Early Advancements in Turbulence-Generated Noise Modelling: A Review

Abstract

Turbulent flows generate a broadband of acoustic noise, which can be extremely important. So, there is need for modelling the generation and propagation of acoustic energy in fluid flows, especially turbulent. This chapter reviews the research work conducted to identify and quantify the noise field generated in turbulent flows. The story starts with the journey of experimental identification and measurement of noise generated from vortices. Various analytical models there were developed, soon after, the popularity of turbulence generated is discussed. The base path-breaking research on quantifying noise generation from conservation laws including Navier–stokes equations is discussed and further used for approximation of acoustic intensity by acoustic analogy with electrostatic quadrupole near-field and far-field. With the development of computational numerical techniques flow field for complex geometries and higher fidelity became possible. The candidates for relevant computational methods are touched and integration with turbulent models is discussed. Finally, a case of simulation of noise generation for turbulent flow over airfoil using acoustic equations and Reynolds-averaged Navier-Stokes (RANS) turbulent model is reviewed.

1. Introduction

Not just the fans, mechanical components and motors in machines make noise. Noise is also generated from turbulent flows. In fact, these noises are quite common as turbulence is quite common by itself. Starting with milder cases, the sound we hear on the beach is purely due to turbulence. The flowing turbulent fluid through air vents found in buildings and vehicles for heating, ventilation, and air conditioning, commonly called HVAC applications, generates irritating noise affecting normal lifestyle as well as sleep. Denoising in HVAC is a wide spread business in cities as every other building creates a broadband humming noise from their piping systems and chimneys. In fact, modern cars are designed for streamlined flows across the vehicle to reduce turbulence, subsequently making it more fuel efficient and silent. Opening the windows or sun roof makes the boundary layer flow turbulent and that is the the cause behind the rhythmic noise formation, which mostly experienced in highways once the vehicle crosses a certain speed. In industries, combustion and flame also create turbulence and, hence, generate a broadband and some time-rattling noise from combusters, furnace, and chimney towers. These can be annoying. But are surely annoying in many cases. The early sirens or sirens used in ships and trains were almost completely turbulence generated obtained by the outflows of high-speed turbofans. The Reynolds number has been a well-known parameter to gauge the extremity or severity of turbulence. So, in cases where we encounter flows with larger Reynolds number, like flow through high-speed fans, impellers and propeller, propulsion, jets, ballistics, rockets, aircrafts, etc., it is not just annoying rather unhealthy for physicality and mentality of living beings. Journalists and activists as well as researchers have been raising concerns about noise from helicopters and flights harming residents around helipads and airports [1, 2]. Recently, drones are new additions to the list of generators of loud noise. Researchers are working on reducing such noise as well as psychoacoustic testing for understanding the effect on human psychology [3, 4].

2. Historical development on flow acoustics pre-1950

Acoustics as a branch of physics has been being studied for over two and a half 1000 years. Most of the eminent physicists have studied acoustics. Starting from Aristotle who concluded wave nature and periodic compression and rarefaction of acoustic energy transmission. Galileo and Mersenne concluded vibration to be the cause behind acoustic waves, and they both independently laid the foundational theory of vibration. Soon Newton gave the relationship for wave velocity in solids. Then, a series of development followed with the works by Euler, Lagrange, d’Alembert, and Hemlholtz, who laid the mathematical foundation for acoustics in solid and fluid mediums. First John Tyndall in 1869 brought together the past work with his incremental work on singing flames [5]. It was not clearly understood turbulent flow could also have been a reason along with resonance. Soon after Lord Rayleigh made an extensive study on acoustic behaviour in a wide range of setups and configurations in fluid and solid, whose work is considered monumental till date and has propelled the research on acoustics dramatically [6, 7].

Physicists had been studying various aspects of fluid flows. It was not known that fluid flows could produce sound. Many experiments were being conducted to study wakes and vortices. Luckily, Strouhal in 1878 and Kohlrausch in 1881, quite around the same time, found out about a faint sound originating from vortices, to which the latter described as “reibungstone” [8, 9]. This began to motivate physicists to think about the acoustic aspects of flowing fluids. For quite a long time, numerous works on the new topic started appearing. Mostly experimental and empirical quantification of noise for different cases of fluid flows, like boundary layers, pipe flow, jets, flow past solids, started getting published. Almost all of them used flow parameters like velocity, viscosity, tip speed, non-dimensional quantities like Reynolds number, Strouhal number, etc. [10, 11, 12, 13]. Though a wide range of theoretical studies were being conducted, somehow, study on understanding and quantifying the noise generation in fluid flow from first principles was missing.

