Sound propagation in a lined duct with flow

Transcription

1 Sound propagation in a lined duct with flow Martien Oppeneer supervisors: Sjoerd Rienstra and Bob Mattheij CASA day Eindhoven, April 7, / 47

2 Outline 1 Introduction & Background 2 Modeling the problem 3 Numerical implementation 4 (Numerical) results 5 Future plans 2 / 47

3 Outline Introduction & Background 1 Introduction & Background Project motivation Acoustic liners General project goal Brush-up: modes 2 Modeling the problem Pridmore-Brown equation (ODE) Boundary conditions 3 Numerical implementation Method 1: bvp4c / BVP SOLVER Method 2: COLSYS Continuation in Z 4 (Numerical) results Impedance wall, no flow Impedance wall, uniform mean flow Some numerical problems 5 Future plans 3 / 47

4 Introduction & Background Project motivation Project motivation APU: Auxiliary Power Unit produces power when main engines are switched off to start main engines, AC,... major source of ramp noise Goal: APU noise reduction Figure: APU on an Airbus A / 47

5 Acoustic liners Introduction & Background Acoustic liners Figure: Locally reacting liner (impedance wall). Figure: Spiralling (non-locally reacting) liner. Figure: Metallic foam (bulk absorber). 5 / 47

6 General project goal Introduction & Background General project goal cool air inlet hard wall resistive sheet liner cavity mean flow profile ū(r) Figure: APU geometry. exhaust Model sound propagation / attenuation sheared flow non-locally reacting liners segmented / non-uniform liners strong temperature gradients (swirling flow) (annular hub) (varying duct radius) Sufficiently fast for liner design calculations semi-analytical model, based on modes 6 / 47

7 Introduction & Background Brush-up: modes Modes Motivation: Direct Navier-Stokes (DNS) not practical / feasible (esp. for design) Eigensolution of a BVP Characterized by: eigenfunction P mµ (r) eigenvalue k mµ C Traveling waves of the form: p mµ (r) = P mµ (r) exp( iωt+ik mµ x+imθ) Total field is superposition (or integral) of modes Figure: ω = 20, m = 1, Z = 3 + 3i, µ = 3. 7 / 47

8 Outline Modeling the problem 1 Introduction & Background Project motivation Acoustic liners General project goal Brush-up: modes 2 Modeling the problem Pridmore-Brown equation (ODE) Boundary conditions 3 Numerical implementation Method 1: bvp4c / BVP SOLVER Method 2: COLSYS Continuation in Z 4 (Numerical) results Impedance wall, no flow Impedance wall, uniform mean flow Some numerical problems 5 Future plans 8 / 47

9 Duct geometry Modeling the problem d r θ h x Figure: Duct geometry, velocity components: v = uˆx + vˆr + w ˆθ. 9 / 47

10 Modeling outline Modeling the problem No viscosity No heat conduction Cylindrical coordinates Small perturbations Time-harmonic solutions Navier- Stokes eqns Euler eqns Linearized Euler eqns ODE for P (r) BVP Boundary conditions 10 / 47

11 Modeling the problem Pridmore-Brown equation (ODE) Navier- Stokes eqns No viscosity No heat conduction Cylindrical coordinates Euler eqns Small perturbations Linearized Euler eqns Time-harmonic solutions ODE for P (r) Navier-Stokes (conservation laws) t ρ + (ρv) = 0 (ρv) + (ρvv) = p + τ t (ρe) + (ρev) = q (pv) + τ v t inviscid: τ = 0 non-heat-conducting: q = 0 11 / 47

12 Modeling the problem Pridmore-Brown equation (ODE) Navier- Stokes eqns No viscosity No heat conduction Cylindrical coordinates Euler eqns Small perturbations Linearized Euler eqns Time-harmonic solutions ODE for P (r) Euler Equations (in primary variables) t ρ + (ρv) = 0 ( ) v ρ t + v v = p p + v p + γp v = 0 t Ideal gas: p = ρrt 12 / 47

