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In this work the turbulence based acoustic sources and the corresponding wave propagation of fluctuating flow values in incompressible fluid flows are considered. Lighthill’s and Curle’s acoustic analogies are implemented in the open source computational fluid dynamics framework OpenFOAM. The main objective of this work is to visualize and localize the dominated sound sources and the resulting values of fluctuating pressure values within the computation domain representing the acoustical near field. This is all done on one mesh and during the iterative computation of the transient fluid flow. Finally the flow field and acoustical results of different simulation cases are presented and the properties of the shown method are discussed.

Products like air intake ducts for combustion engines emit acoustic noise mainly based on turbulences and their interaction with solid surfaces. The reduction of these acoustical emissions of technical products and components are becoming increasingly important. In product development process the acoustic parameters are often considered not before an existing mechanical prototype. Usually any changes of concepts, products or components themselves are hardly feasible. Numerical methods like finite elements for structural mechanics or finite volumes for computational fluid dynamics are commonly used by engineers and researchers. Using the advantages of each numerical methods and already existing tools are extended to compute e.g. the sound pressure level in the acoustical far field at a specific observer point. The resulting data is comparable to the results of common acoustic measurements. Concerning the scale disparity between the fluid dynamic pressure and acoustical pressure and the limitation of computation resources often there are used hybrid approaches for computational aero acoustics. Hybrid computational aero acoustic approaches solve the fluid dynamics by e.g. unsteady RANS (Reynolds Averaged Navier Stokes) simulations on a fine mesh in a small domain and then transform the relevant parameters for computing the sound propagation on a coarse mesh representing the acoustical far field.

Further developments in computing methods and parallel communication software and growing resources in computing hardware especially in High Performance Computing (HPC) widen and improve the possibilities to simulate complex acoustic effects. A popular CFD tool is OpenFOAM (Open Field Operation and Manipulation). It is a free, open source software toolbox for Computational Fluid Dynamics (CFD) applications, based on [

There are different approaches in OpenFOAM for computational acoustics, especially for aero acoustics as mentioned in [

In this work, the presented approach for computational aero acoustics in OpenFOAM 2.1.1 is mainly based on Curle’s Acoustic Analogy and takes the possibility and availability of high performance computing resources into account. Due to HPC resources, the shown method makes it possible to determine acoustic parameters of flow fields and provides the advantage to compute and visualize the fields of fluid dynamics, acoustic sources and the resulting wave propagation of the fluctuating values of the decomposed pressure (sound propagation) in the acoustical near field and on one mesh only.

The implementation of acoustic source terms and the corresponding wave propagation of the fluctuating pressure values within the computation domain according to Lighthill [

with its density

with

In (2.3) the Tensor

The term

if using the decomposed Quantities (2.4)-(2.6). The speed of sound is described by

including the decomposed fluid parameters. Finally the Tensor

and is called the Lighthill-Tensor.

For an incompressible

Following the restrictions for the fluid the density corresponds approximately the density

Lighthill’s acoustic analogy was mainly developed for free jet streams. No rigid objects or surfaces are allowed within the computation domain. Unfortunately a large amount of technical application cases need to take into account such surfaces. Curle’s acoustic analogy takes rigid and stationary surfaces and objects within the computation domain into account. It is based on the extended Lighthill Equation [

The three terms on the right hand side of Equation (2.11) can be interpreted as

Listing 1. Implementaion of Lighthill’s acoustic sources in OpenFOAM, according to [

・ the volume distribution of quadrupoles

・ as surface distribution of monopoles

・ as surface distribution of dipoles

For a rigid object within the computation domain its surface can be described by the auxiliary function

and this yields to an expression which disappears within the volume of the object and becomes one inside the fluid region. As mentioned above this work focuses on the Curle’s Acoustic Analogy, where the considered surfaces are stationary and terms containing velocity components obtain

Additionally for non-porous or rigid control surfaces the following equation is valid:

These assumptions eliminates the surface distribution of monopoles in the extended Lighthill Equation (2.11).

The remaining dipole term

becomes to

if viscous stresses are neglected.

