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Aero-Acoustics, noise estimation and noise reduction strategies
CAA can be explanded into Computer Aided Acoustics (similar to CAD and CAE) or Computational Aero-Acoustics - both mean same. We may regard sound either a sensation or the stimulus which produces the sensation. The physics is mainly concerned with the stimulus which is the phenomena occurring external to the ear. Acoustics is defined as the science of sound and phenomenon of hearing. While the word 'sound' refers to vibrations of all frequencies (audible or otherwise, similar to the word 'light' which represents visible radiations as well as invisible infrared and ultra-violet radiations), acoustics is reserved for the aspects connected with hearing. (Reference: A Textbook of Sound by A. B. Wood). A book available free at migeot.eu/acoustics-textbook is Acoustics: Essential concepts, theory and models of linear acoustics for engineers by Jean-Louis Migeot et al.
Aero-acoustics deals with production, propagation (transmission: air-borne or solid-borne), reflection (as sound moves from one medium to another), absorption, scattering, attenuation and reception (by human ears) of sound waves. The sound is primarily generated by fluid fluctuations forced due to unsteady motion of fluids. Sound is a small perturbation of pressure over a mean (steady state) pressure, p’/p0 which propagates as a wave. Aero-acoustics is different from the study on generation of sound such as organ pipe and loud-speakers. The later is referred as "classical acoustics".
Excerpts from COMSOL user manual: "Acoustics is the physics of sound. Sound is the sensation, as detected by the ear, of very small rapid changes in the acoustic pressure p above and below a static value. This static value is the atmospheric pressure (about 100,000 Pa). The acoustic pressure variations are typically described as pressure waves propagating in space and time." During their propagation, sound waves can be reflected, refracted or attenuated by the medium and can have interference with sounds of different frequencies.
Excerpts from "ACOUSTICAL USES FOR PERFORATED METALS: Principles and Applications" - If we play a single key (or three or four adjacent keys near middle-C, the sound energy will be concentrated around the frequency 250Hz (cycles per second).
Approximate sound speeds for 3 media: air, water and steel are 340 [m/s], 1500 [m/s] and 5000 [m/s] respectively. Note that the speed of sound (which is speed of pressure waves) corresponds to fluid in unconfined condition. The speed of pressure waves in a confined space such as long tube is < that in unconfined space due to elasticity of the walls of the pipe. This phenomena is of practical significance in water hammer calculations.
A linear, time-harmonic wave equation in 3D is given by ∇2p' + k2p' = 0 where p' is the acoustic pressure. This is also known as Helmholtz equation for sound pressure.
For a given pressure field p'(x, t), the particle velocity is given by u'(x, t) = -p'(x, t)/[ρ·c]
p' = S·k·c·ρ·/4πr·sin[k(ct - r)], u' = -S·k/4πr [1/k/r·cos[k(ct - r) - sin[k(ct - r)], S = maximum rate of fluid emission from source, ρ is the density of the medium, r is radial distance from source. For spherical waves, the intensity decreases with the square of the distance but the sound pressure level decreases linearly with distance.
p' = a·k·c2·ρ·sin[k(ct - x)], u' = a·k·c·sin[k(ct - x)], a = particle displacement, k = wave number.
Real sounds or noise generated from a source may contain many component frequencies leading to a mixture of planar and spherical sources. This makes the measurements and calculation of acoustic field complicated. In theory, plane wave pressures do not decrease with distance as they do for spherical waves.
In direct field (like near field situation), sound pressure decreases linearly with distance while there is no decrease in sound pressure with distance in the diffuse field (as observed in plane waves). The distance at which the direct and reverberant sounds are at equal pressure is called the critical distance.
In a free-field that is where no reflected sound waves are present and far away from a noise source, I = p2RMS/ρc where pRMS is Root Mean Square acoustic pressure.
A similar concept is sound source strength which is the maximum rate of volume displacement [m3/s] produced by a source when emitting a harmonic sound wave.
