[Fluent Inc. Logo] return to home search
next up previous contents index

21.2.1 The Ffowcs Williams and Hawkings Model

The Ffowcs Williams and Hawkings (FW-H) equation is essentially an inhomogeneous wave equation that can be derived by manipulating the continuity equation and the Navier-Stokes equations. The FW-H [ 43, 105] equation can be written as:

$\displaystyle \frac{1}{a_0^2} \frac{\partial^2 p'}{\partial t^2} - \nabla^2 p'$ $\textstyle =$ $\displaystyle \frac{\partial^2}{\partial x_i \partial x_j} \left\{T_{ij} H(f) \right\}$  
  $\textstyle -$ $\displaystyle \frac{\partial}{\partial x_i}\left\{\left[P_{ij} n_j + \rho u_i \left(u_n - v_n\right)\right] \delta(f)\right\}$  
  $\textstyle +$ $\displaystyle \frac{\partial}{\partial t}\left\{\left[\rho_0 v_n + \rho \left(u_n - v_n\right)\right] \delta(f)\right\}$ (21.2-1)

  $u_i$ = fluid velocity component in the $x_i$ direction
  $u_n$ = fluid velocity component normal to the surface $f=0$
  $v_i$ = surface velocity components in the $x_i$ direction
  $v_n$ = surface velocity component normal to the surface
  $\delta(f)$ = Dirac delta function
  $H(f)$ = Heaviside function

$p'$ is the sound pressure at the far field ( $p' = p - p_0$). $f = 0$ denotes a mathematical surface introduced to "embed'' the exterior flow problem ( $f > 0$) in an unbounded space, which facilitates the use of generalized function theory and the free-space Green function to obtain the solution. The surface ( $f = 0$) corresponds to the source (emission) surface, and can be made coincident with a body (impermeable) surface or a permeable surface off the body surface. $n_i$ is the unit normal vector pointing toward the exterior region ( $f > 0$), $a_0$ is the far-field sound speed, and $T_{ij}$ is the Lighthill stress tensor, defined as

 T_{ij} = \rho u_i u_j + P_{ij} - a_0^2\left(\rho - \rho_0\right) \delta_{ij} (21.2-2)

$P_{ij}$ is the compressive stress tensor. For a Stokesian fluid, this is given by

 P_{ij} = p \delta_{ij} - \mu \left[\frac{\partial u_i}{\part... ...rac{2}{3} \frac{\partial u_k}{\partial x_k} \delta_{ij}\right] (21.2-3)

The free-stream quantities are denoted by the subscript $0$.

The solution to Equation  21.2-1 is obtained using the free-space Green function ( $\delta(g)/4 \pi r$). The complete solution consists of surface integrals and volume integrals. The surface integrals represent the contributions from monopole and dipole acoustic sources and partially from quadrupole sources, whereas the volume integrals represent quadrupole (volume) sources in the region outside the source surface. The contribution of the volume integrals becomes small when the flow is low subsonic and the source surface encloses the source region. In FLUENT, the volume integrals are dropped. Thus, we have

 p'({\vec x}, t) = p'_T ({\vec x}, t) + p'_L ({\vec x}, t) (21.2-4)


$\displaystyle 4 \pi p'_T ({\vec x}, t)$ $\textstyle =$ $\displaystyle \int_{f=0}\left[\frac{\rho_0 \left(\dot{U_n} + U_{\dot{n}}\right)}{r\left(1 - M_r\right)^2}\right] \, dS$  
  $\textstyle +$ $\displaystyle \int_{f=0}\left[\frac{\rho_0 U_n \left\{r \dot{M}_r + a_0 \left(M_r - M^2\right)\right\}}{r^2\left(1 - M_r\right)^3 }\right] \, dS$ (21.2-5)
$\displaystyle 4 \pi p'_L ({\vec x}, t)$ $\textstyle =$ $\displaystyle \frac{1}{a_0} \int_{f=0}\left[\frac{\dot{L}_r}{r\left(1 - M_r\right)^2}\right] \, dS$  
  $\textstyle +$ $\displaystyle \int_{f=0} \left[\frac{L_r - L_M}{r^2\left(1 - M_r\right)^2 }\right] \, dS$  
  $\textstyle +$ $\displaystyle \frac{1}{a_0} \int_{f=0} \left[\frac{L_r\left\{r \dot{M}_r + a_0 \left(M_r - M^2\right)\right\}}{r^2\left(1 - M_r\right)^3 }\right] \, dS$ (21.2-6)


$\displaystyle U_i$ $\textstyle =$ $\displaystyle v_i + \frac{\rho}{\rho_0} \left(u_i - v_i\right)$ (21.2-7)
$\displaystyle L_i$ $\textstyle =$ $\displaystyle P_{ij}\hat{n}_j + \rho u_i (u_n - v_n)$ (21.2-8)

When the integration surface coincides with an impenetrable wall, the two terms on the right in Equation  21.2-4, $p'_T ({\vec x}, t)$ and $p'_L ({\vec x}, t)$, are often referred to as thickness and loading terms, respectively, in light of their physical meanings. The square brackets in Equations  21.2-5 and 21.2-6 denote that the kernels of the integrals are computed at the corresponding retarded times, $\tau$, defined as follows, given the observer time, $t$, and the distance to the observer, $r$,

 \tau = t - \frac{r}{a_0} (21.2-9)

The various subscripted quantities appearing in Equations  21.2-5 and 21.2-6 are the inner products of a vector and a unit vector implied by the subscript. For instance, $L_r = {\vec L} \cdot \hat{{\vec r}} = L_i r_i$ and $U_n = {\vec U} \cdot {\vec n} = U_i n_i$, where ${\vec r}$ and ${\vec n}$ denote the unit vectors in the radiation and wall-normal directions, respectively. The dot over a variable denotes source-time differentiation of that variable.

Please note the following remarks regarding the applicability of this integral solution:

next up previous contents index Previous: 21.2 Acoustics Model Theory
Up: 21.2 Acoustics Model Theory
Next: 21.2.2 Broadband Noise Source
© Fluent Inc. 2006-09-20