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Application of computational fluid dynamics softwares for 2d acoustical wave

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APPLICATION OF COMPUTATIONAL FLUID DYNAMICS SOFTWARES FOR 2D ACOUSTICAL WAVE PROPAGATION

P. Tóth Ph.D. student, Department of Fluid Dynamics, Budapest University of Technology and Economics,

H-1111 Budapest, Bertalan L. u. 4-6. Tel: (+36-1) 463-2546, e-mail: toth@ara.bme.hu

A. Fritzsch Student in Engineering Science, Berlin Institute of Technology (TU Berlin)

D-10623 Berlin, Strasse des 17. Juni 135, e-mail: attila.fritzsch@daad-alumni.de

M. M. Lohász Assistant professor, Department of Fluid Dynamics, Budapest University of Technology and Economics, H-

1111 Budapest, Bertalan L. u. 4-6. 5. Tel: (+36-1) 463-1560, e-mail: lohasz@ara.bme.hu

Abstract: The direct noise computation approach in aeroacoustics is an important tool to

predict the noise emitted by turbulent flows. A special treatment of the solution algorithms is

needed in this case. This paper attempts to show the capabilities of two general purpose

Computational Fluid Dynamics codes in a 2D acoustical simulation of the propagation of a

pressure pulse. The effect of the time step size, grid resolution, numerical schemes, and

solution algorithms were investigated for this purpose. The test showed that the so called

density-based formulation gives the most reliable results.

Keywords: CFD, CAA, acoustic pulse

1. INTRODUCTION The recent progress in the unsteady flow simulation techniques, namely the direct

numerical and the large-eddy simulations (DNS/LES) allow the direct computation of the

sound generated by turbulent flows. However the orders of magnitudes discrepancy between

the flow and acoustic scales require special treatment in terms of numerical solution methods.

The solution method requires high accuracy spatial and temporal discretisation in order not to

wash out the small acoustical scales from the simulation. In aeroacoustics often special in-

house codes are used to overcome these problems. Therefore the applicability of general

purpose computational fluid dynamics (CFD) software for direct aeroacoustics computations

is not evident. Such codes are mainly designed to solve complex flow physics with difficult

geometry and use robust solution algorithms for arbitrary mesh topology and usually this

comes with the use of smaller accuracy numerical methods. These codes are fitted to solve

industrial fluid dynamics applications. Moreover there is an increasing demand on the noise

emission by industrially relevant flows.

With the computational resources available today only the noise generated by the

simplest flows can be calculated in the acoustical far field by the means of direct noise

computation (DNC). The computational cost of this solution method in an industrially

relevant problem is prohibitively large. Although in some cases (e.g. free shear layer flows),

where the volumetric acoustic source terms need to be taken into account, the combination of

the DNC simulation with other aeroacoustics methods (i.e integral methods) is beneficial. In

those cases the volumetric source data is computed by a DNC simulation in a relatively small

region, where the sound is produced and from the boundary of that source region another,

simpler method is used to obtain the far field acoustic signal [9]. The advantage of this

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procedure is the smaller amount of computer memory and storage capacity requirement,

because in such cases for acoustical post processing it is enough to store the source

information on the boundary of the source region instead of storing the entire volumetric

source region information. This greatly enhances the post processing of the acoustic far field

region.

In the light of the previously mentioned issues the applicability of general purpose

CFD applications in the field of direct sound computations can be an important question. If

these industrially relevant applications enable DNC in the near-field of the acoustic sources,

(with acceptable accuracy) then noise computation of complex turbulent flows (e.g.

aeroacoustics of free jet flow) could be obtained. However there are only a few guidelines (as

far as the authors know) concerning the applicable numerical methods, which can be used in a

general purpose CFD application to compute the aerodynamically generated noise by means

of DNC.

This paper attempts to show the capabilities of two general purpose CFD codes in a

2D acoustical simulation of the propagation of a pressure pulse. This is a basic aeroacoustic

test case used to validate and tune aeroacoustical solvers and boundary conditions [1,8]. The

tested two CFD codes are Fluent6.3.26 and OpenFOAM1.4.1. Below the test case is

introduced, and the two CFD applications and numerical parameters are discussed briefly.

Results obtained by different numerical methods are shown and compared to the analytical

solution of the problem. The conclusions are given at the end.