LW [dB] = 10 × log10W + 120 [dB] where W is in [Watts]. Comparing noise source as a charge, as per Gauss's law: the total electric flux through a closed surface is zero if no charge is enclosed by the surface. Similarly, the total acoustic power through a closed surface is zero if no noise source is enclosed by the surface. Thus, sound power level measurement will not be affected by background noise if the measurement surface encompassing the noise source does not enclose source of background noise.
A-Weighting and Equal Loudness Curve: SPL measurements in terms of dB take no account of how sound pressure / power is distributed with respect to sound frequency as it represents a comparison or ratio between the quantity being measured (such as sound pressure or power) and a reference value (reference pressure or reference power). As the sound pressure level (SPL) reduces, human ear's ability o detect low or high frequency noise also diminishes. On the other hand, almost all noises are generally dependent of frequency and consist of a mix of frequencies. A complete representation is a frequency spectrum (dB vs. f curve) where noise levels are plotted as a function of noise frequencies. For purposes like comparing two noise sources, specifying noise reduction level ... a single value is typically required. A-weighted sound pressure levels in terms of dB(A) are widely used for this purpose.
Weighting comes from Fletcher-Munson Curves where 'A-weighting' is 40 Phon equal loudness contour 'B-weighting' is 70 Phon equal loudness contour and 'C-weighting' is 100 Phon equal loudness contour.
Steps to calculate A-weighting:
Noise Type, Acoustic Impedance and Directivity
The image shown above is taken from "Dell Enterprise Acoustics: A Dell Technical White Paper". In the case of tonal stimulation at varied stimulus frequency (f), the ratio of the sound pressure p(f) to the volume velocity Q(f) defines the acoustic behaviour of a system. In a linear system, a sinusoidal excitation produces sinusoidal response which may lead, lag or be in the phase of the excitation. Similarly, the responses produced by a tonal stimulus can differ from the input in terms of the magnitude which can be quantify by a root-mean-square (RMS) value, and phase (which describes the relative timing of the sinusoidal responses - leading or lagging).
Radiation into a free space such end of a duct is expressed as radiation impedance. Here 'R' is the "radiation reactance" which if the energy radiated away from the open end in the form of sound waves and 'X' is "radiation admittance" representing the mass loading of the fluid just outside the open end. As per Munjal, impedance of an unflanged open duct with plane waves propagating inside it is given by: ZUNFLANGED-END = ρc/ADUCT[0.25k2R2 + j 0.6kR].
The measurement of sound levels are performed in what is called anechoic chambers. These are rooms isolated from external sound or electromagnetic radiation sources, use soundproofing on their walls and prevents the reflection of wave phenomena. Anechoic chambers are widely used for measuring the properties of acoustic instruments, measuring of the transfer functions of electro-acoustic devices, testing microphones and performing psychoacoustics experiments. Anechoic chambers are used primarily for SPL measurements, ranking of sound sources, noise source identification and directivity of sound sources and receivers.
ISO 5801:2017 specifies procedures for the determination of the performance of fans of all types except those designed solely for air circulation, e.g. ceiling fans and table fans. Testing of jet fans is described in ISO 13350. The aeroacoustic measurements are performed using a test rig (the so-called In-Duct method) according to the industry norm ISO 5136 where farfield noise levels are recorded inside a circular duct using three slit-tube microphones. ISO 13347 deals with determination of sound power level in industrial fans under standard laboratory conditions.
Benchmarking of simulation results is also one of the important area of measurements in anechoic chamber. However, there can be deviation in test and simulated results due to omission of few noise sources in simulations. For example, in case of an axial flow fan, the deviation may come from additional noise generated from motor fan casing, noise generated by electric motor and vibrations due to unbalanced rotating masses. All these 3 modes of noise source are structure-borne which are generally omitted in CFD simulations intended to estimate fluid-borne noise. A Benchmark Case for Aerodynamics and Aero-acoustics of a Low Pressure Axia Flow Fan - SAE Technical Paper 2016-01-1805 authored by Zenger et al can be used for numerical simulations for noise in fans.
