Fluid Mechanics & Aeroacoustics of Fans and Compressors
Courtesy: Pollrich DLK
Day 1: Axial Flow Compressors & Fans
Short Course Offered at BCAM− July 2-4, 2013 Farzad Taghaddosi, Ph.D. Course Objective • Provide basic understanding of fluid flow in compressors and fans (axial & centrifugal)
• Understand sources of noise and methods for acoustic analysis
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 2, 2013) − page 2 Definition • Compressors & fans: – Use input mechanical energy to increase fluid total pressure – Fans (Δ푝 ~ 0.01 atm), compressors (Δ푝 ≥ 1 atm) – Configs: axial, radial/centrifugal, or mixed Flow
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 2, 2013) − page 3 Choosing the Right Fan/Compressor
• Performance variables: 푤푠푡푎푔푒, 휂, Δ푝 = 푓(푚 , 𝜌, 푁, 퐷, 휇, 푎)
• 휓, 휂, 푝02 = 푓(휙, 푅푒, 푀) Non-dimensional form: 푝01
푤푠 푚 𝜌 – Where 휓 = 휙 = 푁2퐷2 푁퐷3 loading coefficient flow coefficient
• Specific speed • Specific diameter 1/2 1/2 1/4 1/4 휙 푁(푚 𝜌) 휓 퐷 푤푠 푁푠 = 3/4 = 3/4 퐷푠 = 1/2 = 1/2 휓 푤푠 휙 (푚 𝜌)
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 2, 2013) − page 4 Cordier Line/Diagram
푁 푚 퐷 푤푠 푁푠 ~ 퐷푠 ~ 푤푠 푚
−1.936 8.26 퐷푠
−0.916 2.5 퐷푠
3.23
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 2, 2013) − page 5 Sample 푁푠퐷푠 (Balje) Diagram
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 2, 2013) − page 6 Axial-flow Compressors
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 2, 2013) − page 7 Axial-flow Compressors
• Applications: Industrial gas turbines, aircraft engines
http://www.youtube.com/watch?v=CXSi4GXUojo
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 2, 2013) − page 8 Axial-flow Compressors
• Best for applications with high 푁푠 and low 퐷푠 – High mass flow w/ relatively small Δ푝 per stage – Therefore large number of stages are needed – Each stage: rotor followed by stator single stage
compressor section of a turbofan engine
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 2, 2013) − page 9 Terminology • Cylindrical coordinate system 푟
• Velocity components » 푐푥: axial 푥 » 푐푟: radial 휃 » 푐휃: tangential/circumferential 2 2 » 푐푚 = 푐푥 + 푐푟 : meridional
• Relative frame of reference 푐푥 » 푈: blade tangential velocity = 푟휔 훼 훽 푐휃 » 푤: relative velocity 푐 » 푐 : absolute velocity = 푤 + 푈 푤 푤 푈 휃 • Flow angles 푥 » 훼: absolute » 훽: relative 휃
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 2, 2013) − page 10 Flow through Compressor Stage
• Pressure increase through both 1 rotor and stator
푤휃1 푐휃1 • Moderate pressure rise (flow deceleration) due to adverse pressure gradient 2
푤휃2 푐휃2 • As a result, blades have small curvature/camber; are very thin
• Need multiple stages to create 푝 3 푥 large pressure increase 푐 푥 Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 2, 2013) − page 11 Stage Work (Loading) • Euler’s equation:
푤푠 = 푈2푐휃2 − 푈1푐휃1
푤 = 푈(푐 − 푐 ) at mean radius 푤휃1 푐휃1 푠 휃2 휃1 » All work done in rotor, none in stator
• Loading coefficient 푤휃2 푐휃2 2 휓 = 푤푠/푈 = (푐휃2 − 푐휃1)/푈 = Δ푐휃/푈
» 휓 directly related to flow turning Δ푐휃 Δ푐 » Higher 휓 → reduced no. of stages 휃 » Reducing inlet swirl → higher 휓
» 휓푑푒푠푖푔푛 ~ 0.4
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 2, 2013) − page 12 Flow Coefficient
• Definition: 휙 = 푐푥/푈 » Determines change in flow angles
» Typically 푐푥 const. thru compressor 푤휃1 푐휃1 » Higher 휙 → reduced flow turning
