turbulent jet noise simulation and propagation using a

I NT. C ONF. ON J ETS , WAKES AND S EPARATED F LOWS J UNE 16-18, 2015, KTH M ECHANICS , S TOCKHOLM , S WEDEN

TURBULENT JET NOISE SIMULATION AND PROPAGATION USING A 3RD ORDER MUSCL/CD SCHEME ON UNSTRUCTURED GRID AND FFOWCS-WILLIAMS HAWKINGS. Mirela Caraeni1 , Alastair West2 , Doru Caraeni1 CD-adapco, Lebanon, USA1 ; CD-adapco, London, UK2

SUMMARY

The noise generation by a turbulent high speed jet is simulated using Large Eddy Simulation (LES) and a newly implemented MUSCL/CD 3rd order numerical scheme. The acoustic wave propagation in the far-field is performed using a permeable formulation Ffowcs Williams-Hawkings (FW-H) approach, available in the STAR-CCM+ code. The flow-field results and the predicted sound pressure levels are both in good agreement with experiments. The predicted sound pressure levels using LES are within a 1-2 dB deviation for all measured observer locations. INTRODUCTION

Demanding noise level regulations at national and international airports have determined aerospace engine manufacturers to increase their efforts to reduce jet noise. These efforts have resulted into numerous experimental and computational studies. For computational studies, turbulence modeling using Large Eddy Simulation (LES) is the method of choice for jet noise research. Although most computational studies have used block-structured high-order finite difference methods [1], Spalart, Shur and Strelets [2] have pointed out the need to develop unstructured-grid methods for such simulations in complex geometries. Today there is a growing interest in unstructured methods for high fidelity turbulence and aeroacoustic simulations, including in jet noise research. For jet noise simulations, a typical approach to reduce total computational effort is to couple the numerical simulation of turbulence and noise generation, with integral methods, like Ffowcs-Williams Hawkings, to perform the propagation of sound in the far-field. Although LES is a popular tool for jet noise research, simulations of full-scale jet-nozzle configurations are still unattainable at practical Reynolds numbers due to the viscous sublayer of the turbulent boundary layer, presents a resolution requirement leading to problematic case size. Therefore, typically the nozzle geometry is excluded from the calculation to reduce computational costs. The usual practice is to start the computational domain just downstream of the nozzle exit and to specify a laminar shear layer profile at the inflow. To force the transition to turbulence a variety of methods are used, like the injection of artificial inflow excitations (such as random velocity perturbations) into the jet shear layer [3]. Another popular method is to de-activate the Subgrid-Scale (SGS) model and to rely on the numerical dissipation of a the high-order convection scheme [4]. The latter method is preferred by Spalart, Shur and Strelets who have since updated their solution procedure to include inflow conditions from a precursor RANS simulation including the nozzle geometry [2]. Other researchers [5] [6] simulate the whole jet-nozzle configuration but choose to reduce the Reynolds number to an affordable value. Researchers at ONERA [7] chose to de-activate the Subgrid-Scale (SGS) model (perform Implicit LES) within both the jet and nozzle region. Both of these approaches have the effect of increasing the boundary layer thickness (in a relatively uncontrolled manner) which will affect the shear layer transition to turbulence and alter the high-frequency part of the noise spectrum. The goal of this study is to demonstrate the capability of the fully-unstructured code, STAR-CCM+, to correctly predict jet noise using Large Eddy Simulation. Compressible LES is conducted for a high subsonic jet with the round nozzle geometry included within the computational domain. The experimental conditions of Jordan et al.[8] [9] were matched, such that the time-averaged flow-field and far-field sound pressure levels could be compared with experimental data. The paper will first discuss the computational setup, the numerical scheme, mesh, boundary conditions and the simulation procedure. A detailed analysis of the LES results then compares the flow and far-field sound pressure levels using FW-H integration against experimentally measured values.