"Aerodynamically generated sound is governed by a non-linear process. One class of problems is turbulence generated noise (e.g. jet noise). An accurate turbulence model is usually needed in this case. A second class of problems involves impulsive noise due to moving surfaces (e.g. helicopter rotor noise, propeller noise, fan noise...). In these cases an Euler/Navier-Stokes model or even a full potential model is adequate, because turbulence is not important. Furthermore, because the acoustic fluctuations are usually quite small (about three orders of magnitude less than the flow fluctuations), the use of non-linear equations (whether Navier-Stokes or Euler) could result in errors. One usually has no choice but to separate the computation into two domains, one describing the nonlinear generation of sound, the other describing the propagation of sound."
Similarly, excerpts from "Implementation of Acoustical Analogies in OpenFOAM and CALFEM" by Johan Nilsson: There are several challenges to overcome when attempting to compute aero-acoustical sound. Two of these are the large energy and length scale disparity. The energy in the flow is much greater than the energy in the acoustical waves. To quote Crighton, "If all the acoustic energy radiated in the 45-second take-off of a large jet transport were recovered, that energy would be about enough to fry one egg!". Compare that to the energy in the flow which is used to lift the whole large jet aircraft to get a sense of the energy disparity.
For flow simulations intended to predict noise level, mesh needs to be generated keeping in mind upper frequency limit [say 3000 Hz] and at least 10 points per wavelength. Thus, to capture noise up to 5000 [Hz], the maximum element size should be 340 [m/s]/5000[Hz]/10 = 6.8 [mm]. During the transient flow simulation, the time histories of the instant surface pressure values needs to be written into the acoustic data files [*.asd format in ANSYS FLUENT] with which an operation can be performed for the surface zones [such as fan blades] that were selected as acoustic sources.
Excerpts from STAR-CCM+ User Guide: The physics in aero-acoustics are inherently transient and ultimately, must be solved using transient calculations. The computational run times that are associated with such simulations are high. As such transient simulations take a long time to run, optimize the mesh and settings so that you only capture noise sources and frequencies that are relevant to your analysis. An initial steady state analysis provides you with the data for optimizing the mesh and settings.
Recommended Analysis Procedure: It is recommended that aero-acoustics simulations are set up as three-dimensional cases, using compressible flow rather than constant density and run:
*In situations involving calculations of broadband noise, statistical turbulence quantities computed from [steady state] RANS equations - mean flow field [u, v, w, p], turbulent kinetic energy [k] and turbulent dissipation rate [ε] - is utilized, in conjunction with semi-empirical correlations and acoustic analogies, to estimate source of broadband noise. However, this model is limited to the broadband noise characteristics prediction and does not provide any tonal performance data.
James Lighthill in the 1950s using free flow assumptions (no reflections in solid surfaces) developed a wave equation including a source term caused by the instantaneous velocity and pressure fluctuations in the flow. He used the the same set of governing equations which describe flow field for fluids. The way to solve the acoustic wave equation is by integration of the sources using Green functions. This analogy holds true for cases like noise generated by jet engines in aeroplanes but may not be appropriate for closed volume like automotive passenger cabins. Lighthill established aero-acoustics when he developed an equation for the propagation of sound waves for a turbulent flow. Since then many extensions to Lighthill’s original theory have been made. Some of the renowned are Curle’s analogy that takes into account the presence of solid boundaries and Ffowcs-Williams and Hawkings (FWH) equation that also includes surfaces in motion.
Noise is a sort of very low amplitude pressure wave typically measured at a significant distance away from the source. To model this, a very fine mesh is needed all the way to the measuring or receiver location and beyond that to avoid reflection of sound wave. Since, the wave is a transient phenomena, a time-accurate solution of flow field is required.
The limitation of computational resources are very effectively addressed by aeroacoustic analogies by reducing the computational power needed when solving for sound propagation. Acoustic analogies helps trim the computational domain to the near field where high resolution is required and the sources are computed. All non-linear effects are prescribed to the near-field computation and a linear wave operator is used to compute propagation of the sound in the acoustic domain.
