Aeroacoustic and Aerodynamics of Swirling Flows*

Transcription

1 Aeroacoustic and Aerodynamics of Swirling Flows* Hafiz M. Atassi University of Notre Dame * supported by ONR grant and OAIAC

2 OVERVIEW OF PRESENTATION Disturbances in Swirling Flows Normal Mode Analysis Application to Computational Aeroacoustics Vortical Disturbances Aerodynamic and Acoustic Blade Response Conclusions

3 Swirling Flow in a Fan

4 Issues For Consideration Effect of swirl on aeroacoustics and aerodynamics? Can we consider separately acoustic, vortical and entropic disturbances? How does swirl affect sound propagation (trapped modes)? How do vortical disturbances propagate? How strong is the coupling between pressure, vortical and entropic modes? What are the conditions for flow instability? What are the boundary conditions to be specified?

5 Scaling Analysis Acoustic phenomena: Acoustic frequency: nb Ω Rossby number = Convected Disturbances: Convection Frequency ~ Shaft Frequency Ω Rossby number = nb Ω rt >> c Ωr t O(1) U x 0 Wakes are distorted as they convect at different velocity. Centrifugal and Coriolis accelerations create force imbalance which modifies amplitude and phase and may cause hydrodynamic instability. 1

6 Mathematical Formulation Linearized Euler equations Axisymmetric swirling mean flow r r r r U(x) = U x(x, r)ex + U s(x, r) e Mean flow is obtained from data or computation For analysis the swirl velocity is taken U s = Ω r + The stagnation enthalpy, entropy, velocity and vorticity are related by Crocco s equation Γ r θ H = T S + U ζ

7 Normal Mode Analysis

8 Normal Mode Analysis A normal mode analysis of linearized Euler equations is carried out assuming solutions of the form i( ω t+ mν θ + kmnx) f ( r) e Eigenvalue problem is not a Sturm-Liouville type A combination of spectral and shooting methods is used in solving this problem Spectral method produces spurious acoustic modes Shooting method is used to eliminate the spurious modes and to increase the accuracy of the acoustic modes

9 Comparison Between the Spectral and Shooting Methods M x =0.55, M Γ =0.24, M Ω =0.21, ω=16, and m=-1

10 Effect of Swirl on Eigenmode Distribution M xm= 0.56, M Γ =0.25, M Ω =0.21

11 Pressure Content of Acoustic and Vortical Modes M x =0.5, M Γ =0.2, M Ω =0.2, ω=2π, and m=-1

12 Summary of Normal Mode Analysis Normal Modes Pressure-Dominated Acoustic Modes Vorticity-Dominated Nearly-Convected Modes Propagating Decaying Singular Behavior Nonreflecting Boundary Conditions

13 Nonreflecting Boundary Conditions Accurate nonreflecting boundary conditions are necessary for computational aeroacoustics Inflow Conditions Computational Domain Outflow Conditions Quieting the skies: engine noise reduction for subsonic aircraft Advanced subsonic technology program. NASA Lewis research center, Cleveland, Ohio

14 Formulation Presssure at the boundaries is expanded in terms of the acoustic eigenmodes. r p( x, t) = ω ν = n= 0 c mn p mn ( ω, r) e i ( ω t+ mν θ + kmn x ) dω Only outgoing modes are used in the expansion. Group velocity is used to determine outgoing modes. Causality

15 Nonreflecting Boundary Conditions (Cont.) ) ( ) ( ), ( x k m t i N n mn mn M M mn e r p c t x p + + = = = θ ω ν ν r [ ] [ ][ ] c p R = [ ] [ ] [ ] c p L L R = [ ] [ ] [ ] c p L L 1 1 R = L-1 L [ ] [ ] [ ] [ ] R R = L L L L p p Computational Domain

16 Application to Computational Aeroacoustics

17 Test Problems for Acoustic Waves Acoustic waves and/or a combination of acoustic and vortical waves are imposed upstream of an annular duct with swirling mean flow and nonreflecting boundary condition applied downstream Quieting the skies: engine noise reduction for subsonic aircraft Advanced subsonic technology program. NASA Lewis research center, Cleveland, Ohio Acoustic and /or Vortical Mode Nonreflecting Boundary conditions

18 Acoustic Normal Mode Spectrum M x =0.5, M Γ =0.2, M Ω =0.2, ω=2π, and m=-1

19 Density and Velocity Distribution in Uniform Flow k 1,1 =

20 Density and Velocity Distribution in Swirling Flow First Propagating Acoustic Mode k 1,1 =

21 Density and Velocity Distribution in Swirling Flow Second Propagating Acoustic Mode k 1,2 =

22 Density and Velocity Distribution in Swirling Flow Acoustic & Vortical Modes k 1,1 k 1,3 = =

23 Sensitivity of Numerical Solutions to Accuracy of Eigenvalue M x =0.5, M Γ =0.2, M Ω =0.2, ω=2π, and m=-1

24 Vortical Disturbances

25 Initial Value Solution u ( x, r, θ, t ) = + m = + m = A m ( x, r )e i ( α x + m θ ω t ) d ω D o = Dt ( α x + m θ ω t ) 0

26 Wake Distortion by Swirl

27 U Accelerating axial flow x L ( x, r ) γ e x + e, m = 10, ω = r = θ γ = 0 γ > 0 γ < 0

28 Effect of viscosity Small scales are most affected by viscosity. For large modal number m (equivalent to wavenumber), viscous effects are large. Rapid-distortion theory assumes viscosity as a source term modifying the evolution process. Slip/Non-slip boundary conditions were tested.

