Ffowcs Williams-Hawkings Model

The Ffowcs Williams-Hawkings (FW-H) acoustics integral formulation is the preferred strategy for far-field noise prediction such as overhead aircraft noise. This model calculates the far-field sound signal that is radiated from near-field flow data from a CFD solution. The goal is to predict small amplitude acoustic pressure fluctuations at the locations of each receiver.

Index

Flow direction
Impermeable surface that is located on the sources of sound body surface (e.g. cylinder surface)
Permeable surface enclosing the sound sources
Computational domain boundary / Limit of Direct Noise Simulation near-field solution
Receivers in the far field
Quadrupoles noise visualization

The FW-H acoustic model for stationary motion can simulate wind tunnel aeroacoustic measurements with stationary sources such as aircraft landing gear or an airfoil trailing edge. The FW-H acoustic model is used only to predict the propagation of sound in free space. It does not include effects such as sound reflections, refraction, or material property change.

The Ffowcs Williams-Hawkings formulations are based on Farassat's Formulation 1A ( [880], [888]) and Dunn Farassat Padula 1A ( [897]), both for the non-convective form of FW-H. Therefore, numerical accuracy is not an issue of great concern. With the FW-H approach, a second order discretization scheme, both spatial and temporal, together with judicious choices for mesh resolution and time-step size, can give accurate far-field noise prediction.

The Ffowcs Williams-Hawkings Model

Ffowcs Williams and Hawkings [892] extended the work of Lighthill [901] and Curle [884] to the formulation of aerodynamic sound generated by a surface in arbitrary motion.

Similar as Lighthill equation, they recast the equations of fluid motion in the form of an inhomogeneous wave equation. An important aspect in defining the FW-H equation was the casting of certain flow variables as generalized functions.

The result obtained by Ffowcs Williams and Hawkings (see [892]) was formally presented for an impermeable surface, but they understood that mathematical surface in their formulation does not have to coincide with the solid body.

Both impermeable and permeable surfaces can be used for Ffowcs Williams-Hawkings acoustic analogy model where the formulation is given below.

Ffowcs Williams-Hawkings Formulation

The Ffowcs Williams-Hawkings (FW-H) formulation is based on Farassat’s Formulation 1A and the Dunn Farassat Padula Formulation 1A.

Farassat’s Formulation 1A (see [879]) is the default and preferred formulation, a non-convective form of FW-H for general subsonic source regions, including the Impermeable formulation for transient rotating motion, in both whole and periodic domains.
Dunn Farassat Padula 1A (see [890]) is retained for backwards compatibility, a non-convective form of FW-H, for subsonic rotating sections representing the Impermeable formulation for transient rotating motion, whole and periodic domains.

The FW-H equation is an exact rearrangement of the continuity and the momentum equations into the form of an inhomogeneous wave equation. The FW-H equation gives accurate results even if the surface of integration lies in the nonlinear flow region. It is based on the free-space Green’s function to compute the sound pressure at the observer location, $x$ . The FW-H equation for pressure that is radiated into a medium at rest by a flow in a region or a set of surfaces is:

图 1. EQUATION_DISPLAY

p′ (x, t) = {p′}_{T} (x, t) + {p′}_{L} (x, t) + {p′}_{Q} (x, t)

(4750)

The monopole term is:

图 2. EQUATION_DISPLAY

{p′}_{T} (x, t) = \frac{1}{4 π} ((\frac{\partial}{\partial t}) \int_{S}^{} {[\frac{Q}{(r (1 - M_{r}))}]}_{r e t} d S)

(4751)

The dipole term is:

图 3. EQUATION_DISPLAY

{p′}_{L} (x, t) = \frac{1}{4 π} ((- \frac{\partial}{\partial x_{i}}) \int_{S}^{} {[\frac{L_{i}}{(r (1 - M_{r}))}]}_{r e t} d S)

(4752)

The quadrupole term is:

图 4. EQUATION_DISPLAY

{p′}_{Q} (x, t) = \frac{1}{4 π} ((\frac{\partial^{2}}{(\partial x_{i}) (\partial x_{j})}) \int_{V}^{} {[\frac{T_{i j}}{(r (1 - M_{r}))}]}_{r e t} d V)

