http://www.cfd-online.com/W/index.php?title=Special:Contributions/Jhseo&feed=atom&limit=50&target=Jhseo&year=&month=CFD-Wiki - User contributions [en]2015-08-03T14:43:10ZFrom CFD-WikiMediaWiki 1.16.5http://www.cfd-online.com/Wiki/Hydrodynamic/acoustic_splittingHydrodynamic/acoustic splitting2008-07-30T22:26:27Z<p>Jhseo: /* Linearized Perturbed Compressible Equations */</p>
<hr />
<div>A hydrodynamic/acoustic splitting technique is a good resolving method to the low Mach number aeroacoustics.<br />
In the hydrodynamic/acoustic splitting method, the total flow variables are decomposed into the incompressible variables and the <br />
perturbed compressible ones as,<br />
<br />
<br />
<math>\rho(\vec{x},t)=\rho_0+\rho'(\vec{x},t) </math><br />
<br />
<math>\vec{u}(\vec{x},t)=\vec{U}(\vec{x},t)+\vec{u'}(\vec{x},t)</math><br />
<br />
<math>p(\vec{x},t)=P(\vec{x},t)+p'(\vec{x},t)</math><br />
<br />
<br />
The incompressible variables represent the hydrodynamic flow field, while the acoustic fluctuations and other compressibility <br />
effects are resolved by the perturbed quantities denoted by ('). The original full perturbed compressible equations (PCE) <br />
to calculate the perturbed quantities can be derived by subtracting the incompressible Navier-Stokes equations from the <br />
compressible ones. The PCE recently revised by Seo and Moon are written in a vector form as,<br />
<br />
<br />
<math>\frac{\partial\rho'}{\partial t}+(\vec{u}\cdot\nabla)\rho'+\rho(\nabla\cdot\vec{u'})=0</math><br />
<br />
<math>\frac{\partial\vec{u'}}{\partial<br />
t}+(\vec{u}\cdot\nabla)\vec{u'}+(\vec{u'}\cdot\nabla)\vec{U}+\frac{1}{\rho}\nabla<br />
p'+\frac{\rho'}{\rho}\frac{D\vec{U}}{Dt}=\frac{1}{\rho}\vec{f'_{vis}}</math><br />
<br />
<math>\frac{{\partial p'}}{{\partial t}}\!+\!(\vec u \cdot \nabla )p'\!+\!\gamma p(\nabla \cdot \vec u') + (\vec<br />
u' \cdot \nabla )P = - \frac{{DP}}{{Dt}} + (\gamma - 1)\left( {\Phi - \nabla \cdot \vec q} \right)</math><br />
<br />
<br />
where <math>D/Dt = \partial /\partial t + (\vec U \cdot \nabla )</math>, <math>\vec f'_{vis}</math> is the perturbed viscous force<br />
vector, <math>\Phi</math> and <math>\vec q</math> represent thermal viscous dissipation and heat flux vector, respectively. <br />
The PCE are the mixed-scales, non-linear equations, in which terms are involved with coupling effects between the acoustic <br />
fluctuations and the incompressible flow field. Through the coupling effects, a non-radiating vortical component, so-called <br />
'perturbed vorticity', is generated in the perturbed system. The perturbed vorticity may be regarded as the modification of <br />
hydrodynamic vorticity due to acoustic fluctuations, but it just reside in the perturbed system by this 'one-way' splitting method, <br />
in which backscattering of acoustic fluctuations onto the incompressible flow field is prohibited.<br />
<br />
<br />
<br />
== Linearized Perturbed Compressible Equations ==<br />
<br />
Although the effects of perturbed vorticity on noise generation is actually negligible at low Mach numbers, it is important to <br />
realize that the perturbed vorticity field can easily become unstable for various reasons. If so, it will get self-excited and <br />
grow unphysically, which then affects the acoustic field and causes inconsistency in the acoustic solutions. Recently, <br />
it is observed that the PCE exhibits grid dependant self-excited errors caused by the instability of perturbed vorticity when <br />
applied to the vortex sound prediction at high Reynolds numbers <br />
<br />
In order to resolve the aforementioned matter, the linearized perturbed compressible equations (LPCE) have been proposed by Seo and Moon. The LPCE is a modified version of the original PCE for very low Mach number aeroacoustics. In the LPCE <br />
formulation, the evolution of perturbed vorticity is firmly suppressed by dropping the coupling terms that generate vortical <br />
components in the perturbed system. This is for excluding possible errors caused by its instability, deliberating the fact that <br />
perturbed vorticity is not an important acoustic source for low Mach number flows. A set of LPCE can be written in a vector form as,<br />
<br />
<br />
<math>\frac{{\partial \rho '}}{{\partial t}} + (\vec U \cdot \nabla )\rho ' + \rho _0 (\nabla \cdot \vec u') = 0</math><br />
<br />
<math>\frac{{\partial \vec u'}}{{\partial t}} + \nabla (\vec u' \cdot \vec U) + \frac{1}{{\rho _0 }}\nabla p' = 0</math><br />
<br />
<math>\frac{{\partial p'}}{{\partial t}} + (\vec U \cdot \nabla )p' + \gamma P(\nabla \cdot \vec u') + (\vec u'<br />
\cdot \nabla )P = - \frac{{DP}}{{Dt}}</math><br />
<br />
<br />
The LPCE is linearized to a base incompressible flow field and its formulation is much simpler than the original PCE. <br />
The perturbed vorticity evolution is surely suppressed in the LPCE formulation, since the curl of linearized perturbed momentum <br />
equations, yields <br />
<br />
<math> \frac{{\partial \vec \omega'}}{{\partial t}} = 0 </math><br />
<br />
Consequently, the LPCE prevent any further changes (generation, convection, and decaying) of perturbed vorticity in time and a <br />
physical base for large grid spacing near the wall is also established, since they are an invisicd form of equations. The details <br />
about derivation of LPCE and the characteristics of perturbed vorticity can be found in Ref.(Seo and Moon, 2006). <br />
The left hand sides of LPCE represent the effects of acoustic wave propagation and refraction in the unsteady, inhomogeneous flows, <br />
while the right hand side only contains the acoustic source term, which will be projected from the hydrodynamic flow solution. <br />
For very low Mach number flows, it is interesting to note that the total change of hydrodynamic pressure <math>DP/Dt</math> is considered as <br />
the only explicit noise source term. It agrees with the result of Goldstein in his generalized acoustic analogy, <br />
specifically on the linearized Navier-Stokes equations to a 'non-radiating' base flow field. Ewert and Schroder have <br />
also shown that the dominant acoustic source from the flows at the incompressible limit can be represented by the hydrodynamic <br />
pressure. It is also considerable for the turbulent flow noise. The static pressure field of incompressible LES presents the <br />
features of turbulent flow and their noise can be predicted by the LPCE through the time changes of this pressure field. Therefore, <br />
the LPCE could exclude the uncertainty of modeling the source term for flow noise by turbulence.<br />
<br />
<br />
=== Example ===<br />
<br />
Here are an example of the hydrodynamic/acoustic splitting method. The following figures show Aeolian tone generated by cross flow over a circular cylinder at Re = 180 and Ma = 0.1. The first image is the result of DNS and the next one is the result of Hydrodynamic/acoustic splitting method (incompressible NS/LPCE). The LPCE are computed on the four-times coarser grid. As one can see, the results are almost the same. <br />
<br />
[[Image:cyl_180_DNS.jpg|Aeolian tone by cross flow around a circular cylinder at Re=180, Ma=0.1]]<br />
[[Image:cyl_180_LPCE.jpg|Aeolian tone by cross flow around a circular cylinder at Re=180, Ma=0.1]]<br />
<br />
== References ==<br />
{{reference-paper|author=Seo, J. H. and Moon, Y. J. |year=2005|title=The Perturbed Compressible Equations for Aeroacoustic Noise Prediction at Low Mach Numbers|rest=AIAA Journal, Vol. 43, No. 8, pp. 1716-1724}} <br />
<br />
{{reference-paper|author=Seo, J. H. and Moon, Y. J. |year=2006|title=Linearized Perturbed Compressible Equations for Low Mach number Aeroacoustics|rest=Journal of Computational Physics, Vol. 218, pp. 702-719}}</div>Jhseohttp://www.cfd-online.com/Wiki/File:Cyl_180_LPCE.jpgFile:Cyl 180 LPCE.jpg2008-07-30T22:22:12Z<p>Jhseo: Aeolian tone by cross flow around a circular cylinder at Re=180 and Ma=0.1, LPCE result</p>
<hr />
<div>Aeolian tone by cross flow around a circular cylinder at Re=180 and Ma=0.1, LPCE result</div>Jhseohttp://www.cfd-online.com/Wiki/Hydrodynamic/acoustic_splittingHydrodynamic/acoustic splitting2008-07-30T22:20:01Z<p>Jhseo: /* Example */</p>
<hr />
<div>A hydrodynamic/acoustic splitting technique is a good resolving method to the low Mach number aeroacoustics.<br />
In the hydrodynamic/acoustic splitting method, the total flow variables are decomposed into the incompressible variables and the <br />
perturbed compressible ones as,<br />
<br />
<br />
<math>\rho(\vec{x},t)=\rho_0+\rho'(\vec{x},t) </math><br />
<br />
<math>\vec{u}(\vec{x},t)=\vec{U}(\vec{x},t)+\vec{u'}(\vec{x},t)</math><br />
<br />
<math>p(\vec{x},t)=P(\vec{x},t)+p'(\vec{x},t)</math><br />
<br />
<br />
The incompressible variables represent the hydrodynamic flow field, while the acoustic fluctuations and other compressibility <br />
effects are resolved by the perturbed quantities denoted by ('). The original full perturbed compressible equations (PCE) <br />
to calculate the perturbed quantities can be derived by subtracting the incompressible Navier-Stokes equations from the <br />
compressible ones. The PCE recently revised by Seo and Moon are written in a vector form as,<br />
<br />
<br />
<math>\frac{\partial\rho'}{\partial t}+(\vec{u}\cdot\nabla)\rho'+\rho(\nabla\cdot\vec{u'})=0</math><br />
<br />
<math>\frac{\partial\vec{u'}}{\partial<br />
t}+(\vec{u}\cdot\nabla)\vec{u'}+(\vec{u'}\cdot\nabla)\vec{U}+\frac{1}{\rho}\nabla<br />
p'+\frac{\rho'}{\rho}\frac{D\vec{U}}{Dt}=\frac{1}{\rho}\vec{f'_{vis}}</math><br />
<br />
<math>\frac{{\partial p'}}{{\partial t}}\!+\!(\vec u \cdot \nabla )p'\!+\!\gamma p(\nabla \cdot \vec u') + (\vec<br />
u' \cdot \nabla )P = - \frac{{DP}}{{Dt}} + (\gamma - 1)\left( {\Phi - \nabla \cdot \vec q} \right)</math><br />
<br />
<br />
where <math>D/Dt = \partial /\partial t + (\vec U \cdot \nabla )</math>, <math>\vec f'_{vis}</math> is the perturbed viscous force<br />
vector, <math>\Phi</math> and <math>\vec q</math> represent thermal viscous dissipation and heat flux vector, respectively. <br />
The PCE are the mixed-scales, non-linear equations, in which terms are involved with coupling effects between the acoustic <br />
fluctuations and the incompressible flow field. Through the coupling effects, a non-radiating vortical component, so-called <br />
'perturbed vorticity', is generated in the perturbed system. The perturbed vorticity may be regarded as the modification of <br />
hydrodynamic vorticity due to acoustic fluctuations, but it just reside in the perturbed system by this 'one-way' splitting method, <br />
in which backscattering of acoustic fluctuations onto the incompressible flow field is prohibited.<br />
<br />
<br />
<br />
== Linearized Perturbed Compressible Equations ==<br />
<br />
Although the effects of perturbed vorticity on noise generation is actually negligible at low Mach numbers, it is important to <br />
realize that the perturbed vorticity field can easily become unstable for various reasons. If so, it will get self-excited and <br />
grow unphysically, which then affects the acoustic field and causes inconsistency in the acoustic solutions. Recently, <br />
it is observed that the PCE exhibits grid dependant self-excited errors caused by the instability of perturbed vorticity when <br />
applied to the vortex sound prediction at high Reynolds numbers <br />
<br />
In order to resolve the aforementioned matter, the linearized perturbed compressible equations (LPCE) have been proposed by Seo and Moon. The LPCE is a modified version of the original PCE for very low Mach number aeroacoustics. In the LPCE <br />
formulation, the evolution of perturbed vorticity is firmly suppressed by dropping the coupling terms that generate vortical <br />
components in the perturbed system. This is for excluding possible errors caused by its instability, deliberating the fact that <br />
perturbed vorticity is not an important acoustic source for low Mach number flows. A set of LPCE can be written in a vector form as,<br />
<br />
<br />
<math>\frac{{\partial \rho '}}{{\partial t}} + (\vec U \cdot \nabla )\rho ' + \rho _0 (\nabla \cdot \vec u') = 0</math><br />
<br />
<math>\frac{{\partial \vec u'}}{{\partial t}} + \nabla (\vec u' \cdot \vec U) + \frac{1}{{\rho _0 }}\nabla p' = 0</math><br />
<br />
<math>\frac{{\partial p'}}{{\partial t}} + (\vec U \cdot \nabla )p' + \gamma P(\nabla \cdot \vec u') + (\vec u'<br />
\cdot \nabla )P = - \frac{{DP}}{{Dt}}</math><br />
<br />
<br />
The LPCE is linearized to a base incompressible flow field and its formulation is much simpler than the original PCE. <br />
The perturbed vorticity evolution is surely suppressed in the LPCE formulation, since the curl of linearized perturbed momentum <br />
equations, yields <br />
<br />
<math> \frac{{\partial \vec \omega'}}{{\partial t}} = 0 </math><br />
<br />
Consequently, the LPCE prevent any further changes (generation, convection, and decaying) of perturbed vorticity in time and a <br />
physical base for large grid spacing near the wall is also established, since they are an invisicd form of equations. The details <br />
about derivation of LPCE and the characteristics of perturbed vorticity can be found in Ref.(Seo and Moon, 2006). <br />
The left hand sides of LPCE represent the effects of acoustic wave propagation and refraction in the unsteady, inhomogeneous flows, <br />
while the right hand side only contains the acoustic source term, which will be projected from the hydrodynamic flow solution. <br />
For very low Mach number flows, it is interesting to note that the total change of hydrodynamic pressure $DP/Dt$ is considered as <br />
the only explicit noise source term. It agrees with the result of Goldstein in his generalized acoustic analogy, <br />
