Curle Analogy Function object

AlbertoArtoni · October 22, 2021, 06:33

I was trying to understand why the Curle analogy is implement as it has been implemented:

$p' = \frac{1}{4\pi} \frac{\mathbf{r}}{|\mathbf{r}|^2}\large( \frac{\mathbf{F}}{|\mathbf{r}|} + \frac{1}{c_0}\frac{d\mathbf{F}}{dt} \large) .$

The actual formulation by Curle can be seen as:

$p' = \frac{1}{4\pi}\nabla\cdot\nabla\cdot \int_{V} \frac{\mathbf{T}}{r} dV - \frac{1}{4\pi}\nabla\cdot \int_{\partial V} (\frac{p'\mathbf{I}}{r}) \cdot \mathbf{n} d\partial V$

I can see that if we are far enough, we can replace the divergence with $-\frac{1}{c_0}\frac{\mathbf{r}}{|\mathbf{r}|}\frac{\partial}{\partial t}$ .

I don't understand:
1) why was the volume term omitted
2) from where the term $\frac{\mathbf{F}}{|\mathbf{r}|}$ comes from.

Thanks!

cmiked · June 15, 2022, 05:21

The derivation of the OpenFOAM formulation can be found in the book "Aeroacoustics of Low Mach Number Flows" by Glegg et. al. pg 83.
OpenFOAM accounts only for the dipoles in this formulation.

Furthermore, when calculating the force, OpenFOAM uses only the pressure at given input surfaces since, by default, the convection term is zero at walls for viscous flows. (see attached).

October 22, 2021, 06:33	Curle Analogy Function object	#1
AlbertoArtoni New Member Alberto Artoni Join Date: Oct 2019 Posts: 7 Rep Power: 7	I was trying to understand why the Curle analogy is implement as it has been implemented: $p' = \frac{1}{4\pi} \frac{\mathbf{r}}{\|\mathbf{r}\|^2}\large( \frac{\mathbf{F}}{\|\mathbf{r}\|} + \frac{1}{c_0}\frac{d\mathbf{F}}{dt} \large) .$ The actual formulation by Curle can be seen as: $p' = \frac{1}{4\pi}\nabla\cdot\nabla\cdot \int_{V} \frac{\mathbf{T}}{r} dV - \frac{1}{4\pi}\nabla\cdot \int_{\partial V} (\frac{p'\mathbf{I}}{r}) \cdot \mathbf{n} d\partial V$ I can see that if we are far enough, we can replace the divergence with $-\frac{1}{c_0}\frac{\mathbf{r}}{\|\mathbf{r}\|}\frac{\partial}{\partial t}$ . I don't understand: 1) why was the volume term omitted 2) from where the term $\frac{\mathbf{F}}{\|\mathbf{r}\|}$ comes from. Thanks!

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