Engineering Acoustics/Noise from turbine blades

Introduction
Sound prediction in fluid flows is hard to predict because of the non-linearity of governing equations. The sound production occurs at high Reynolds number, where the nonlinear inertial terms are much higher than the viscous terms. The sound production is a very small portion of energy in fluid flow, especially in open space for subsonic flows. Aeroacoustics provide approximations of such flows, and the difference between the actual flow and the reference flow is identified as the source of sound. The sound field is obtained through the Green's function, in which the Green's function is the linear response of the fluid flow to an impulsive sound source, expressed in delta function of space and time. The Green's function is as below: $$\frac{1}{c_0^2}\frac{d^2G}{dt^2} =\delta(x-y)\delta(t-\tau)$$ Aeroacoustics is a field of study that focuses on sound from fluid flow, and is often used to predict sound in turbine flows.

Blade displacement noise(monopole)
Blade displacement noise is a monopole source of sound, and can be severe for turbomachinery and helicopter blades. The simplest model of a monopole is a radially expanding sphere. In an infinite homogenous medium, a pulsating sphere will produce a spherical wave as below: $$p(r,t) = (A/r)e^{j(wt-kr)}$$ where A is determined by an approximate boundary condition. In a sphere of average radius a, vibrating radially in a complex speed $$U_0{exp(jwt)}$$ The specific acoustic impedance for the spherical wave is $$z(a) = \rho_0{c}{cos\theta_a}e^{j\theta_a}$$ Where $$cot{\theta_a} = ka$$. The pressure of the surface is then $$p(a,t) = {\rho_0}{cU_0}cos\theta_a}{e^{i(wt-ka+\theta_a)}$$ Then A becomes $$A = {\rho_0}{cU_0}cos\theta_a}{e^{i(ka+\theta_a)}$$ So, the pressure at any distance r>a is $$p(r,t) = {\rho_0}{cU_0}cos\theta_a}{e^{i(wt-k(r-a)+\theta_a)}$$

Tonal noise(Dipole)
Tonal noise at blade passing frequency(BPF) is an example of a dipole source. While volume displacement is a monopole source, fluctuating pressures is a dipole source, and unsteady Reynolds stress or transport of momentum would be a quadrupole source. The fluctuating blade pressures (dipoles) are always an important source of sound for rotating machinery. Steady rotating forces and unsteady rotating forces would classify as dipole blade forces, and examples of these are uniform stationary inflow and non-uniform stationary inflow, non-uniform unstationary inflow, vortex shedding and secondary flows. If two monopole sources of equal strength but opposite direction are close enough, it resembles a dipole. A rigid sphere whose center is oscillating back and forth is another example of a dipole. The net force exerted on the fluid by the sphere, in accordance to Newton’s third law, is the surface integral of $$ p(a,e,t)e_r$$. Symmetry requires that this force have only a z component, so the force is  $$F(t) = F_z(t)e_z = e_z{a^2}\int_{0}^{2\pi}\int_{0}^{\pi}p(a,\theta,t)\cos{\theta}\sin{\theta}{d\theta}dz$$

Noise from wind turbine blade(Flutter)
Flutter has been a problem traditionally related to compressor and fan blades. Over the years fan blades has decreased in blade and disc thickness and increased in aspect ratio, in efforts to increase it lift coefficient. This leads to the decrease in blade stiffness of the bladed disc assembly and its natural frequencies, and as a result can lead to flutter motion. Flutter boundaries are very sensitive to mode-shapes, and reduced frequencies play a secondary role. Flutter would bring pressure fluctuations and would be a source of dipole sound.



Noise from gas turbines
In a gas turbine, there are three main sources of noise such as intake, exhaust, and casing. Intake noise is created by the interaction of the axial air compressor rotor and stator, and is a function of blade number, tip speed, and pressure increase. Intake noise is less than exhaust noise in overall, but its high frequency content sounds are much larger than that of the exhaust noise. Exhaust noise has higher amplitude and has lower frequency due to combustion process. Typically, the inlet and exhaust sound power levels range from 120 dB to over 155 dB. Casing noise is generated through high speed misaligned mechanical components in the turbine housing radiating to the outer casing. In principle, gas turbine noise come from aerodynamic sources. High aerodynamic turbulence are combustion are present in the operation of gas turbine. Combustion would be a monopole source of sound along with rotating shock waves. Dipole sources of sound would mainly be from fluctuating forces on blades and guide vanes, and free jets would be a quadrupole noise.

External References

 * 1) Pierce, A. D., & Beyer, R. T. (1990). Acoustics: An Introduction to Its Physical Principles and Applications. 1989 Edition. 2.	Kinsler, L. E., Frey, A. R., Coppens, H. B., Sanders, J. V., & Saunders, H. (1983). Fundamentals of acoustics.
 * 2) Kinsler, L. E., Frey, A. R., Coppens, H. B., Sanders, J. V., & Saunders, H. (1983). Fundamentals of acoustics.
 * 3) http://www.sandia.gov/
 * 4) http://www.sonobex.com/gas-turbines/