specifically on the linearized Navier-Stokes equations to a 'non-radiating' base flow field. Ewert and Schroder have <br />
also shown that the dominant acoustic source from the flows at the incompressible limit can be represented by the hydrodynamic <br />
pressure. It is also considerable for the turbulent flow noise. The static pressure field of incompressible LES presents the <br />
features of turbulent flow and their noise can be predicted by the LPCE through the time changes of this pressure field. Therefore, <br />
the LPCE could exclude the uncertainty of modeling the source term for flow noise by turbulence.<br />
<br />
<br />
=== Example ===<br />
<br />
Here are an example of the hydrodynamic/acoustic splitting method. The following figures show Aeolian tone generated by cross flow over a circular cylinder at Re = 180 and Ma = 0.1. The first image is the result of DNS and the next one is the result of Hydrodynamic/acoustic splitting method (incompressible NS/LPCE). The LPCE are computed on the four-times coarser grid. As one can see, the results are almost the same. <br />
<br />
[[Image:cyl_180_DNS.jpg|Aeolian tone by cross flow around a circular cylinder at Re=180, Ma=0.1]]<br />
[[Image:cyl_180_LPCE.jpg|Aeolian tone by cross flow around a circular cylinder at Re=180, Ma=0.1]]<br />
<br />
== References ==<br />
{{reference-paper|author=Seo, J. H. and Moon, Y. J. |year=2005|title=The Perturbed Compressible Equations for Aeroacoustic Noise Prediction at Low Mach Numbers|rest=AIAA Journal, Vol. 43, No. 8, pp. 1716-1724}} <br />
<br />
{{reference-paper|author=Seo, J. H. and Moon, Y. J. |year=2006|title=Linearized Perturbed Compressible Equations for Low Mach number Aeroacoustics|rest=Journal of Computational Physics, Vol. 218, pp. 702-719}}</div>Jhseohttp://www.cfd-online.com/Wiki/File:Cyl_180_DNS.jpgFile:Cyl 180 DNS.jpg2008-07-30T22:13:59Z<p>Jhseo: Aeolian tone by cross flow over a circular cylinder at Re=180, Ma=0.1, DNS result</p>
<hr />
<div>Aeolian tone by cross flow over a circular cylinder at Re=180, Ma=0.1, DNS result</div>Jhseohttp://www.cfd-online.com/Wiki/Hydrodynamic/acoustic_splittingHydrodynamic/acoustic splitting2008-07-30T22:10:50Z<p>Jhseo: /* Example */</p>
<hr />
<div>A hydrodynamic/acoustic splitting technique is a good resolving method to the low Mach number aeroacoustics.<br />
In the hydrodynamic/acoustic splitting method, the total flow variables are decomposed into the incompressible variables and the <br />
perturbed compressible ones as,<br />
<br />
<br />
<math>\rho(\vec{x},t)=\rho_0+\rho'(\vec{x},t) </math><br />
<br />
<math>\vec{u}(\vec{x},t)=\vec{U}(\vec{x},t)+\vec{u'}(\vec{x},t)</math><br />
<br />
<math>p(\vec{x},t)=P(\vec{x},t)+p'(\vec{x},t)</math><br />
<br />
<br />
The incompressible variables represent the hydrodynamic flow field, while the acoustic fluctuations and other compressibility <br />
effects are resolved by the perturbed quantities denoted by ('). The original full perturbed compressible equations (PCE) <br />
to calculate the perturbed quantities can be derived by subtracting the incompressible Navier-Stokes equations from the <br />
compressible ones. The PCE recently revised by Seo and Moon are written in a vector form as,<br />
<br />
<br />
<math>\frac{\partial\rho'}{\partial t}+(\vec{u}\cdot\nabla)\rho'+\rho(\nabla\cdot\vec{u'})=0</math><br />
<br />
<math>\frac{\partial\vec{u'}}{\partial<br />
t}+(\vec{u}\cdot\nabla)\vec{u'}+(\vec{u'}\cdot\nabla)\vec{U}+\frac{1}{\rho}\nabla<br />
p'+\frac{\rho'}{\rho}\frac{D\vec{U}}{Dt}=\frac{1}{\rho}\vec{f'_{vis}}</math><br />
<br />
<math>\frac{{\partial p'}}{{\partial t}}\!+\!(\vec u \cdot \nabla )p'\!+\!\gamma p(\nabla \cdot \vec u') + (\vec<br />
u' \cdot \nabla )P = - \frac{{DP}}{{Dt}} + (\gamma - 1)\left( {\Phi - \nabla \cdot \vec q} \right)</math><br />
<br />
<br />
where <math>D/Dt = \partial /\partial t + (\vec U \cdot \nabla )</math>, <math>\vec f'_{vis}</math> is the perturbed viscous force<br />
vector, <math>\Phi</math> and <math>\vec q</math> represent thermal viscous dissipation and heat flux vector, respectively. <br />
The PCE are the mixed-scales, non-linear equations, in which terms are involved with coupling effects between the acoustic <br />
fluctuations and the incompressible flow field. Through the coupling effects, a non-radiating vortical component, so-called <br />
'perturbed vorticity', is generated in the perturbed system. The perturbed vorticity may be regarded as the modification of <br />
hydrodynamic vorticity due to acoustic fluctuations, but it just reside in the perturbed system by this 'one-way' splitting method, <br />
in which backscattering of acoustic fluctuations onto the incompressible flow field is prohibited.<br />
<br />
<br />
<br />
== Linearized Perturbed Compressible Equations ==<br />
<br />
Although the effects of perturbed vorticity on noise generation is actually negligible at low Mach numbers, it is important to <br />
realize that the perturbed vorticity field can easily become unstable for various reasons. If so, it will get self-excited and <br />
grow unphysically, which then affects the acoustic field and causes inconsistency in the acoustic solutions. Recently, <br />
it is observed that the PCE exhibits grid dependant self-excited errors caused by the instability of perturbed vorticity when <br />
applied to the vortex sound prediction at high Reynolds numbers <br />
<br />
In order to resolve the aforementioned matter, the linearized perturbed compressible equations (LPCE) have been proposed by Seo and Moon. The LPCE is a modified version of the original PCE for very low Mach number aeroacoustics. In the LPCE <br />
formulation, the evolution of perturbed vorticity is firmly suppressed by dropping the coupling terms that generate vortical <br />
components in the perturbed system. This is for excluding possible errors caused by its instability, deliberating the fact that <br />
perturbed vorticity is not an important acoustic source for low Mach number flows. A set of LPCE can be written in a vector form as,<br />
<br />
<br />
<math>\frac{{\partial \rho '}}{{\partial t}} + (\vec U \cdot \nabla )\rho ' + \rho _0 (\nabla \cdot \vec u') = 0</math><br />
<br />
<math>\frac{{\partial \vec u'}}{{\partial t}} + \nabla (\vec u' \cdot \vec U) + \frac{1}{{\rho _0 }}\nabla p' = 0</math><br />
<br />
<math>\frac{{\partial p'}}{{\partial t}} + (\vec U \cdot \nabla )p' + \gamma P(\nabla \cdot \vec u') + (\vec u'<br />
\cdot \nabla )P = - \frac{{DP}}{{Dt}}</math><br />
<br />
<br />
The LPCE is linearized to a base incompressible flow field and its formulation is much simpler than the original PCE. <br />
The perturbed vorticity evolution is surely suppressed in the LPCE formulation, since the curl of linearized perturbed momentum <br />
equations, yields <br />
<br />
<math> \frac{{\partial \vec \omega'}}{{\partial t}} = 0 </math><br />
<br />
Consequently, the LPCE prevent any further changes (generation, convection, and decaying) of perturbed vorticity in time and a <br />
physical base for large grid spacing near the wall is also established, since they are an invisicd form of equations. The details <br />
about derivation of LPCE and the characteristics of perturbed vorticity can be found in Ref.(Seo and Moon, 2006). <br />
The left hand sides of LPCE represent the effects of acoustic wave propagation and refraction in the unsteady, inhomogeneous flows, <br />
while the right hand side only contains the acoustic source term, which will be projected from the hydrodynamic flow solution. <br />
For very low Mach number flows, it is interesting to note that the total change of hydrodynamic pressure $DP/Dt$ is considered as <br />
the only explicit noise source term. It agrees with the result of Goldstein in his generalized acoustic analogy, <br />
specifically on the linearized Navier-Stokes equations to a 'non-radiating' base flow field. Ewert and Schroder have <br />
also shown that the dominant acoustic source from the flows at the incompressible limit can be represented by the hydrodynamic <br />
pressure. It is also considerable for the turbulent flow noise. The static pressure field of incompressible LES presents the <br />
features of turbulent flow and their noise can be predicted by the LPCE through the time changes of this pressure field. Therefore, <br />
the LPCE could exclude the uncertainty of modeling the source term for flow noise by turbulence.<br />
<br />
<br />
=== Example ===<br />
<br />
[[Image:cyl_180_DNS.jpg|Aeolian tone by cross flow around a circular cylinder at Re=180, Ma=0.1]]<br />
<br />
== References ==<br />
{{reference-paper|author=Seo, J. H. and Moon, Y. J. |year=2005|title=The Perturbed Compressible Equations for Aeroacoustic Noise Prediction at Low Mach Numbers|rest=AIAA Journal, Vol. 43, No. 8, pp. 1716-1724}} <br />
<br />
{{reference-paper|author=Seo, J. H. and Moon, Y. J. |year=2006|title=Linearized Perturbed Compressible Equations for Low Mach number Aeroacoustics|rest=Journal of Computational Physics, Vol. 218, pp. 702-719}}</div>Jhseohttp://www.cfd-online.com/Wiki/Hydrodynamic/acoustic_splittingHydrodynamic/acoustic splitting2008-07-30T22:07:39Z<p>Jhseo: /* References */</p>
<hr />
<div>A hydrodynamic/acoustic splitting technique is a good resolving method to the low Mach number aeroacoustics.<br />
In the hydrodynamic/acoustic splitting method, the total flow variables are decomposed into the incompressible variables and the <br />
perturbed compressible ones as,<br />
<br />
<br />
<math>\rho(\vec{x},t)=\rho_0+\rho'(\vec{x},t) </math><br />
<br />
<math>\vec{u}(\vec{x},t)=\vec{U}(\vec{x},t)+\vec{u'}(\vec{x},t)</math><br />
<br />
<math>p(\vec{x},t)=P(\vec{x},t)+p'(\vec{x},t)</math><br />
<br />
<br />
The incompressible variables represent the hydrodynamic flow field, while the acoustic fluctuations and other compressibility <br />
effects are resolved by the perturbed quantities denoted by ('). The original full perturbed compressible equations (PCE) <br />
to calculate the perturbed quantities can be derived by subtracting the incompressible Navier-Stokes equations from the <br />
compressible ones. The PCE recently revised by Seo and Moon are written in a vector form as,<br />
<br />
<br />
<math>\frac{\partial\rho'}{\partial t}+(\vec{u}\cdot\nabla)\rho'+\rho(\nabla\cdot\vec{u'})=0</math><br />
<br />
<math>\frac{\partial\vec{u'}}{\partial<br />
t}+(\vec{u}\cdot\nabla)\vec{u'}+(\vec{u'}\cdot\nabla)\vec{U}+\frac{1}{\rho}\nabla<br />
p'+\frac{\rho'}{\rho}\frac{D\vec{U}}{Dt}=\frac{1}{\rho}\vec{f'_{vis}}</math><br />
<br />
<math>\frac{{\partial p'}}{{\partial t}}\!+\!(\vec u \cdot \nabla )p'\!+\!\gamma p(\nabla \cdot \vec u') + (\vec<br />
u' \cdot \nabla )P = - \frac{{DP}}{{Dt}} + (\gamma - 1)\left( {\Phi - \nabla \cdot \vec q} \right)</math><br />
<br />
<br />
where <math>D/Dt = \partial /\partial t + (\vec U \cdot \nabla )</math>, <math>\vec f'_{vis}</math> is the perturbed viscous force<br />
vector, <math>\Phi</math> and <math>\vec q</math> represent thermal viscous dissipation and heat flux vector, respectively. <br />
The PCE are the mixed-scales, non-linear equations, in which terms are involved with coupling effects between the acoustic <br />
fluctuations and the incompressible flow field. Through the coupling effects, a non-radiating vortical component, so-called <br />
'perturbed vorticity', is generated in the perturbed system. The perturbed vorticity may be regarded as the modification of <br />
hydrodynamic vorticity due to acoustic fluctuations, but it just reside in the perturbed system by this 'one-way' splitting method, <br />
in which backscattering of acoustic fluctuations onto the incompressible flow field is prohibited.<br />
<br />
<br />
<br />
== Linearized Perturbed Compressible Equations ==<br />
<br />
Although the effects of perturbed vorticity on noise generation is actually negligible at low Mach numbers, it is important to <br />
realize that the perturbed vorticity field can easily become unstable for various reasons. If so, it will get self-excited and <br />
grow unphysically, which then affects the acoustic field and causes inconsistency in the acoustic solutions. Recently, <br />
it is observed that the PCE exhibits grid dependant self-excited errors caused by the instability of perturbed vorticity when <br />
applied to the vortex sound prediction at high Reynolds numbers <br />
<br />
In order to resolve the aforementioned matter, the linearized perturbed compressible equations (LPCE) have been proposed by Seo and Moon. The LPCE is a modified version of the original PCE for very low Mach number aeroacoustics. In the LPCE <br />
formulation, the evolution of perturbed vorticity is firmly suppressed by dropping the coupling terms that generate vortical <br />
components in the perturbed system. This is for excluding possible errors caused by its instability, deliberating the fact that <br />
perturbed vorticity is not an important acoustic source for low Mach number flows. A set of LPCE can be written in a vector form as,<br />
<br />
<br />
<math>\frac{{\partial \rho '}}{{\partial t}} + (\vec U \cdot \nabla )\rho ' + \rho _0 (\nabla \cdot \vec u') = 0</math><br />
<br />
<math>\frac{{\partial \vec u'}}{{\partial t}} + \nabla (\vec u' \cdot \vec U) + \frac{1}{{\rho _0 }}\nabla p' = 0</math><br />
<br />
<math>\frac{{\partial p'}}{{\partial t}} + (\vec U \cdot \nabla )p' + \gamma P(\nabla \cdot \vec u') + (\vec u'<br />
\cdot \nabla )P = - \frac{{DP}}{{Dt}}</math><br />
<br />
<br />
The LPCE is linearized to a base incompressible flow field and its formulation is much simpler than the original PCE. <br />
The perturbed vorticity evolution is surely suppressed in the LPCE formulation, since the curl of linearized perturbed momentum <br />