Reference: Prediction of the flow and acoustic fields generated by an isolated propeller by da Silva et. al. "Direct methods perform the noise computation in the same domain as the fluid dynamics, without any modeling for the sound. The full set of equations, Navier-Stokes or Euler, is solved in the domain of interest for both the flow and acoustic fields. This requires a domain sufficiently large in order to calculate noise propagation up to the receptor points." Further about use of libAcoustic library in OpenFOAM: "For the far-field noise computations we used a dynamic library called libAcoustics, which is integrated with OpenFOAM and was developed by Epikhin et al. and llya Evdokimov et al. This code contains acoustic analogies to be used in conjunction with CFD computations. In this work, the Ffwocs-Willians and Hawkings analogy (FW-H) based on permeable surfaces was used to compute the far-field noise. In such approach, near field pressure and velocity data are stored on an arbitrary control surface, which should encompass the most significant sound-generating regions. This surface information is used as an input for the far-field noise computations."
Turbulent flow is considered in the source region and used as excitation to the acoustic propagation. The sources are categorised as quadrupoles (within the field such as in turbulent wakes - derived based on time history of velocity field), dipoles (on stationary surfaces with flow-induced fluctuating pressure - derived based on time history of the static pressure on the surfaces), monopoles (mass-flow fluctuation across a boundary such as wall or when mass is added at an unsteady rate: in turbomachines it is called thickness noise due displacement of fluid caused by motion blades having finite thickness) and rotating dipoles (fluctuating pressure on moving surfaces such as blades of a fan and turbines, also known as loading noise - derived based on time history of the static pressure on the surfaces). A monopole source [1/s2] can be thought as point source which has a equal intensity in all directions - a radially oscillating sphere. A dipole source [N/m3] is stronger in two opposite directions and can be visualized as translational oscillating sphere.
Thickness noise: As the rotor rotates, the volume of each blade displaces fluid volume and consequently causes pressure fluctuates in near field thus noise is generated. This noise is tonal at the running frequency and generally weak for fans running at low speed say 1000-2000 [RPM].
Direct Method: Any acoustic calculation approach requires at least 2 steps: generation and propagation of sound. In direct method, both generation and propagation of sound waves are directly computed by solving the fluid dynamics equations modified to capture sound waves. This method requires very fine meshes encompassing receivers as well as acoustically non-reflecting boundary conditions.
Hybrid (CFD/CAA) Approach: This is called so because noise predictions are a two-step approach due to the large difference in requirements on the flow field and acoustic propagation. Ffowcs Williams-Hawkings formulation (bounded flow) and Lighthill's analogy (unbounded field) are examples of hybrid approach. A distinction will be made between calculating the flow field and using the data from the flow field to predict the sound field. Full-fledged CFD tools are used to find the source term and Linearized Euler Equations (LEE) is used to compute the sound propagation.
Here, unsteady near-field flow may be simulated using LES or DES solver and then acoustic analogy gives the propagation of sound into far-field (such as receiver location) using near-field sound sources as input.
CFD simulation can be either of the followings. ANSYS FLUENT recommends LES turbulence model because it resolves all eddies with scales larger than grid scale and hence wide-band aeroacoustic noise can be predicted correctly using LES model. Similarly, to calculate pressure fluctuations on the wall, PRESTO! is recommended for spatial discretization of pressure as it is more accurate scheme for interpolating face pressure values from cell pressures. FLUENT also recommends "Bounded Central Differencing" for momentum in case of LES on unstructured mesh.
Exterior Aero-acoustic Setup is subdivided in three 1)Source zone where the Actran SNGR module will compute the aeroacoustic sources. 2)Buffer zone which is the zone without sources acting as a buffer region before applying the non-reflecting boundary condition, and 3) Non-reflecting boundary condition wrapped around the buffer region by a Perfectly Matched Layer (PML) which ensures a free propagation of the acoustic solution in far field at reduced computational cost.
Reference: Towards a low-noise axial fan for automotive applications by Nicola Casari et. al. "The blade or fan self-noise component is the one generated by blades operating in a clean undisturbed flow, thus it represents the minimum noise a fan would emit, even when no installation effects are involved. Another contribution is represented by the noise due to the upstream turbulence ingested by the rotor, shed for instance by other parts of the automotive cooling module. An important mechanism involved is the trailing-edge noise, caused by the interaction of the blade turbulent boundary layer with the geometrical discontinuity represented by the "trailing edge". In addition, blade tip vortices and leakage flows may also contribute significantly to fan noise."