» 휙푑푒푠푖푔푛 ~ 0.4-0.8 » 휓 and 휙 are directly related:
푤 푐 휓 = (푐휃2 − 푐휃1)/푈 휃2 휃2
⋮
휓 = 휙 tan 훼2 − tan 훼1 = 휙(tan 훽1 − tan 훽2)
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 2, 2013) − page 13
Stage Reaction
rotor stator
Δℎ푟표푡표푟 ℎ2 − ℎ1 Δ푝 푟표푡표푟 푅 = = ≈ , 0 ≤ 푅 ≤ 1 Δℎ푠푡푎푔푒 ℎ3 − ℎ1 Δ푝 푠푡푎푔푒 1 2 3 • Impacts asymmetry of velocity triangles hence blade shapes
• Typical range: 푅푑푒푠푖푔푛 ≈ 0.5-0.8
• Relationship with 휓 and 휙: 휓 = 2 1 − 푅 − 휙 tan 훼1 » Higher reaction tends to reduce stage loading
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 2, 2013) − page 14 Stage Thermodynamics • Work:
푤푠 = 푈2푐휃2 − 푈1푐휃1 = ℎ02 − ℎ01
→ ℎ01 − 푈1푐휃1 = ℎ02 − 푈2푐휃2
퐼 ≡ ℎ0 − 푈푐휃 rothalpy 2 2 푤 푈 2 퐼 ≡ ℎ + − → 퐼 ≡ ℎ0,푟푒푙 − 푈 /2 2 2
• Enthalpy change across rotor:
» 퐼1 = 퐼2 or at mean radius (푈1 = 푈2): rotor 2 stator ℎ01,푟푒푙 = ℎ02,푟푒푙 ℎ0,푟푒푙 = ℎ + 푤 /2 • Enthalpy change across stator (푈 = 0): 1 2 3 2 ℎ02 = ℎ03 ℎ0 = ℎ + 푐 /2
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 2, 2013) − page 15 Stage Losses & Efficiency • Efficiency of the compressor is impacted by the losses in each stage (rotor+stator) • Losses are typically quantified using
correlations obtained from experimental tests
annulus/clearance Losses
secondary
profile
휙
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 2, 2013) − page 16 • Profile/annulus losses • Tip leakage losses – BL drag & wake mixing » Tip vortex mixing
• Secondary flow losses – Corner stalls, 3D effects • Shock-induced losses
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 2, 2013) − page 17 Stage Loss Metrics • Enthalpy loss coefficients:
ℎ2 − ℎ2푠 ℎ3 − ℎ3푠 휁푅 = 2 휁푁 = 2 rotor 푤2 /2 stator 푐3 /2
• Stagnation pressure loss coefficients:
푝01,푟푒푙 − 푝02,푟푒푙 푝02 − 푝03 푌푅 = 푌푁 = 푝01,푟푒푙 − 푝1 푝02 − 푝2
• Loss coefficients are related: – At low Mach numbers: 푌 ≈ 휁 – But at higher Mach numbers: 푌 > 휁
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 2, 2013) − page 18 Stage Efficiency • Efficiency
훾 − 1 푌푅 1 − 푝1/푝01,푟푒푙 + 푌푁 1 − 푝2/푝02 휂 ≅ 1 − 푡푡 훾 1 − 푇 /푇 1 03 Or 2 2 휁푅푤2 푇3 푇2 +휁푁푐3 휂푡푡 ≅ 1 − 2 ℎ03−ℎ01
» For low-speed/incompressible machines:
푇03Δ푠푠푡푎푔푒 Δ푝0,푅 + Δ푝0,푁 휂푡푡 ≅ 1 − = 1 − ℎ03 − ℎ01 𝜌 ℎ03 − ℎ01
Or 2 2 (푤1 푌푅 + 푐2 푌푁) 휂푡푡 ≅ 1 − 2(ℎ03 − ℎ01)
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 2, 2013) − page 19 Measuring Losses: Cascade Flow Analysis • Sample Cascade Tunnel
• Main objective: – Characterize losses – Measure exit flow angle • Based on simplified 2D, steady flow • Both design and off-design conditions are tested
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 2, 2013) − page 20 Cascade Nomenclature
c1
c1 c2
c2 b = axial chord c/s = solidity
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 2, 2013) − page 21 Some Design Criteria • Diffusion factor
푤 −푤 Δ푐 푠 DF = 1 2 + 휃 ≈ 0.45
푤1 2푤1 푐 Losses deceleration turning
» Helps determine space-chord ratio (푠/푐) » For given DF, higher turning requires reduced blade spacing to avoid separation » Typical values: 푠/푐 ≈ 0.8 − 1.2
• Inlet swirl angle: 훼1 ≈ 20° − 30° » Helps reduce relative inlet Mach number H » Reduces flow turning hence stage loading
• Blade aspect ratio: 퐻 푐 ≈ 1 − 2 s
• Blade spacing: 푠 푏 ≈ 0.5
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 2, 2013) − page 22 Multi-stage Compressors • Effective annulus area is reduced because of BL growth
– Axial velocity is adversely impacted Impact reduced after ~ 4th stage
