18 a 21 de novembro de 2014 , Caldas Novas-Goiás ACOUSTIC PROPAGATION EVALUATION USING TRADITIONAL FINITE DIFFERENCES AND DISPERSION RELATION PRESERVING ( DRP )

1 Federal University of Uberlandia – Mechanical Engineering Faculty – Fluid Mechanics Laboratory Av. Joao Naves de Avila 2121 – Campus Santa Monica – Uberlandia – MG – 38408-100 2 Federal University of Uberlandia – Mechanical Engineering Faculty – Fluid Mechanics Laboratory Av. Joao Naves de Avila 2121 – Campus Santa Monica – Uberlandia – MG – 38408-100 3 Federal University of Uberlandia – Mechanical Engineering Faculty – Fluid Mechanics Laboratory Av. Joao Naves de Avila 2121 – Campus Santa Monica – Uberlandia – MG – 38408-100


INTRODUCTION
The stream noise generated by the air over trees, rocks and other obstacles always interested researchers along the centuries.Older studies of the phenomenon goes back to the ancient Greece and the term eolian is used to describe that behavior.In modern age, names as Vincent Strouhal left a great legacy on the development of this science, studying the sound generated by air streams through wires (aeolian tones).
A few decades later, the aeroacoustic emerges as a branch of fluid mechanics interested in sound generation an its propagation of air flowing over flat surfaces.Once more, a practical problem guides the research efforts to this area.In this case the primary motivation was the start operation of the noise turbojets after second world war, in the decades of 1950 and 1960, mostly in north america and europe continents.
In fact, the fast growing of the aviation using reaction engines (also called turbojets), the problem of the noise generated by high speed exhaust gas flow became an inconvenience for the society, a hard psychological and physiological impact in the people living close by large airports, for example, Heatrow, in London, and JFK, in New York.
The intense engine noise is created by the interface of exhaust gas flow and the atmospheric air, due a strong shear strength of the velocity gradient between the two flows, like the sound produced when two smooth surfaces are rubbed one against the other.
In this scenario of crisis born the aeroacoustic, with the first works of Sir James Lighthill.Lighthill deduced a transport equation from Navier-Stokes equations describing the specific mass fluctuation in an outflow.Under certain conditions the analytical solution of this equation enable to determine the sound propagation to a far field, away from the sound source.Moreover, following his researches, Lighthill also demonstrate that the intense of the aerodynamic noise is proportional to the relative velocity between the exhaust gas flow and the atmospheric air, elevated to the power of 8.
As a result of these first studies the aeronautic industry created the turbofan engines, that were nothing more than the turbojet engines now equipped with a huge inlet air to supply the engine, and also to cover the exhaust gas flow with a coat, faster than static (or quasi static) atmospheric air and slower than the exhaust gas, decreasing the shear stress between those flows.
Surpassed the first obstacles, the daily coexistence shows other noise sources apart of the engines.The airflow passing through the aircraft`s body, the high lift devices, the landing gear and the wing tips, proved to be sources of noise so intense as the turbofan engines.
In-depth studies revealed that the turbojets generated noises in order of 115 dB (decibel).The air pressure gradient due sound propagation, even close to the engine, was very small, about of the atmospheric pressure.Although the sound generation and its propagation depend on the same variables, the order of magnitude of those variables for the same phenomena are too disparate.To solve the sound generation together with its propagation requires a refined control error, a very high computational cost even in these days, and for that time, unthinkable.To solve the sound generation means to solve the flow (aerodynamics), to solve the turbulence present in the flow.From the point of view of the fluid mechanics, the best way to do this is by DNS (Direct Numerical Simulation), solving Navier-Stokes equations completely.But, even through the powerful moderns computers, this task is only possible for very simple cases, for very low Reynolds numbers.To solve the turbulence without solve all its scales (RANS, URANS, LES...), is still under discussion because of the required filtration, which eliminates a very large band of sound frequencies.To solve the turbulence in all scales requires a very refined mesh, even in a little domain, is still impractical for most of engineering problems.To add to the solution the sound propagation, in a far field (extremely large mesh), elevates the computational efforts beyond the current technological capability.
Lighthill`s research brought a perception that was possible to separate sound propagation from its generation if the flow had a very low Mach number.His studies revealed also the low influence of air viscosity for propagation purposes, far from source sound generation (low Mach number).This fact reduced the costs of the heavy computational processing, turning the Navier-Stokes equations into Euler equations.
Many sound propagation schemes have been proposed over the years to solve the first order spatial partial derivative of the sound equation.Minimize the error under a low computational cost is the main objective of all schemes, but some of them has add a very simple implementation to its profile.In computational aeroacoustic (CAA) the finite differences technique has been detached.
Based on Taylor series this simple technique has reached good efficiency.The focus of these techniques is to minimize the sound dissipation error (energy loss) but, for the fact that it is an approximation of the real value of the variable, the imprecision of the scheme stimulates the growing of an artificial sound dispersion.So it has been created by Tam and Webb (1993) a finite differences schemes to inhibit this dispersion growing, among them stands out the DRP (Dispersion Relation Preserving).
On the basis of the above, this work has the main objective to evaluate the performance of these two groups of spatial schemes (finite differences and DRP) together with temporal schemes (Euler method, 2nd and 4th order Runge-Kutta and the CAA optimized methods: LDDRK and RK46-NL) on 1D (one dimension) acoustic pulse propagation, comparing results with analytical data and show a brief deduction of DRP schemes.