2. THE ACOUSTIC TEST CASE The quality of a numerical solution can be verified by the comparison of the results to

the analytical solution of that problem. However this analytical solution exists only for some

particular cases. In this study a 2D acoustic pulse propagation case in a uniform flow was

chosen as the test simulation. The analytical solution of this case is obtained from the

linearized Euler equations presented in [8]. The two dimensional domain is a square. A

uniform flow with Mach number M=0.5 is prescribed in one direction of the domain. The

boundary conditions are defined in order to maintain this uniform flow in the domain. An

initial perturbation, which is a Gaussian pressure distribution, is imposed at the centre of the

domain at t=0.

2

( , ,0)

( , ,0)

( , ,0) 0

p x y e

u x y M

v x y

αηρ ε −= =

=

=

(1)

, where α is related to the half width of the Gaussian profile b, by 2ln 2 / bα = , furthermore

( )1/ 2

2 2x Mt yη = − +

, which is equal to the radial coordinate in the case of t=0. (p and ρ

denotes perturbation values around the average) The Gaussian half width was b=3, and the

amplitude was ε=0.01 in this study (this small ε is needed for the lineralized equations to be a

good approximation of the nonlinear ones). The simulation domain with the initial pressure

pulse can be seen in Fig. 1 left. This kind of initial condition induces the acoustic wave mode

[2] of the underlying equations. The analytical solution of the problem for the pressure and

density can be given with a Bessel function J0 order of zero:

2 / 4

0

0

( , , ) ( , , ) cos( ) ( )2

p x y t x y t e t J dξ αε

ρ ξ ξη ξ ξα

∞−= = ∫ (2)

This solution can be seen in Fig. 1 right at t=20. The accuracy of the numerical solution was

evaluated by comparing it to this particular solution of the lineralized equations. The

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simulation variables are nondimensionalized for the post processing. For this purpose the

ambient sound velocity c0, the ambient density ρ0 and the grid spacing ∆x=∆y are used for the

velocity, density and length scale respectively. Therefore the time scale is ∆x/a0 and the

pressure scale is ρ0 c02.

Fig.1. Left: Initial pressure pulse imposed in the domain with uniform mean flow. Right:

Pressure disturbance at t=20 from the analytical solution.

3. THE SOLVERS As it was previously mentioned in this study two general purpose CFD codes are

examined for acoustic wave propagation. One of them is the commercial code Fluent 6.3.26.

The other one is an open source CFD application called OpenFOAM 1.4.1. Both codes use

collocated variable arrangement and unstructured mesh handling. However the former

contains two different solution methods: the so called “pressure-based” and the “density-

based” algorithms for the solution of the compressible Navier-Stokes (N-S) system of

equations. The OpenFOAM environment provides only “pressure-based” algorithm with the

PISO pressure velocity coupling method. The available discretisation schemes are different

except some basic methods. In both codes implicit and explicit spatial and temporal schemes

are available. An important difference between the codes is that the open source code

provides only a sequential solution method for the governing equations meaning that the

coupling between the momentum, pressure correction and energy equation is not satisfied

accurately. This introduces a so called splitting error in the solution procedure [3]. However

the open source type of the code gives the opportunity to modify the existing solver

algorithms, therefore this type of error can be reduced with the expense of higher

computational cost. The boundary conditions are different on the applied solvers. In the

present investigation attention is devoted to simulate the wave propagation accurately.

Therefore the boundary conditions are handled in the simplest way. In all simulation cases

presented here reflective boundary conditions were applied and placed at a distance where

they cannot influence the results obtained at t=20.

4. NUMERICAL SETUP The computational domain can be seen in Fig.2 left. This is a square region meshed

with uniformly distributed quadrilateral cells. The grid spacing is ∆x=∆y. The domain extends

in the 55 55x− ≤ ≤ , 55 55y− ≤ ≤ region. Mass flow inlet and velocity inlet boundary

conditions are imposed at x=-55 for the Fluent and OpenFOAM solvers, respectively. The

inlet is a uniform flow with prescribing the value of M=0.5 for the velocity and the related

total temperature. On the outflow boundary a constant pressure condition is applied which

defines the static temperature as well. The other boundaries are treated with symmetry

condition. For the time period examined it can be assumed that this does not influence the

solution, since no wave reaches this boundary. The initial conditions for the flow variables

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were prescribed by expression (1). In the underlying solvers the density distribution cannot be

directly imposed as an initial condition. The density field can be prescribed through the

constitutive equation by setting the temperature field properly. The ideal gas law is applied as

a constitutive equation. The uniform flow (u=M) was imposed towards the x direction. In all

simulations of N-S a fluid with dynamic viscosity ]ms/kg[107894.1 5−⋅=µ , specific heat

constant cp=1006.43[J/kgK], specific gas constant R=287.038[J/kgK] and thermal

conductivity λ=0.0242[W/mK] was used. The gradient formulation in the solvers was similar

based on the Gauss theorem. In the following section the results obtained by the two different

solvers and by the different numerical parameters are presented.