29 Effect of Reynolds number on the modes U 50 r ( x, r ) 85 e x + e, m = 10 and m = 20, ω = 5000 = θ Re=10,000 Damping β = Re t 1 ρ o U χ o O exp β m 2 r 2 x

30 Aerodynamic and Acoustic Blade Response

31 Aerodynamic and Acoustic Blade Response Linearized Euler model Rapid distortion theory disturbance propagation Source term on blades Swirling Mean flow + disturbance Blade unsteady loading & radiated sound field Normal mode analysis construction of nonreflecting boundary conditions Non-reflecting boundary conditions

32 Linearized Euler Model Two schemes are developed: Primitive variable approach Pseudo Time Formulation. Lax-Wendroff Scheme. Splitting velocity field approach Help understand physics. Computational time requirements reduced. No singularity at leading edge. Implicit scheme leads to large number of equations which must be solved using an iterative method. Parallelization significantly reduces computational time.

33 Benchmark Test Problem ( ) π = α = θ θ + + ω θ hub tip hub ) r ( h B x i x r r r r B q 2 h ( r ) e U x ), ( r, v

34 Parameters for Benchmark Test Problem Narrow Annulus Full Annulus Data r tip /r hub 1.0/0.98 r tip /r hub 1.0/0.5 M x (mach number) 0.5 ω 6.17 ω 5.64 α (disturbance) B (rotor blades) V (stator blades) C (chord) 2π/V 3c L (length)

35 Primitive Variable Approach Linearized Euler Equations Pseudo Time Formulation Lax-Wendroff Scheme [] [ ] [ ] [ ] [ ] 0 Y D r C r 1 B x A i t I r x = + + θ + + ω θ [ ] T r x p u u u Y ρ = θ

36 Unsteady Pressure Jump Across the Blade for q=1 at Different Spanwise Locations Primitive Variable Approach ND: real part -, imaginary part --; Schulten: real part -.-, imaginary part

37 Unsteady Pressure Jump Across the Blade for q=3 at Different Chordwise Locations Primitive Variable Approach ND: real part -, imaginary part --; Schulten: real part -.-, imaginary part

38 Acoustic Coefficients for Mode (1,0) at Different Gust Spanwise Wavenumbers Primitive Variable Approach Upstream Downstream

39 Acoustic Coefficients for Mode (1,1) at Different Gust Spanwise Wavenumbers Primitive Variable Approach Upstream Downstream

40 Magnitude of the Downstream Acoustic Coefficients Primitive Variable Approach q k μ Namba Schulten ND E E E E E E E E E E E E E E E E E E E E E E E E-03

41 Magnitude of the Upstream Acoustic Coefficients Primitive Variable Approach q k μ Namba Schulten ND E E E E E E E E E E E E E E E E E E E E E E E E-03

42 Splitting Velocity Approach ( ) ( ) ( ) ( ) ( ) 0 s u Dt S D c s Dt D U U u Dt u D 2 c t s u 1 1 Dt D c 1 Dt D o o p o o R R o p R o o o o o 2 o o = + φ φ = + ρ ρ = φ ρ ρ φ r r r r v r φ + = R u u + = φ ρ = o o o o t Dt D where Dt D ' p U

43 Narrow Annulus ω rm =7.55 Grid sensitivity study Pressure difference compared to LINC. Upstream m=-8 Namba Schulten ND Downstream m=-8 Namba Schulten ND 7.58x x10-3 i 7.36x x10-3 i 7.03x x10-3 i -1.12x x10-3 i -9.95x x10-3 i -9.67x x10-3 i

44 Full Annulus Case: Pressure jump for q=0, ω rm = Comparison with Schulten Splitting Approach

45 Spanwise Pressure Jump for q=3, ω rm = Comparison with Schulten Splitting Approach

46 Upstream & Downstream Acoustic Coefficients. Splitting Approach

47 Lift Coefficient for q=0, 3 versus ω and radius Splitting Approach

48 Full Annulus Lift Distribution Comparison with Strip Theory Splitting Approach

49 Meridian Plane Approximation for Mean Flow (2D Cascade) Actual Meanflow 20 o stagger, M=0.3 Meridianal Meanflow

50 Unsteady Lift Comparison Actual and Meridianal Meanflows Low Loading C l =0.20 High Loading C l =0.92

51 Conclusions For swirling flows, two families of normal modes exist: pressure-dominated nearly-sonic, and vorticitydominated nearly-convected modes. Nonreflecting boundary conditions were derived, implemented, and tested for a combination of acoustic and vorticity waves. An initial-value formulation is used to calculate incident gusts. Two schemes (primitive variable and splitting) have been developed for the high frequency aerodynamic and acoustic blade response. Results are in good agreement with boundary element codes. A meridian approximation of the mean flow gives surprising good unsteady results for 2D cascades.

52 Future Work The numerical code will be used to study unloaded annular cascades in swirling flows. Method is under development for loaded annular cascades in swirling flows using a meridian approach. Parallelization will significantly reduce computational time making it possible to treat broadband noise. Express results in term of the acoustic power radiated.