(4753)

with:

图 5. EQUATION_DISPLAY

Q = ρ_{0} U_{i} n_{i}

(4754)

图 6. EQUATION_DISPLAY

U_{i} = (1 - \frac{ρ}{ρ_{0}}) v_{i} + \frac{ρ u_{i}}{ρ_{0}}

(4755)

图 7. EQUATION_DISPLAY

L_{i} = P_{i j} n_{i} + ρ u_{i} (u_{n} - v_{n})

(4756)

图 8. EQUATION_DISPLAY

P_{i j} = (p - p_{0}) δ_{i j} - σ_{i j}

(4757)

图 9. EQUATION_DISPLAY

T_{i j} = ρ u_{i} u_{j} + δ_{i j} [(p - p_{0}) - {c_{0}}^{2} (ρ - ρ_{0})] - σ_{i j}

(4758)

where:

$u_{i}$ represents fluid velocity component in the $i$ direction.
$u_{n}$ is the fluid velocity component normal to the surface.
$v_{i}$ represents surface velocity component in the $i$ direction.
$v_{n}$ is the surface velocity component normal to the surface.
$n_{i}$ is the surface normal vector.
$p_{0}$ is the atmospheric/far field pressure, the fluid pressure in quiescent medium.
$σ_{i j}$ is the viscous stress tensor.
$ρ_{0}$ is the far field density.
$P_{i j}$ is the compressive stress tensor.
$T_{i j}$ is the Lighthill stress tensor.

The subscript $r e t$ indicates retarded time, which is the time of emission.

The space derivatives from Eqn. (4751) and Eqn. (4752) are transformed into time derivatives. Afterwards, the time derivatives at the observer locations are moved into the integrals.

Eqn. (4760) describes the monopole source term using the advanced-time formulation. Eqn. (4761) describes the dipole source term using the advanced-time formulation.

Automatic Signal Pruning

Before the computed FW-H sound pressure is exported, the FW-H receiver data is automatically pruned—data is clipped from both ends of the time signal, because the sound source is incomplete in those intervals.

Based on the receiver position, the minimum and maximum travel times ${T D}_{MIN}$ and ${T D}_{MAX}$ for the signal to move from the FW-H surfaces to the receiver are:

{T D}_{MIN} = \frac{r_{min}}{c_{0}}, {T D}_{MAX} = \frac{r_{max}}{c_{0}}

where:

$r_{\min}$ is the shortest distance between the FW-H source surfaces and the receiver location.
$r_{\max}$ is the longest distance between the sound source surfaces and the receiver location.
$c_{0}$ is the speed of sound.

The far-field noise predicted at the receiver location is tabulated from time $t_{0}$ to $t_{n}$ :

\begin{matrix} t_{0} = T_{0} + {T D}_{MIN} \\ t_{n} = T_{n} + {T D}_{MAX} \end{matrix}

where:

$t_{0}$ is the time at which the data set starts accumulating at the receiver, before pruning.
$t_{n}$ is the time at which the data set stops accumulating at the receiver, before pruning.
$T_{0}$ is the start time property for the On-the-Fly FW-H solver or first physical time recorded in .simh file format for the Post FW-H solver.
$T_{n}$ is the end time of the simulation for On-the-Fly FW-H solver or the last physical time recorded in .simh file format for Post FW-H solver.

The receiver does not receive the sound pressure signal from the farthest point from the FW-H source surfaces ( $r_{\max}$ ) until the beginning reception time becomes:

t_{b} = T_{0} + {T D}_{MAX}

From time $t_{0}$ to $t_{b}$ , the sound signal accumulated on the receiver register does not include the contribution from the whole FW-H surface area, and thus the sound pressure data received during that time is not complete. The identical pruning process occurs during the period from end time $t_{e}$ to $t_{n}$ :

\begin{matrix} t_{e} = T_{n} + {T D}_{MIN} \end{matrix}

Pruning removes the signal on the incomplete ends, from $t_{0}$ to $t_{b}$ , a time interval lasting ${T D}_{MAX} - {T D}_{MIN}$ , and again from $t_{e}$ to $t_{n}$ , another interval of ${T D}_{MAX} - {T D}_{MIN}$ , for a total period of $2 \cdot ({T D}_{MAX} - {T D}_{MIN})$ . The pruned signal spans the time from $t_{b}$ , when pruned data begins, to $t_{e}$ , when pruned data ends.