Fig.2. Left: Simulation mesh with the boundary conditions. Right: CFL number effect on the

results with Fluent pressure-based solver (Pressure profiles extracted at the line y/∆y=0.)

5. RESULTS The simulation parameters and computed errors are summarized in Table1-3. The

simulations evaluated by the Fluent’s pressure-based solver can be seen in Table1, the

density-based solver parameters in Table 2 and the details of the OpenFOAM simulations

with coodles solver in Table 3. A unique identifier is assigned to every simulation which is

depicted in the first column of the tables. The discretisation issues are divided into two parts

the spatial discretisation and the temporal one. Below the spatial discretisation tag the grid

size parameter ( / 6bδ ≅ ) and the name of the discretisation schemes used for the convective

and pressure terms of the equations can be found. In Table 2 only one scheme is depicted for

spatial discretisation because of the density-based approach. Under the temporal discretisation

tag the flow CFL number (see Eqn. 4) and the name of the scheme are indicated. In the next

column the pressure velocity coupling for the pressure-based solver and the flux formulation

method for the density-based solver are presented. The solution algorithm column refers to

the previously mentioned iterative (ITA) or non-iterative (NITA) method. For the density-

based explicit formulation nothing is presented in Table 2 indicating the explicit treatment. In

the last columns of the tables the simulation error computed as a comparison to the analytical

solution is depicted.

( )

2

2

ana

ana

p p dAErr

p dA

−=∫∫∫∫

(3)

Here pana is the analytical solution on the domain, and p is the numerically computed value.

The integration is taken on the whole 2D computational domain.

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Spatial discretisation Temporal discretisation

Case

Grid

∆x,∆y

Convective

term

Pressure CFL

Scheme

Pressure velocity coupling

Solution algorithm

Err

Fp1 δ BCD PRESTO 0.075 imp.

Gear

FSM NITA 1.10844

Fp2 δ BCD PRESTO 0.15 imp.

Gear

FSM NITA 0.32915

Fp3 δ BCD PRESTO 0.24 imp.

Gear

FSM NITA 0.33541

Fp4 δ BCD PRESTO 0.3 imp.

Gear

FSM NITA 0.34655

Fp5 δ BCD PRESTO 0.6 imp.

Gear

FSM NITA 0.80521

Fp6 δ BCD Linear 0.24 imp.

Gear

FSM NITA 0.28625

Fp7 δ BCD second

ord.

0.24 imp.

Gear

FSM NITA 0.24071

Fp8 δ BCD standard 0.24 imp.

Gear

FSM NITA 0.90848

Fp9 δ Den., En.:

SOU

Mom.: BCD

PRESTO 0.24 imp.

Gear

FSM NITA 0.37289

Fp10 δ Den., En.:

QUICK

Mom.: BCD

PRESTO 0.24 imp.

Gear

FSM NITA 0.37051

Fp11 δ Den., En.:

MUSCL

Mom.: BCD

PRESTO 0.24 imp.

Gear

FSM NITA 0.37025

Fp12 δ BCD PRESTO 0.24 imp.

Gear

coup. ITA 0.21625

Fp13 δ BCD PRESTO 0.24 imp.

Gear

PISO ITA 0.21625

Fp14 δ BCD PRESTO 0.24 imp.

Gear

Simple ITA 0.21625

Fp15 δ/2 BCD PRESTO 0.24 imp

Gear

FSM NITA 0.16657

Fp16 δ/2 BCD second

order

0.24 imp

Gear

FSM NITA 0.11680

Fp17 δ/4 BCD PRESTO 0.24 imp.

Gear

FSM NITA 0.13031

Tab.1. Comparison table for the simulations with Fluent’s pressure-based solver.

