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21.2.2 Broadband Noise Source Models

Proudman's Formula

Proudman [ 293], using Lighthill's acoustic analogy, derived a formula for acoustic power generated by isotropic turbulence without mean flow. More recently, Lilley [ 210] rederived the formula by accounting for the retarded time difference which was neglected in Proudman's original derivation. Both derivations yield acoustic power due to unit volume of isotropic turbulence (in W/m $^3$) as

 P_A = \alpha \rho_0 \left(\frac{u^3}{\ell}\right) \frac{u^5}{a_0^5} (21.2-10)

where $u$ and $\ell$ are the turbulence velocity and length scales, respectively, and $a_0$ is the speed of sound. $\alpha$ in Equation  21.2-10 is a model constant. In terms of $k$ and $\varepsilon$, Equation  21.2-10 can be rewritten as

 P_A = \alpha_{\varepsilon} \rho_0 \varepsilon M_t^5 (21.2-11)


 M_t = \frac{\sqrt{2 k}}{a_0} (21.2-12)

The rescaled constant, $\alpha_{\epsilon}$, is set to 0.1 in FLUENT based on the calibration of Sarkar and Hussaini [ 316] using direct numerical simulation of isotropic turbulence.

FLUENT can also report the acoustic power in dB, which is computed from

 L_P = 10 \log \left(\frac{P_A}{P_{\rm ref}}\right) (21.2-13)

where $P_{\rm ref}$ is the reference acoustic power ( $P_{\rm ref} = 10^{-12} W/m^3$ by default).

The Proudman's formula gives an approximate measure of the local contribution to total acoustic power per unit volume in a given turbulence field. Proper caution, however, should be taken when interpreting the results in view of the assumptions made in the derivation, such as high Reynolds number, small Mach number, isotropy of turbulence, and zero mean motion.

The Jet Noise Source Model

This source model for axisymmetric jets is based on the works of Goldstein [ 125] who modified the model originally proposed by Ribner [ 305] to better account for anisotropy of turbulence in axisymmetric turbulent jets.

In Goldstein's model, the total acoustic power emitted by the unit volume of a turbulent jet is computed from

$\displaystyle P_A(\vec{y})$ $\textstyle =$ $\displaystyle \int_0^{2 \pi} \int_0^{\pi} I(r, \theta; \vec{y}) r^2 \sin \theta d\theta \, d\psi$  
  $\textstyle =$ $\displaystyle 2 \pi r^2 \int_0^{\pi} I(r, \theta; \vec{y}) \sin \theta \, d\theta$ (21.2-14)

where $r$ and $\theta$ are the radial and angular coordinates of the receiver location, and $I(r, \theta; \vec{y})$ is the directional acoustic intensity per unit volume of a jet defined by

 I(r, \theta; \vec{y}) = \frac{12 \, \rho_0 \; \omega_f^4 \, ... ...frac{\partial U}{\partial r}\right)^2\frac{D_{\rm shear}}{C^5} (21.2-15)

$C$ in Equation  21.2-15 is the modified convection factor defined by

 C = 1 - M_c \cos \theta (21.2-16)

$\displaystyle D_{\rm self}$ $\textstyle =$ $\displaystyle 1 + 2(\frac{M}{9} - N) \cos^2 \theta \, \sin^2 \theta$  
  $\textstyle +$ $\displaystyle \frac{1}{3}\left[\frac{M^2}{7} + M - 1.5 N (3 - 3 N + \frac{1.5}{\Delta^2} - \frac{\Delta^2}{2})\right] \, \sin^4 \theta$ (21.2-17)
$\displaystyle D_{\rm shear}$ $\textstyle =$ $\displaystyle \cos^2 \theta \left[\cos^2 \theta + \frac{1}{2}\left(\frac{1}{\Delta^2} - 2 N\right) \sin^2 \theta\right]$ (21.2-18)