equations, yields <br />
<br />
<math> \frac{{\partial \vec \omega'}}{{\partial t}} = 0 </math><br />
<br />
Consequently, the LPCE prevent any further changes (generation, convection, and decaying) of perturbed vorticity in time and a <br />
physical base for large grid spacing near the wall is also established, since they are an invisicd form of equations. The details <br />
about derivation of LPCE and the characteristics of perturbed vorticity can be found in Ref.(Seo and Moon, 2006). <br />
The left hand sides of LPCE represent the effects of acoustic wave propagation and refraction in the unsteady, inhomogeneous flows, <br />
while the right hand side only contains the acoustic source term, which will be projected from the hydrodynamic flow solution. <br />
For very low Mach number flows, it is interesting to note that the total change of hydrodynamic pressure $DP/Dt$ is considered as <br />
the only explicit noise source term. It agrees with the result of Goldstein in his generalized acoustic analogy, <br />
specifically on the linearized Navier-Stokes equations to a 'non-radiating' base flow field. Ewert and Schroder have <br />
also shown that the dominant acoustic source from the flows at the incompressible limit can be represented by the hydrodynamic <br />
pressure. It is also considerable for the turbulent flow noise. The static pressure field of incompressible LES presents the <br />
features of turbulent flow and their noise can be predicted by the LPCE through the time changes of this pressure field. Therefore, <br />
the LPCE could exclude the uncertainty of modeling the source term for flow noise by turbulence.<br />
<br />
<br />
=== Example ===<br />
<br />
[[Image:Example.jpg]]<br />
<br />
<br />
== References ==<br />
{{reference-paper|author=Seo, J. H. and Moon, Y. J. |year=2005|title=The Perturbed Compressible Equations for Aeroacoustic Noise Prediction at Low Mach Numbers|rest=AIAA Journal, Vol. 43, No. 8, pp. 1716-1724}} <br />
<br />
{{reference-paper|author=Seo, J. H. and Moon, Y. J. |year=2006|title=Linearized Perturbed Compressible Equations for Low Mach number Aeroacoustics|rest=Journal of Computational Physics, Vol. 218, pp. 702-719}}</div>Jhseohttp://www.cfd-online.com/Wiki/Hydrodynamic/acoustic_splittingHydrodynamic/acoustic splitting2008-07-30T22:06:56Z<p>Jhseo: </p>
<hr />
<div>A hydrodynamic/acoustic splitting technique is a good resolving method to the low Mach number aeroacoustics.<br />
In the hydrodynamic/acoustic splitting method, the total flow variables are decomposed into the incompressible variables and the <br />
perturbed compressible ones as,<br />
<br />
<br />
<math>\rho(\vec{x},t)=\rho_0+\rho'(\vec{x},t) </math><br />
<br />
<math>\vec{u}(\vec{x},t)=\vec{U}(\vec{x},t)+\vec{u'}(\vec{x},t)</math><br />
<br />
<math>p(\vec{x},t)=P(\vec{x},t)+p'(\vec{x},t)</math><br />
<br />
<br />
The incompressible variables represent the hydrodynamic flow field, while the acoustic fluctuations and other compressibility <br />
effects are resolved by the perturbed quantities denoted by ('). The original full perturbed compressible equations (PCE) <br />
to calculate the perturbed quantities can be derived by subtracting the incompressible Navier-Stokes equations from the <br />
compressible ones. The PCE recently revised by Seo and Moon are written in a vector form as,<br />
<br />
<br />
<math>\frac{\partial\rho'}{\partial t}+(\vec{u}\cdot\nabla)\rho'+\rho(\nabla\cdot\vec{u'})=0</math><br />
<br />
<math>\frac{\partial\vec{u'}}{\partial<br />
t}+(\vec{u}\cdot\nabla)\vec{u'}+(\vec{u'}\cdot\nabla)\vec{U}+\frac{1}{\rho}\nabla<br />
p'+\frac{\rho'}{\rho}\frac{D\vec{U}}{Dt}=\frac{1}{\rho}\vec{f'_{vis}}</math><br />
<br />
<math>\frac{{\partial p'}}{{\partial t}}\!+\!(\vec u \cdot \nabla )p'\!+\!\gamma p(\nabla \cdot \vec u') + (\vec<br />
u' \cdot \nabla )P = - \frac{{DP}}{{Dt}} + (\gamma - 1)\left( {\Phi - \nabla \cdot \vec q} \right)</math><br />
<br />
<br />
where <math>D/Dt = \partial /\partial t + (\vec U \cdot \nabla )</math>, <math>\vec f'_{vis}</math> is the perturbed viscous force<br />
vector, <math>\Phi</math> and <math>\vec q</math> represent thermal viscous dissipation and heat flux vector, respectively. <br />
The PCE are the mixed-scales, non-linear equations, in which terms are involved with coupling effects between the acoustic <br />
fluctuations and the incompressible flow field. Through the coupling effects, a non-radiating vortical component, so-called <br />
'perturbed vorticity', is generated in the perturbed system. The perturbed vorticity may be regarded as the modification of <br />
hydrodynamic vorticity due to acoustic fluctuations, but it just reside in the perturbed system by this 'one-way' splitting method, <br />
in which backscattering of acoustic fluctuations onto the incompressible flow field is prohibited.<br />
<br />
<br />
<br />
== Linearized Perturbed Compressible Equations ==<br />
<br />
Although the effects of perturbed vorticity on noise generation is actually negligible at low Mach numbers, it is important to <br />
realize that the perturbed vorticity field can easily become unstable for various reasons. If so, it will get self-excited and <br />
grow unphysically, which then affects the acoustic field and causes inconsistency in the acoustic solutions. Recently, <br />
it is observed that the PCE exhibits grid dependant self-excited errors caused by the instability of perturbed vorticity when <br />
applied to the vortex sound prediction at high Reynolds numbers <br />
<br />
In order to resolve the aforementioned matter, the linearized perturbed compressible equations (LPCE) have been proposed by Seo and Moon. The LPCE is a modified version of the original PCE for very low Mach number aeroacoustics. In the LPCE <br />
formulation, the evolution of perturbed vorticity is firmly suppressed by dropping the coupling terms that generate vortical <br />
components in the perturbed system. This is for excluding possible errors caused by its instability, deliberating the fact that <br />
perturbed vorticity is not an important acoustic source for low Mach number flows. A set of LPCE can be written in a vector form as,<br />
<br />
<br />
<math>\frac{{\partial \rho '}}{{\partial t}} + (\vec U \cdot \nabla )\rho ' + \rho _0 (\nabla \cdot \vec u') = 0</math><br />
<br />
<math>\frac{{\partial \vec u'}}{{\partial t}} + \nabla (\vec u' \cdot \vec U) + \frac{1}{{\rho _0 }}\nabla p' = 0</math><br />
<br />
<math>\frac{{\partial p'}}{{\partial t}} + (\vec U \cdot \nabla )p' + \gamma P(\nabla \cdot \vec u') + (\vec u'<br />
\cdot \nabla )P = - \frac{{DP}}{{Dt}}</math><br />
<br />
<br />
The LPCE is linearized to a base incompressible flow field and its formulation is much simpler than the original PCE. <br />
The perturbed vorticity evolution is surely suppressed in the LPCE formulation, since the curl of linearized perturbed momentum <br />
equations, yields <br />
<br />
<math> \frac{{\partial \vec \omega'}}{{\partial t}} = 0 </math><br />
<br />
Consequently, the LPCE prevent any further changes (generation, convection, and decaying) of perturbed vorticity in time and a <br />
physical base for large grid spacing near the wall is also established, since they are an invisicd form of equations. The details <br />
about derivation of LPCE and the characteristics of perturbed vorticity can be found in Ref.(Seo and Moon, 2006). <br />
The left hand sides of LPCE represent the effects of acoustic wave propagation and refraction in the unsteady, inhomogeneous flows, <br />
while the right hand side only contains the acoustic source term, which will be projected from the hydrodynamic flow solution. <br />
For very low Mach number flows, it is interesting to note that the total change of hydrodynamic pressure $DP/Dt$ is considered as <br />
the only explicit noise source term. It agrees with the result of Goldstein in his generalized acoustic analogy, <br />
specifically on the linearized Navier-Stokes equations to a 'non-radiating' base flow field. Ewert and Schroder have <br />
also shown that the dominant acoustic source from the flows at the incompressible limit can be represented by the hydrodynamic <br />
pressure. It is also considerable for the turbulent flow noise. The static pressure field of incompressible LES presents the <br />
features of turbulent flow and their noise can be predicted by the LPCE through the time changes of this pressure field. Therefore, <br />
the LPCE could exclude the uncertainty of modeling the source term for flow noise by turbulence.<br />
<br />
<br />
=== Example ===<br />
<br />
[[Image:Example.jpg]]<br />
<br />
<br />
== References ==<br />
{{reference-paper|author=Seo, J. H. and Moon, Y. J. |year=2005|title=The Perturbed Compressible Equations for Aeroacoustic Noise Prediction at Low Mach Numbers|rest=AIAA Journal, Vol. 43, No. 8, pp. 1716-1724}} <br />
{{reference-paper|author=Seo, J. H. and Moon, Y. J. |year=2006|title=Linearized Perturbed Compressible Equations for Low Mach number Aeroacoustics|rest=Journal of Computational Physics, Vol. 218, pp. 702-719}}</div>Jhseohttp://www.cfd-online.com/Wiki/Hydrodynamic/acoustic_splittingHydrodynamic/acoustic splitting2008-07-30T21:53:09Z<p>Jhseo: </p>
<hr />
<div>A hydrodynamic/acoustic splitting technique is a good resolving method to the low Mach number aeroacoustics.<br />
In the hydrodynamic/acoustic splitting method, the total flow variables are decomposed into the incompressible variables and the <br />
perturbed compressible ones as,<br />
<br />
<br />
<math>\rho(\vec{x},t)=\rho_0+\rho'(\vec{x},t) </math><br />
<br />
<math>\vec{u}(\vec{x},t)=\vec{U}(\vec{x},t)+\vec{u'}(\vec{x},t)</math><br />
<br />
<math>p(\vec{x},t)=P(\vec{x},t)+p'(\vec{x},t)</math><br />
<br />
<br />
The incompressible variables represent the hydrodynamic flow field, while the acoustic fluctuations and other compressibility <br />
effects are resolved by the perturbed quantities denoted by ('). The original full perturbed compressible equations (PCE) <br />
to calculate the perturbed quantities can be derived by subtracting the incompressible Navier-Stokes equations from the <br />
compressible ones. The PCE recently revised by Seo and Moon are written in a vector form as,<br />
<br />
<br />
<math>\frac{\partial\rho'}{\partial t}+(\vec{u}\cdot\nabla)\rho'+\rho(\nabla\cdot\vec{u'})=0</math><br />
<br />
<math>\frac{\partial\vec{u'}}{\partial<br />
t}+(\vec{u}\cdot\nabla)\vec{u'}+(\vec{u'}\cdot\nabla)\vec{U}+\frac{1}{\rho}\nabla<br />
p'+\frac{\rho'}{\rho}\frac{D\vec{U}}{Dt}=\frac{1}{\rho}\vec{f'_{vis}}</math><br />
<br />
<math>\frac{{\partial p'}}{{\partial t}}\!+\!(\vec u \cdot \nabla )p'\!+\!\gamma p(\nabla \cdot \vec u') + (\vec<br />
u' \cdot \nabla )P = - \frac{{DP}}{{Dt}} + (\gamma - 1)\left( {\Phi - \nabla \cdot \vec q} \right)</math><br />
<br />
<br />
where <math>D/Dt = \partial /\partial t + (\vec U \cdot \nabla )</math>, <math>\vec f'_{vis}</math> is the perturbed viscous force<br />
vector, <math>\Phi</math> and <math>\vec q</math> represent thermal viscous dissipation and heat flux vector, respectively. <br />
The PCE are the mixed-scales, non-linear equations, in which terms are involved with coupling effects between the acoustic <br />
fluctuations and the incompressible flow field. Through the coupling effects, a non-radiating vortical component, so-called <br />
'perturbed vorticity', is generated in the perturbed system. The perturbed vorticity may be regarded as the modification of <br />
hydrodynamic vorticity due to acoustic fluctuations, but it just reside in the perturbed system by this 'one-way' splitting method, <br />
in which backscattering of acoustic fluctuations onto the incompressible flow field is prohibited.<br />
<br />
<br />
<br />
== Linearized Perturbed Compressible Equations ==<br />
<br />
Although the effects of perturbed vorticity on noise generation is actually negligible at low Mach numbers, it is important to <br />
realize that the perturbed vorticity field can easily become unstable for various reasons. If so, it will get self-excited and <br />
grow unphysically, which then affects the acoustic field and causes inconsistency in the acoustic solutions. Recently, <br />
it is observed that the PCE exhibits grid dependant self-excited errors caused by the instability of perturbed vorticity when <br />
applied to the vortex sound prediction at high Reynolds numbers <br />
<br />
In order to resolve the aforementioned matter, the linearized perturbed compressible equations (LPCE) have been proposed by Seo and Moon. The LPCE is a modified version of the original PCE for very low Mach number aeroacoustics. In the LPCE <br />
formulation, the evolution of perturbed vorticity is firmly suppressed by dropping the coupling terms that generate vortical <br />
components in the perturbed system. This is for excluding possible errors caused by its instability, deliberating the fact that <br />
perturbed vorticity is not an important acoustic source for low Mach number flows. A set of LPCE can be written in a vector form as,<br />
<br />
<br />
<math>\frac{{\partial \rho '}}{{\partial t}} + (\vec U \cdot \nabla )\rho ' + \rho _0 (\nabla \cdot \vec u') = 0</math><br />
<br />
<math>\frac{{\partial \vec u'}}{{\partial t}} + \nabla (\vec u' \cdot \vec U) + \frac{1}{{\rho _0 }}\nabla p' = 0</math><br />
<br />
<math>\frac{{\partial p'}}{{\partial t}} + (\vec U \cdot \nabla )p' + \gamma P(\nabla \cdot \vec u') + (\vec u'<br />
\cdot \nabla )P = - \frac{{DP}}{{Dt}}</math><br />
<br />
<br />
The LPCE is linearized to a base incompressible flow field and its formulation is much simpler than the original PCE. <br />