– The effect is taken into account by introducing work-done factor (휆): 푤 = 휆푈(푐 − 푐 ) 푠 휃2 휃1 – American design practice: apply blockage factor to account for reduced annulus area
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 2, 2013) − page 23 Radial Flow Variations
• 2D flow assumption only • For 푟ℎ푢푏/푟푡푖푝 ≈ 0.4-0.8, valid when 푟ℎ푢푏/푟푡푖푝 is blade speed (푈) & flow large (≥ 0.8) – typically angles will significantly last stages blades vary from hub to tip » Blades require significant twist
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 2, 2013) − page 24 Radial Flow Variations
• Change in annulus shape means 푐푟 cannot be ignored, although still smaller than 푐푥 and 푐휃
• Pressure increase from hub to tip to counter centrifugal forces acting on the fluid will cause slight variation in the radial direction
• Radial flow variation is taken into account by solving “radial equilibrium equation”
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 2, 2013) − page 25 Flow Instability - Surge • It is caused by drop in delivery pressure due to reduction in 푚
• If 푝푒푥푖푡 does not drop fast enough, air will reverse direction and flow upstream due to resulting pressure gradient. This will cause sudden drop of compressor exit pressure, reversing air flow direction… • The cycle can then continue at high frequency • Surge characterized by vibration in “axial” direction, causes excessive blade vibration, and can lead to flame-out (flame extinction)
http://www.youtube.com/watch?v=osAT6mwkr94 http://www.youtube.com/watch?v=9KhZwsYtNDE
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 2, 2013) − page 26 Flow Instability – Rotating Stall • An instability usually observed at low operating speed (푁) • Is caused by blade stall (due to increased loading, tip vortex or corner stall), leading to flow blockage and change in angle-of-attack of neighboring blades (increase on one side and decrease on another side) • This causes neighboring blade stall an recover creating stall patches that will travel around compressor annulus • Rotating stall can exist in normal operating conditions; both part-span and full-span stall has been observed • It causes vibration in circumferential direction
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 2, 2013) − page 27 Low-speed Ducted Fans
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 2, 2013) − page 28 Introduction • Ducted fans are essentially single-stage compressors but with low pressure ratio
• Two configurations may be used: a) IGV – Rotor b) Rotor – OGV
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 2, 2013) − page 29 Introduction • Ducted fans have typically higher space-chord ratio (low solidity) compared to compressors
• Isolated airfoil theory is often used since the influence of neighboring blades is small
푘 approaches one for high 푠/푐
푠/푐 is much greater
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 2, 2013) − page 30 Fluid Mechanics & Aeroacoustics of Fans and Compressors
Courtesy: NASA
Day 2: Centrifugal Compressors & Fans
Short course offered at BCAM− July 2-4, 2013 Farzad Taghaddosi, Ph.D. Introduction • Efficiency of axial flow compressors sharply drops at low flow rates » Increased losses due to larger surface/volume ratio of annulus » Manufacturing of small parts, high maintenance cost, etc.