Finally the acoustic source terms of Curle’s Acoustic Analogy are simplified to (2.19)

which are implemented in OpenFOAM acoustic solver according Listing 2.

Concerning the decomposition of fluid parameters into mean and fluctuating values as described in chapter 2.1 and according the statements in linear acoustics [

Listing 2. Acoustic source terms according Curle’s Acoustic Analogy were implemented in OpenFOAM 2.1.1.

where the pressure fluctuations are described by

The wave operator (2.21) with its second time derivative of the fluctuating pressure values is implemented in OpenFOAM as an additional header file shown in Listing 3 where the fluctuating pressure values are defined by the variable pa.

The header files for computation of acoustic sources and for sound pressure propagation respectively are included in the pisoFoam. C-file within the runtime-loop, but after the PSIO (Pressure-Implicit Split-Operator)- loop itself. This provides the computation of the additional acoustic fields during every time step of the pisoFoam-Solver and based on the latest corrected pressure and velocity fields including the computed turbulences of the chosen turbulence model.

The proposed “sound” pressure wave propagation is strongly dependent on the way the computed pressure is decomposed into mean and fluctuating parts. In the presented work the pressure is decomposed into the computed pressure subtracting by the ambient pressure of the initial fluid flow setup. This strongly results in a propagating pressure wave similar to hydrodynamic pressure waves.

Actually there are no specific boundary condition in the OpenFOAM distribution which can perform acoustic behaviors like non-reflecting or absorbing effects neither for incompressible nor for compressible flows. The boundary condition advective works roughly like a non-reflecting boundary condition for incompressible case set-ups. It is based on the OpenFOAM-class mixed FvPatch Field [

and its principle behavior is similar to boundary conditions for compressible fluid flow, which is often used for transonic or supersonic simulations. The advective or waveTransmissive boundary condition are applicable on patches defined as outlet only. For a lot of cases these might be insufficient. To overcome this limitation the in this work the computation domain boundary patches are set up as stated in [

In

Proving the presented principal concept of implementing a computational aero acoustic approach in OpenFOAM several simulations are done and will be presented in the following chapter.

For verifying the quality of a numerical solution the results might be compared with the results of the analytical solution of a specific problem, provided that the analytical solution exists. A typical benchmark test case for a novel implemented computational aero acoustic application is the two dimensional acoustic pulse propagation presented by [

Listing 3. Computation of fluctuating pressure wave propagation in OpenFOAM 2.1.1 related to [

The two dimensional computation domain is a square as showed in

Due to the geometric simplicity of the case, the computation domain is meshed by blockMesh, the meshing tool distributed by OpenFOAM. The mesh consists of 40,000 hexahedra elements. The grid spacing of the two dimensional structural mesh is

On the left hand side of the domain the boundary conditions are set as a velocity inlet, however there is no background flow

The two dimensional initial velocity distribution is shown in

The Gaussian distribution for the initial values of the velocity field was set up at the domain center into x- and y-direction to obtain an almost homogeneous initial velocity distribution. If matching both initial fields and analyze and visualize the velocity field values along the cross section line, mentioned above, this results in _{x} and U_{y} cannot be equal along this cross section line.

The described initial set up is chosen to compute the transport of the calculated pressure fluctuations based on the acoustic sources. The used application solver was acousticFoam to compute the acoustic sources and the corresponding wave propagation. In this work for all presented incompressible cases air is used as fluid with its kinematic viscosity^{−3} at a temperature of 25˚C. The speed of sound is declared as a constant with 346.3 m∙s^{−1}. The settings are done in an adapted transport Proper ties―file of a common OpenFOAM case structure.

The Acoustic Test Case was set to laminar where all the turbulence parameters in OpenFOAM are switched off.

The boundary conditions for the acoustic fields, acousticSources, pSurfaceSources and pa are set each according to

To simulate the transient Acoustic Test Case the acousticFoam-Solver based on pisoFOAM-Solver according to chapter 2 is used. The simulation time is set to 0.1 s with a time step of 1.6×10^{−4} s. The simulation run was done in parallel on 4 processors. The parallel execution time is about 1691 s.