COMPUTATIONAL SET-UP Results from simulations of a Mach 0.75 jet exiting a round nozzle configuration will be presented. The flow properties are summarized in Table 1. The Reynolds number was artificially reduced in the same vein as Andersson et al. [5], with the assumption that the flow is only weakly Reynolds number dependent. Thus, the Reynolds number based on nozzle exit diameter and jet velocity at the nozzle exit plane, ReD is 50,000 (that of the experiment is ReD 106 ). Mach 0.75

Tj =T1 1.0

P1 (P a) 99,670

(kg=m3 ) 1.225

c1 (m=s) 337.75

U1 (m=s) 0.0

T1 (K) 283.15

Pj (P a) 144,400

ReD 50,000

(Table 1.)

Computational domain The nozzle dimensions are those of the JEAN (Jet Exhaust aerodynamics and Noise) project nozzle [8] [9] which has a nozzle length of 0:38 m and an exit diameter of 0:05 m. This axial length of the domain is 80 Dj and the radial distance extends from 80 Dj to 120 Dj at the far-field inlet and outlet boundaries respectively. An unstructured grid topology using trimmed hexahedra and boundary prismatic cells was constructed using 6 successive 8:1 volumetric refinements towards the jet nozzle exit and initial shear-layer area (see Figure 1). The total number of cells is 23 million. Boundary conditions The flow conditions at the nozzle inlet : Mach1 = 0:16, Pressure = 44800 P a, Temperature = 315 K, have been specified as constant profiles, to match the experimental condition at nozzle jet exit, e.g. Mach = 0.75. All external fluid boundaries use non-reflecting free-stream boundary conditions (based on characteristics theory) and are positioned far enough from the nozzle jet exit such as to minimize the acoustic reflections produced by turbulence vortical structures traveling through the exit boundary, see Figure 2. To completely eliminate this spurious acoustic reflections the Acoustic Suppression Zone model in STAR-CCM+ has been used to complement the freestream characteristics based boundary conditions. This model adds an appropriate damping source term to the solved equation system: max

2

(1 + cos(

d ))(Q - Q) w

(1)

where Q is the vector of conserved variables ( ; Ui ; E) and Q is their time-average, max = 1000 sec 1 is the inverse of the relaxation time-scale, w is the suppression zone thickness, w = 5 Dj ; and d is the distance from the external boundary to the cell-centroid. The effect of the introduced source term is to exponentially damp the unsteady flow-field towards a (time-mean) steady field give by Q, as it approaches the external fluid boundaries, to completely eliminate any acoustic reflection. NUMERICAL METHOD A cell-centered finite-volume algorithm on unstructured-mesh with arbitrary polyhedral shape has been employed. An implicit unsteady Pressure-based algorithm is used using Rhie-Chow dissipation, to avoid spurious pressure oscillations [10]. The temporal integration scheme is 2nd order with 5 inner iterations per time step (Under-relaxations factors of 0:9 and 0:8 were used for the velocity and pressure solvers, respectively). The Least-Squares gradient reconstruction method, with the Min-Mod gradient reconstruction limiter is used, with Total Variation Bounded (T V B) gradient limiting, with the acceptable field variation set to 0:1 (10%). The 3rd order MUSCL/CD scheme was used for convection, with Upwind Bounding Factor of 0:15, and 2nd order for diffusion. The ideal gas law, = p=RT is used for the constitutive equation. Turbulence modeling For Large-Eddy Simulation modeling the Dynamic Smagorinsky SGS model proposed by Germano et al. [11] was used, with the least-squares minimization procedure proposed by Lilly [12]. Hybrid MUSCL 3rd-order/CD For convection scheme , the hybrid MUSCL 3rd order /CD is recommended for high-fidelity turbulence (LES/DES), aeroacoustic and aerodynamic simulations. The MUSCL 3rd order/CD can be used for both steady and unsteady

simulations, where one model parameter is used to control the numerical dissipation in the scheme. Similar as the Bounded Central-Differencing scheme, this hybrid MUSCL 3rd order /CD uses a Normalized-Variable Diagram (NVD) value & to ensure the boundedness of the scheme by switching to the first-order scheme in regions of nonsmooth flows. When smooth local flow conditions are detected, the hybrid MUSCL 3rd order /CD is constructed as a blend between a MUSCL 3rd order upwind and the 3rd order central-differencing reconstruction schemes. The convective flux is computed as following: :