The perturbed vorticity evolution is surely suppressed in the LPCE formulation, since the curl of linearized perturbed momentum <br />
equations, yields <br />
<br />
<math> \frac{{\partial \vec \omega'}}{{\partial t}} = 0 </math><br />
<br />
Consequently, the LPCE prevent any further changes (generation, convection, and decaying) of perturbed vorticity in time and a <br />
physical base for large grid spacing near the wall is also established, since they are an invisicd form of equations. The details <br />
about derivation of LPCE and the characteristics of perturbed vorticity can be found in Ref.(Seo and Moon, 2006). <br />
The left hand sides of LPCE represent the effects of acoustic wave propagation and refraction in the unsteady, inhomogeneous flows, <br />
while the right hand side only contains the acoustic source term, which will be projected from the hydrodynamic flow solution. <br />
For very low Mach number flows, it is interesting to note that the total change of hydrodynamic pressure $DP/Dt$ is considered as <br />
the only explicit noise source term. It agrees with the result of Goldstein in his generalized acoustic analogy, <br />
specifically on the linearized Navier-Stokes equations to a 'non-radiating' base flow field. Ewert and Schroder have <br />
also shown that the dominant acoustic source from the flows at the incompressible limit can be represented by the hydrodynamic <br />
pressure. It is also considerable for the turbulent flow noise. The static pressure field of incompressible LES presents the <br />
features of turbulent flow and their noise can be predicted by the LPCE through the time changes of this pressure field. Therefore, <br />
the LPCE could exclude the uncertainty of modeling the source term for flow noise by turbulence.<br />
<br />
<br />
<br />
== References ==<br />
Seo, J. H., and Moon, Y. J., The Perturbed Compressible Equations for Aeroacoustic Noise Prediction at Low Mach Numbers, AIAA Journal, Vol. 43, No. 8, pp. 1716-1724, 2005.<br />
<br />
[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WHY-4JS1TKR-4&_user=145269&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000012078&_version=1&_urlVersion=0&_userid=145269&md5=a8e366ac6c8196b49ff4a4e016bd62b8 Seo, J. H., and Moon, Y. J., Linearized Perturbed Compressible Equations for Low Mach number Aeroacoustics, Journal of Computational Physics, Vol. 218, pp. 702-719, 2006.]</div>Jhseohttp://www.cfd-online.com/Wiki/Hydrodynamic/acoustic_splittingHydrodynamic/acoustic splitting2008-07-30T21:52:36Z<p>Jhseo: /* References */</p>
<hr />
<div>A hydrodynamic/acoustic splitting technique is a good resolving method to the low Mach number aeroacoustics.<br />
In the hydrodynamic/acoustic splitting method, the total flow variables are decomposed into the incompressible variables and the <br />
perturbed compressible ones as,<br />
<br />
<br />
<math>\rho(\vec{x},t)=\rho_0+\rho'(\vec{x},t) </math><br />
<br />
<math>\vec{u}(\vec{x},t)=\vec{U}(\vec{x},t)+\vec{u'}(\vec{x},t)</math><br />
<br />
<math>p(\vec{x},t)=P(\vec{x},t)+p'(\vec{x},t)</math><br />
<br />
<br />
The incompressible variables represent the hydrodynamic flow field, while the acoustic fluctuations and other compressibility <br />
effects are resolved by the perturbed quantities denoted by ('). The original full perturbed compressible equations (PCE) <br />
to calculate the perturbed quantities can be derived by subtracting the incompressible Navier-Stokes equations from the <br />
compressible ones. The PCE recently revised by Seo and Moon are written in a vector form as,<br />
<br />
<br />
<math>\frac{\partial\rho'}{\partial t}+(\vec{u}\cdot\nabla)\rho'+\rho(\nabla\cdot\vec{u'})=0</math><br />
<br />
<math>\frac{\partial\vec{u'}}{\partial<br />
t}+(\vec{u}\cdot\nabla)\vec{u'}+(\vec{u'}\cdot\nabla)\vec{U}+\frac{1}{\rho}\nabla<br />
p'+\frac{\rho'}{\rho}\frac{D\vec{U}}{Dt}=\frac{1}{\rho}\vec{f'_{vis}}</math><br />
<br />
<math>\frac{{\partial p'}}{{\partial t}}\!+\!(\vec u \cdot \nabla )p'\!+\!\gamma p(\nabla \cdot \vec u') + (\vec<br />
u' \cdot \nabla )P = - \frac{{DP}}{{Dt}} + (\gamma - 1)\left( {\Phi - \nabla \cdot \vec q} \right)</math><br />
<br />
<br />
where <math>D/Dt = \partial /\partial t + (\vec U \cdot \nabla )</math>, <math>\vec f'_{vis}</math> is the perturbed viscous force<br />
vector, <math>\Phi</math> and <math>\vec q</math> represent thermal viscous dissipation and heat flux vector, respectively. <br />
The PCE are the mixed-scales, non-linear equations, in which terms are involved with coupling effects between the acoustic <br />
fluctuations and the incompressible flow field. Through the coupling effects, a non-radiating vortical component, so-called <br />
'perturbed vorticity', is generated in the perturbed system. The perturbed vorticity may be regarded as the modification of <br />
hydrodynamic vorticity due to acoustic fluctuations, but it just reside in the perturbed system by this 'one-way' splitting method, <br />
in which backscattering of acoustic fluctuations onto the incompressible flow field is prohibited.<br />
<br />
<br />
<br />
== Linearized Perturbed Compressible Equations ==<br />
<br />
Although the effects of perturbed vorticity on noise generation is actually negligible at low Mach numbers, it is important to <br />
realize that the perturbed vorticity field can easily become unstable for various reasons. If so, it will get self-excited and <br />
grow unphysically, which then affects the acoustic field and causes inconsistency in the acoustic solutions. Recently, <br />
it is observed that the PCE exhibits grid dependant self-excited errors caused by the instability of perturbed vorticity when <br />
applied to the vortex sound prediction at high Reynolds numbers <br />
<br />
In order to resolve the aforementioned matter, the linearized perturbed compressible equations (LPCE) have been proposed by Seo and Moon. The LPCE is a modified version of the original PCE for very low Mach number aeroacoustics. In the LPCE <br />
formulation, the evolution of perturbed vorticity is firmly suppressed by dropping the coupling terms that generate vortical <br />
components in the perturbed system. This is for excluding possible errors caused by its instability, deliberating the fact that <br />
perturbed vorticity is not an important acoustic source for low Mach number flows. A set of LPCE can be written in a vector form as,<br />
<br />
<br />
<math>\frac{{\partial \rho '}}{{\partial t}} + (\vec U \cdot \nabla )\rho ' + \rho _0 (\nabla \cdot \vec u') = 0</math><br />
<br />
<math>\frac{{\partial \vec u'}}{{\partial t}} + \nabla (\vec u' \cdot \vec U) + \frac{1}{{\rho _0 }}\nabla p' = 0</math><br />
<br />
<math>\frac{{\partial p'}}{{\partial t}} + (\vec U \cdot \nabla )p' + \gamma P(\nabla \cdot \vec u') + (\vec u'<br />
\cdot \nabla )P = - \frac{{DP}}{{Dt}}</math><br />
<br />
<br />
The LPCE is linearized to a base incompressible flow field and its formulation is much simpler than the original PCE. <br />
The perturbed vorticity evolution is surely suppressed in the LPCE formulation, since the curl of linearized perturbed momentum <br />
equations, yields <br />
<br />
<math> \frac{{\partial \vec \omega'}}{{\partial t}} = 0 </math><br />
<br />
Consequently, the LPCE prevent any further changes (generation, convection, and decaying) of perturbed vorticity in time and a <br />
physical base for large grid spacing near the wall is also established, since they are an invisicd form of equations. The details <br />
about derivation of LPCE and the characteristics of perturbed vorticity can be found in Ref.(Seo and Moon, 2006). <br />
The left hand sides of LPCE represent the effects of acoustic wave propagation and refraction in the unsteady, inhomogeneous flows, <br />
while the right hand side only contains the acoustic source term, which will be projected from the hydrodynamic flow solution. <br />
For very low Mach number flows, it is interesting to note that the total change of hydrodynamic pressure $DP/Dt$ is considered as <br />
the only explicit noise source term. It agrees with the result of Goldstein in his generalized acoustic analogy, <br />
specifically on the linearized Navier-Stokes equations to a 'non-radiating' base flow field. Ewert and Schroder have <br />
also shown that the dominant acoustic source from the flows at the incompressible limit can be represented by the hydrodynamic <br />
pressure. It is also considerable for the turbulent flow noise. The static pressure field of incompressible LES presents the <br />
features of turbulent flow and their noise can be predicted by the LPCE through the time changes of this pressure field. Therefore, <br />
the LPCE could exclude the uncertainty of modeling the source term for flow noise by turbulence.<br />
<br />
<br />
<br />
[http://www.example.com link title]== References ==<br />
Seo, J. H., and Moon, Y. J., The Perturbed Compressible Equations for Aeroacoustic Noise Prediction at Low Mach Numbers, AIAA Journal, Vol. 43, No. 8, pp. 1716-1724, 2005.<br />
<br />
[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WHY-4JS1TKR-4&_user=145269&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000012078&_version=1&_urlVersion=0&_userid=145269&md5=a8e366ac6c8196b49ff4a4e016bd62b8 Seo, J. H., and Moon, Y. J., Linearized Perturbed Compressible Equations for Low Mach number Aeroacoustics, Journal of Computational Physics, Vol. 218, pp. 702-719, 2006.]</div>Jhseohttp://www.cfd-online.com/Wiki/Hydrodynamic/acoustic_splittingHydrodynamic/acoustic splitting2008-07-30T21:51:28Z<p>Jhseo: </p>
<hr />
<div>A hydrodynamic/acoustic splitting technique is a good resolving method to the low Mach number aeroacoustics.<br />
In the hydrodynamic/acoustic splitting method, the total flow variables are decomposed into the incompressible variables and the <br />
perturbed compressible ones as,<br />
<br />
<br />
<math>\rho(\vec{x},t)=\rho_0+\rho'(\vec{x},t) </math><br />
<br />
<math>\vec{u}(\vec{x},t)=\vec{U}(\vec{x},t)+\vec{u'}(\vec{x},t)</math><br />
<br />
<math>p(\vec{x},t)=P(\vec{x},t)+p'(\vec{x},t)</math><br />
<br />
<br />
The incompressible variables represent the hydrodynamic flow field, while the acoustic fluctuations and other compressibility <br />
effects are resolved by the perturbed quantities denoted by ('). The original full perturbed compressible equations (PCE) <br />
to calculate the perturbed quantities can be derived by subtracting the incompressible Navier-Stokes equations from the <br />
compressible ones. The PCE recently revised by Seo and Moon are written in a vector form as,<br />
<br />
<br />
<math>\frac{\partial\rho'}{\partial t}+(\vec{u}\cdot\nabla)\rho'+\rho(\nabla\cdot\vec{u'})=0</math><br />
<br />
<math>\frac{\partial\vec{u'}}{\partial<br />
t}+(\vec{u}\cdot\nabla)\vec{u'}+(\vec{u'}\cdot\nabla)\vec{U}+\frac{1}{\rho}\nabla<br />
p'+\frac{\rho'}{\rho}\frac{D\vec{U}}{Dt}=\frac{1}{\rho}\vec{f'_{vis}}</math><br />
<br />
<math>\frac{{\partial p'}}{{\partial t}}\!+\!(\vec u \cdot \nabla )p'\!+\!\gamma p(\nabla \cdot \vec u') + (\vec<br />
u' \cdot \nabla )P = - \frac{{DP}}{{Dt}} + (\gamma - 1)\left( {\Phi - \nabla \cdot \vec q} \right)</math><br />
<br />
<br />
where <math>D/Dt = \partial /\partial t + (\vec U \cdot \nabla )</math>, <math>\vec f'_{vis}</math> is the perturbed viscous force<br />
vector, <math>\Phi</math> and <math>\vec q</math> represent thermal viscous dissipation and heat flux vector, respectively. <br />
The PCE are the mixed-scales, non-linear equations, in which terms are involved with coupling effects between the acoustic <br />
fluctuations and the incompressible flow field. Through the coupling effects, a non-radiating vortical component, so-called <br />
'perturbed vorticity', is generated in the perturbed system. The perturbed vorticity may be regarded as the modification of <br />
hydrodynamic vorticity due to acoustic fluctuations, but it just reside in the perturbed system by this 'one-way' splitting method, <br />
in which backscattering of acoustic fluctuations onto the incompressible flow field is prohibited.<br />
<br />
<br />
<br />
== Linearized Perturbed Compressible Equations ==<br />
<br />
Although the effects of perturbed vorticity on noise generation is actually negligible at low Mach numbers, it is important to <br />
realize that the perturbed vorticity field can easily become unstable for various reasons. If so, it will get self-excited and <br />
grow unphysically, which then affects the acoustic field and causes inconsistency in the acoustic solutions. Recently, <br />
it is observed that the PCE exhibits grid dependant self-excited errors caused by the instability of perturbed vorticity when <br />
applied to the vortex sound prediction at high Reynolds numbers <br />
<br />
In order to resolve the aforementioned matter, the linearized perturbed compressible equations (LPCE) have been proposed by Seo and Moon. The LPCE is a modified version of the original PCE for very low Mach number aeroacoustics. In the LPCE <br />
formulation, the evolution of perturbed vorticity is firmly suppressed by dropping the coupling terms that generate vortical <br />
components in the perturbed system. This is for excluding possible errors caused by its instability, deliberating the fact that <br />
perturbed vorticity is not an important acoustic source for low Mach number flows. A set of LPCE can be written in a vector form as,<br />
<br />
<br />
<math>\frac{{\partial \rho '}}{{\partial t}} + (\vec U \cdot \nabla )\rho ' + \rho _0 (\nabla \cdot \vec u') = 0</math><br />
<br />
<math>\frac{{\partial \vec u'}}{{\partial t}} + \nabla (\vec u' \cdot \vec U) + \frac{1}{{\rho _0 }}\nabla p' = 0</math><br />
<br />
<math>\frac{{\partial p'}}{{\partial t}} + (\vec U \cdot \nabla )p' + \gamma P(\nabla \cdot \vec u') + (\vec u'<br />