Solving the generated linear equation systems the PCG (Preconditioned Conjugated Gradient)-Solver of OpenFOAM is used for pressure and acoustic fields and the PBiCG (Preconditioned Bi-conjugated Gradient)- Solver for asymmetric matrices like the velocity field or fields relevant for turbulence modelling as turbulent kinetic energy k or specific turbulence dissipation^{−6} and the solver relative tolerance relTol was set to 0.

To compute the wave propagation according chapter 2.3 the second time derivative of the sound pressure field has to be calculated numerically. In OpenFOAM for the second order of time derivative there is the Euler scheme available only.

The novel implementation of an acoustic solver in OpenFOAM 2.1.1 was tested by running the OpenFOAM distributed tutorial case pitzDaily. This case is a two dimensional set-up of a transient incompressible flow over a backward facing step which is shown in

Inlet | Outlet | Wall | Front and Back | |
---|---|---|---|---|

Acoustic Sources | Zero Gradient | Zero Gradient | Zero Gradient | Empty |

pSurface Sources | Zero Gradient | Zero Gradient | Zero Gradient | Empty |

pa | Fixed Value | Fixed Value | Zero Gradient | Empty |

Length L | 310.6 mm |
---|---|

Length l | 20.6 mm |

Height H | 50.8 mm |

Height h | 25.4 mm |

To mesh the computation domain the OpenFOAM-meshing tool blockMesh was chosen. The computation mesh consists of about 12,225 hexahedra elements. This results in an element size of about 0.5 mm concerning the given dimensions in

The physical properties of the fluid are set up as described before in chapter 2.4.2. The velocity u of the fluid flow is set to

To calculate the turbulences the turbulence model kOmegaSST (k-ω-SST) of OpenFOAM is used with the initial values of the turbulent kinetic energy

The setup of the boundary conditions of the acoustic fields like acousticSources, pSurfaceSources and pa is identical to the settings of the Acoustic Test Case above.

The transient pitzDaily―case is computed by the developed application solver acousticFoam with the already mentioned solver settings for linear equation systems. The simulation time is set to 0.1 s with a time step of 1 ×10^{−6} s. The small geometric dimensions and the relative short simulation time make it possible to run this case in serial on one processor. The serial execution time is about 670 s.

In the quadratic brace 2D simulation case as presented in

The meshing was done with blockMesh. Concerning the dimensions of

Length L | 300 mm |
---|---|

Length l | 50 mm |

Height H | 60 mm |

Width D | 100 mm |

In the quadratic brace 2D simulation case the velocity

Turbulences are calculated by the turbulence model kEpsilon (k-ε) of OpenFOAM with the initial values of the turbulent kinetic energy

In difference to the previous cases the boundary conditions for the acoustic field pa are set according to

To simulate the transient quadratic brace 2D―case acousticFoam with the same solver settings for linear equation systems is used as in both other simulation cases before. The simulation time in this case is limited to 0.1 s with a time step of 1 × 10^{−6} s. The fine mesh granularity and the high time resolution make it necessary to run this case in parallel as well. To reduce the influence of the initial error in acoustic fields the solution of a steady state case is used as initial values for the transient simulation.

The numerical solution of acousticFoam is compared with the analytical solution (3.1) presented in [

The analytical solution of pressure distribution according to (3.1) for a certain time step

is compared with the numerical solution of acousticFoam after certain time step along the cross section line. The shape of the numerical solution fits to the analytical solution curve (

Inlet | Outlet | Wall | Brace | Front and Back | |
---|---|---|---|---|---|

pa | Advective | Advective | Fixed Value | Zero Gradient | Empty |

between the numerical and analytical absolute values of the pressure values in the peaks due to the numerical errors of the used linear solvers.