Compressor chart will be similar
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 3, 2013) − 32 Cordier Diagram
푁 푚 퐷 푤푠 푁푠 ~ 퐷푠 ~ 푤푠 푚
−1.936 8.26 퐷푠
−0.916 Centrifugal compressors best choice for 2.5 퐷푠 high pressure rise for small flow rate
3.23
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 3, 2013) − 33 Introduction • Centrifugal compressors » Smaller number of components » More compact design » Pressure ratio’s as high as 8:1
• Centrifugal fans/blowers: Δ푝 small, about a few inches of water (𝜌푒/𝜌푖 ≤ 1.05) » Usually treated as incompressible
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 3, 2013) − 34 First Jet Engine (Frank Whittle) - 1930 • Used a centrifugal compressor
• Soon became apparent that they are not suitable for higher mass flows – Larger frontal area, lower efficiency, etc.)
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 3, 2013) − 35 Some Applications • Automobile turbochargers • Auxiliary Power Units (APU’s)
• Gas pipeline, refrigeration, process plants Honeywell APU
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 3, 2013) − 36 Components & Operation
• Impeller: pressure rise due to centrifugal action & diffusion
• Diffuser (vaned / vaneless): pressure rise by diffusion (velocity almost reduced to inlet value)
• Design practice: 50-50 pressure rise across impeller & diffuser
• Scroll or Volute: collects and delivers the air
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 3, 2013) − 37 Flow Path
퐶휃2
퐶푥1 푤1
• Air enters through impeller eye in axial direction • Unless inlet guide-vanes (IGV’s) are used, vanes must be curved to allow smooth inflow • Air leaves impeller tip with absolute velocity 푐2 • Some impellers have shroud to reduce leakage & losses
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 3, 2013) − 38 Stage Work
• Impeller: 퐼1 = 퐼2 (constant rothalpy) 푤2 푈2 퐼 ≡ ℎ + − 2 2 푈2 > 푈1 1 1 → ℎ − ℎ = 푈2 − 푈2 + (푤2 − 푤2) 2 1 2 2 1 2 1 2
centrifugal action flow deceleration 휔 ~ 75% ~ 25%
» Δℎ directly related to Δ푝
• Diffuser: ℎ02 = ℎ03 (constant stagnation enthalpy) 푐2 1 ℎ ≡ ℎ + → ℎ − ℎ = (푐2 − 푐2) Δ푝 due to flow deceleration 0 2 3 2 2 2 3
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 3, 2013) − 39 Stage Thermodynamics • Impeller – Rothalpy: 푤2 푈2 퐼 ≡ ℎ + − 2 2
• Diffuser – Stagnation enthalpy: 푐2 ℎ ≡ ℎ + 0 2
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 3, 2013) − 40 Stage Thermodynamics • Work: 푤 = 푈 푐 − 푈 푐 2 휃2 1 휃1 퐶휃2
푐휃1 = 0 for axial inflow
→ 푤 = 푈 푐 = ℎ − ℎ 2 휃2 02 01 퐶푥1 푤1
• Slip factor
» Ideally: 푐휃2 = 푈2, but in reality: 푐휃2 < 푈2 due to less than perfect guidance received because of finite no. of vanes
» Define 𝜎푠 = 푐휃2/푈2 as slip factor:
0.63휋 2 𝜎푠 ≈ 1 − ≈ 1 − Stanitz formula 푁푣푎푛푒 푁푣푎푛푒
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 3, 2013) − 41 Stage Thermodynamic • Power input factor (휆) » Correction factor to account for losses in the impeller only » 휆 ≈ 1.035 – 1.04
• Overall stagnation pressure ratio:
2 훾/(훾−1) 푝03 휂푐휆 𝜎푠푈 − 푈1푐휃1 = 1 + 2 푝01 푐푝푇01
휂푐 : isentropic efficiency 푇01: inlet stagnation temperature 푐푝 : specific heat
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 3, 2013) − 42 Impeller Design Considerations
Backward Swept Vanes 퐶휃2
» Radial impeller designs lead to high exit 푤2 velocity 푐2, which may lead to flow separation in diffuser » Backward swept vanes will reduce (increase 푤2) hence reduce diffusion in both impeller & diffuser » Because of more controlled diffusion in impeller & diffuser both overall efficiency and operating margin improve » To maintain pressure ratio, however, rpm has to be increased. Therefore, centrifugal stresses will be higher » Swept vanes will also experience bending stresses » Typical bend angles: 훽 = 30°-40°