Fluent pressure-based solver The first test is devoted to clarify the effect of the CFL number. The CFL number

takes the local convection speed of a disturbance in the flow, the grid spacing, and the time

step into account. It determines conditions for the maximum time step value in order to reach

reasonable temporal accuracy:

( )u c t

CFLx

+ ∆=

∆ (4)

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Five simulations with the same numerical parameters except the time step were accomplished

on the test case. The result is presented in Table 1. Fp1-Fp5. The Bounded Central

Differencing scheme (BCD) is used for the convective terms [5]. For the pressure term the

Pressure Staggering Option (PRESTO) is applied [4]. The pressure velocity coupling is based

on the Fractional Step method (FSM) [3] with the non-iterative time advancement solution

method (NITA) [4]. It can be seen in the table that the simulation error has minimum value at

CFL=0.15. Decreasing the time step further resulted in higher error and spurious oscillations

in the resulting flow field Fig 2 right. It will be matter of further investigation if this error is

due to the increased impact of the rounding errors on the simulation (i.e more time-steps are

needed to reach the end of the simulation which can be result in the accumulation of the

rounding errors [7]). The dispersion and dissipation error [3] of the pressure wave is higher at

the downstream positions. This can be seen in Fig.2 right, where the impulse at the

downstream location has smaller amplitude and it is wider than the one located upstream.

The CFL=0.24 value seemed to be an optimum in the sense of computational resource and

accuracy considerations. This value was chosen for the further investigations.

With the pressure-based solver the effect of the interpolation schemes were also investigated.

For the momentum equations the BCD scheme [5] was used in every test case due to its low

diffusion property required for the accurate simulation of the N-S equation. In terms of the

pressure interpolation schemes (Fp6-Fp8) the second order scheme [4] performed the best

(Table 1 Fp7). The standard scheme [4] is significantly worse than the others (Table 1. Fp8).

The linear scheme for the pressure is slightly better than the PRESTO scheme (Table 1. Fp6).

Changing the interpolation scheme for the convective terms in the energy and density

equations from BCD to MUSCL, Second order upwind (SOU) or QUICK [4] is resulting in

the increase of the simulation error. This can be seen on simulations Fp9-Fp11.

Results obtained by the iterative time advancement method using the coupled, PISO and

SIMPLE [4], pressure velocity coupling methods can be seen in Table 1. Fp12 Fp13, Fp14

respectively. The iterative solution procedure reduced the solution error with the same value

independently from the coupling used between the pressure and velocity comparing to the

reference case (Fp3). This result shows the importance of the coupling between the flow

equations. The iterative solver can provide the variables are satisfying all of the equations

with good accuracy. The difference between the NITA and ITA time advancement procedures

is demonstrated on Fig 3, where the pressure and density profiles are plotted together for the

Fp3 simulation with NITA and Fp13 simulation with ITA procedure. As previously

mentioned in this test case the pressure and density have analytically the same value in each

point. It can be seen on the chart that the NITA solution procedure does not exactly satisfy

this criteria in the whole domain. The pressure and density profiles do not coincide in the

region 0<x/∆x<15 (see Fig.3 left). With the ITA method the profiles have a better overlap.

However some spurious oscillations can be found (see Fig.3 right).

The application of a twice better grid resolution (Fp15, Fp16) can help to

approximately halve the solution error. Using a four times finer grid ( Fp17) in every direction

does not reduce the error significantly indicating that mainly the splitting error in the solution

procedure influences the results, and not the spatial resolution of the underlying numerical

scheme. According to the previous tests the accuracy reached by the pressure-based solver

with the non-iterative time advancement is not satisfactory. However the computational cost

of this procedure with NITA is reasonable, which is also an important parameter. The

simulation cost for the four times smaller cell size is greatly increased, and it is highly

unpractical to use. The ITA procedure can reduce the simulation error with much higher

computational cost.

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Fig.3. Left: Pressure and density profiles of the simulation Fp3 extracted at the line y/∆y=0.

Right: Pressure and density profiles of the simulation Fp13 extracted at the line y/∆y=0.

Results obtained by the density-based solver The other simulation method offered by the Fluent solver is the density-based

algorithm. This method was originally designed for solving compressible flow problems [4].

It solves the equations in a fully coupled manner, which requires significantly greater

computer memory and usually more computation (CPU) time if implicit discretisation is used.

Whereas the explicit spatial and temporal discretisation method provides comparable

computational time requirement as the pressure-based NITA solver. The simulation results

with the density-based solver are summarized in Table 2. The results obtained by the “low

diffusion Roe” type [4] flux calculation (Fd2, Fd3) provided the lowest simulation errors. The

other methods (Roe, AUSM [4]) (Fd1, Fd4 respectively) performed worse. Considering the

simulations with the low diffusion Roe scheme the third order MUSCL (Fd3) discretisation

for the flow variables provides smaller simulation error comparing to the simulation using

first order upwind (FOU) scheme (Fd2). This is in agreement with the expectations.