:

(m )f =

m : m(

f ou muscl3 + (1

)

for & < 0 or 1 < & & 1 cd3 ) for 0

(2)

The blending factor is user-controlled and must be decided based on the physical problem or model (and mesh quality considerantions.) The MUSCL 3rd order upwind reconstructed value muscl3 is limited at high speeds so as to not affect the formal order of accuracy while preventing spurious oscillations. The high speed limiting is based on WENO (Weighted Essentially Non-Oscillatory) principles and is only applied to the quadratic part of the face-value reconstruction. WENO-based averaging is computed based on three different stencils. In the regions next to the strong shocks, the muscl3 accuracy is reduced to second order. The switch between the unlimited and the WENO-limited muscl3 is done based on a Mach-number-based flux-limiter. For incompressible simulations, muscl3 is unlimited (the flux limiter is 1), thus only relying on the use of the NVD (Normalized-Variable Diagram) value & to ensure the boundedness of the scheme. The hybrid MUSCL 3rd order /CD scheme provides improved (reduced) dissipation when compared with both secondorder and Bounded Central Differencing (BCD) schemes. It is robust (due to its boundedness) and capable of simulating steady and unsteady flows from incompressible to high-speed compressible regimes. Ffowcs-Williams Hawkings algorithm Due to the high computational cost of direct simulations, a hybrid method is proposed to evaluate the far-field sound propagation. Firstly, all near-field sources of sound are simulated using LES modeling. Then the Ffowcs-Williams Hawkings (FW-H) acoustic analogy gives the propagation of sound into far field using the near-field sound sources as the input. This aeroacoustic model is based on the Farassat’s Formulation 1A of the FW-H analogy [14] which uses an advancedtime formulation (or source time dominant algorithm) proposed by Casalino [15]. This yields the far-field acoustic pressure fluctuations computed using the FW-H surface integral formula:

4 p(x; t)

=

Z " S

1 + c0

: 0 (U n

r(1 Z " S

+ Un: ) + M r )2 :

: 0 Un (r M r + c0 Mr r2 (1 Mr )3 :

c0 M 2 )

#

dS

(3)

ret

2

Lr Lr (rM r + c0 Mr c0 M ) + r(1 Mr )2 r2 (1 Mr )3

#

ret

dS +

Z S

Lr r2 (1

LM Mr )2

dS ret

where the first term is the thickness noise Ui = ui = 0 + vi (1 = 0 ) and the second and third terms are the loading noise Li = Pij nj + ui (un vn ), where Pij = (p p0) ij ij is the compressive stress tensor and ij is the viscous stress tensor. Here, 0 is the far-field reference density, vi is the velocity at the surface and Mi = vi =c0 is the surface Mach number vector based on the far-field speed of sound c0 . Dots on quantities denote time derivative with respect to the source time. Subscripts denote a scalar product with either the surface outer normal, unit vector ni , the unit radiation direction vector rbi or the Mach number vector Mi . This formulation explicitly omits the quadrupole volume integral, which reduces the On-the-Fly FW-H model computational time. The quadrupole volume integral omission does not mean that the quadrupole generated noise is not captured. By using permeable FW-H surface integrals proposed by di’Francescantonio [16], non-negligible volumetric sources can be taken into account. The FW-H permeable formulation describes the noise source terms distributed on the surface, as monopole and dipole terms, and the contribution to the total noise of volumetric sources enclosed by the integration surface. This formulation assumes that the contribution to the total noise of volumetric sources outside a chosen integration surface is negligible. For more details about the implementation the reader is referred to Caraeni et al. [17]. The permeable FW-H surface used is a open cone shape of length 62 Dj , shown in Figure 3. The open shape results in some contamination at the very high frequency range which has been reported by Shur et al. [4] and Mendez et al. [18] (among others). For this reason a band-pass filter is used to cut off the lower frequencies when calculating overall sound directivity (0:05