Cavitation Inception and Dynamics in Vortical Flows

Transcription

1 Cavitation Inception and Dynamics in Vortical Flows Steve L. Ceccio Mechanical Engineering Naval Architecture & Marine Engineering University of Michigan Ann Arbor, MI, American Physical Society 62nd Annual Meeting Division of Fluid Dynamics 1

2 Vortex Cavitation Regions of concentrated vorticity can form in many flows: Tip Vortices Leakage Vortices Wake Vortices Turbulent Shear Flows The pressure in the core of the vortex is lower than that of the surrounding fluid. Small bubbles (nuclei) in the core of the vortex grow when exposed to the low pressure. Reviewed by Arndt (2002), Annual Reviews of Fluid Mechanics 2

3 Vortex Cavitation in Turbomachinery Issue: Incipient Tip Leakage Cavitation can be detected acoustically well before visual inception on ducted rotors. Issue: TLC Inception occurs at higher than expected cavitation numbers. Objective: Understand the physical processes leading to TLV inception by studying a canonical open and ducted rotors up to 1 meter in diameter. NSWC-CD 36 Water Tunnel 3

4 Tip Leakage Cavitation Rotation 4

5 PIV Measurements of Vortical Flow In Tip Leakage Region S = Identification of High Vorticity Regions Can Be Lost Through Phase Averaging! Oweis, Fry, Chesnakas, Jessup, and Ceccio, JFE, (2005a) (2005b)! S =

6 Identification of High Vorticity Regions Can Be Lost Through Phase Averaging! PIV reveals time variation in vortex patterns, spatial positions, and properties! Oweis and Ceccio, EIF, (2005).! s/c Circulation 438 RPM" Core radius 6

7 Sometimes TLV Cavitation Made a Popping or Chirping Sound Pop! Hydrophone Signal Chirp! time (ms) Squeak! Incipient TLV cavitation and an accompanying noise signal. (Chesnakas and Jessup, Proc. ASME, 2003) 7

8 Prediction of Inception Pressure and Location is Difficult Measured Vortex Core Properties: Instantaneous circulations. Γ O, and core sizes, a, are measured. The typical (average) pressure drop in the core is determined: Γ ΔP = p C p = η 3 ρ O f 2πa 2 Minimum Average Pressure Location of SVC Inception Pressure coefficient is determined as a function of downstream distance, s/c. Unexpected Findings Inception occurred downstream of the minimum mean pressure location on the 5206 and 5407 rotors. Inception occurs at much higher pressures. Inception controlled by unsteady pressures σi 10 Prop 5407 Prop Range of -C p at s/c ~ Re tip (10 6 ) 8

9 Basic Experiments Were Conducted to Address Basic TVC Flows How are nuclei captured by the vortex? What causes the core pressure of the cavitating vortex to be suddenly reduced? (Merging, stretching, axial core jet/wakes in the vortex?) How does the TVC bubble make noise? (Is noise produced during growth, convection, and/or collapse?) Choi and Ceccio, JFM, (2007) 9

10 Vortex Cavitation Bubbles in a Single Vortex Video signal computer High speed video camera Trigger Time Delay Vortex Flow direction Hydrofoil Q- switch output Hydrophone Master trigger Storage O-Scope Frequency filer Charge amplifier Nd-YAG Laser (IR 1064 nm) 9- Inch Cavitation Tunnel at the University of Michigan 10

11 Inlet Vortex Properties for Four Conditions α=4 O (1) α=4.5 O (2) Γ O, m 2 /s r C, mm Γ O, m 2 /s r C, mm Foil With with Trip trip (T) / / / /-0.35 Foil with Without roughness Trip (R) (T) / / / /-0.23 SPIV Measurements of Mean Flow. Axial jet/wake in the vortex core was not present. 11

12 Gaussian Vortex With Axial Jet u θ (r) = Γ O 2πr (1 2 e α (r / r C ) ) Tangential Velocity u θ (r C ) = β Γ O 2πr C P(r) P = P C = P ηρ Γ O 2πr C 0 ρu 2 θ (r) dr = ρ Γ O r 2πr C 2 Maximum Tangential Velocity (β = 0.715) e α (r / r C ) 2 e 2α (r / r C ) 2 1 2α(r / r 2(r / r C ) 2 C ) 2 Ei(α(r / r C ) 2 ) +2α(r / r C ) 2 Ei(2α(r / r C ) 2 ) Radial Pressure Distribution Minimum Core Pressure (η = 0.870) σ = (P P V )/ 1 ρu 2 2 σ C = 2η β + 2πr CU 2 βγ O 2 1 U C U 2 + σ Cavitation Number Cavitation Number on Vortex Axis 12