Spatial discretisation Temporal

discretisation

Case

Grid

∆x,∆y

Scheme CFL Scheme

Flux type Solution

algorithm

Err

Fd1 δ SOU 0.24 Gear roe ITA 0.18759

Fd2 δ FOU 0.24 Gear low diff

roe

ITA 0.12756

Fd3 δ MUSCL 0.24 Gear low diff

roe

ITA 0.10094

Fd4 δ SOU 0.24 Gear AUSM ITA 0.16655

Fd5 δ/2 MUSCL 0.24 Gear low diff

roe

ITA 0.01762

Fd6 δ/2 FOU 0.24 Gear low diff

roe

ITA 0.02705

Fd7 δ Exp SOU 1 Rk4 Roe-FDS - 0.19249

Fd8 δ Exp SOU 0.3 Rk4 Roe-FDS - 0.17526

Fd9 δ Exp

MUSCL

1 Rk4 Roe-FDS - 0.22729

Tab.2. Comparison table for the simulations with Fluent’s density-based solver.

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Using the mesh with halved grid spacing the simulation error is reduced by one order of

magnitude. This can be seen in Table 2 Fd5, Fd6. This solution error can be accepted for

simulation of acoustic wave propagation though only on a short distance.

There is a possibility to use explicit discretisation schemes for both the spatial and the

temporal terms in the density-based solver of Fluent. The explicit solution procedure is

computationally efficient but there is a stability limit for the time step size in the simulation.

This restricts the CFL number below the value of one [4]. As previously seen the CFL

number about 0.2 is required for the accurate simulation even with the implicit solver.

Therefore this criterion can be satisfied without any increase in the required computer

resource. Simulation results (Fd7, Fd8, Fd9) obtained with fully explicit density-based solver

using the four stage Runge-Kutta time discretisation scheme (Rk4) can be found in Table 2

also. It can be observed that with the SOU spatial discretisation and with CFL=0.3 the

simulation error is smaller than the simulation results obtained with the pressure-based solver

on the same mesh. The simulation setup with CFL=1 and with the SOU or MUSCL scheme

provided almost the same simulation error that was found in the best pressure-based solver

results.

Fig.4. Left: Pressure profiles of the best simulation results with the pressure-based solver

extracted at the line y/∆y=0. Right: Pressure profiles of the best simulation results with the

density-based solver extracted at the line y/∆y=0.

The profiles of the best simulation result with the density-based solver can be seen in Fig. 4

right. The reasonable good agreement of the Fd5 simulation with the analytical solution can

be observed.

This test clearly shows the superior performance of the density-based solution method

considering that better accuracy can be achieved with lower computational cost comparing to

the pressure-based solver. If the limit for the simulation error is not very high the density-

based solver with the explicit formulation is recommended due to its low computational cost.

The OpenFOAM results The results obtained by OpenFOAM solver are summarized in Table 3 In this

environment only a pressure-based solver (called coodles) with PISO corrector loop is

available for solving the N-S equations. Basically in this solver the equations are solved

sequentially, similarly to the method of NITA used in Fluent. In the present study mainly the

effect of the temporal discretisation scheme, the CFL number and the grid resolution is

considered. The Filtered linear scheme is used for the discretisation of the convective term.

This is a modified second order central differencing scheme to prevent spurious oscillations

[6]. The pressure term is discretised with the linear scheme, which is a second order central

differencing scheme [6].

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For the simulations O6, O10 the solution algorithm is modified in order to resemble the

Fluent’s iterative algorithm.