To measure the accuracy of the applied numerical method the error between the simulated solution and the presented analytical solution is calculated. The local error (3.2) at a given time step is the absolute value of the difference between the exact analytical solution and the numerical solution, calculated as following

The Global Error (3.3) is calculated according to the Euclidean norm to

In the presented benchmark test case the global error between the numerical and the analytical solution is computed to 0.38. This global error corresponds to results presented by [

The fluid dynamic results of the pitzDaily computation case are well known by OpenFOAM users as a common tutorial case in the OpenFOAM distribution. The first three pictures (a-c) of

The results of transient simulation are a snapshot of the developing steady fluid flow over the backward facing step. The developing velocity profile moves downstream towards the outlet and the beginning of the vortex on the ground behind the step is captured in

The quadratic brace 2D-simulation case was run in parallel on 8 processors. For domain decomposition the scotch method was chosen. The computation domain was decomposed in 8 subdomains according to

The solution of the initial pisoFoam-Simulation according to [

In

model of OpenFOAM and are represented by the results of the turbulent kinetic energy field and turbulent viscosity field.

In

The propagation of the fluctuation pressure values in

For investigations of acoustic parameters like “sound pressure”, “sound pressure level” or their frequency spectra a point located at x y (0.06 0) within the computation domainis considered as a specific probe location. This probe location represents a microphone positioned at a distance of 1.5 x (Width D) behind the obstacle along the x-axis. At this probe location the values of the pressure fluctuations are written out every time step of the computation run. During the post processing the time signal of the pressure fluctuations are analyzed by using the GNU Octave library.

The time signal in

ence of the numerical boundary conditions, which are not totally suitable for acoustic simulations. This effect is obvious in

The main target of the presented work was to realize acoustic simulations within the open source computational framework OpenFOAM. The Acoustic Analogies according to Lighthill and Curle were implemented in an application solver of OpenFOAM. These modified application solver, called acousticFoam, is developed to compute and solve incompressible fluid dynamic simulation cases regarding turbulence inducted noise. The solver also takes existing walls and objects in the computation domain into account. Using acousticFoam, acoustic sources due to turbulences can be computed and visualized. Based on this computed acoustic sources the propagation of fluctuating values can be also determined and visualized within the “acoustical” near field. The novel implemented method to compute acoustical fields has been verified and validated on the basis of common computational aero acoustical and computational fluid dynamic benchmark cases. The presented work also includes a parallelized computation case. In this simulation case the time signal on a specific measurement point, represented as a probe location within the computed fluid flow is recorded during simulation time. This time signal is used for further acoustic analysis such as frequency analysis or calculating the sound pressure level over time at this specific observer point.

The presented results clarify the principal applicability of the shown method for computing the sound sources of incompressible turbulent flows. The estimation of sound propagation within OpenFOAM is also feasible. The correctness of the presented results are highly sensitive to the used linear solver, chosen mesh granularity and taken simulation time step. The main error concerning the propagation of the fluctuating pressure values occurs mainly because of the second order time derivative in the wave operator on the RHS of Equation (2.10). Up to now there are no boundary conditions concerning any special acoustical behavior like absorbing or non-reflec- tive characteristics in the OpenFOAM distributions implemented yet. Despite to the numerical and methodical limitations similar to [

Possible applications of the shown method might be the acoustical investigations of air intake systems for combustion engines, HVAC-systems for heating, ventilation and air condition of vehicles or aircrafts, or any other product or system which causes turbulence based acoustic sources. This investigations might be done in an early stage of development process. Hence the results might be used for the optimization of the investigated geometry concerning the aero acoustic emissions and fluid dynamic properties of technical parts or entire systems.

Further development works have to aim at the enhancement of the presented aero acoustic solver in OpenFOAM. One of the most important development steps might be detailed investigations on acoustically suitable boundary conditions with non-reflecting or absorbing characteristics in OpenFOAM. The implementation of the Ffowcs-Williams-Hawkings acoustic analogy will also enlarge the principal applicability of the novel application solver on more realistic simulation cases. Using a compressible application solver as a basis for an improved novel aero acoustic solver will improve it mainly regarding its wave propagation ability. The implementation of acoustic analogies in a compressible application solver of the OpenFOAM-framework will also emphasize the plausibility of the shown method.

The authors want to acknowledge the Project “FEToL―Fault Tolerant Framework for peta-scale MPI-Solvers” supported by the German Federal Ministry of Education and Research.

^{3}] Density of fluid

^{3}] Density (mean value)