle modelling in LES of turbulent free jets. AIAA Paper 2005-2883. 3 Bodnoy, D.J. & Lele, S.K. Review of the current status of jet noise predictions using large-eddy simulation. AIAA Paper 2006-0468. 4 Goldstein, M.E. Hybrid Reynolds-Averaged Navier-Stokes/Large Eddy Simulation approach for predicting jet noise. AIAA J., 44:3136-3142, 2006. 5 Ffowcs-Williams, J. & Hawkings, D. Sound generation by turbulence and surfaces in arbitrary motion. Phil Trans. Roy. Soc., 264:321-342, 1969. 6 Yan, J., Mockett, C. & F. Thiele, F. Investigation of alternative length scale substitutions in detached-eddy simulation. Flow Turbulence and Combustion. 74:85-102, 2005. 7 Spalart, P., Deck, S., Shur, M., Squires, K., Strelets M., Travin A. A new version of detached-eddy simulation, resistant to ambiguous grid densities. Theor. Comp. Fluid Dynamics, 20:181-195, 2006. 8 Shur, M., Spalart, P., Strelets, M., Garbaruk. A. Further steps in LES-based noise prediction for complex jets. AIAA Paper 2006-485. 9 Bogey, C. & Bailly, C. Three-dimensional non-reflective boundary conditions for acoustic simulation: far field formulation and validation test cases. ACTA Acustica united with Acustica, 88:463-471, 2002. 10 Rhie, C. & Chow, W. Numerical study of the turbulent flow past an airfoil with trailing edge separation AIAA J., 21: 1525-1532, 1983. 11 Strelets, M. Detached Eddy Simulation of massively separated flows. AIAA Paper 2001-0879. 12 Wu, X., Page, G.J. & McGuirk, J. An approach to improve high frequency noise prediction in LES of jets. AIAA Paper 2006-2442. 13 Andersson, N., Eriksson, L.E. & Davidson, L. LES prediction of flow and acoustic field of a coaxial jet. AIAA Paper 2005-2884. 14 Tinney, C.E., Jordan, P., Guitton, A., Delville, J. & Coiffet, F. A study in the near pressure field of coaxial subsonic jets. AIAA Paper 2006-2589. 15 Michel, U. Irrotational fluctuations associated with axisymmetric shear layers. Eur. J. Mech.,B/Fluids, 8 :269-281, 1989. 16 Phillips, O.M. The irrotational motion outside a free turbulent boundary, Proc. Camb. Phil. Soc., 51 :220-229, 1955. 17 Arndt, R.E.A., Long, D.F. & Glauser, M. N. The proper orthogonal decomposition of pressure fluctuations surrounding a turbulent jet. J. Fluid Mech., 340:1-33, 1997.
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