\cdot \nabla )P = - \frac{{DP}}{{Dt}}</math><br />
<br />
<br />
The LPCE is linearized to a base incompressible flow field and its formulation is much simpler than the original PCE. <br />
The perturbed vorticity evolution is surely suppressed in the LPCE formulation, since the curl of linearized perturbed momentum <br />
equations, yields <br />
<br />
<math> \frac{{\partial \vec \omega'}}{{\partial t}} = 0 </math><br />
<br />
Consequently, the LPCE prevent any further changes (generation, convection, and decaying) of perturbed vorticity in time and a <br />
physical base for large grid spacing near the wall is also established, since they are an invisicd form of equations. The details <br />
about derivation of LPCE and the characteristics of perturbed vorticity can be found in Ref.(Seo and Moon, 2006). <br />
The left hand sides of LPCE represent the effects of acoustic wave propagation and refraction in the unsteady, inhomogeneous flows, <br />
while the right hand side only contains the acoustic source term, which will be projected from the hydrodynamic flow solution. <br />
For very low Mach number flows, it is interesting to note that the total change of hydrodynamic pressure $DP/Dt$ is considered as <br />
the only explicit noise source term. It agrees with the result of Goldstein in his generalized acoustic analogy, <br />
specifically on the linearized Navier-Stokes equations to a 'non-radiating' base flow field. Ewert and Schroder have <br />
also shown that the dominant acoustic source from the flows at the incompressible limit can be represented by the hydrodynamic <br />
pressure. It is also considerable for the turbulent flow noise. The static pressure field of incompressible LES presents the <br />
features of turbulent flow and their noise can be predicted by the LPCE through the time changes of this pressure field. Therefore, <br />
the LPCE could exclude the uncertainty of modeling the source term for flow noise by turbulence.<br />
<br />
<br />
<br />
== References ==<br />
Seo, J. H., and Moon, Y. J., The Perturbed Compressible Equations for Aeroacoustic Noise Prediction at Low Mach Numbers, AIAA Journal, Vol. 43, No. 8, pp. 1716-1724, 2005.<br />
<br />
[Seo, J. H., and Moon, Y. J., Linearized Perturbed Compressible Equations for Low Mach number Aeroacoustics, Journal of Computational Physics, Vol. 218, pp. 702-719, 2006.]</div>Jhseohttp://www.cfd-online.com/Wiki/Hydrodynamic/acoustic_splittingHydrodynamic/acoustic splitting2008-07-30T21:49:50Z<p>Jhseo: </p>
<hr />
<div>A hydrodynamic/acoustic splitting technique is a good resolving method to the low Mach number aeroacoustics.<br />
In the hydrodynamic/acoustic splitting method, the total flow variables are decomposed into the incompressible variables and the <br />
perturbed compressible ones as,<br />
<br />
<br />
<math>\rho(\vec{x},t)=\rho_0+\rho'(\vec{x},t) </math><br />
<br />
<math>\vec{u}(\vec{x},t)=\vec{U}(\vec{x},t)+\vec{u'}(\vec{x},t)</math><br />
<br />
<math>p(\vec{x},t)=P(\vec{x},t)+p'(\vec{x},t)</math><br />
<br />
<br />
The incompressible variables represent the hydrodynamic flow field, while the acoustic fluctuations and other compressibility <br />
effects are resolved by the perturbed quantities denoted by ('). The original full perturbed compressible equations (PCE) <br />
to calculate the perturbed quantities can be derived by subtracting the incompressible Navier-Stokes equations from the <br />
compressible ones. The PCE recently revised by Seo and Moon are written in a vector form as,<br />
<br />
<br />
<math>\frac{\partial\rho'}{\partial t}+(\vec{u}\cdot\nabla)\rho'+\rho(\nabla\cdot\vec{u'})=0</math><br />
<br />
<math>\frac{\partial\vec{u'}}{\partial<br />
t}+(\vec{u}\cdot\nabla)\vec{u'}+(\vec{u'}\cdot\nabla)\vec{U}+\frac{1}{\rho}\nabla<br />
p'+\frac{\rho'}{\rho}\frac{D\vec{U}}{Dt}=\frac{1}{\rho}\vec{f'_{vis}}</math><br />
<br />
<math>\frac{{\partial p'}}{{\partial t}}\!+\!(\vec u \cdot \nabla )p'\!+\!\gamma p(\nabla \cdot \vec u') + (\vec<br />
u' \cdot \nabla )P = - \frac{{DP}}{{Dt}} + (\gamma - 1)\left( {\Phi - \nabla \cdot \vec q} \right)</math><br />
<br />
<br />
where <math>D/Dt = \partial /\partial t + (\vec U \cdot \nabla )</math>, <math>\vec f'_{vis}</math> is the perturbed viscous force<br />
vector, <math>\Phi</math> and <math>\vec q</math> represent thermal viscous dissipation and heat flux vector, respectively. <br />
The PCE are the mixed-scales, non-linear equations, in which terms are involved with coupling effects between the acoustic <br />
fluctuations and the incompressible flow field. Through the coupling effects, a non-radiating vortical component, so-called <br />
'perturbed vorticity', is generated in the perturbed system. The perturbed vorticity may be regarded as the modification of <br />
hydrodynamic vorticity due to acoustic fluctuations, but it just reside in the perturbed system by this 'one-way' splitting method, <br />
in which backscattering of acoustic fluctuations onto the incompressible flow field is prohibited.<br />
<br />
<br />
<br />
== Linearized Perturbed Compressible Equations ==<br />
<br />
Although the effects of perturbed vorticity on noise generation is actually negligible at low Mach numbers, it is important to <br />
realize that the perturbed vorticity field can easily become unstable for various reasons. If so, it will get self-excited and <br />
grow unphysically, which then affects the acoustic field and causes inconsistency in the acoustic solutions. Recently, <br />
it is observed that the PCE exhibits grid dependant self-excited errors caused by the instability of perturbed vorticity when <br />
applied to the vortex sound prediction at high Reynolds numbers <br />
<br />
In order to resolve the aforementioned matter, the linearized perturbed compressible equations (LPCE) have been proposed by Seo and Moon. The LPCE is a modified version of the original PCE for very low Mach number aeroacoustics. In the LPCE <br />
formulation, the evolution of perturbed vorticity is firmly suppressed by dropping the coupling terms that generate vortical <br />
components in the perturbed system. This is for excluding possible errors caused by its instability, deliberating the fact that <br />
perturbed vorticity is not an important acoustic source for low Mach number flows. A set of LPCE can be written in a vector form as,<br />
<br />
<br />
<math>\frac{{\partial \rho '}}{{\partial t}} + (\vec U \cdot \nabla )\rho ' + \rho _0 (\nabla \cdot \vec u') = 0</math><br />
<br />
<math>\frac{{\partial \vec u'}}{{\partial t}} + \nabla (\vec u' \cdot \vec U) + \frac{1}{{\rho _0 }}\nabla p' = 0</math><br />
<br />
<math>\frac{{\partial p'}}{{\partial t}} + (\vec U \cdot \nabla )p' + \gamma P(\nabla \cdot \vec u') + (\vec u'<br />
\cdot \nabla )P = - \frac{{DP}}{{Dt}}</math><br />
<br />
<br />
The LPCE is linearized to a base incompressible flow field and its formulation is much simpler than the original PCE. <br />
The perturbed vorticity evolution is surely suppressed in the LPCE formulation, since the curl of linearized perturbed momentum <br />
equations, yields <br />
<br />
<math> \frac{{\partial \vec \omega'}}{{\partial t}} = 0 </math><br />
<br />
Consequently, the LPCE prevent any further changes (generation, convection, and decaying) of perturbed vorticity in time and a <br />
physical base for large grid spacing near the wall is also established, since they are an invisicd form of equations. The details <br />
about derivation of LPCE and the characteristics of perturbed vorticity can be found in Ref.(Seo and Moon, 2006). <br />
The left hand sides of LPCE represent the effects of acoustic wave propagation and refraction in the unsteady, inhomogeneous flows, <br />
while the right hand side only contains the acoustic source term, which will be projected from the hydrodynamic flow solution. <br />
For very low Mach number flows, it is interesting to note that the total change of hydrodynamic pressure $DP/Dt$ is considered as <br />
the only explicit noise source term. It agrees with the result of Goldstein in his generalized acoustic analogy, <br />
specifically on the linearized Navier-Stokes equations to a 'non-radiating' base flow field. Ewert and Schroder have <br />
also shown that the dominant acoustic source from the flows at the incompressible limit can be represented by the hydrodynamic <br />
pressure. It is also considerable for the turbulent flow noise. The static pressure field of incompressible LES presents the <br />
features of turbulent flow and their noise can be predicted by the LPCE through the time changes of this pressure field. Therefore, <br />
the LPCE could exclude the uncertainty of modeling the source term for flow noise by turbulence.<br />
<br />
<br />
<br />
== References ==<br />
Seo, J. H., and Moon, Y. J., The Perturbed Compressible Equations for Aeroacoustic Noise Prediction at Low Mach Numbers, AIAA Journal, Vol. 43, No. 8, pp. 1716-1724, 2005.<br />
<br />
Seo, J. H., and Moon, Y. J., Linearized Perturbed Compressible Equations for Low Mach number Aeroacoustics, Journal of Computational Physics, Vol. 218, pp. 702-719, 2006.</div>Jhseohttp://www.cfd-online.com/Wiki/Hydrodynamic/acoustic_splittingHydrodynamic/acoustic splitting2008-07-30T21:48:34Z<p>Jhseo: </p>
<hr />
<div>A hydrodynamic/acoustic splitting technique is a good resolving method to the low Mach number aeroacoustics.<br />
In the hydrodynamic/acoustic splitting method, the total flow variables are decomposed into the incompressible variables and the <br />
perturbed compressible ones as,<br />
<br />
<math>\rho(\vec{x},t)=\rho_0+\rho'(\vec{x},t) </math><br />
<br />
<math>\vec{u}(\vec{x},t)=\vec{U}(\vec{x},t)+\vec{u'}(\vec{x},t)</math><br />
<br />
<math>p(\vec{x},t)=P(\vec{x},t)+p'(\vec{x},t)</math><br />
<br />
The incompressible variables represent the hydrodynamic flow field, while the acoustic fluctuations and other compressibility <br />
effects are resolved by the perturbed quantities denoted by ('). The original full perturbed compressible equations (PCE) <br />
to calculate the perturbed quantities can be derived by subtracting the incompressible Navier-Stokes equations from the <br />
compressible ones. The PCE recently revised by Seo and Moon are written in a vector form as,<br />
<br />
<math>\frac{\partial\rho'}{\partial t}+(\vec{u}\cdot\nabla)\rho'+\rho(\nabla\cdot\vec{u'})=0</math><br />
<br />
<math>\frac{\partial\vec{u'}}{\partial<br />
t}+(\vec{u}\cdot\nabla)\vec{u'}+(\vec{u'}\cdot\nabla)\vec{U}+\frac{1}{\rho}\nabla<br />
p'+\frac{\rho'}{\rho}\frac{D\vec{U}}{Dt}=\frac{1}{\rho}\vec{f'_{vis}}</math><br />
<br />
<math>\frac{{\partial p'}}{{\partial t}}\!+\!(\vec u \cdot \nabla )p'\!+\!\gamma p(\nabla \cdot \vec u') + (\vec<br />
u' \cdot \nabla )P = - \frac{{DP}}{{Dt}} + (\gamma - 1)\left( {\Phi - \nabla \cdot \vec q} \right)</math><br />
<br />
where <math>D/Dt = \partial /\partial t + (\vec U \cdot \nabla )</math>, <math>\vec f'_{vis}</math> is the perturbed viscous force<br />
vector, <math>\Phi</math> and <math>\vec q</math> represent thermal viscous dissipation and heat flux vector, respectively. <br />
The PCE are the mixed-scales, non-linear equations, in which terms are involved with coupling effects between the acoustic <br />
fluctuations and the incompressible flow field. Through the coupling effects, a non-radiating vortical component, so-called <br />
'perturbed vorticity', is generated in the perturbed system. The perturbed vorticity may be regarded as the modification of <br />
hydrodynamic vorticity due to acoustic fluctuations, but it just reside in the perturbed system by this 'one-way' splitting method, <br />
in which backscattering of acoustic fluctuations onto the incompressible flow field is prohibited.<br />
<br />
<br />
<br />
== Linearized Perturbed Compressible Equations ==<br />
<br />
Although the effects of perturbed vorticity on noise generation is actually negligible at low Mach numbers, it is important to <br />
realize that the perturbed vorticity field can easily become unstable for various reasons. If so, it will get self-excited and <br />
grow unphysically, which then affects the acoustic field and causes inconsistency in the acoustic solutions. Recently, <br />
it is observed that the PCE exhibits grid dependant self-excited errors caused by the instability of perturbed vorticity when <br />
applied to the vortex sound prediction at high Reynolds numbers <br />
<br />
<br />
In order to resolve the aforementioned matter, the linearized perturbed compressible equations (LPCE) have been proposed by Seo and Moon. The LPCE is a modified version of the original PCE for very low Mach number aeroacoustics. In the LPCE <br />
formulation, the evolution of perturbed vorticity is firmly suppressed by dropping the coupling terms that generate vortical <br />
components in the perturbed system. This is for excluding possible errors caused by its instability, deliberating the fact that <br />
perturbed vorticity is not an important acoustic source for low Mach number flows. A set of LPCE can be written in a vector form as,<br />
<br />
<math>\frac{{\partial \rho '}}{{\partial t}} + (\vec U \cdot \nabla )\rho ' + \rho _0 (\nabla \cdot \vec u') = 0</math><br />
<br />
<math>\frac{{\partial \vec u'}}{{\partial t}} + \nabla (\vec u' \cdot \vec U) + \frac{1}{{\rho _0 }}\nabla p' = 0</math><br />
<br />
<math>\frac{{\partial p'}}{{\partial t}} + (\vec U \cdot \nabla )p' + \gamma P(\nabla \cdot \vec u') + (\vec u'<br />
\cdot \nabla )P = - \frac{{DP}}{{Dt}}</math><br />
<br />
The LPCE is linearized to a base incompressible flow field and its formulation is much simpler than the original PCE. <br />
The perturbed vorticity evolution is surely suppressed in the LPCE formulation, since the curl of linearized perturbed momentum <br />