3. Lighthill’s acoustic analogy

Sir James Lighthill, in 1950s, discovered the theoretical connection between fluid flow and acoustics from the conservation laws to derive the wave equation for acoustics [14, 15]. Rayleigh had initiated something in his “Theory of Sound” regarding the origin of sound with which Lighthill could conclude on three ways in which kinetic energy could transform to acoustic energy. They are as follows:

1. By forcing the mass in a fixed region of space to fluctuate, as in a loudspeaker.

2. By forcing the momentum in a fixed region of space to fluctuate, or, which is the same thing, forcing the rates of mass flux across fixed surfaces to vary; both these occur when a solid object vibrates after being struck.

3. By forcing the rates of momentum flux across fixed surfaces to vary, as when sound is generated aerodynamically with no motion of solid boundaries.

Based on these three statements, Lighthill could recognise that if flow field generates acoustic energy, then their sources could be derived from Navier-Stokes equation. He creates an experiment assuming a patch of turbulent flowing fluid surrounded by a large domain of surrounding stationary fluid. Let turbulent flow produces noise; however, the noise would transmit to the surrounding fluid at rest. By rearranging and comparing terms in the conservation equations for stationary fluid, the resulting equation could be written as a forced bidirectional wave equation. The forcing term or the source term is what generates noise in flows. Here is the derived wave equation in Einstein’s notations,

$$\frac{{\partial }^2\rho }{{\partial }^{2}t}-c_o^2{\nabla }^2\rho =\frac{{\partial }^2{T}_{\mathit{ij}}}{\partial x_i\partial x_j}$$E1

$$T_{\mathit{ij}}=\rho v_i{v}_j-{\sigma }_{\mathit{ij}}+\left(p-c_o^2\rho \right){\delta }_{\mathit{ij}}$$E2

where T is called Lighthill’s turbulence stress tensor and has three components or three sources for noise generation. $\rho v_i{v}_j$ is the convection of momentum fluctuation, ${\sigma }_{\mathit{ij}}$ is the viscous stress, and $\left(p-c_o^2\rho \right){\delta }_{\mathit{ij}}$ is the difference in exact p and approximated thermodynamic pressure, $c_o^2\rho .\rho$ is the density, $c_o$ is the speed of sound, t is the the time dimension, x is the spatial dimension, and v is the velocity. This equation quantifies sound sources from flowing fluid where it takes care of turbulent fluctuations, and viscous dissipation thermodynamics jumps. Now the scientific community is enabled to find acoustic intensity field very accurately from the first principle. This is a remarkable breakthrough in the field of fluid dynamics, which also made Lighthill the father of aeroacoustics. It had been a couple of years with industrial aircrafts in use and both propellers and jet engines produced strong broadband noise. It was a big concern for aerospace researchers and manufacturers, especially during world wars and also after the commercialization of air transport that required more powerful engines to make them faster and carry heavier payloads. In a time of high demand, Lighthill’s quantification showed everyone how to use the quantified noise source to find the intensity field right away.

3.1 Lighthill’s eighth power law

The major benefit of getting this governing equation, which in fact made it so popular by the name acoustic analogy, is the analogy with electrostatics. In a turbulent flow, what vortex stretching creates are acoustic dipoles as the shear creates compression on one side, while rarefaction on the other side. In fact, these dipoles often occur in pairs as vortex stretching creates circulations in couple to balance forces, momentum, and angular momentum. Lighthill could safely assume a turbulent flow field to be distributed acoustic quadrupoles of different powers based on the velocities and vorticity, which typically increases as the size of eddies decreases during vortex stretching due to the conservation of angular momentum. Lighthill could exploit the existing formulations for electrostatic monopoles, dipoles, quadrupoles, etc. That is the base for the eighth power law (Figure 1).

Figure 1.

Theoretical directivity patterns for far-field sound pressure levels radiated from lateral quadrupole, commonly found in vortices [16].