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 3, 2013) − 43 Impeller Design Considerations
• Inlet pre-whirl » Without pre-swirling inflow (using inlet guide vanes), and hence relative Mach no. will be 퐶휃1 very high 푈 » The flow can become supersonic, creating 푤1 shock waves, which in interaction with the BL 푤1 may cause flow separation » Adding pre-whirl will help reduce inlet Mach number » But, as a result, 푐휃1 will no longer be zero, which mean more work is needed to create the same pressure ratio: 2 훾/(훾−1) 푝03 휂푐휆 𝜎푠푈2 − 푈1푐휃1 = 1 + 푝01 푐푝푇01
» Since 푤1 will be highest at the tip of the eye (highest 푈1), one can minimize the impact by adding pre-whirl near tip of the eye only
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 3, 2013) − 44 Diffuser Design Considerations
• Can further increase pressure in diffuser by reducing 푐2 (or 푐휃2) – Since past impeller exit, angular momentum stays constant:
푟푐휃 = const. increasing radius will achieve this
• Vaneless diffuser: reduce 푐휃 by increasing radius
• Vaned diffuser: use vanes to reduce 푐휃 faster
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 3, 2013) − 45 Diffuser Design Considerations • Volute or Scroll » Collects and delivers the flow » Spiral-shaped channel of increasing cross-sectional area
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 3, 2013) − 46 Performance Characteristics
• Centrifugal compressors can also suffer from instabilities such as rotating stall & surge
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 3, 2013) − 47 Fluid Mechanics & Aeroacoustics of Fans and Compressors
Day 3: Introduction to Aeroacoustics
Short course offered at BCAM− July 2-4, 2013 Farzad Taghaddosi, Ph.D. Introduction • What is aero-acoustics? » Study of sound generated by aerodynamic sources
• Examples:
Courtesy NASA
Courtesy ISVR
Courtesy NASA
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 4, 2013) − page 49 Introduction • There is obviously a need to reduce man-made noise
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 4, 2013) − page 50 Physical Nature of Sound • Sound: pressure disturbances/fluctuations of very small amplitude (푝′ ≔ 푝 − 푝 ) » Sound waves require a medium to travel
• For any 푝′, there is an associated fluctuations of velocity particles (푣′)
• Speed of sound: speed of sound propagation in a ′ ′ medium; in undisturbed medium 푐0 = 휕푝 휕𝜌 푠 – Note that 푣′ and 푐0 are not the same
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 4, 2013) − page 51 Noise Signal Analysis • Noise signals are measured in the “time domain” but are analyzed in the “frequency domain” using Fourier transform • Any complex signal can be decomposed this way
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 4, 2013) − page 52 Noise Signal Analysis • Human ear can hear a sound, if the frequency content of the signal is in the range: 20 Hz – 20 kHz, provided the signal amplitude is higher than threshold of hearing. The amplitude is measured using sound pressure level (SPL)
• Amplitude is usually weighted within above freq. range to replicate human ear sensitivity » A-weighting (dBA) is most commonly used