equations, yields <br />
<br />
<math> \frac{{\partial \vec \omega'}}{{\partial t}} = 0 </math><br />
<br />
Consequently, the LPCE prevent any further changes (generation, convection, and decaying) of perturbed vorticity in time and a <br />
physical base for large grid spacing near the wall is also established, since they are an invisicd form of equations. The details <br />
about derivation of LPCE and the characteristics of perturbed vorticity can be found in Ref.(Seo and Moon, 2006). <br />
The left hand sides of LPCE represent the effects of acoustic wave propagation and refraction in the unsteady, inhomogeneous flows, <br />
while the right hand side only contains the acoustic source term, which will be projected from the hydrodynamic flow solution. <br />
For very low Mach number flows, it is interesting to note that the total change of hydrodynamic pressure $DP/Dt$ is considered as <br />
the only explicit noise source term. It agrees with the result of Goldstein in his generalized acoustic analogy, <br />
specifically on the linearized Navier-Stokes equations to a 'non-radiating' base flow field. Ewert and Schroder have <br />
also shown that the dominant acoustic source from the flows at the incompressible limit can be represented by the hydrodynamic <br />
pressure. It is also considerable for the turbulent flow noise. The static pressure field of incompressible LES presents the <br />
features of turbulent flow and their noise can be predicted by the LPCE through the time changes of this pressure field. Therefore, <br />
the LPCE could exclude the uncertainty of modeling the source term for flow noise by turbulence.<br />
<br />
<br />
<br />
== References ==<br />
Seo, J. H., and Moon, Y. J., The Perturbed Compressible Equations for Aeroacoustic Noise Prediction at Low Mach Numbers, AIAA Journal, Vol. 43, No. 8, pp. 1716-1724, 2005.<br />
Seo, J. H., and Moon, Y. J., Linearized Perturbed Compressible Equations for Low Mach number Aeroacoustics, Journal of Computational Physics, Vol. 218, pp. 702-719, 2006.</div>Jhseohttp://www.cfd-online.com/Wiki/Hydrodynamic/acoustic_splittingHydrodynamic/acoustic splitting2008-07-30T21:43:58Z<p>Jhseo: </p>
<hr />
<div>A hydrodynamic/acoustic splitting technique is a good resolving method to the low Mach number aeroacoustics.<br />
In the hydrodynamic/acoustic splitting method, the total flow variables are decomposed into the incompressible variables and the <br />
perturbed compressible ones as,<br />
<br />
<math>\rho(\vec{x},t)=\rho_0+\rho'(\vec{x},t) </math><br />
<br />
<math>\vec{u}(\vec{x},t)=\vec{U}(\vec{x},t)+\vec{u'}(\vec{x},t)</math><br />
<br />
<math>p(\vec{x},t)=P(\vec{x},t)+p'(\vec{x},t)</math><br />
<br />
The incompressible variables represent the hydrodynamic flow field, while the acoustic fluctuations and other compressibility <br />
effects are resolved by the perturbed quantities denoted by ('). The original full perturbed compressible equations (PCE) <br />
to calculate the perturbed quantities can be derived by subtracting the incompressible Navier-Stokes equations from the <br />
compressible ones. The PCE recently revised by Seo and Moon are written in a vector form as,<br />
<br />
<math>\frac{\partial\rho'}{\partial t}+(\vec{u}\cdot\nabla)\rho'+\rho(\nabla\cdot\vec{u'})=0</math><br />
<br />
<math>\frac{\partial\vec{u'}}{\partial<br />
t}+(\vec{u}\cdot\nabla)\vec{u'}+(\vec{u'}\cdot\nabla)\vec{U}+\frac{1}{\rho}\nabla<br />
p'+\frac{\rho'}{\rho}\frac{D\vec{U}}{Dt}=\frac{1}{\rho}\vec{f'_{vis}}</math><br />
<br />
<math>\frac{{\partial p'}}{{\partial t}}\!+\!(\vec u \cdot \nabla )p'\!+\!\gamma p(\nabla \cdot \vec u') + (\vec<br />
u' \cdot \nabla )P = - \frac{{DP}}{{Dt}} + (\gamma - 1)\left( {\Phi - \nabla \cdot \vec q} \right)</math><br />
<br />
where <math>D/Dt = \partial /\partial t + (\vec U \cdot \nabla )</math>, <math>\vec f'_{vis}</math> is the perturbed viscous force<br />
vector, <math>\Phi</math> and <math>\vec q</math> represent thermal viscous dissipation and heat flux vector, respectively. <br />
The PCE are the mixed-scales, non-linear equations, in which terms are involved with coupling effects between the acoustic <br />
fluctuations and the incompressible flow field. Through the coupling effects, a non-radiating vortical component, so-called <br />
'perturbed vorticity', is generated in the perturbed system. The perturbed vorticity may be regarded as the modification of <br />
hydrodynamic vorticity due to acoustic fluctuations, but it just reside in the perturbed system by this 'one-way' splitting method, <br />
in which backscattering of acoustic fluctuations onto the incompressible flow field is prohibited.<br />
<br />
<br />
<br />
== Linearized Perturbed Compressible Equations ==<br />
<br />
Although the effects of perturbed vorticity on noise generation is actually negligible at low Mach numbers, it is important to <br />
realize that the perturbed vorticity field can easily become unstable for various reasons. If so, it will get self-excited and <br />
grow unphysically, which then affects the acoustic field and causes inconsistency in the acoustic solutions. Recently, <br />
it is observed that the PCE exhibits grid dependant self-excited errors caused by the instability of perturbed vorticity when <br />
applied to the vortex sound prediction at high Reynolds numbers <br />
<br />
<br />
In order to resolve the aforementioned matter, the linearized perturbed compressible equations (LPCE) have been proposed by Seo and Moon. The LPCE is a modified version of the original PCE for very low Mach number aeroacoustics. In the LPCE <br />
formulation, the evolution of perturbed vorticity is firmly suppressed by dropping the coupling terms that generate vortical <br />
components in the perturbed system. This is for excluding possible errors caused by its instability, deliberating the fact that <br />
perturbed vorticity is not an important acoustic source for low Mach number flows. A set of LPCE can be written in a vector form as,<br />
<br />
<math>\frac{{\partial \rho '}}{{\partial t}} + (\vec U \cdot \nabla )\rho ' + \rho _0 (\nabla \cdot \vec u') = 0</math><br />
<br />
<math>\frac{{\partial \vec u'}}{{\partial t}} + \nabla (\vec u' \cdot \vec U) + \frac{1}{{\rho _0 }}\nabla p' = 0</math><br />
<br />
<math>\frac{{\partial p'}}{{\partial t}} + (\vec U \cdot \nabla )p' + \gamma P(\nabla \cdot \vec u') + (\vec u'<br />
\cdot \nabla )P = - \frac{{DP}}{{Dt}}</math><br />
<br />
The LPCE is linearized to a base incompressible flow field and its formulation is much simpler than the original PCE. <br />
The perturbed vorticity evolution is surely suppressed in the LPCE formulation, since the curl of linearized perturbed momentum <br />
equations, yields <br />
<math> \frac{{\partial \vec \omega'}}{{\partial t}} = 0 </math><br />
<br />
Consequently, the LPCE prevent any further changes (generation, convection, and decaying) of perturbed vorticity in time and a <br />
physical base for large grid spacing near the wall is also established, since they are an invisicd form of equations. The details <br />
about derivation of LPCE and the characteristics of perturbed vorticity can be found in Ref.(Seo and Moon, 2005). <br />
The left hand sides of LPCE represent the effects of acoustic wave propagation and refraction in the unsteady, inhomogeneous flows, <br />
while the right hand side only contains the acoustic source term, which will be projected from the hydrodynamic flow solution. <br />
For very low Mach number flows, it is interesting to note that the total change of hydrodynamic pressure $DP/Dt$ is considered as <br />
the only explicit noise source term. It agrees with the result of Goldstein in his generalized acoustic analogy, <br />
specifically on the linearized Navier-Stokes equations to a 'non-radiating' base flow field. Ewert and Schroder have <br />
also shown that the dominant acoustic source from the flows at the incompressible limit can be represented by the hydrodynamic <br />
pressure. It is also considerable for the turbulent flow noise. The static pressure field of incompressible LES presents the <br />
features of turbulent flow and their noise can be predicted by the LPCE through the time changes of this pressure field. Therefore, <br />
the LPCE could exclude the uncertainty of modeling the source term for flow noise by turbulence.</div>Jhseohttp://www.cfd-online.com/Wiki/Hydrodynamic/acoustic_splittingHydrodynamic/acoustic splitting2008-07-30T21:35:11Z<p>Jhseo: </p>
<hr />
<div>In the hydrodynamic/acoustic splitting method, the total flow variables are decomposed into the incompressible variables and the <br />
perturbed compressible ones as,<br />
<br />
<math>\rho(\vec{x},t)=\rho_0+\rho'(\vec{x},t) </math><br />
<br />
<math>\vec{u}(\vec{x},t)=\vec{U}(\vec{x},t)+\vec{u'}(\vec{x},t)</math><br />
<br />
<math>p(\vec{x},t)=P(\vec{x},t)+p'(\vec{x},t)</math><br />
<br />
The incompressible variables represent the hydrodynamic flow field, while the acoustic fluctuations and other compressibility <br />
effects are resolved by the perturbed quantities denoted by ('). The original full perturbed compressible equations (PCE) <br />
to calculate the perturbed quantities can be derived by subtracting the incompressible Navier-Stokes equations from the <br />
compressible ones. The PCE recently revised by Seo and Moon are written in a vector form as,<br />
<br />
<math>\frac{\partial\rho'}{\partial t}+(\vec{u}\cdot\nabla)\rho'+\rho(\nabla\cdot\vec{u'})=0</math><br />
<br />
<math>\frac{\partial\vec{u'}}{\partial<br />
t}+(\vec{u}\cdot\nabla)\vec{u'}+(\vec{u'}\cdot\nabla)\vec{U}+\frac{1}{\rho}\nabla<br />
p'+\frac{\rho'}{\rho}\frac{D\vec{U}}{Dt}=\frac{1}{\rho}\vec{f'_{vis}}</math><br />
<br />
<math>\frac{{\partial p'}}{{\partial t}}\!+\!(\vec u \cdot \nabla )p'\!+\!\gamma p(\nabla \cdot \vec u') + (\vec<br />
u' \cdot \nabla )P = - \frac{{DP}}{{Dt}} + (\gamma - 1)\left( {\Phi - \nabla \cdot \vec q} \right)</math><br />
<br />
where <math>D/Dt = \partial /\partial t + (\vec U \cdot \nabla )</math>, <math>\vec f'_{vis}</math> is the perturbed viscous force<br />
vector, <math>\Phi</math> and <math>\vec q</math> represent thermal viscous dissipation and heat flux vector, respectively. <br />
The PCE are the mixed-scales, non-linear equations, in which terms are involved with coupling effects between the acoustic <br />
fluctuations and the incompressible flow field. Through the coupling effects, a non-radiating vortical component, so-called <br />
'perturbed vorticity', is generated in the perturbed system. The perturbed vorticity may be regarded as the modification of <br />
hydrodynamic vorticity due to acoustic fluctuations, but it just reside in the perturbed system by this 'one-way' splitting method, <br />
in which backscattering of acoustic fluctuations onto the incompressible flow field is prohibited.</div>Jhseohttp://www.cfd-online.com/Wiki/Hydrodynamic/acoustic_splittingHydrodynamic/acoustic splitting2008-07-30T21:30:41Z<p>Jhseo: </p>
<hr />
<div>In the hydrodynamic/acoustic splitting method, the total flow variables are decomposed into the incompressible variables and the <br />
perturbed compressible ones as,<br />
<br />
<math>\rho(\vec{x},t)=\rho_0+\rho'(\vec{x},t) </math><br />
<br />
<math>\vec{u}(\vec{x},t)=\vec{U}(\vec{x},t)+\vec{u'}(\vec{x},t)</math><br />
<br />
<math>p(\vec{x},t)=P(\vec{x},t)+p'(\vec{x},t)</math><br />
<br />
The incompressible variables represent the hydrodynamic flow field, while the acoustic fluctuations and other compressibility <br />
effects are resolved by the perturbed quantities denoted by ('). The original full perturbed compressible equations (PCE) <br />
to calculate the perturbed quantities can be derived by subtracting the incompressible Navier-Stokes equations from the <br />
compressible ones. The PCE recently revised by Seo and Moon are written in a vector form as,<br />
<br />
<math>\frac{\partial\rho'}{\partial t}+(\vec{u}\cdot\nabla)\rho'+\rho(\nabla\cdot\vec{u'})=0</math><br />
<br />
<math>\frac{\partial\vec{u'}}{\partial<br />
t}+(\vec{u}\cdot\nabla)\vec{u'}+(\vec{u'}\cdot\nabla)\vec{U}+\frac{1}{\rho}\nabla<br />
p'+\frac{\rho'}{\rho}\frac{D\vec{U}}{Dt}=\frac{1}{\rho}\vec{f'_{vis}}</math><br />
<br />
<math>\frac{{\partial p'}}{{\partial t}}\!+\!(\vec u \cdot \nabla )p'\!+\!\gamma p(\nabla \cdot \vec u') + (\vec<br />
u' \cdot \nabla )P = - \frac{{DP}}{{Dt}} + (\gamma - 1)\left( {\Phi - \nabla \cdot \vec q} \right)</math></div>Jhseohttp://www.cfd-online.com/Wiki/Hydrodynamic/acoustic_splittingHydrodynamic/acoustic splitting2008-07-30T21:30:04Z<p>Jhseo: </p>
<hr />
<div>In the hydrodynamic/acoustic splitting method, the total flow variables are decomposed into the incompressible variables and the <br />
perturbed compressible ones as,<br />
<br />
<math>\rho(\vec{x},t)=\rho_0+\rho'(\vec{x},t) \\ </math><br />
<math>\vec{u}(\vec{x},t)=\vec{U}(\vec{x},t)+\vec{u'}(\vec{x},t)</math><br />
<math>p(\vec{x},t)=P(\vec{x},t)+p'(\vec{x},t)</math><br />
<br />
The incompressible variables represent the hydrodynamic flow field, while the acoustic fluctuations and other compressibility <br />
effects are resolved by the perturbed quantities denoted by ('). The original full perturbed compressible equations (PCE) <br />
to calculate the perturbed quantities can be derived by subtracting the incompressible Navier-Stokes equations from the <br />
compressible ones. The PCE recently revised by Seo and Moon are written in a vector form as,<br />
<br />
<math>\frac{\partial\rho'}{\partial t}+(\vec{u}\cdot\nabla)\rho'+\rho(\nabla\cdot\vec{u'})=0</math><br />
<math>\frac{\partial\vec{u'}}{\partial<br />
t}+(\vec{u}\cdot\nabla)\vec{u'}+(\vec{u'}\cdot\nabla)\vec{U}+\frac{1}{\rho}\nabla<br />