A well-established study on electrostatics and electromagnetic radiation theories could be applied to a great extent. A result of that was the Lighthill’s Eighth Power Law. Due to the mathematical similarity with electromagnetic radiation theory, the similarity with Stefan-Boltzmann’s can also be established where intensity is proportional to the fourth power of temperature. Also, thermodynamic speed is proportional to the root of temperature. So, essentially that gives the intensity of sound radiation is proportional to the eighth power of speed. In aeroacoustics, Lighthill’s eighth power law states that the power of the sound created by a turbulent motion, far from the turbulence, is proportional to the eighth power of the characteristic turbulent velocity. We could confer the participating variables, and using dimensional analysis the relation could be found out as

$$W=K\frac{{\rho }_o}{c_o^5}{L}^2{U}^8$$E3

where W is the acoustic power in the far-field, K is the proportionality constant (or Lighthill’s constant), ${\rho }_o$ is the uniform fluid density, $c_o$ is the speed of sound, L is the characteristic length scale of the turbulent source, and U is the characteristic velocity scale of the turbulent source. The value of K could be determined experimentally. Depending on a case like hot or cold jet or Mach number range or flow restrictions, the values for K was to be determined experimentally. The eighth power is experimentally verified and found to be accurate for low speed flows, i.e., Mach number is small, $M<1$. And also, the source has to be compact to apply this law.

3.2 Statistical extension for turbulent jets

After sometime, Lighthill released his statistical work on theories for turbulent cases as an extension to the previous publication [15]. Soon after the theoretical work, acoustic power coefficient (K) and acoustic efficiency $\left(\eta \right)$ were identified from cold jet and hot jet experiments. The experiments as mentioned in Refs. [17, 18, 19, 20, 21] were appreciated and used by Lighthill for finding the K and $\eta$.

$$K=\frac{W}{{\rho }_o{L}^2{U}^8{c}_o^{-5}}$$E4

$$\eta =\frac{W}{\frac{1}{2}{\rho }_o{U}^3\frac{1}{4}\pi L^2}$$E5

An important inference from the latter work was order determination for K and $\eta$. The K for sub-sonic jet is roughly in the order of ${10}^{-4}$ and $\eta$ for sub-sonic cold jet is roughly in the order of ${10}^{-4}{M}^5$ where M is the orifice Mach number.

4. Sixth power law and later developments

It was understood that once fluid flow becomes chaotic, or turbulent, sharp temporal fluctuation in pressure is often evident due to vortex stretching. Since fluid is a material medium, there is a temporal fluctuation in pressure, and acoustic noise is born. Turbulence happens with eddies of various scales of length and time; hence, an evidence of a broad band of noise is very much logical. With higher Reynolds number, the loudness scales up nonlinearly. These acoustic waves can transmit through walls and can be a real problem, particularly, if it is loud which is the case for missiles, rockets, submarines, and aircraft. Hence, the study and analysis of noise generated from turbulence has been a topic of growing interest.

The vortex theory of aerodynamic sound generation explains how sound is produced by fluid flows through the action of vorticity. This theory is supported by rigorous mathematics and is consistent with Lighthill’s theory. At low speeds, the theory predicts sound-power output proportional to the sixth power of the flow velocity with the similarity method and simplification, while at higher speeds, an eighth-power law is observed [22, 23]. The theory has practical applications in engineering, and it is relatively easy to calculate the sound field of a flow based on the motion of the vorticity alone. The theory could be successfully applied to a wide range of flow problems, including aeolian tones, oscillatory flow, spinning-vortex problems, and turbulent shear and jet flows. Almost all the cases in hydrodynamics fall in this regime.

Lighthill’s work helped build models for different cases of turbulent jets. Validation of physics using Lighthill’s analogy became popular. Yet the most influential research in subsequent years in the field was built on Lighthill’s analogy. Its applications on simplified cases like, incompressible jet, inviscid jet etc. [24] Curle-Lighthill [25] Ffowcs-Hawking [26] Analogies for solid boundaries (stationary and moving, respectively) became the base for integral methods in computational aeroacoustics.

5. Computational aeroacoustics

Potential field-based velocity field by linearity assumption is insufficient for complex geometries and inaccurate. Computational models are essential for high-fidelity simulations, especially for reacting flows, multi-physics, multi-phase, complex constraints cases, etc. Also no new pathbreaking theoretical development was seen ahead of quadrupole analogy and acoustic source quantification. Nonlinear problems were tough to solve (shocks, reacting flows), yet were valid for finite approximation as the equations are derived from first principle. These methods boomed after mid 1980s as computational fluid dynamics using finite volume and finite element boomed during the same time.