13 Vortex Is passed Through a Venturi to Lower and Recover Surrounding Static Pressure Measured pressure along the Venturi with 6 pressure taps. Pressures used to estimate the loss coefficients for the throat, straight passage, and diffuser. The vortex core axial velocity estimated using relation derived by Darmfil et al. (2001). T2 T1 2 2 Ω 1 + U /U (1 a / A ) C, a(z)/ A a(z)/ A A(z)/ A a(z)/ A 2 = Ω 2 1+ U U C, The vortex core pressure is then computed. C P =(P C (z)-p )/(ρ(βγ o /(2πr C ) 2 /2) 13

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15 Time Series of Individual Cavitation Bubbles U 4.5mm 2r C =9.52mm 6 Hydrophone signal (Pa) Lb/rC bubble length bubble radius z/l rb/rc T1 Time (msec) U mm 2r C =9.52mm 6 Hydrophone signal (Pa) Lb/rC bubble length bubble radius z/l rb/rc Time (msec) 15

16 Elongated Bubble Radius versus σ c σ =1.54 σ =1.64 σ = σ =1.74 σ =1.84 σ =1.94 rb/rc 0.08 rb/rc σ C (a) Τ σ C (b) Τ2 rb/rc σ =2.65 σ =2.75 σ =2.85 rb/rc σ =2.90 σ =2.95 σ = σ C (c) R σ C (d) R2 Results are very sensitive to small variations in vortex properties. 16

17 Acoustic Impulse of Collapse versus Equivalent Radius 10 1 P A ~ 2 Q t 2 Î Elongated R1 R2 σ = 1.73 σ = σ = 2.82 No Flow, P =10kPa * No Flow, P =31kPa Near Spherical Detectable noise is produced by growth, fission (infrequently), and collapse. No chirps were detected The acoustic impulse is 2 orders of magnitude less than that produced by nearly spherical bubbles. r b,m /r C 17

18 Size Scaling of 2-Dimensional Elongated Vortex Cavitation Bubble Radius uθ,b (r)/uθ, M σ C Bubble radius r/r C σ C = uθ,b (r)/uθ,m σ C Bubble radius r/r C σ C = Maximum bubble angular velocity (a) γ = 0 γ =0 γ (d) =1 γ = 1.0 Minimum bubble angular velocity Conserve angular momentum and kinetic energy Vapor pressure on the bubble surface Flow field is modified Gaussian. γ is related to the tangential fluid velocity at the bubble interface. r b 18

19 Dynamics of Two-Dimensional and Axisymmetric Vortex Bubble Dynamics Choi, Hsiao, Chahine, and Ceccio, JFM, (2009) Continuity: 1 r r ( r u ) 0 r = r D >1000r C Cylindrical Bubble in an Initially Gaussian Vortex u r Momentum in r-direction: ρ u r t + u r u r r u 2 θ = P r r + µ 2 u r r r u r r + u r r 2 r b r Momentum in θ-direction: ρ u θ u + u θ r t r u u r θ = µ 2 u θ r r 2 + u r u θ r u θ r 2 P B = 1 ρ P R V + P G 2µ b S R b R b Boundary Conditions: u r = r= r b dr dt b P = P r= r b V P (t) P 1000rc (t) du r dr r= rb not specified σ C 0 ΔP P G = P GO ΔT r o R b 2γ 19

20 Scaling of Two-Dimensional and Vortex Bubble Dynamics r C /r O (core size / nucleus size) r C >>r O r C /r D (core size / domain size) r D >>r C Reynolds Number Re Γ = Γ/ν Cavitation Number σ C = (P O + ΔP ) P B 2η 1 ρ β Γ 2 β = 2 O 2 2πr C ΔP 1 ρ β Γ O 2 2πr C 2 Externally Applied Tension P O = 1 P V 2S R bo ΔP σ C 1 ρ β Γ + 2η 2 β 2 O 2 2πr c Time Scale ΔT τ V = ΔT βγ O 4π 2 r c 2 20

21 Vorticity Redistribution Leads To Dynamic Compliance Increasing -σ C rb,eq/rc rb/rc 0.1 ΔT/τ v = t/ τ v (a) t / τ v (b) (a) Growth of the two-dimensional bubble as a function of time for Γ 0 = 0.2, r C = 4 mm, with ΔT/ τ v = 1.0. The results for nine cavitation numbers are shown, σ C = 0.05,-0.1,-0.2,-0.4,-0.5,-0.6,-0.8,-1.0; (b) Growth of the bubble for σ C = -0.3, with ΔT/ τ v = 0.25, 0.6, and