The remaining parameters are defined as

$\displaystyle \Delta$ $\textstyle =$ $\displaystyle \frac{L_2}{L_1}$ (21.2-19)
$\displaystyle M$ $\textstyle =$ $\displaystyle \left[\frac{3}{2}\left(\Delta -\frac{1}{\Delta}\right)\right]^2$ (21.2-20)
$\displaystyle N$ $\textstyle =$ $\displaystyle 1 - \frac{\left(\overline{u_{t2}^2}\right)}{\left(\overline{u_{t1}^2}\right)}$ (21.2-21)

$\displaystyle L_1$ $\textstyle =$ $\displaystyle \frac{\left(\overline{u_{t1}^2}\right)^{3/2}}{\epsilon}$ (21.2-22)
$\displaystyle L_2$ $\textstyle =$ $\displaystyle \frac{\left(\overline{u_{t2}^2}\right)^{3/2}}{\epsilon}$ (21.2-23)
$\displaystyle \omega_f$ $\textstyle =$ $\displaystyle 2 \pi \frac{\epsilon}{k}$ (21.2-24)

where $\overline{u_{t1}^2}$ and $\overline{u_{t2}^2}$ are computed differently depending on the turbulence model chosen for the computation. When the RSM is selected, they are computed from the corresponding normal stresses. For all other two-equation turbulence models, they are obtained from
$\displaystyle \overline{u_{t1}^2}$ $\textstyle =$ $\displaystyle \frac{8}{9} k$ (21.2-25)
$\displaystyle \overline{u_{t2}^2}$ $\textstyle =$ $\displaystyle \frac{4}{9} k$ (21.2-26)

FLUENT reports the acoustic power both in the dimensional units ( $W/m^3$) and in dB computed from

 L_P = 10 \log \left(\frac{P_A}{P_{\rm ref}}\right) (21.2-27)

where $P_{\rm ref}$ is the reference acoustic power ( $P_{\rm ref} = 10^{-12} W/m^3$ by default).

The Boundary Layer Noise Source Model

Far-field sound generated by turbulent boundary layer flow over a solid body at low Mach numbers is often of practical interest. The Curle's integral [ 71] based on acoustic analogy can be used to approximate the local contribution from the body surface to the total acoustic power. To that end, one can start with the Curle's integral

 p' ({\vec x},t) = \frac{1}{4\pi a_0} \int_S \frac{(x_i - y_i... ...\frac{\partial p}{\partial t} ({\vec y}, \tau) \, dS({\vec y}) (21.2-28)

where $\tau$ denotes the emission time ( $\tau = t - r/a_0$), and $S$ the integration surface.

Using this, the sound intensity in the far field can then be approximated by

 \overline{p'^2} \approx \frac{1}{16 \pi^2 a_0^2} \int_S \fra... ...{\vec y},\tau)\right]^2} {\cal A}_c ({\vec y}) \; dS({\vec y}) (21.2-29)

where ${\cal A}_c$ is the correlation area, $r \equiv \vert{\vec x} - {\vec y}\vert$, and $\cos \theta$ is the angle between $\vert{\vec x} - {\vec y}\vert$ and the wall-normal direction ${\vec n}$.

The total acoustic power emitted from the entire body surface can be computed from

$\displaystyle P_A$ $\textstyle =$ $\displaystyle \frac{1}{\rho_0 a_0} \int_0^{2\pi} \int_0^{\pi} \overline{p^{\prime^2}} r^2 \sin \theta \; d\theta d\psi$  
  $\textstyle =$ $\displaystyle \int_S I(\vec{y}) \; dS(\vec{y})$ (21.2-30)

 I(\vec{y}) \equiv \frac{{\cal A}_c(\vec{y})}{12 \rho_0 \pi a_0^3} \overline{\left[\frac{\partial p}{\partial t}\right]^2} (21.2-31)

which can be interpreted as the local contribution per unit surface area of the body surface to the total acoustic power. The mean-square time derivative of the surface pressure and the correlation area are further approximated in terms of turbulent quantities like turbulent kinetic energy, dissipation rate, and wall shear.