Acoustic Propagation Equations
As shown by Mainieri et al. (2013), the set of equations of the acoustic propagation for a quasi-static flow with Reynolds number < 1000 are presented by eq. ( 2) for 1D (one dimension) case: where u is the velocity in x direction, p is the pressure, t is the time and is the velocity propagation of the acoustic pulse in relation to the sound speed.
That set of equations were nondimensionalized as described by Mainieri et al. (2013).

Spatial Derivatives
The spatial derivatives were solved by finite differences or by DRP.The deduction of both schemes is explained in detail in Mainieri et al. (2013).The coefficients of the finite differences schemes are shown in Tab. 1 and for DRP schemes in Tab. 2. In this work it was adopted central schemes due its looseness property.
The Euler and Runge-Kutta methods requires no detailed comments because of its widespread use and well known by specialized literature.LDDRK (Low Dispersion low Dissipation Runge-Kutta) is a Runge-Kutta optimized method for aeroacoustic developed by Hu et al. (1996).RK46-NL is also a Runge-Kutta optimized method for aeroacoustic (46-NL -4 th order and 6 steps NonLinear problems) developed by Berland et al. (2006).

Absolute Error
As the main purpose of this work is to present in a clear and unambiguous manner concepts and ideas of acoustic propagation, it was chosen the simpler kind of error, the absolute error, defined as: It is important to remember that there is no damping function inside the computer code in order to not minimize numeric fluctuations.

( ( √
)) With the value of it obtains .With and it obtains , and so on.Table 2, in previous section, shows some DRP coefficients.
DRP was idealized as a centered scheme.To construct a sided scheme with the same integration limits (from = ⁄ to = ⁄ ) results in the appearance of complex numbers on the error integral, not leading application of the scheme on the border of the domain or on obstacles immersed in the flow.However, using symmetric integration limits the complex coefficients are cancelled.
For a sided scheme 4-2 (four points behind and two points forward) the interior of the error integral (Eq.( 19)) becomes: (23) For an integration limit , Eq. ( 23) can be written as: (24) Remember that now the scheme is not centered, than .Taking the symmetric integration limit, , the Eq. ( 24) is written as: (25) Elevating to the square the complete expression it will appear a huge expression of complex numbers that will cancel itself for those integration limits .To illustrate these ideas, suppose that the expression to be square elevated is: The initial conditions for 1D case were: 5. RESULTS

Spatial Scheme Variation -Mash of 201 Points
The graphics were constructed in order to enable a better overview of the schemes performance.The pressure absolute error in all points of the domain, in all times, until the pressure pulse reaches the border of the domain, were overlaid.It can be observed an asymptotic behavior of the error growth in this mesh of 201 points.It is possible to see that the DRP69 (sixth order) has a performance almost equal to the FD12 (twelveth order), six order above.

Spatial Scheme Variation -Mesh of 401 Points
In this mesh the performance of DRP schemes decreased considerably, as shown below.

Temporal Scheme Variation -Mesh of 201 Points
It can be seen in the next two graphics the superior performance of RK2 (2nd order Runge-Kutta) over RK4 (4th order Runge-Kutta).In the mesh of 201 points RK2 revealed an outstanding superior performance over the optimized aeroacoustic temporal schemes (LDDRK and RK46-NL).

Temporal Scheme Variation -Mesh of 401 Points
Under this mesh the graphics show that the optimized aeroacoustic temporal schemes (LDDRK and RK46-NL) achieved much better performance than RK2, but RK2 still got superior performance over RK4.Note that the graphics for RK2 and RK4 have different full scales.RK2 is one order of magnitude lower.The mesh refining increased the performance of the methodology.

CONCLUSION
This work has shown the need for use high order schemes in CAA, even using optimized schemes for acoustic propagation.A refined grid contributed significantly to reduce the spurious waves within the numerical procedure.The Runge-Kutta 2nd order shows an outstanding efficiency over Runge-Kutta 4th order in all simulations.This behavior must be investigated for futher work in this issue.When the mesh was refined DRP schemes performance dropped in comparison to finite differences schemes, but in general terms, both DRP and finite differences schemes performed well in acoustic propagation.
Comparing the graphics of absolute error growth of LDDRK, RK46-NL, and RK2 it shows that RK2 has the lower error and the lower growing error velocity (inclination of the tangent).