# federico piscaglia slides

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Develpment of NSCBC for compressible Navier-Stokes equations in OpenFOAM : Subsonic Non-Reecting OutowF. Piscaglia, A. MontorfanoDipartimento di Energia, P OLITECNICODI

MILANO

http://www.engines.polimi.it

ContentIntroduction Theory - The NSCBC strategy for Navier-Stokes equations - The Local One Dimensional Inviscid (LODI) relations - Subsonic Non-Reecting Outow - Unsteady ows and numerical waves control - Shock-tube: test Application: non linear acoustic simulation of silencers - Reverse ow chambers - Single-plug perforated mufer Conclusions

http://www.engines.polimi.it

IntroductionAn ideal mufer for a ICE should work as a low pass lter: attenuation should be applied to the uctuating ow which is associated with the acoustic pressure uctuation, while the steady or mean ow should be allowed to pass unimpeded through the mufer.

- The performance of a silencer can be described by two key parameters: - the attenuation of the pressure uctuations crossing the mufer over a pre-dened wide frequency range, usually 20-2000 Hz - the pressure drop associated with convective and dissipative effects of the mean ow: it affects both the acoustic attenuation performance and the back pressure seen by the engine

http://www.engines.polimi.it

Acoustic performance of silencers

The acoustic performance of acoustic silencers is determined by the Transmission Loss, dened as the ratio of Sound Pressure Level spectrums (Lp ) of incident waves: TL(fn ) = 20log10 pups + (fn ) pdws + (fn )

- all acoustic frequencies of the eld of interest must be excited: large-band acoustic sources - anechoic termination is needed Since the gas pressure in the domain is the result of superimposition of incident waves coming from the source and reections from the boundaries, it is impossible to directly measure the incident components of the pressure pulsation. Hence, algorithms for data postprocessing based on the general hypotheses of the linear acoustics are used (two-sensor method).

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MotivationThe objective of this study is: - to develop a three-dimensional time-domain approach based on the CFD simulation to evaluate the transmission loss of silencers and resonators without and with mean ow - to examine the inuence of mean ow on the acoustic attenuation performance of complex silencers ... but: - an inlet b.c. to model different acoustic sources is needed - acoustic simulations of compressible ows require an accurate control of wave reections from the computational domain boundaries. Acoustic waves are often modied by numerical dissipation - the waveTransmisive b.c. in OpenFOAM is not perfectly non reecting; small acoustic waves are reected to the inner domain - there is the need for sophisticated boundary conditions (NSCBC), which can handle correctly the transmission and the reection of acoustic waves on boundaries

http://www.engines.polimi.it

acousticSourceFvPatchFieldA boundary condition acousticSourceFvPatchField to model different types of acoustic sources has been developed in the OpenFOAM technology.S INGLE101.4 101.38 101.36

SINUSOID101.8 101.6

P ULSE101.4

W HITE101.4 101.35

NOISE101.4 101.38 101.36

F REQUENCY

SWEEP

Pressure [kPa]

Pressure [kPa]

Pressure [kPa]

101.2 102 101.8 101.6

101.34 101.32 101.3 101.28 101.26 0

101.3

Pressure [kPa]0.02 0.04 0.06 0.08 0.1

101.34 101.32 101.3

101.25

101.2 101.4 0.02 0.04 0.06 0.08 0.1 101.2 0 0.02 0.04 0.06 0.08 0.1 101.15 0 101.28 101.26 0

0.02

0.04

0.06

0.08

0.1

time [s]

time [s]

time [s]

time [s]

inlet { type sourceType U phi rho psi gamma refPressure f0 fn step amplitude value }

acousticSourceTotalPressure; "whiteNoise"; U; phi; rho; none; 1.4; 100000; 10; 2000; 10; 50; uniform 100000;

Different kind of time-varying perturbations are applied at the inlet boundary patch Ad-hoc developed run time controls ensure correct case setup and avoid aliasing due to poor frequency resolution or to non physical frequency signals distorting the spectrum in the chosen range (Oppenheim and Schaffer)

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Non-reecting (anechoic) outlet b.c.THEORY - Implementation of a true non-reecting outlet based on the NSCBC theory: variables which are not imposed by physical boundary conditions are computed on the boundaries by solving the conservation equations as in the domain - Wave propagation is assumed to be associated only with the hyperbolic part of the NavierStokes equations, waves associated with the diffusion process are neglected - In characteristics analysis, absence of reection is enforced by correcting the amplitude of the ingoing characteristic (wave reected by the boundary) to zero. Partial reection is needed for a well-posed problem - Local viscous terms and tranverse terms have been neglected in the formulation of the governing equations (LODI) - Also: - the method allows a control of the different waves crossing the boundaries - no extrapolation procedure is used VALIDATION - Shock Tube - Engine Silencers