13 Modeling the problem Pridmore-Brown equation (ODE) Small perturbations Sound is due to small pressure perturbations Assumptions: main sound source: turbine engine (rotor / stator interaction) negligible sound source: turbulence Total field = mean flow + perturbations: (u, v, w, ρ, p) = (ū, v, w, ρ, p) + (ũ, ṽ, w, ρ, p) Linearize: neglect quadratic terms (since perturbations are small) 13 / 47

14 Modeling the problem Pridmore-Brown equation (ODE) Mean flow Mean flow assumptions: independent of x: ū v w x = 0, x = 0, x = 0 radial velocity v = 0 circumferential velocity independent of θ: w θ = 0 14 / 47

15 Modeling the problem Pridmore-Brown equation (ODE) Navier- Stokes eqns No viscosity No heat conduction Cylindrical coordinates Euler eqns Small perturbations Linearized Euler eqns Time-harmonic solutions ODE for P (r) Linearized Euler Equations ρ t + ū ρ x + 1 (r ρṽ) + w ( ρ 1 r r r θ + ρ w r θ + ũ ) = 0 x ( ṽ ρ t + ū ṽ x + w ṽ r θ 2 w ) r w w2 p ρ = r r ( w w ρ + ū t x + w w r θ + d w dr ṽ + w ) r ṽ = 1 p r θ ( ũ ρ t + ū ũ x + w ũ r θ + ṽ ū r + w ) ū = p r θ x p t + ū p x + w ( p ρ w2 1 + r θ r ṽ + γ p (rṽ) + 1 w r r r θ + ũ ) = 0 x 15 / 47

16 Modeling the problem Pridmore-Brown equation (ODE) Navier- Stokes eqns No viscosity No heat conduction Cylindrical coordinates Euler eqns Small perturbations Linearized Euler eqns Time-harmonic solutions ODE for P (r) We seek time-harmonic solutions: ODE in P (r): (ũ, ṽ, w, ρ, p) = (U, V, W, R, P ) exp( iωt + ikx + imθ) P + β(r, k)p + γ(r, k)p = 0, on h r d where β(r, k) and γ(r, k) are functions of: mean flow parameters: ū(r), w(r), ρ(r), p(r) m, ω (given) r, k 16 / 47

17 Modeling the problem Pridmore-Brown equation (ODE) Navier- Stokes eqns No viscosity No heat conduction Cylindrical coordinates Euler eqns Small perturbations Linearized Euler eqns Time-harmonic solutions ODE for P (r) Simplifications: No swirl: w(r) = 0, p(r) constant ρ(r) constant Pridmore-Brown equation where P + β(r, k)p + γ(r, k)p = 0, on h r d β(r, k) = 1 r + γ(r, k) = 2kū ω kū (ω kū)2 c 2 k 2 m2 r 2 17 / 47

18 Modeling the problem Boundary conditions Three types of conditions We need 3 types of conditions 1 Impedance wall BC at r = d (and r = h when h 0 2 Regularization condition at r = 0 (when h = 0) 3 Normalization condition 18 / 47

19 1. Impedance wall BC Modeling the problem Boundary conditions Assume: locally reacting liner with impedance Z Due to vanishing mean-flow boundary layer: ( iωṽ n = iω + ū x + w r (Ingard-Myers condition) θ ) ( ) p Z Resulting boundary condition for locally reacting liner: P + κ h (r, k)p = 0 P + κ d (r, k)p = 0 at r = h at r = d where κ h (k) = i ρ (ω kū)2 i ρ (ω kū)2, κ d (k) =. ωz h ωz d 19 / 47

20 2. Regularization BC Modeling the problem Boundary conditions No mean flow (ū(r) = 0): Pridmore-Brown Bessel s equation P + 1 ) r P + (α 2 m2 r 2 P = 0, α 2 = ω2 c 2 k General solution: P = AJ m (αr) + BY m (αr) Note: Y m (αr) is singular at r = 0. Make sure P (r) < at r = 0 P (0) = 0, for m 1 P (0) = 0, for m = (a) J m(x) (b) J m(x) 20 / 47

21 Modeling the problem Boundary conditions 3. Normalization General solution: P = AJ m (αr) + BY m (αr) Every solution P (r) can be scaled P (r) can become 0 at r = 0 Choose P (r) = 1 at r = d (c) J m(x) (d) J m(x) 21 / 47