22 t/τ v = 0.25 t/τ v = 0.5 t/τ v = 0.75 t/τ v = 1 (a) t/τ v = 0.25 t/τ v = 0.38 t/τ v = 0.5 t/τ v = 0.75 (b) t/τ v = 0.06 t/τ v = 0.19 t/τ v = 0.3 t/τ v = 0.5 (c) Volume history of three dimensional bubble geometry during growing for (a) Re Γ = 2 x 10 5 and σ C =-0.4, for and ΔΤ/τ v = 0.46 (b) ΔΤ/τ v = Also shown is the volume history for (c) Re Γ = 4 x 10 5 and, σ C = -0.4, and ΔΤ/τ v =

23 Higher static pressure region τ/τ * = 6, Re Γ = 4x10 5, L b /D b = 2 and σ C =

24 Comparison of the Predicted and Measured Radii of Elongated Vortex Bubbles Dynamic 3-D Model rb/rc Dynamic 2-D Model σ C Single Vortex Experiment

25 Summary Of Single Vortex Results Single vortex scaling methods will often not capture the physics responsible for TVC cavitation, especially when multiple interacting vortices are present. There were no experimental observations of chirping bubbles in the single vortex experiment. Analysis and numerical modeling suggest that chirping bubbles occur when there is a variation in angular momentum due to the change in the elongated bubble radius that couples into the pressure at the bubble interface. The analysis suggests that the rate of pressure changes surrounding the nucleus is important. 25

26 Summary Of Single Vortex Results We therefore need to vary the rate of pressure reduction and recovery surrounding the bubble. This can be achieved with vortex-vortex interactions.

27 UM Visualizations of cavitating co-rotating vortices. 27

28 UM Visualizations of cavitating counter-rotating vortices. Secondary Vortex Primary Vortex 28

29 Counter-Rotating Vortex Pairs Examined run # Γ O,P, m 2 /S Γ O,S, m 2 /S r C,P, mm r C,S, mm Γ O,S /Γ O,P b, mm (r C,S +r C,P )/(2 b ) Re Γ,P C P,C,P C P,C,S Ratio Incepting Vortex Instability T T R R D Primary O D Simultaneous O D Primary O D Secondary O D Secondary O D Secondary O D Secondary O D Primary O D Secondary O D Primary O D Secondary O D Primary X D Primary X D Primary X D Primary X D Secondary O D Secondary O Some combinations are unstable, and the weaker secondary vortex cavitates first 29

30 Vortex Inception in the Secondary Vortex 152 mm x 61 mm 2600 FPS 30

31 Growing Case Makes a Pop hydrophone msec msec 1.402msec 3.710msec 31

32 Collapsing Case Makes a Pop msec msec msec msec 32

33 Chirping After Inception msec 0.446msec 1.601msec 3.524msec Γ S = 0.06 m 2 /s, r C,S = 0.6 mm (after stretching) τ V = 3 khz! 33

34 Chirping After Inception msec 1.387msec 3.309msec 5.232msec 34

35 Acoustic Pressure: Inferred and Measured Acoustic Pressure P est ρ w 4π R 2 Q t 2 1 cm P = 170kPa DO = 25% σ = 3.3 Acoustic Pressure Computed From Measured Volume Acceleration 35

36 1 cm P = 170kPa DO = 25% σ = 3.3 f ~ 4 khz 36

37 Probability of a Chirp Is Related to Nuclei Size p r i h c % Probability Chirp % Dissolved Oxygen σ DO 14% DO 25% DO 44% Small Nuclei Can Still Chirp 12 Large Nuclei Do Not Need High Tensions to Cavitate and Chirp Small Nuclei Need High Tensions to Cavitate and Pop Cavitation Number 37

38 Some Conclusions About TVC Inception and Acoustics Single vortex scaling methods will often not capture the physics responsible for TVC cavitation, especially when multiple interacting vortices are present. Bubble-Vortex Interactions are important for understanding the dynamics and noise emissions of elongated vortex bubbles. A combination of experimental, analytical, and numerical tools are available to understand and (possibly) predict TVC cavitation since there is likely no single scaling method which captures all the necessary physics. The bubble could be used as a flow diagnostic (!) 38

39 Some Acknowledgements Dr. Stuart Jessup (NSWC-CD) Dr. David Fry (NSWC-CD) Dr. Chris Chesnakas (NSWC-CD) Dr. Michael Billet (PSU-ARL) Mr. William Straka (PSU-ARL) Dr. George Chahine (Dynaflow) Dr. Hsiao (Dynaflow) Dr. Ki-Han Kim (ONR) Prof. David R. Dowling (UM) Prof. Ghanem Oweis (AUB) Dr. Natasha Chang (NSWC-CD) Dr. Jaehyung Choi 39

40 Prof. Roger Arndt Prof. Chris Brennen Thank you! 40