Spatial discretisation Temporal discretisation

Case

Grid

∆x,∆y

Convective

term

Pressure CFL scheme

Pressure velocity coupling

Solution algorithm

Err

O1 δ filtered

linear

linear 0.075 Crank-

Nicholson

0.6

PISO NITA 0.27008

O2 δ filtered

linear

linear 0.15 Crank-

Nicholson

0.6

PISO NITA 0.2675

O3 δ filtered

linear

linear 0.24 Crank-

Nicholson

0.6

PISO NITA 0.27139

O4 δ filtered

linear

linear 0.3 Crank-

Nicholson

0.6

PISO NITA 0.28309

O5 δ filtered

linear

linear 0.75 Crank-

Nicholson

0.6

PISO NITA 0.37984

O6 δ filtered

linear

linear 0.15 Crank-

Nicholson

0.6

PISO ITA 0.25169

O7 δ filtered

linear

linear 0.15 Backward

Euler

PISO NITA 0.30534

O8 δ/2 filtered

linear

linear 0.3 Backward

Euler

PISO NITA 0.04916

O9 δ/2 filtered

linear

linear 0.12 Backward

Euler

PISO NITA 0.10753

O10 δ/2 filtered

linear

linear 0.12 Backward

Euler

PISO ITA 0.05486

O11 δ/2 filtered

linear

linear 0.12 Crank-

Nicholson

0.6

PISO NITA 0.05051

O12 δ/2 filtered

linear

linear 0.3 Crank-

Nicholson

0.6

PISO NITA 0.03568

O13 δ/2 filtered

linear

linear 0.12 Crank-

Nicholson 1

PISO NITA 0.10559

O14 δ/2 filtered

linear

linear 0.3 Crank-

Nicholson 1

PISO NITA 0.04551

Tab.1. Comparison table for the simulations with OpenFOAM solver.

The O1-O5 test cases are devoted to determine the optimal CFL number for the

simulation. The simulation error does not change significantly between CFL numbers from

0.075 to 0.3. The CFL value around 0.15 seems to be an optimal choice. The effect of the

CFL number on the pressure profile can be examined on Fig. 5. Overshoots can be observed

at the downstream pressure peaks. It has to be denoted, that this optimum is determined for

the implicit Crank Nicholson discretisation scheme with blending factor of 0.6 and with mesh

size of ∆x=∆y=δ. The blending factor blends the Crank Nicholson scheme into the Euler

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scheme. A coefficient of 1 corresponds to pure Crank Nicholson and 0 corresponds to pure

Euler scheme. The implicit backward Euler scheme has approximately the same error as the

Crank Nicholson scheme (see O7 in Table 3). Comparing the simulation results O8 and O9

obtained on the refined grid the effect of the CFL number is different as before. The result

evaluated by CFL=0.3 (O8) has smaller error than the one calculated using CFL=0.12 (O9).

However the O8 has an overshot at the downstream peaks of the pressure pulse as it can be

seen in Fig 5 right. The same CFL effect can be observed for the simulation cases with Crank

Nicholson scheme O11, O12 and O13, O14. Such behaviour can be disadvantageous on a

simulation with nonuniform mesh spacing and velocity distribution. The smallest simulation

error was achieved on the finer mesh with Crank Nicholson scheme with blending factor of

0.6 (simulation O12). Use of the pure Crank Nicholson (blending factor 1) scheme O14 is

resulted in a bit higher simulation error. The iterative (ITA) coupling between the equations

does not come with significant accuracy increase using the Crank Nicholson scheme (O6).

The situation is slightly better for the simulation with the Euler scheme (O10), where the ITA

procedure approximately halved the error.

Fig.5. Left: Pressure profiles showing the effect of the CFL number with OpenFOAM solver

extracted at the line y/∆y=0. Right: Pressure profiles of the best simulation results with the

OpenFOAM solver extracted at the line y/∆y=0.

6. CONCLUDING REMARKS In this paper the performance of two Computational Fluid Dynamics software packages

were investigated in an acoustic pulse propagation test case. The commercial Fluent 6.3.26

software provides a wide range of solution algorithms. From our experience the solution

results with pressure-based algorithm are showing higher dispersion and dissipation error than

the density-based solution method. For the case of the pressure-based solution these errors are

especially pronounced using the non-iterative time advancement. In the OpenFOAM1.4.1

environment only pressure-based solver is available. Comparing the Fluent pressure-based

and the OpenFOAM coodles solver the same order of error is arisen when comparing to the

analytical solution. The error reduction by grid refinement was found to be higher using

OpenFOAM than the one gain with Fluent pressure-based approach. Calculations with the

OpenFOAM solver also showed smaller dissipation and dispersion error in the test, but some

overshoot in the results can be observed. The Fluent’s density-based solver results show the

best dispersion and dissipation error properties and together with the ability for significant

error reduction when refining the grid. Further on using the explicit formulation of the

density-based algorithm the computational cost of the simulation can be acceptable as well.

As the final conclusion the use of the density-based formulation is recommended for accurate

acoustic wave propagation calculations.

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REFERENCES

[1] Bogey C., Bailly C. Three-dimensional non-reflective boundary conditions for acoustic

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