Broadband Noise Source Model: The broadband noise is created by turbulence effects and non-linearities in the flow and originates from various sources, including, rotor self-noise, unsteady pressure fluctuations of the blade surface, the turbulent leading or trailing edge of the boundary layer, and separation and wake effects.
Tonal Noise: The tonal noise is usually split into loading noise and thickness noise and is characterized by sharp noise band at the Blade-Passing-Frequency (BPF) and its harmonics. At high tip Mach numbers > 0.65, the thickness noise dominates the total noise levels. At lower Mach numbers, the relative importance of broadband noise becomes prevalent. The loading noise is attributed to be primarily generated by the convective amplification of the flow towards upstream direction, as well as the steady pressure differentials responsible to generate thrust such as in propellers.
End correction at neck-cavity interface
End correction at neck-duct interface: as per "Ji, Z.L. 2005, Acoustic length correction of a closed cylindrical side-branched tube, Journal of Sound and Vibration, 283, 1180-1186"
Ref: D. W. Herrin, Ph.D., P.E. University of Kentucky, Department of Mechanical Engineering: Transmission loss of a HR with anechoic termination is specified as:
HR frequency and TL calculation (with anechoic termination downstream the HR neck) is described below. The Excel file can be downloaded from here.
There are two ways to attenuate the level of sound: reflective systems - here the incident sound is scattered and cancelled by destructive interference and dissipative system - here the incident sound energy is absorbed and hence has to be converted into the heat. Expansion-contraction chambers, resonators and Herschel-Quincke tube fall under the category of reflective systems.
Transmission and Insertion Loss for a Muffler: As described in the schematic below, the gas pressure in the duct is the result of superposition of waves coming from the source and reflections from the muffler cavity, it is impossible to directly measure the incident components of the pressure pulsation into the muffler. At the same time, all acoustic frequencies of interest (typically in the range 20 ~ 2000 Hz) must be excited which requires a need of large-band acoustic source during the test. Anechoic termination is also known as reflection free termination.
TL is defined as difference in sound power between the incident wave entering and the transmitted wave exiting the muffler when termination is anechoic (no reflecting waves present - i.e. the acoustic pressure of reflective wave at the outlet is 0 - p'outlet = p'transmitted). However, inline with other definition, it is calculated as logarithm of the ratio of sound powers. Attenuation is the difference in incident and transmitted sound powers through the muffler but the termination need not be anechoic. Thus, attenuation is dependent on outlet pipe length. TL is independent of the inlet and outlet pipe length and depends only on the geometry of muffler.
Similarly, pressure loss (PL) is an important performance parameter which depends only on the shape and size of the muffler and is defined as difference between the (static) pressure drop across the muffler. The sound (or acoustic) pressure level difference across the muffler is termed as Noise Reduction (NR).
Excerpts from build.com.au/reflection-diffusion-and-absorption-sound: Reflection is often used to redirect noise from outside - consider highway barriers, which reflect traffic noise into the sky. If you can always control the way sound is reflected then this type of soundproofing can be effective. Reflective barriers are a good way to block out exterior noise.
Diffusion is a great way to prevent echoes, dispersing the sound wave in all directions when it hits an irregular surface. Think about how much of a difference carpet or a wall rug can make in a brick or concrete room. This method is very effective for high to medium frequencies, as the vibration strength is less than that of a low frequency sound, and therefore easier to disperse.
Absorption performance varies a lot based on the frequency of sound and the absorptive capabilities of the material. A commonly used sound absorber is the underlay in carpet; this works to draw energy from the sound wave and convert it into a tiny amount of heat, creating that ‘deadening’ of a sound. Absorption works best in mid to high frequencies - lower frequency sounds can push through with more force.
Broadband noise source calculated in ANSYS FLUENT for flow over a bluff body is shown below.