22 Outline Numerical implementation 1 Introduction & Background Project motivation Acoustic liners General project goal Brush-up: modes 2 Modeling the problem Pridmore-Brown equation (ODE) Boundary conditions 3 Numerical implementation Method 1: bvp4c / BVP SOLVER Method 2: COLSYS Continuation in Z 4 (Numerical) results Impedance wall, no flow Impedance wall, uniform mean flow Some numerical problems 5 Future plans 22 / 47

23 Numerical implementation Numerical solution of BVP Why numerics? Sheared flow / temperature gradients Important: good initial guess for k and P (r) Handle singularity at r = 0 Handle unknown parameter k 23 / 47

24 Numerical implementation Method 1: bvp4c / BVP SOLVER bvp4c / BVP SOLVER Based on Runge-Kutta (MIRKDC) (damped) Newton root-finder Mesh adaptation based on error estimation ( more refinement for boundary layers) Can handle parameters Can handle 1/r type singularities bvp4c: Matlab, BVP SOLVER: Fortran 24 / 47

25 Numerical implementation Method 1: bvp4c / BVP SOLVER Transformation to remove 1/r 2 singularity By introducing: P (r) = r m φ(r), Pridmore-Brown transforms into: [ ( ) 2m + 1 r m φ + φ + r β(r, ( m k) + φ r β(r, k) + γ(r, k)) ] = 0, where β(r, k) = 2kū ω kū (ω kū)2 γ(r, k) = c 2 k 2 Convert to first order system, φ(r) = φ 1 (r) and φ (r) = φ 2 (r): [ ] φ1 = 1 [ ] [ ] [ ] [ ] 0 0 φ1 0 1 φ1 r βm + (2m + 1) γ β φ 2 φ 2 φ 2 25 / 47

26 Numerical implementation Method 1: bvp4c / BVP SOLVER Handeling the 1/r singularity First order system: φ (r) = 1 Sφ(r) + Aφ(r) r Use: Make sure that Sφ(0) = 0, then: φ(r) φ(0) lim S = Sφ (0) r 0 r 0 φ (0) = Sφ (0) + Aφ(0) 26 / 47

27 COLSYS Numerical implementation Method 2: COLSYS Robust BVP solver: COLNEW / COLSYS (Fortran) [1] [2] Based on collocation at Gaussian points ( no evaluation in singular point r = 0) B-splines (piecewise polynomial functions) (damped) Newton root-finder Mesh adaptation based on error estimation ( more refinement for boundary layers) r = 0 } {{ } subinterval r 27 / 47

28 Numerical implementation Method 2: COLSYS Problem formulation for COLSYS Add dif. eq. for parameter k: k = 0 k = 0 Split into real and imaginary parts P = β(r, k)p γ(r, k)p k R = 0, k I = 0, P R = β R (r, k R, k I )P R + β I (r, k R, k I )P I γ R (r, k R, k I )P R + γ I (r, k R, k I )P I, P I = β I (r, k R, k I )P R β R (r, k R, k I )P I γ I (r, k R, k I )P R γ R (r, k R, k I )P I COLSYS solves for {k R, k I, P R, P R, P I, P I } Calculate Jacobians Similarly for BCs 28 / 47

29 Hard wall, no flow Numerical implementation Continuation in Z No flow: P (r) = AJ m (αr), α 2 = ω 2 k 2 Hard walls: P (1) = J m(α) = right running left running Im(k) Re(k) Figure: h = 0, d = 1, m = 3, ω = 20. Here: using p = P (r) exp(+iωt ikx imθ) convention 29 / 47

30 Numerical implementation Continuation in Z Continuation in Z Good initial guess is important continuation Im i Re Z i Z = R + ix Keep R constant Vary X from to (from hard wall to hard wall) 30 / 47

31 Outline (Numerical) results 1 Introduction & Background Project motivation Acoustic liners General project goal Brush-up: modes 2 Modeling the problem Pridmore-Brown equation (ODE) Boundary conditions 3 Numerical implementation Method 1: bvp4c / BVP SOLVER Method 2: COLSYS Continuation in Z 4 (Numerical) results Impedance wall, no flow Impedance wall, uniform mean flow Some numerical problems 5 Future plans 31 / 47