A few popular computational techniques had linearity assumption that was suitable for fluid at almost rest. Hence, fast and lightweight techniques that were popular in general acoustic simulations like ray tracing, finite difference time domain (FDTD), and boundary element method (BEM) did not prove to be very effective in typical turbulent aeroacoustics. Other fast and lightweight techniques are machine learning based reduced order models, which are catching attention these days. There are linear models like proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) that are being used for turbulent combustion. Nonlinear models like deep neural networks are also very efficient. However, these machine learning models require a bulky lot of appreciable quality simulated or experimental data to develop the model and training (regression) of the model takes a lot of computational time and resources. So, justified use of machine-learning based reduced order model is a hot debate. Hybridised full-order discretized numerical techniques using finite volume (FV), finite element (FE), spectral methods, etc., with turbulence (with URANS, LES, DES, DNS, etc.) and acoustic models (analogy-based integral methods) have been the best choice till date.

Later decomposition scheme is by Ewert and Schröder [27], an essential technique in hybrid computational aeroacoustics. They describe an approach for simulating acoustic fields in space and time by deriving a family of acoustic perturbation equations (APEs) that predict acoustic sources based on unsteady flow simulation. Their APE formulations are shown to be stable and effective for predicting convection effects in mean flows with vorticity. Different source term formulations have been derived for incompressible and compressible flow solutions, with the vortex sound source being the time derivative of the incompressible pressure or the perturbation Lamb vector. The accuracy of the APE system in predicting convection effects has been tested against solutions of the linearized Euler equations for a monopole in a sheared mean flow and for a spinning vortex pair, showing good agreement. The hybrid approach could be compared with highly resolved unsteady CFD simulation for a laminar flow over a cylinder, and the APE systems yield convincing results for the structure of the pressure contours and decay of pressure with increasing distance from the cylinder.

6. Shen-sørensen aeroacoustic modelling

A numerical algorithm for acoustic noise generation is extended to handle turbulent flows [28]. It comprises of two steps: one for viscous incompressible flow and an other for acoustics of inviscid system. The acoustic part could start at any time during the incompressible computation. The flow is split into Reynolds-averaged component and a component corresponding to the turbulent small-scale fluctuations solved by an eddy viscosity model. The eddy viscosity model called Baldwin-Barth one equation model is used to approximate the noise source.

Case-1: Two-dimensional laminar flow past a NACA 0015 airfoil at an angle of attack equal to 20 degrees at Re = 300 and Mach number 0.2 (Figures 2, 3).

Figure 2.

Directivity pattern of NACA0015 airfoil noise radiation at Re = 300, M = 0.2, and distance = 12.

Figure 3.

FFT spectrum of the incompressible vorticity signal for laminar flow past a NACA0015 airfoil Re = 300.

Case-2: Two-dimensional turbulent flow past a NACA 0015 airfoil at an angle of attack equal to 20 degrees at $Re=1.5\mathrm{x}{10}^6$ Mach number 0.2 (Figures 4, 5).

Figure 4.

Directivity pattern of NACA0015 airfoil noise radiation at $\mathit{Re}=1.5x{10}^6,$ M = 0.2, and distance = 12.

Figure 5.

FFT spectrum of the incompressible vorticity signal for turbulent flow past a NACA0015 airfoil $\mathit{Re}=1.5x{10}^6$.

Comparing the two cases of flow across an airfoil, it could be noticed that acoustic noise is dominated by the Strouhal frequency and its harmonics. The laminar flow case could be validated with Lighthill acoustic analogy. Regarding the noise level comparison, although turbulent flow is noisier, however both turbulent and laminar cases have the same order of noise level. Only one frequency and its associated higher harmonics were observed for RANS-based model. So, it could be inferred, that needs LES or DNS that captures more frequencies.

7. Conclusions

In conclusion, the study of aerodynamic noise generation has made significant progress in recent years, with a focus on understanding the sources of noise and developing effective computational models for predicting and mitigating noise. Through the analysis of conservation laws, researchers have gained insight into the mechanisms behind both broadband noise and sharp pulses and their harmonics. Lighthill’s pioneering work laid the foundation for current computational turbulence acoustic modelling, and ongoing research has led to useful constraints and treatments for modelling complex cases. Despite the annoyance of turbulence-induced noise, these advances bring hope for developing more efficient and quieter technologies in various fields.

Conflict of interest

The authors declare no conflict of interest.