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 4, 2013) − page 53 Metrics • Strength of acoustic signal is measured using rms ′ 2 (root-mean-square) value, defined as: 푝푟푚푠 = √ (푝′) ′ -5 » Threshold of hearing: 푝푟푚푠 ≈ 10 Pa ′ 2 » Threshold of pain: 푝푟푚푠 ≈ 10 Pa » Because of large range of values, logarithmic scale is used » Acoustic signal strength is measured using sound pressure level (SPL)
• Sound pressure level (푆푃퐿 or 퐿푝):
′2 2 ′ 푆푃퐿 = 10 Log ( 푝푟푚푠 푝푟푒푓) = 20 Log( 푝푟푚푠 푝푟푒푓) (dB)
−5 −6 where 푝푟푒푓 = 2 × 10 Pa for air and 10 Pa in other media. ′ » Doubling 푝푟푚푠 will increase the noise by only ~ 6dB (= 20 Log 2)
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 4, 2013) − page 54 Typical Noise Levels
Loud noise of short duration is less annoying to human ear than a persistent noise of lower amplitude
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 4, 2013) − page 55 Metrics • Sound intensity Level (퐼퐿) – Sound intensity (energy flux): 퐼 (풙) = 푝′푣′; or time-averaged: 퐼 = 푝′푣′ – The direction of the intensity is the average direction in which the acoustic energy is flowing
퐼퐿 = 10 Log( 퐼 퐼 ) (dB) where 퐼 = 10−12 W/m2 푟푒푓 푟푒푓
• Sound power level (퐿푊) – Is the power of sound sources enclosed within an area, 퐴 – Sound power is thus obtained by integrating intensity over the area – It is independent of integration area as long as 퐴 encloses all sources
−12 퐿푊 = 10 Log( 푃 푃푟푒푓) (dB) where 푃푟푒푓 = 10 W
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 4, 2013) − page 56 Directivity • In general, noise is a directional phenomena, i.e., it radiates more intensely in certain direction(s) • Examples:
Angle 휃 Typical directivity plot
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 4, 2013) − page 57 Wave Equation • General form of governing equations:
휕푣 휕𝜌 + 훻 ∙ 𝜌푣 = 0 𝜌 + 푣 ∙ 훻푣 = −훻푝 + 훻 ∙ 휏 푖푗 + 푓 continuity 휕푡 momentum 휕푡
• Assumptions:
» Neglect body (푓) and viscous forces (휏 푖푗) » Small perturbations : 𝜌 = 𝜌0 + 𝜌′, 푝 = 푝0 + 푝′, etc. » Stagnant fluid (푣 0 = 0) with uniform properties (𝜌0 = const) at observer
• Combine continuity & momentum equations :
휕2𝜌′ 휕2𝜌′ 휕2푝′ 휕2푝′ 2 2 2 − 푐0 2 = 0 or 2 − 푐0 2 = 0 휕푡 휕푥 휕푡 휕푥
» This is the homogenous wave equation
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 4, 2013) − page 58 More on Wave Equation
2 2 2 2 휕 𝜌′ 2 휕 𝜌′ 휕 푝′ 2 휕 푝′ 2 − 푐0 2 = 0 or 2 − 푐0 2 = 0 휕푡 휕푥 휕푡 휕푥 • It is both linear and homogeneous • Solutions can be sought using Green’s function • Only governs sound propagation without any references to sound sources
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 4, 2013) − page 59 Lighthill’s Equation • General form of governing equations:
휕푣 휕𝜌 + 훻 ∙ 𝜌푣 = 0 𝜌 + 푣 ∙ 훻푣 = −훻푝 + 훻 ∙ 휏 푖푗 + 푓 continuity 휕푡 momentum 휕푡
• Assumptions:
» DO NOT neglect body (푓) and viscous forces (휏 푖푗) » Small perturbations : 𝜌 = 𝜌0 + 𝜌′, 푝 = 푝0 + 푝′, etc. » Stagnant fluid (푣 0 = 0) with uniform properties (𝜌0 = const) at observer
• Combine continuity & momentum equations:
2 2 ′ 2 ′ 2 ′ 휕 𝜌′ 휕 𝜌 휕 푇푖푗 휕푓푖 푇푖푗 = 𝜌푢푖푢푗 − 휏푖푗 + 푝 − 푐0 𝜌 훿푖푗 − 푐2 = − 휕푡2 0 휕푥2 휕푥 푥 휕푥 푖 푖 푗 푖 Lighthill stress tensor
» This is called Lighthill’s equation (1952)
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 4, 2013) − page 60 Lighthill’s Acoustic Analogy
2 ′ 2 ′ 2 2 ′ 푇푖푗 = 𝜌푢푖푢푗 − 휏푖푗 + 푝 − 푐 𝜌 훿푖푗 휕 𝜌′ 2 휕 𝜌 휕 푇푖푗 휕푓푖 0 − 푐 = − 휕푡2 0 휕푥2 휕푥 푥 휕푥 푖 푖 푗 푖 Lighthill stress tensor
• Lighthill’s equation is exact (based on Navier-Stokes eqs)
• The word “analogy” refers to the fact that we can determine the sound field of a complex noise generating phenomena by treating it as source terms of the wave eq.