For multiple receivers, the receiver data are pruned using the minimum of ${T D}_{MIN}$ and the maximum of ${T D}_{MAX}$ over all receivers.

FW-H Surface Terms

When the integration surface coincides with the surface of the solid body, the monopole term Eqn. (4751), the dipole term Eqn. (4752), and the quadrupole term Eqn. (4753) are called:

The Thickness Surface Term, ${p′}_{T} (x, t)$ , resulting from the displacement of fluid as the body passes. The term is defined in Eqn. (4760) for general flows and in Eqn. (4762) for flows with rigid body motion or moving reference frames.
The Loading Surface Term, ${p′}_{L} (x, t)$ , resulting from transient variation of the force distribution on the body surface. The term is defined in Eqn. (4761) for general flows and in Eqn. (4763) for flows with rigid body motion or moving reference frames.
The Total Surface Term, ${p′}_{S} (x, t)$ , resulting from the sum of the Thickness Surface Term and the Loading Surface Term. The term is defined as:

图 10. EQUATION_DISPLAY

{p′}_{S} (x, t) = {p′}_{T} (x, t) + {p′}_{L} (x, t)

(4759)

Farrassat’s Formulation 1A

This formulation is for general subsonic source regions, for general far-field noise prediction. See [879] and [880].

图 11. EQUATION_DISPLAY

{p′}_{T} (x, t) = \frac{1}{4 π} (\int_{(f = 0)}^{} {[\frac{ρ_{0} ({\dot{U}}_{n} + U_{\bar{n}})}{r {(1 - M_{r})}^{2}}]}_{r e t} d S + \int_{(f = 0)}^{} {[\frac{ρ_{0} U_{n} [r {\dot{M}}_{r} + a_{0} (M_{r} - M^{2})]}{r^{2} {(1 - M_{r})}^{3}}]}_{r e t} d S)

(4760)

图 12. EQUATION_DISPLAY

{p′}_{L} (x, t) = \frac{1}{4 π} (\frac{1}{a_{0}} \cdot \int_{(f = 0)}^{} {[\frac{{\dot{L}}_{r}}{r {(1 - M_{r})}^{2}}]}_{r e t} d S + \int_{(f = 0)}^{} {[\frac{(L_{r} - L_{M})}{r^{2} {(1 - M_{r})}^{2}}]}_{r e t} d S + \frac{1}{a_{0}} \cdot \int_{(f = 0)}^{} {[\frac{L_{r} [r {\dot{M}}_{r} + a_{0} (M_{r} - M^{2})]}{r^{2} {(1 - M_{r})}^{3}}]}_{r e t} d S)

(4761)

where:

$L_{i} = P_{i j} \cdot n_{i} + ρ \cdot u_{i} (u_{n} - v_{n})$ where $P_{i j} = (p - p_{0}) δ_{i j}$
$M_{i} = v_{i} / a_{0}$
$r = x_{observer} - y_{face}$

and where:

$f = 0$ denotes a mathematical surface to embed the exterior flow problem $f > 0$ in an unbounded space.
$f = 0$ represents the emission surface and is made coincident with a body, impermeable surface, or permeable surface.

If the data surface coincides with a solid surface, then the normal velocity of the fluid is the same as the normal velocity of the surface: $u_{n} = v_{n}$

In this case, Eqn. (4760) and Eqn. (4761) correspond to the Impermeable FW-H Surface type and some of the terms are eliminated.

Dunn-Farrassat-Padula Formulation 1A

This formulation is based on modification of Farassat’s Formulation 1A, which was used in both ANOPP (Aircraft Noise Prediction Program) and DFP-ATP (Dunn-Farassat-Padula Advanced Turboprop Prediction) for subsonic rotating blades to compute the acoustic pressure, based on Hubbard [897].