p'+\frac{\rho'}{\rho}\frac{D\vec{U}}{Dt}=\frac{1}{\rho}\vec{f'_{vis}}</math><br />
<math>\frac{{\partial p'}}{{\partial t}}\!+\!(\vec u \cdot \nabla )p'\!+\!\gamma p(\nabla \cdot \vec u') + (\vec<br />
u' \cdot \nabla )P = - \frac{{DP}}{{Dt}} + (\gamma - 1)\left( {\Phi - \nabla \cdot \vec q} \right)</math></div>Jhseohttp://www.cfd-online.com/Wiki/Hydrodynamic/acoustic_splittingHydrodynamic/acoustic splitting2008-07-30T21:29:18Z<p>Jhseo: </p>
<hr />
<div>In the hydrodynamic/acoustic splitting method, the total flow variables are decomposed into the incompressible variables and the <br />
perturbed compressible ones as,<br />
<br />
<math>\rho(\vec{x},t)=\rho_0+\rho'(\vec{x},t) </math><br />
<math>\vec{u}(\vec{x},t)=\vec{U}(\vec{x},t)+\vec{u'}(\vec{x},t)</math><br />
<math>p(\vec{x},t)=P(\vec{x},t)+p'(\vec{x},t)</math><br />
<br />
The incompressible variables represent the hydrodynamic flow field, while the acoustic fluctuations and other compressibility <br />
effects are resolved by the perturbed quantities denoted by ('). The original full perturbed compressible equations (PCE) <br />
to calculate the perturbed quantities can be derived by subtracting the incompressible Navier-Stokes equations from the <br />
compressible ones. The PCE recently revised by Seo and Moon are written in a vector form as,<br />
<br />
<math>\frac{\partial\rho'}{\partial t}+(\vec{u}\cdot\nabla)\rho'+\rho(\nabla\cdot\vec{u'})=0</math><br />
<math>\frac{\partial\vec{u'}}{\partial<br />
t}+(\vec{u}\cdot\nabla)\vec{u'}+(\vec{u'}\cdot\nabla)\vec{U}+\frac{1}{\rho}\nabla<br />
p'+\frac{\rho'}{\rho}\frac{D\vec{U}}{Dt}=\frac{1}{\rho}\vec{f'_{vis}}</math><br />
<math>\frac{{\partial p'}}{{\partial t}}\!+\!(\vec u \cdot \nabla )p'\!+\!\gamma p(\nabla \cdot \vec u') + (\vec<br />
u' \cdot \nabla )P = - \frac{{DP}}{{Dt}} + (\gamma - 1)\left( {\Phi - \nabla \cdot \vec q} \right)</math></div>Jhseohttp://www.cfd-online.com/Wiki/Hydrodynamic/acoustic_splittingHydrodynamic/acoustic splitting2008-07-30T21:26:00Z<p>Jhseo: </p>
<hr />
<div>In the hydrodynamic/acoustic splitting method, the total flow variables are decomposed into the incompressible variables and the <br />
perturbed compressible ones as,<br />
<br />
<math>\rho(\vec{x},t)=\rho_0+\rho'(\vec{x},t) </math></div>Jhseohttp://www.cfd-online.com/Wiki/Hydrodynamic/acoustic_splittingHydrodynamic/acoustic splitting2008-07-30T21:24:56Z<p>Jhseo: New page: In the hydrodynamic/acoustic splitting method, the total flow variables are decomposed into the incompressible variables and the perturbed compressible ones as, <math>\rho(\vec{x},t)=\rho...</p>
<hr />
<div>In the hydrodynamic/acoustic splitting method, the total flow variables are decomposed into the incompressible variables and the <br />
perturbed compressible ones as,<br />
<math>\rho(\vec{x},t)=\rho_0+\rho'(\vec{x},t) \\<br />
\vec{u}(\vec{x},t)=\vec{U}(\vec{x},t)+\vec{u'}(\vec{x},t) \\ <br />
p(\vec{x},t)=P(\vec{x},t)+p'(\vec{x},t)</math></div>Jhseohttp://www.cfd-online.com/Wiki/Aero-acoustics_and_noiseAero-acoustics and noise2008-07-30T21:23:55Z<p>Jhseo: /* Different Methods */</p>
<hr />
<div>== Introduction ==<br />
Sound can be understood as the pressure fluctuation in a medium. Acoustics is the study of sound propagation in a medium; AeroAcoustics deals with the study of noise generated by air. Examples include the flow around the landing gear of an aircraft, or the buffeting noise caused when driving along with the window/sunroof open. As a result of the stringent conditions imposed on the Aircraft industries to limit noise pollution, focus is now shifting towards predicting the noise generated by a given aerodynamic flow. Similarly, in the automotive industry, passenger comfort is of great importance, so OEMs are keen to minimise unnecessary noise sources.<br />
<br />
AeroAcoustics is an advanced field of fluid dynamics in which the flow scale is removed to the acoustic levels. The first advance in the field of AeroAcoustics was made by Sir James Lighthill when he presented an &quot;Acoustic Analogy&quot;. With proper manipulation of the Euler equations, he derived a wave equation based on pressure as the fluctuating variable, and the flow variables contributing to the source of fluctuation. The resulting wave equation can then be integrated with the help of Green's Function, or can be integrated numerically. Thus, this equation can represent the sound propagation from a source in an ambient condition. With the success of the acoustic analogy, many improvements were made on the derivation of the wave equation. Two common form of the equation used in the acoustic analogy are the Ffowcs Williams - Hawkins equation and the Kirchoff's Equation. <br />
<br />
Although the Acoustic Analogy solves the problem of noise prediction to a great extent, focus is now shifting towards direct computation, in which noise is computed directly by the flow solver. Of course the acoustic analogy is still applied in far field propagation, but near field sound generation is resolved to a large extent. Large Eddy Simulation is widely used for these studies. DNS is still unuseable for problems of practical dimensions; industries require a code that can provide them results in a day, not a month. Hence, RANS based models (like JET3D by NASA) are widely used in industry.<br />
<br />
One of the main difficulties in Computational AeroAcoustics is the scale of the problem. Acoustic waves have a high velocity relative to the flow structures and, at the same time, are nearly 10 orders of magnitude smaller. Also, due to the propagation to long distances, the numerical scheme should be less dissipative and less dispersive. The CFD solvers have inherent dissipation to ensure stability. This makes most robust CFD solvers incapable of simulating acoustic flows. Advanced schemes such as Dispersion Relation Preserving (DRP) schemes, compact schemes etc., aim at a less dispersive solution. Still, given the limits of current computational capability, acoustic computation for a problem of practical interest is still out of reach.<br />
<br />
The solution adopted by the main code vendors (STAR-CD, Fluent, CFX) is to de-couple the problem: solve for the acoustic sources in the CFD code, then couple to an acoustic propagation code (SYSNoise, Actran) to discover noise levels some distance from the source.<br />
<br />
== Different Methods ==<br />
=== DNS ===<br />
<br />
=== Green's Function ===<br />
=== [[Hydrodynamic/acoustic splitting]] ===<br />
The hydrodynamic/acoustic splitting method (also known as viscous/acoustic splitting) has been originally proposed by Hardin and Pope (1994) for resolving the issue of scale disparity in low Mach number aeroacoustics. This method splits the direct numerical simulation (DNS) into the viscous-hydrodynamic and inviscid-acoustic calculations. The viscous flow field is computed by solving the incompressible Navier-Stokes equations, while the acoustic field is obtained by the perturbed compressible equations (PCE). This splitting method has further been modified by Shen and Sorenson (1999) and Slimon et al (1999). <br />
Recently, Seo and Moon (2005) proposed the Linearized Perturbed Compressible Equation (LPCE). The LPCE <br />
simulates the noise generation and propagation from the incompressible flow field solution in a natural way, and also could secure <br />
a consistent acoustic solution with suppressing the evolution of unstable vortical mode in the perturbed system. Since this method <br />
is based on the incompressible flow solution, it is very effective for the flows at low Mach numbers. Moreover, computational <br />
efficiency can further be enhanced, if grid system for the flow and acoustics are treated separately for resolving the scale <br />
disparity at low Mach numbers.<br />
<br />
==Higher Order Schemes for Aero-acoustics==<br />
=== Finite Difference ===<br />
<br />
=== Finite Volume ===<br />
<br />
==Boundary Conditions ==<br />
== Reference ==<br />
<br />
{{stub}}</div>Jhseohttp://www.cfd-online.com/Wiki/Aero-acoustics_and_noiseAero-acoustics and noise2008-07-30T21:23:01Z<p>Jhseo: /* Different Methods */</p>
<hr />
<div>== Introduction ==<br />
Sound can be understood as the pressure fluctuation in a medium. Acoustics is the study of sound propagation in a medium; AeroAcoustics deals with the study of noise generated by air. Examples include the flow around the landing gear of an aircraft, or the buffeting noise caused when driving along with the window/sunroof open. As a result of the stringent conditions imposed on the Aircraft industries to limit noise pollution, focus is now shifting towards predicting the noise generated by a given aerodynamic flow. Similarly, in the automotive industry, passenger comfort is of great importance, so OEMs are keen to minimise unnecessary noise sources.<br />
<br />
AeroAcoustics is an advanced field of fluid dynamics in which the flow scale is removed to the acoustic levels. The first advance in the field of AeroAcoustics was made by Sir James Lighthill when he presented an &quot;Acoustic Analogy&quot;. With proper manipulation of the Euler equations, he derived a wave equation based on pressure as the fluctuating variable, and the flow variables contributing to the source of fluctuation. The resulting wave equation can then be integrated with the help of Green's Function, or can be integrated numerically. Thus, this equation can represent the sound propagation from a source in an ambient condition. With the success of the acoustic analogy, many improvements were made on the derivation of the wave equation. Two common form of the equation used in the acoustic analogy are the Ffowcs Williams - Hawkins equation and the Kirchoff's Equation. <br />
<br />
Although the Acoustic Analogy solves the problem of noise prediction to a great extent, focus is now shifting towards direct computation, in which noise is computed directly by the flow solver. Of course the acoustic analogy is still applied in far field propagation, but near field sound generation is resolved to a large extent. Large Eddy Simulation is widely used for these studies. DNS is still unuseable for problems of practical dimensions; industries require a code that can provide them results in a day, not a month. Hence, RANS based models (like JET3D by NASA) are widely used in industry.<br />
<br />
One of the main difficulties in Computational AeroAcoustics is the scale of the problem. Acoustic waves have a high velocity relative to the flow structures and, at the same time, are nearly 10 orders of magnitude smaller. Also, due to the propagation to long distances, the numerical scheme should be less dissipative and less dispersive. The CFD solvers have inherent dissipation to ensure stability. This makes most robust CFD solvers incapable of simulating acoustic flows. Advanced schemes such as Dispersion Relation Preserving (DRP) schemes, compact schemes etc., aim at a less dispersive solution. Still, given the limits of current computational capability, acoustic computation for a problem of practical interest is still out of reach.<br />
<br />
The solution adopted by the main code vendors (STAR-CD, Fluent, CFX) is to de-couple the problem: solve for the acoustic sources in the CFD code, then couple to an acoustic propagation code (SYSNoise, Actran) to discover noise levels some distance from the source.<br />
<br />
== Different Methods ==<br />
=== DNS ===<br />
<br />
=== Green's Function ===<br />
=== Hydrodynamic/acoustic splitting ===<br />
[[The hydrodynamic/acoustic splitting method]] (also known as viscous/acoustic splitting) has been originally proposed by Hardin and Pope (1994) for resolving the issue of scale disparity in low Mach number aeroacoustics. This method splits the direct numerical simulation (DNS) into the viscous-hydrodynamic and inviscid-acoustic calculations. The viscous flow field is computed by solving the incompressible Navier-Stokes equations, while the acoustic field is obtained by the perturbed compressible equations (PCE). This splitting method has further been modified by Shen and Sorenson (1999) and Slimon et al (1999). <br />
Recently, Seo and Moon (2005) proposed the Linearized Perturbed Compressible Equation (LPCE). The LPCE <br />
simulates the noise generation and propagation from the incompressible flow field solution in a natural way, and also could secure <br />
a consistent acoustic solution with suppressing the evolution of unstable vortical mode in the perturbed system. Since this method <br />
is based on the incompressible flow solution, it is very effective for the flows at low Mach numbers. Moreover, computational <br />
efficiency can further be enhanced, if grid system for the flow and acoustics are treated separately for resolving the scale <br />
disparity at low Mach numbers.<br />
<br />
==Higher Order Schemes for Aero-acoustics==<br />
=== Finite Difference ===<br />
<br />
=== Finite Volume ===<br />
<br />
==Boundary Conditions ==<br />
== Reference ==<br />
<br />
{{stub}}</div>Jhseohttp://www.cfd-online.com/Wiki/Aero-acoustics_and_noiseAero-acoustics and noise2008-07-30T21:22:06Z<p>Jhseo: /* incompressible/acoustic splitting */</p>
<hr />
<div>== Introduction ==<br />
Sound can be understood as the pressure fluctuation in a medium. Acoustics is the study of sound propagation in a medium; AeroAcoustics deals with the study of noise generated by air. Examples include the flow around the landing gear of an aircraft, or the buffeting noise caused when driving along with the window/sunroof open. As a result of the stringent conditions imposed on the Aircraft industries to limit noise pollution, focus is now shifting towards predicting the noise generated by a given aerodynamic flow. Similarly, in the automotive industry, passenger comfort is of great importance, so OEMs are keen to minimise unnecessary noise sources.<br />