32 (Numerical) results R is large: close to hard wall Impedance wall, no flow Figure: Trajectories of k for R = 2, X runs from to, h = 0, d = 1, m = 3, ω = 20, no mean flow. 32 / 47

33 (Numerical) results Impedance wall, no flow R becomes smaller: trajectories join Figure: Trajectories of k for R = 1.5, X runs from to, h = 0, d = 1, m = 3, ω = 20, no mean flow. 33 / 47

34 (Numerical) results Impedance wall, no flow R becomes smaller: acoustic surface waves arise Figure: Trajectories of k for R = 1, X runs from to, h = 0, d = 1, m = 3, ω = 20, no mean flow. 34 / 47

35 Surface wave (Numerical) results Impedance wall, no flow k far away from hard wall value magnitude of Im(α) < 0 is large Then: P (r) = J m (αr) J m (α) eim(α)(1 r) r P (r) decays away from wall: surface wave 35 / 47

36 (Numerical) results Mean flow: trajectories shift Impedance wall, uniform mean flow Figure: Trajectories of k for R = 2, X runs from to, h = 0, d = 1, m = 3, ω = 5, ū = / 47

37 (Numerical) results Mean flow: poles go to Impedance wall, uniform mean flow Figure: Trajectories of k for R = 0.5, X runs from to, h = 0, d = 1, m = 3, ω = 5, ū = / 47

38 (Numerical) results Impedance wall, uniform mean flow Mean flow: hydrodynamic surface waves arise Figure: Trajectories of k for R = 0.2, X runs from to, h = 0, d = 1, m = 3, ω = 5, ū = / 47

39 (Numerical) results Impedance wall, uniform mean flow Mean flow: hydrodynamic surface waves arise Figure: Trajectories of k for R = 0.1, X runs from to, h = 0, d = 1, m = 3, ω = 5, ū = / 47

40 (Numerical) results Some numerical problems Some numerical problems bvp4c: no problems, but slow COLSYS: some convergence problems BVP SOLVER: currently working on it 40 / 47

41 Everything ok (Numerical) results Some numerical problems 10 intermediate hard wall soft wall Figure: Paths of wave number k for several modes, where ω = 5, m = 1, Ma = 0.08, and Z = 1 + iz i where Z i runs from -100 to / 47

42 Convergence problems (Numerical) results Some numerical problems 10 intermediate hard wall soft wall Figure: Paths of wave number k for several modes, where ω = 5, m = 1, Ma = 0.09, and Z = 1 + iz i where Z i runs from -100 to / 47

43 (Numerical) results More convergence problems Some numerical problems intermediate hard wall soft wall Figure: Paths of wave number k for several modes, where ω = 5, m = 1, Ma = 0.3, and Z = 1 + iz i where Z i runs from -100 to / 47

44 (Numerical) results More convergence problems Some numerical problems intermediate hard wall soft wall Figure: Paths of wave number k for several modes, where ω = 5, m = 5, Ma = 0.3, and Z = 1 + iz i where Z i runs from -100 to / 47

45 Outline Future plans 1 Introduction & Background Project motivation Acoustic liners General project goal Brush-up: modes 2 Modeling the problem Pridmore-Brown equation (ODE) Boundary conditions 3 Numerical implementation Method 1: bvp4c / BVP SOLVER Method 2: COLSYS Continuation in Z 4 (Numerical) results Impedance wall, no flow Impedance wall, uniform mean flow Some numerical problems 5 Future plans 45 / 47

46 Future plans Future plans 1 Create fast and robust solver 2 Add non-uniform flow 3 Add non-locally reacting liners 4 Add segmented liners 5 Add temperature gradients 46 / 47

47 Future plans Thank you for your attention 47 / 47

48 Appendix Bibliography I U. Ascher, J. Christiansen, and R.D. Russel. Collocation software for boundary-value odes. ACM Transaction on Mathematical Software, 7(2): , June Uri M. Ascher, Robert M.M. Mattheij, and Robert D. Russel. Numerical solution of Boundary Value Problems for Ordinary Differential Equations. Computational Mathematics. Prentice Hall, / 47