For the Permeable FW-H Surface type, the FW-H Surface integrals are computed on the internal interface boundary, from an in-place interface.

图 13. EQUATION_DISPLAY

{p′}_{T} (x, t) = \frac{1}{4 π} {(\int_{(f = 0)}^{} [\frac{ρ_{0} v_{n} [(r {\dot{M}}_{r}) + a_{0} (M_{r} - M^{2})]}{r^{2} {(1 - M_{r})}^{3}}] d S)}_{r e t}

(4762)

图 14. EQUATION_DISPLAY

{p′}_{L} (x, t) = \frac{1}{4 π} {(\frac{1}{a_{0}} \cdot \int_{(f = 0)}^{} [\frac{{\dot{L}}_{i} r_{i}}{r {(1 - M_{r})}^{2}}] d S + \int_{(f = 0)}^{} [\frac{(L_{r} - L_{i} M_{i})}{r^{2} {(1 - M_{r})}^{2}}] d S + \frac{1}{a_{0}} \cdot \int_{(f = 0)}^{} [\frac{L_{r} [(r {\dot{M}}_{r}) + a_{0} (M_{r} - M^{2})]}{r^{2} {(1 - M_{r})}^{3}}] d S)}_{r e t}

(4763)

and:

图 15. EQUATION_DISPLAY

{\dot{M}}_{r} = {\dot{M}}_{i} r_{i}

(4764)

where:

$L_{i}$ is the blade load vector. $L_{i} = P_{i j} \cdot n_{i} + ρ \cdot u_{i} (u_{n} - v_{n})$ where $P_{i j} = (p - p_{0}) δ_{i j}$
${\dot{L}}_{i}$ is the derivative of the blade load vector with respect to source time.
$v_{i}$ is the local velocity of the blade surface with respect to the quiescent fluid.
$v_{n}$ is the surface normal velocity.
$r_{i}$ is the distance from a source point to the observer.
$M_{r}$ is the Mach number of the source toward the observer.

The subscript $r e t$ indicates retarded time, which is the time of emission.

Accounting for Time-Lag

The major task in evaluating the FW-H integrals is how to account for the time-lag between emission and reception times.

The advanced time algorithm looks forward in time to see when the observer perceives the currently generated sound waves:

The procedure starts with a sequence of emission times (conveniently taken as the flow times).
The source strengths are calculated (thickness surface noise and loading surface noise) at all source elements (faces of the integration surfaces) for a given emission time.
The contribution of the sources is interpolated in the far-field time domain to build the sound signal.

The total sound pressure the observer perceives consists of the contribution from all source elements. The sound pressure at the receiver is obtained by accumulating the arriving signals in time slots. The overall observer acoustic signal is found from the summation of the acoustic signal from each source element of the FW-H surface during the same source time.

FW-H Volume Terms

The quadrupole noise is a volume distribution of sources, which accounts for nonlinearities in the flow.

These nonlinearities are of two types [902]. First, the local speed of sound is not constant, but varies due to particle acceleration. Second, the finite particle velocity near the body (for example blade) influences the velocity of sound propagation. When strong shear layers exist in the flow or when the Mach number increases, the quadrupole term is not negligible.

Farassat and Brentner [879] have shown that the noise contribution from the quadrupole, ${p′}_{Q} (x, t)$ , can be expressed as a “collapsing-sphere” formulation. Using this formulation, the space derivatives from Eqn. (4753) are transformed into time derivatives:

图 16. EQUATION_DISPLAY

{p′}_{Q} (x, t) = \frac{1}{4 π} ((\frac{1}{c}) (\frac{\partial^{2}}{{\partial t}^{2}}) \int_{- \infty}^{t} [\int_{(f > 0)}^{} \frac{T_{r r}}{r} d Ω] d τ + (\frac{\partial}{\partial t}) \int_{- \infty}^{t} [\int_{(f > 0)}^{} \frac{{3 T}_{r r} - T_{i i}}{r^{2}} d Ω] d τ) + (c \int_{- \infty}^{t} [\int_{(f > 0)}^{} \frac{{3 T}_{r r} - T_{i i}}{r^{3}} d Ω] d τ)

(4765)

where:

$T_{r r}$ is the double contraction of $T_{i j}$ .
$r_{i}$ and $r_{i}$ are the components of the unit vector in the direction of radiation.
$T_{i j}$ is the Lighthill stress tensor. $T_{i j} = ρ u_{i} u_{j} + δ_{i j} [(p - p_{0}) - {c_{0}}^{2} (ρ - ρ_{0})] - σ_{i j}$ and $σ_{i j}$ is the viscous stress tensor.