<br />
AeroAcoustics is an advanced field of fluid dynamics in which the flow scale is removed to the acoustic levels. The first advance in the field of AeroAcoustics was made by Sir James Lighthill when he presented an &quot;Acoustic Analogy&quot;. With proper manipulation of the Euler equations, he derived a wave equation based on pressure as the fluctuating variable, and the flow variables contributing to the source of fluctuation. The resulting wave equation can then be integrated with the help of Green's Function, or can be integrated numerically. Thus, this equation can represent the sound propagation from a source in an ambient condition. With the success of the acoustic analogy, many improvements were made on the derivation of the wave equation. Two common form of the equation used in the acoustic analogy are the Ffowcs Williams - Hawkins equation and the Kirchoff's Equation. <br />
<br />
Although the Acoustic Analogy solves the problem of noise prediction to a great extent, focus is now shifting towards direct computation, in which noise is computed directly by the flow solver. Of course the acoustic analogy is still applied in far field propagation, but near field sound generation is resolved to a large extent. Large Eddy Simulation is widely used for these studies. DNS is still unuseable for problems of practical dimensions; industries require a code that can provide them results in a day, not a month. Hence, RANS based models (like JET3D by NASA) are widely used in industry.<br />
<br />
One of the main difficulties in Computational AeroAcoustics is the scale of the problem. Acoustic waves have a high velocity relative to the flow structures and, at the same time, are nearly 10 orders of magnitude smaller. Also, due to the propagation to long distances, the numerical scheme should be less dissipative and less dispersive. The CFD solvers have inherent dissipation to ensure stability. This makes most robust CFD solvers incapable of simulating acoustic flows. Advanced schemes such as Dispersion Relation Preserving (DRP) schemes, compact schemes etc., aim at a less dispersive solution. Still, given the limits of current computational capability, acoustic computation for a problem of practical interest is still out of reach.<br />
<br />
The solution adopted by the main code vendors (STAR-CD, Fluent, CFX) is to de-couple the problem: solve for the acoustic sources in the CFD code, then couple to an acoustic propagation code (SYSNoise, Actran) to discover noise levels some distance from the source.<br />
<br />
== Different Methods ==<br />
=== DNS ===<br />
<br />
=== Green's Function ===<br />
=== incompressible/acoustic splitting ===<br />
[[The hydrodynamic/acoustic splitting method]] (also known as viscous/acoustic splitting) has been originally proposed by Hardin and Pope (1994) for resolving the issue of scale disparity in low Mach number aeroacoustics. This method splits the direct numerical simulation (DNS) into the viscous-hydrodynamic and inviscid-acoustic calculations. The viscous flow field is computed by solving the incompressible Navier-Stokes equations, while the acoustic field is obtained by the perturbed compressible equations (PCE). This splitting method has further been modified by Shen and Sorenson (1999) and Slimon et al (1999). <br />
Recently, Seo and Moon (2005) proposed the Linearized Perturbed Compressible Equation (LPCE). The LPCE <br />
simulates the noise generation and propagation from the incompressible flow field solution in a natural way, and also could secure <br />
a consistent acoustic solution with suppressing the evolution of unstable vortical mode in the perturbed system. Since this method <br />
is based on the incompressible flow solution, it is very effective for the flows at low Mach numbers. Moreover, computational <br />
efficiency can further be enhanced, if grid system for the flow and acoustics are treated separately for resolving the scale <br />
disparity at low Mach numbers.<br />
<br />
==Higher Order Schemes for Aero-acoustics==<br />
=== Finite Difference ===<br />
<br />
=== Finite Volume ===<br />
<br />
==Boundary Conditions ==<br />
== Reference ==<br />
<br />
{{stub}}</div>Jhseohttp://www.cfd-online.com/Wiki/Aero-acoustics_and_noiseAero-acoustics and noise2008-07-30T21:17:25Z<p>Jhseo: /* incompressible/acoustic splitting */</p>
<hr />
<div>== Introduction ==<br />
Sound can be understood as the pressure fluctuation in a medium. Acoustics is the study of sound propagation in a medium; AeroAcoustics deals with the study of noise generated by air. Examples include the flow around the landing gear of an aircraft, or the buffeting noise caused when driving along with the window/sunroof open. As a result of the stringent conditions imposed on the Aircraft industries to limit noise pollution, focus is now shifting towards predicting the noise generated by a given aerodynamic flow. Similarly, in the automotive industry, passenger comfort is of great importance, so OEMs are keen to minimise unnecessary noise sources.<br />
<br />
AeroAcoustics is an advanced field of fluid dynamics in which the flow scale is removed to the acoustic levels. The first advance in the field of AeroAcoustics was made by Sir James Lighthill when he presented an &quot;Acoustic Analogy&quot;. With proper manipulation of the Euler equations, he derived a wave equation based on pressure as the fluctuating variable, and the flow variables contributing to the source of fluctuation. The resulting wave equation can then be integrated with the help of Green's Function, or can be integrated numerically. Thus, this equation can represent the sound propagation from a source in an ambient condition. With the success of the acoustic analogy, many improvements were made on the derivation of the wave equation. Two common form of the equation used in the acoustic analogy are the Ffowcs Williams - Hawkins equation and the Kirchoff's Equation. <br />
<br />
Although the Acoustic Analogy solves the problem of noise prediction to a great extent, focus is now shifting towards direct computation, in which noise is computed directly by the flow solver. Of course the acoustic analogy is still applied in far field propagation, but near field sound generation is resolved to a large extent. Large Eddy Simulation is widely used for these studies. DNS is still unuseable for problems of practical dimensions; industries require a code that can provide them results in a day, not a month. Hence, RANS based models (like JET3D by NASA) are widely used in industry.<br />
<br />
One of the main difficulties in Computational AeroAcoustics is the scale of the problem. Acoustic waves have a high velocity relative to the flow structures and, at the same time, are nearly 10 orders of magnitude smaller. Also, due to the propagation to long distances, the numerical scheme should be less dissipative and less dispersive. The CFD solvers have inherent dissipation to ensure stability. This makes most robust CFD solvers incapable of simulating acoustic flows. Advanced schemes such as Dispersion Relation Preserving (DRP) schemes, compact schemes etc., aim at a less dispersive solution. Still, given the limits of current computational capability, acoustic computation for a problem of practical interest is still out of reach.<br />
<br />
The solution adopted by the main code vendors (STAR-CD, Fluent, CFX) is to de-couple the problem: solve for the acoustic sources in the CFD code, then couple to an acoustic propagation code (SYSNoise, Actran) to discover noise levels some distance from the source.<br />
<br />
== Different Methods ==<br />
=== DNS ===<br />
<br />
=== Green's Function ===<br />
=== incompressible/acoustic splitting ===<br />
[[The hydrodynamic/acoustic splitting method]] (also known as viscous/acoustic splitting) has been originally proposed by Hardin and Pope for resolving the issue of scale disparity in low Mach number aeroacoustics. This method splits the direct numerical simulation (DNS) into the viscous-hydrodynamic and inviscid-acoustic calculations. The viscous flow field is computed by solving the incompressible Navier-Stokes equations, while the acoustic field is obtained by the perturbed compressible equations (PCE). This splitting method has further been modified by Shen and Sorenson and Slimon et al. <br />
Recently, Seo and Moon proposed the Linearized Perturbed Compressible Equation (LPCE). The LPCE <br />
simulates the noise generation and propagation from the incompressible flow field solution in a natural way, and also could secure <br />
a consistent acoustic solution with suppressing the evolution of unstable vortical mode in the perturbed system. Since this method <br />
is based on the incompressible flow solution, it is very effective for the flows at low Mach numbers. Moreover, computational <br />
efficiency can further be enhanced, if grid system for the flow and acoustics are treated separately for resolving the scale <br />
disparity at low Mach numbers.<br />
<br />
==Higher Order Schemes for Aero-acoustics==<br />
=== Finite Difference ===<br />
<br />
=== Finite Volume ===<br />
<br />
==Boundary Conditions ==<br />
== Reference ==<br />
<br />
{{stub}}</div>Jhseohttp://www.cfd-online.com/Wiki/Aero-acoustics_and_noiseAero-acoustics and noise2008-07-30T21:09:33Z<p>Jhseo: /* incompressible/acoustic splitting */</p>
<hr />
<div>== Introduction ==<br />
Sound can be understood as the pressure fluctuation in a medium. Acoustics is the study of sound propagation in a medium; AeroAcoustics deals with the study of noise generated by air. Examples include the flow around the landing gear of an aircraft, or the buffeting noise caused when driving along with the window/sunroof open. As a result of the stringent conditions imposed on the Aircraft industries to limit noise pollution, focus is now shifting towards predicting the noise generated by a given aerodynamic flow. Similarly, in the automotive industry, passenger comfort is of great importance, so OEMs are keen to minimise unnecessary noise sources.<br />
<br />
AeroAcoustics is an advanced field of fluid dynamics in which the flow scale is removed to the acoustic levels. The first advance in the field of AeroAcoustics was made by Sir James Lighthill when he presented an &quot;Acoustic Analogy&quot;. With proper manipulation of the Euler equations, he derived a wave equation based on pressure as the fluctuating variable, and the flow variables contributing to the source of fluctuation. The resulting wave equation can then be integrated with the help of Green's Function, or can be integrated numerically. Thus, this equation can represent the sound propagation from a source in an ambient condition. With the success of the acoustic analogy, many improvements were made on the derivation of the wave equation. Two common form of the equation used in the acoustic analogy are the Ffowcs Williams - Hawkins equation and the Kirchoff's Equation. <br />
<br />
Although the Acoustic Analogy solves the problem of noise prediction to a great extent, focus is now shifting towards direct computation, in which noise is computed directly by the flow solver. Of course the acoustic analogy is still applied in far field propagation, but near field sound generation is resolved to a large extent. Large Eddy Simulation is widely used for these studies. DNS is still unuseable for problems of practical dimensions; industries require a code that can provide them results in a day, not a month. Hence, RANS based models (like JET3D by NASA) are widely used in industry.<br />
<br />
One of the main difficulties in Computational AeroAcoustics is the scale of the problem. Acoustic waves have a high velocity relative to the flow structures and, at the same time, are nearly 10 orders of magnitude smaller. Also, due to the propagation to long distances, the numerical scheme should be less dissipative and less dispersive. The CFD solvers have inherent dissipation to ensure stability. This makes most robust CFD solvers incapable of simulating acoustic flows. Advanced schemes such as Dispersion Relation Preserving (DRP) schemes, compact schemes etc., aim at a less dispersive solution. Still, given the limits of current computational capability, acoustic computation for a problem of practical interest is still out of reach.<br />
<br />
The solution adopted by the main code vendors (STAR-CD, Fluent, CFX) is to de-couple the problem: solve for the acoustic sources in the CFD code, then couple to an acoustic propagation code (SYSNoise, Actran) to discover noise levels some distance from the source.<br />
<br />
== Different Methods ==<br />
=== DNS ===<br />
<br />
=== Green's Function ===<br />
=== incompressible/acoustic splitting ===<br />
The hydrodynamic/acoustic(incompressible/acoustic) splitting method (also known as viscous/acoustic splitting) has been originally proposed by Hardin and Pope\cite{Hardin} for resolving the issue of scale disparity in low Mach number aeroacoustics. This method splits the direct numerical simulation (DNS) into the viscous-hydrodynamic and inviscid-acoustic calculations. The viscous flow field is computed by solving the incompressible Navier-Stokes equations, while the acoustic field is obtained by the perturbed compressible <br />
equations (PCE). This splitting method has further been modified by Shen and Sorenson\cite{Shen} and Slimon et al.\cite{Slimon}.<br />
<br />
==Higher Order Schemes for Aero-acoustics==<br />
=== Finite Difference ===<br />
<br />
=== Finite Volume ===<br />
<br />
==Boundary Conditions ==<br />
== Reference ==<br />
<br />
{{stub}}</div>Jhseo