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Non-reecting (anechoic) outlet b.c.For each cell face at the boundary end, the governing equations written in a local reference frame (, , ) are:Continuity: u2 u3 + d1 + + =0 t Momentum: u1 u1 u2 u1 u3 + u1 d1 + d3 + + = t u2 u2 u2 u3 u2 + u2 d1 + d4 + + = t u3 u3 u2 u3 u3 + u3 d1 + d5 + + = t Energy: E 1 d2 [(E + p)u2 ] [(E + p)u3 ] + (uk uk )d1 + + u1 d3 + u2 d4 + u3 d5 + + = q t 2 1 0y

p p

z

x

Each reference frame has its origin in the cell face center and the vector is set as perpendicular to the cell face.

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Extension to local Cartesian coordinates- For each cell face at the boundary end, a local reference frame (, , ) has been dened: x = x(, , ) y = y (, , ) z = z(, , ) - Each reference frame has its origin in the cell face center and the vector is set as perpendicular to the cell face. - The governing equations for the global reference frame take the form: where U = F 1 F 2 F 3 U + + + = p t x y z U J F 1 x + F 2 x + F 3 x F 1 = x J F 1 y + F 2 y + F 3 y F 2 = y J F 1 z + F 2 z + F 3 z F 3 = z J

y x

z

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NSCBC approach for b.c.The vector d given by characteristic analysis (Thompson) can be written as: d1 d2 d = d3 = d4 d5 where: L1 L2 L = L3 = L4 L5 1 p c u1 c 2 p 2 3 u2 4 u3 5 p + c u1 1 = u1 c 2 = 3 = 4 = u1 5 = u1 + c i is the characteristic velocity associated to Li m1

c 2 m1 p + u1 u u1 1 + 1 p = u1 u2 u3 u1

1 c2

L2 +1 2 1 c

1 2

(L5 + L1 )

(L5 + L1 ) (L5 L1 ) L3 L4

- Li is the amplitude variation of the ith characteristic wave crossing the boundary - L1 is the incoming characteristic reected by the boundary

http://www.engines.polimi.it

Subsonic non-reecting outow- A perfectly subsonic non-reecting outow (L1 =0) might lead to an ill-posed problem (mean pressure at the outlet would result to be undetermined) - Corrections must be added to the treatment of the b.c. to make the problem well posed. The amplitude of the incoming wave is then set as: L1 = K (p p ) that in global coordinates becomes: |1 M2 | L1 = 2Jl - M is the max. Mach number dened over the patch - is a constant leading the pressure drift. 0.1 < < (Strickwerda) - l is a characteristic size of the domain - J is the Jacobian marix The resulting formulation makes the b.c. partially non reecting and the problem well-posed.

http://www.engines.polimi.it

Numerical solutionGoverning equations have been solved by a multistage time stepping scheme in t n,k : t n,k t n + k t = t n + k t K k [1; K ] (1)

where t n,k is a variable local fractional time-step. The method consists of the iteration of two main steps:k 1) Evaluation of backward spatial derivatives at t n and of the uxes at time t n + K t; conservation equations are solved sequentially. The solution is rst order in time.

2) Fluxes and source terms calculated at the previous step are used to nd the solution at time t n + k+1 t. The time accuracy of this method is of the second order at this stage. K The process is iterated until the solution at the new time t n+1 t + t is calculated. The time stepping algorithm: - requires a relatively small amount of memory storage - it is more stable and accurate - it allows for larger global time steps in the simulation than a traditional explicit method.

http://www.engines.polimi.it

NonReecting NSCBC vs waveTransmissiveSHOCK TUBE simulation: 40500 hexahedral cells pmax = 1.2 bar p0 = 1.0 bar, T0 = 293 Kxclosed end nonreecting open end

p1.2 bar

1 bar

OpenFOAM wave transmissive BC

LODI nonreecting BC

http://www.engines.polimi.it

Case study: reverse-ow silencers

b d

e1

D

d

b

e2

L

Silencer RC-l1 RC-l2 RC-m RC-s

l [mm] 494 494 377 127

w [mm] 197 197 197 197

d [mm] 50 50 50 50

b [mm] 17 17 17 17

e1 [mm] 17 257 167 17

s

s

e2 [mm] 17 17 17 17

s [mm] 50 50 50 50

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Case setup- solver: pisoFoam - temporal discretisation: Crank-Nicholson scheme - differential operators: standard nite volume discretisation of Gaussian integration - working uid: air - boundary conditions: - inlet : pressure pulse with frequency content f[20;2000] Hz (step 20 Hz) - outlet: non-reective NSCBC anechoic boundary conditio

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