FLUENT reports the acoustic surface power defined by Equation  21.2-31 both in physical ( $W/m^2$) and dB units.

Source Terms in the Linearized Euler Equations

The linearized Euler equations (LEE) can be derived from the Navier-Stokes equations starting from decompositions of the flow variables into mean, turbulent, and acoustic components, and by assuming that the acoustic components are much smaller than the mean and turbulent components. The resulting linearized Euler equations for the acoustic velocity components can be written as

\frac{\partial u_{ai}}{\partial t} + U_j\frac{\partial u_{ai}... ...c{\rho_a}{\overline{\rho}^2 }\frac{\partial P}{\partial x_i} =

 \underbrace{- U_j\frac{\partial u'_i}{\partial x_j} - u'_j \... ...partial t} + \frac{\partial}{\partial x_j}\overline{u'_j u'_i} (21.2-32)

where the subscript " $a$'' refers to the corresponding acoustic components, and the prime superscript refers to the turbulent components.

The right side of Equation  21.2-32 can be considered as effective source terms responsible for sound generation. Among them, the first three terms involving turbulence are the main contributors. The first two terms denoted by $L_{sh}$ are often referred to as "shear-noise" source terms, since they involve the mean shear. The third term denoted by $L_{se}$ is often called the "self-noise" source term, for it involves turbulent velocity components only.

The turbulent velocity field needed to compute the LEE source terms is obtained using the method of stochastic noise generation and radiation (SNGR) [ 30]. In this method, the turbulent velocity field and its derivatives are computed from a sum of $N$ Fourier modes.

 {\vec u}\left({\vec x}, t\right) = 2 \sum_{n=1}^{N} \tilde{u... ...left({\vec k}_n \cdot {\vec x} + \psi_n\right) {\vec \sigma}_n (21.2-33)

where $\tilde{u}_n$, $\psi_n$, ${\vec \sigma}_n$ are the amplitude, phase, and directional (unit) vector of the $n^{\rm th}$ Fourier mode associated with the wave-number vector ${\vec k}_n$.

Note that the source terms in the LEE are vector quantities, having two or three components depending on the dimension of the problem at hand.

Source Terms in Lilley's Equation

Lilley's equation is a third-order wave equation that can be derived by combining the conservation of mass and momentum of compressible fluids. When the viscous terms are omitted, it can be written in the following form:

 \frac{D}{Dt}\left[\frac{D^2\Pi}{Dt^2} - \frac{\partial}{\par... ...{\partial u_j}{\partial x_k} \frac{\partial u_i}{\partial x_j} (21.2-34)

where $\Pi = (1/\gamma) \ln \frac{p}{p_o}$.

Lilley's equation can be linearized about the underlying steady flow as

 u_i({\vec x}, t) = U_i({\vec x}) + u'_i({\vec x}, t) (21.2-35)

where $u'({\vec x}, t)$ is the turbulent velocity component.

Substituting Equation  21.2-35 into the source term of Equation  21.2-34, we have

$\displaystyle S$ $\textstyle \equiv$ $\displaystyle -2 \frac{\partial u_k}{\partial x_i} \frac{\partial u_j}{\partial x_k} \frac{\partial u_i}{\partial x_j}$  
  $\textstyle =$ $\displaystyle -2 \frac{\partial U_k}{\partial x_i} \frac{\partial U_j}{\partial... ..._k} \frac{\partial U_i}{\partial x_j}}_{\mbox{Shear-Noise Terms}} \phantom{XXX}$ (21.2-36)

The resulting source terms in Equation  21.2-36 are evaluated using the mean velocity field and the turbulent (fluctuating) velocity components synthesized by the SNGR method. As with the LEE source terms, the source terms in Equation  21.2-36 are grouped depending on whether the mean velocity gradients are involved ( shear noise or self noise), and reported separately in FLUENT.

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