OpenFOAM-dev has a library called libAcoustics far-field noise computation which can be accessed from here. incompressible\ pimpleFoam\ LES\ vortexShed contains example of noise calculation. Martin Heinrich from Institute of Mechanics and Fluid Dynamics, Freiberg University of Mining and Technology has implement the modified Curle acoustic analogy presented by Larsson et al. as functionObject for OpenFOAM 3.0.x. The method is implemented for incompressible cases including the surface and volume integrals. The library is available at https://github.com/Kiiree/curleAnalogy The description of this implementation as per slides prepared by Aya Aihara from Division of Electricity, Uppsala University, Sweden can be found here.
Both ANSYS Mechanical and COMSOL uses PML (perfectly matched layer) boundary type which emulates a non-reflecting boundary condition independent of the shape and frequency of the incident wave front.
It also has a FSI module for (vibro-acoustics) acoustic-structure interaction to model multiphysics phenomenon where the acoustic pressure causes a fluid pressure on the solid surface and the structural acceleration (surface velocity) acts on the fluid domain as a normal acceleration across the fluid-solid boundary.
STAR-CCM+ has FW-H model that can be used with transient flow field and requires FW-H surfaces and locations of the FW-H receivers (points at specific distances from the noise sources). Then FFT (Fast Fourier Transforms) is used to determine sound pressure as a function of frequency at each of the receiver(s). This is described as "Point Time Fourier Transform G[p]". Similar conversion from time domain to frequency domain can be performed for a line and surface using "Line Time Fourier Transform G[h[l]]" and "Surface Time Fourier Transform G[h[s]]" respectively.
A generic approach is described for an automotive engine. Note that this method is based on a weak fluid-structure coupling that is one way coupling in structural vibration and acoustic field. Thus, this is not a typical FSI (Fluid-Structure Interaction) approach.
One dimensional, plane wave propagation is assumed and hence method is accurate only below cut-on frequency (defined as the frequency below which only plane waves propagate inside a duct). For a duct of diameter D, fCUT-ON = 1.8412 × c/πD [ref: A. D. Pierce: Acoustics – An Introduction to its Physical Principles and Applications]. Similarly, cut-off frequency can be defined as the frequency above which no plane waves propagate inside a duct. For elliptical ducts, fCUT-OFF = β × c/2πa where 2a = major diameter, 2b = minor diameter, β = f(e), e = sqrt(1-b2/a2)
TMM for a duct:
Sudden expansion and contraction: from continuity and equality of pressure on the two sides of the expansion or contraction plane, p(0) = p(L) and u(0)A(0)=u(L)A(L). Thus, transfer matrix is given by:
Frequency bands for octave and 1/3-octaves
The frequency weighting correction value in tabulate form is as follows:
The continuity or mass conservation equation is given by:
X-Momentum equation:
Y-Momentum equation:
Z-Momentum equation:
Other variants of this equations have evolved such as Ffowcs-Williams and Hawkings equation that includes surfaces in motion. Thus, FWH analogy is used in solid surface interactions that are directly involved in the generation of flow sound, for example those due to rotors of a helicopter, propellers driving aeroplanes and marine vessels and prime movers like fans, compressors and turbines.
Similarly, a special version of FWH analogy is the Curle’s analogy that takes into account presence of solid rigid boundaries where only the dipole source terms takes care of the sound scattering caused by the stationary surface.
[6]In addition to analogies presented above, numerous other analogies have been developed: Phillips’ analogy and Lilley’s analogy are based on scaled logarithmic pressure Π = 1/γ·ln[p'/p0]. These are also known as phi-based analogy and have been used as a starting point for aero-acoustics calculations of jet engines.
Like any other acoustic analogy, here also the different scales of hydrodynamics and acoustics are handled by spliting the flow field into incompressible mean flow field and compressible acoustic part, also known as HAS (Hydrodynamics Acoustics Splitting) method. Thus: p = pIC + pAC, u = uIC + uAC and ρ = ρ0 + ρAC where speed of sound 'c' is defined as ρAC = c02pAC.
Explanation on Green's function
Flow chart of OpenFOAM-Acoustic-Solver acousticFoam for incompressible simulations: reference - Implementation of Acoustic Analogies in OpenFOAM for Computation of Sound Fields by J. Schmalz, W. Kowalczyk, Chair of Mechanics and Robotics, University of Duisburg-Essen, Germany
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