Eqn. (4765) is transformed from a collapsing-sphere formulation to an advanced time formulation using the following four equations. In these equations, the time derivatives at the observer are moved into the integrals to prevent numerical time differentiation of the integrals. The “source-time-dominant” algorithm from [881] is used to allow the estimation of the ${p′}_{Q} (x, t)$ volume term of the FW-H equation as follows:

图 17. EQUATION_DISPLAY

{p′}_{Q} (x, t) = \frac{1}{4 π} (\int_{(f > 0)}^{} {[\frac{K_{1}}{c^{2} r} + \frac{K_{2}}{c r^{2}} + \frac{K_{3}}{r^{3}}]}_{r e t} d V)

(4766)

图 18. EQUATION_DISPLAY

K_{1} = K_{11} + K_{12} + K_{13} = [\frac{{\ddot{T}}_{r r}}{{(1 - M_{r})}^{3}}] + [\frac{{{\ddot{M}}_{r} T}_{r r} + 3 {\dot{M}}_{r}_{\bar{T}}}{{(1 - M_{r})}^{4}}] + [\frac{3 {{\dot{M}}_{r}^{2} T}_{r r}}{{(1 - M_{r})}^{5}}]

(4767)

图 19. EQUATION_DISPLAY

K_{2} = K_{21} + K_{22} + K_{23} + K_{24} = [\frac{- {\dot{T}}_{i i}}{{(1 - M_{r})}^{2}}] + [\frac{4 {\dot{T}}_{M_{r}} + 2 T_{{\dot{M}}_{r}} + {\dot{M}}_{r} T_{i i}}{{(1 - M_{r})}^{3}}] + [\frac{3 [(1 - M^{2}) {\dot{T}}_{r r} - 2 {\dot{M}}_{r} T_{M r} - M_{i} {\dot{M}}_{i} T_{r r}]}{{(1 - M_{r})}^{4}}] + [\frac{6 {\dot{M}}_{r} (1 - M^{2}) T_{r r}}{{(1 - M_{r})}^{5}}]

(4768)

图 20. EQUATION_DISPLAY

K_{3} = K_{31} + K_{32} + K_{33} = [\frac{2 T_{M M} - (1 - M^{2}) T_{i i}}{{(1 - M_{r})}^{3}}] - [\frac{6 (1 - M^{2}) T_{M r}}{{(1 - M_{r})}^{4}}] + [\frac{3 {(1 - M^{2})}^{2} T_{r r}}{{(1 - M_{r})}^{5}}]

(4769)

where:

$T_{r r} = T_{i j} r_{i} r_{j}$	$T_{M M} = T_{i j} M_{i} M_{j}$	$T_{M r} = T_{i j} M_{i} r_{j}$	$T_{\dot{M r}} = T_{i j} {\dot{M}}_{i} r_{j}$
${\dot{T}}_{M r} = {\dot{T}}_{i j} M_{i} r_{j}$	${\dot{T}}_{r r} = {\dot{T}}_{i j} r_{i} r_{j}$	${\ddot{T}}_{r r} = {\ddot{T}}_{i j} r_{i} r_{j}$	$M_{i} = \frac{v_{i}}{c}$
$M_{r} = M_{i} r_{i}$	${\dot{M}}_{r} = \dot{M} r_{i}$	${\ddot{M}}_{r} = {\ddot{M}}_{i} r_{i}$	${\ddot{M}}_{r r} = \ddot{M} r_{i} r_{i}$
$r_{i} = \frac{x_{i} - y_{i}}{r}$

where:

$r_{i}$ denotes the unit vector in the direction of radiation.
A dot above a variable denotes the derivative with respect to source time of that variable.