• It is a non-homogeneous equation where the right-hand side represents aeroacoustic sources
• Solution can be sought using Green’s function, if the source terms can be suitably modeled
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 4, 2013) − page 61 Sources of Aerodynamic Sound
2 ′ 2 ′ 2 2 ′ 푇푖푗 = 𝜌푢푖푢푗 − 휏푖푗 + 푝 − 푐 𝜌 훿푖푗 휕 𝜌′ 2 휕 𝜌 휕 푇푖푗 휕푓푖 0 2 − 푐0 2 = − 휕푡 휕푥푖 휕푥푖푥푗 휕푥푖 Lighthill stress tensor
The right-hand side represents the sources: – Monopole: ′ ′ 2 ′ » Any changes in the entropy (where s = 푝 − 푐0 𝜌 will be non- zero) or deviation from uniform speed of sound (푐0) – Dipole:
» Acoustic field due to external forces exerted on the flow (휕푓푖 휕푥푖) – Quadrupole: » Induced by non-linear convective forces represented by the Reynolds stress tensor (𝜌푢푖푢푗), such as turbulence » Due to viscous forces (휏푖푗)
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 4, 2013) − page 62 Modeling Acoustic Sources • Monopole – Thickness noise
• Dipole – Loading noise
Courtesy ISVR • Quadrupole – BL/viscous effects
http://www.acs.psu.edu/drussell/demos/rad2/mdq.html
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 4, 2013) − page 63 FW-H Equation • Ffowcs Williams–Hawkings equation is a generalization of the Lighthill analogy to add sound field associated with sources in arbitrary motion • Like Lighthill’s equation, FW-H equation is derived using full Navier-Stokes equations w/o simplifying assumptions • FW-H vs. Lighthill:
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 4, 2013) − page 64 FW-H Equation
• The first term on the RHS (volume integral) is the Lighthill tensor.
• Surface integrals are associated with moving source assumption. So, in absence of moving sources, FW-H reduces to Lighthill eqn
• If the source term represented by 푇푖푗 is moved inside the control surface 휕푉퐻, volume integral will vanish b/c of Heaviside function. This has very important practical implications
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 4, 2013) − page 65 FW-H Equation • Practical applications:
FW-H surface
et al. et
Souliez Courtesy: Courtesy:
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 4, 2013) − page 66 Kirchhoff Integral • Based on solution of the homogeneous wave equation using the free-space Green’s function
• Is equivalent to the FW-H integral, if integration surface is placed in the linear region of the flow
• FW-H is superior because is valid in both linear and nonlinear flow regions
• It can yield wrong answers if homogenous wave equation not satisfied on the control surface
• Linear assumption usually valid in a region far enough from the sources
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 4, 2013) − page 67 Noise Prediction Using CAA • Computational Aero-Acoustics (CAA) refers to the numerical simulation of sound propagation/radiation, with noise sources either modeled or resolved as part of the simulation • Although CAA relies on existing CFD methodology, it requires special treatment in certain aspects of simulation
Courtesy: stanford.edu
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 4, 2013) − page 68 Computational Aero-Acoustics • Two main issues arise in acoustic simulation: 1. Acoustic perturbations (푝′, 𝜌′, 푢′) are usually several orders of magnitude (~10−6) smaller than background flow variables (푝, 𝜌, 푢) Therefore, discretization method used should be able to resolve such disparity, while maintaining amplitude & phase characteristics of the waves This requires the use of high-order methods, which are computationally expensive (compact FD schemes, DRP schemes, spectral methods) 2. Boundary conditions should be non-reflective to avoid contamination of
interior solution
Courtesy: tacc.utexas.edu &Khorrami) (Lockard NASA Courtesy:
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 4, 2013) − page 69 Noise Prediction – Hybrid Methods • Typically, it is desired to calculate the noise at the far-field • Using CAA, it is generally impractical or impossible to extend the computational domain to the far-field • A hybrid approach is therefore the best (often only) choice: – Use CAA in the near-field – Use FW-H equation for far-field noise propagation. Note that accuracy of far-filed predictions will heavily depend on accuracy of predicted sources on the FW-H surface FW-H surface
Courtesy: NASA (Lockard &Khorrami) (Lockard NASA Courtesy: Far-field observer
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 4, 2013) − page 70 Turbofan Engine Noise • Different sources: – Fan/OGV interaction (tone & broadband) – Core noise (broadband) – Jet noise (broadband)
Fan/OGV broadband Fan/OGV Tone
Core & Jet
Courtesy: Sjoerd W. Rienstra
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 4, 2013) − page 71 Fan Noise Mechanism Caused by interaction of rotor with downstream stator/OGV. It consists of: – Tone noise associated with periodic aerodynamic interactions » Tonal noise is radiated at multiples of blade-passage frequency (BPF) – Broadband noise associated with turbulence
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 4, 2013) − page 72 Fan Noise Mechanism • It creates a pressure field, locked to the rotor, which is made of 푚-lobe patterns each rotating at the speed 푛퐵Ω/푚: – 푛: harmonics of the blade-passing frequency – 퐵 and 푉 are the number of rotor and stator blades, respectively – Ω is the rotor’s angular speed – 푚 = 푛퐵 ± 푘푉, where 푘 is a positive integer
• According to Tyler-Sofrin theory, the pressure field at the fan face for a circular duct is then given by:
푖(푚휃+푘푥푥−휔푡) 푝′푚푠 푥, 푟, 휃, 푡 = 퐴푚푠 퐽푚 푘푚푠푟 푒 푠 – 퐽푚 : Bessel function of the first kind and order 푚 ′ – 푘푚푠 : eigenvalues defined by 퐽 푚 푘푚푠푅 = 0; – 푠 : radial mode number – 푘푥 : axial wave number – 휔 = Ω푅/푐0 : non-dimensional frequency with 푅 being the duct radius
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 4, 2013) − page 73 Fan Noise Mechanism • 푚 is also called engine order rotor-locked pressure
fluctuations
2876
-
: Schuster, AIAA 2005 AIAA Schuster, : Courtesy
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 4, 2013) − page 74 Fan Noise Mechanism • Acoustic field is made up of • For rotors with spinner, radial 푚-lobe patterns variation is given by:
퐴푚푠 퐽푚 푘푚푠푟 + 푌푚 푘푚푠푟
푌푚 : Bessel function of second kind
Courtesy: Sjoerd W. Rienstra
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 4, 2013) − page 75 Fan Noise Propagation • The acoustic waves given by
푖(푚휃+푘푥푥−휔푡) 푝′푚푠 푥, 푟, 휃, 푡 = 퐴푚푠 퐽푚 푘푚푠푟 푒
푠 will propagate down the duct only if 푘푥 is real- valued, which happens if 휔/푘푚푠 > 1 (assuming no mean flow). Otherwise, 푘푥 will be complex and the corresponding mode will be damped and not propagate (cut-off mode)
• The pressure field given by the above equation could alternately be obtained by direct simulation of rotor/stator flow interaction
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 4, 2013) − page 76 Fan Noise Propagation • Typically, a hybrid approach is used to determine far-field noise of the fan. CAA is used (LEE, potential flow) to simulate flow propagation inside the duct and in a small region surrounding duct exit, where FW-H is located.
• More complex CAA analysis should include the effect of duct boundary layer and sound refraction
• The effect of liners on duct wall can be simulated by defining impedance boundary conditions
Fluid Mechanics & Aeroacoustics of Fans and Compressors Farzad Taghaddosi (July 4, 2013) − page 77