Duct Acoustics: Fundamentals and Applications to Mufflers and Silencers

Cover
Half-title
Title page
Copyright information
Dedication
Contents
Preface
1 Some Preliminaries
1.1 Introduction to the Linear Theory of Sound Wave Motion
1.1.1 Linearization Hypothesis
1.1.2 Partitioning Turbulent Fluctuations
1.1.3 Linearization of Inviscid Fluid Flow
1.1.4 Evolution of Non-Linear Waves
1.2 Representation of Acoustic Waves in the Frequency Domain
1.2.1 Fourier Transform
1.2.2 Periodic Functions
1.2.3 Impulse Sampling
1.2.4 Power Spectral Density
1.3 Representation of Waves in the Wavenumber Domain
1.3.1 Spatial Fourier Transform
1.3.2 Briggs' Criterion
1.4 Intensity and Power of Sound Waves
1.5 Introduction to the Linear System View of Duct Acoustics
References
2 Introduction to Acoustic Block Diagrams
2.1 Introduction
2.2 Classification of Acoustic Models of Ducts
2.2.1 Classification by Number of Ports
2.2.2 Classification by Type of Port
2.2.2.1 One-Dimensional Elements
2.2.2.2 Modal Elements
2.3 Mathematical Models of Acoustic Elements
2.3.1 One-Port Elements
2.3.2 Two-Port Elements
2.3.3 Multi-Port Elements
2.4 Assembly of Blocks
2.4.1 Assembly of Two-Ports
2.4.2 Assembly of Multi-Ports
2.4.3 Optimization of Global Matrix Size
2.4.4 Contraction of Assembled Modal Two-Ports
2.5 Acoustic Elements Based on Numerical Methods
2.5.1 The Finite Element Method
2.5.2 The Boundary Element Method
2.6 Programming Considerations
References
3 Transmission of Low-Frequency Sound Waves in Ducts
3.1 Introduction
3.2 One-Dimensional Theory of Sound Propagation in Ducts
3.2.1 Unsteady Flow Equations
3.2.2 Equations Governing Acoustic Wave Motion
3.3 Solution of Linearized Acoustic Equations
Homogeneous Ducts (ε'=0)
3.3.1.1 Uniform Ducts
3.3.1.2 Homogeneous Non-Uniform Ducts
Inhomogeneous Ducts (ε'≠0)
3.3.3 Numerical Matrizant Method
3.4 Time-Averaged Power of One-Dimensional Acoustic Waves
3.5 Hard-Walled Uniform Ducts
3.5.1 Wave Transfer Matrix
3.5.2 Traveling Waves and Direction of Propagation
3.5.3 Reflection Coefficient and Standing Waves
3.5.4 Lumped Acoustic Elements
3.6 Hard-Walled Homogeneous Ducts with Non-Uniform Cross Section
3.6.1 Wave Transfer Matrix
3.6.2 High Frequency Approximation
3.7 Ducts Packed with Porous Material
3.8 Acoustic Boundary Conditions on Duct Walls
3.8.1 Impermeable Walls
3.8.1.1 No-Slip Model
3.8.1.2 Full-Slip Model
3.8.1.3 Partial-Slip Model
3.8.1.4 Rough-Wall Model
3.8.1.5 Unified Boundary Condition
3.8.2 Permeable Walls
3.9 Homogeneous Ducts with Impermeable Finite Impedance Walls
3.9.1 Non-Uniform Duct
3.9.2 Uniform Duct
3.9.2.1 Direction of Propagation
3.9.2.2 Wave Equation
3.9.2.3 Impedance Eduction Formula
3.9.2.4 Peripherally Non-Uniform Wall Impedance
3.9.3 Finite Wall Impedance Models
3.9.3.1 Lined Impermeable Walls
3.9.3.2 Permeable Rigid Porous Walls
3.9.3.3 Perforated Rigid Walls
3.9.3.4 Turbulent Boundary Layer Over Rigid Walls
3.9.3.5 Elastic Walls
3.10 Inhomogeneous Ducts
3.10.1 Linearized Energy Equation
3.10.2 Matrizant of a Duct with Finite Impedance Walls
3.10.3 Hard-Walled Ducts with Mean Temperature Gradient
3.11 Ducts with Two-Phase Flow
3.12 Ducts with Time-Variant Mean Temperature
References
4 Transmission of One-Dimensional Waves in Coupled Ducts
4.1 Introduction
4.2 Quasi-Static Theory of Wave Transmission at Compact Junctions
4.2.1 Quasi-Static Conservation Laws
4.2.2 Transformation to Pressure Wave Components
4.3 Two Ducts Coupled by Forming Sudden Area Change
4.3.1 Open Area Change
4.3.1.1 The Case of Zero Mean Flow
4.3.1.2 Core-Flow Model
4.3.1.3 Effect of Inner Duct Wall Thickness
4.3.2 Closed Area Changes
4.3.3 End-Correction
4.4 Sudden Area Changes Formed by Multiple Ducts
4.4.1 Identical Inner Ducts
4.4.2 Staggered Inner Duct Extensions
4.4.3 Duct Splits
4.5 Wave Transmission Through a Perforated Rigid Baffle
4.5.1 Area-Change Model
4.5.2 Lumped Impedance Model
4.6 Wave Transmission in Junction Cavities
4.6.1 Multi-Duct Junction
4.6.2 Two-Duct Junction
4.7 Continuously Coupled Perforated Ducts
4.7.1 Single-Coupled Perforated Ducts
4.7.1.1 Identical Perforated Ducts
4.7.2 Double-Coupled Perforated Ducts
4.8 Row-Wise Coupled Perforated Ducts
4.8.1 Wave Transfer Across a Row of Apertures
4.8.2 Single-Coupled n-Duct Section
4.8.3 Double-Coupled n-Duct Section
4.8.4 Wave Transfer Matrix of n-Duct Element
4.9 Dissipative Units and Lined Ducts
4.10 Wave Transfer Across Adiabatic Pressure-Loss Devices
References
5 Resonators, Expansion Chambers and Silencers
5.1 Introduction
5.1.1 Mufflers and Silencers
5.1.2 The System and Its Environment
5.2 Transmission Loss
5.2.1 Single-Frequency Analysis
5.2.2 Overall Transmission Loss
5.3 Duct Resonances
5.3.1 Resonance and Anti-Resonance Frequencies
5.3.2 Resonators
5.3.3 Single-Duct Resonator
5.3.4 Resonators with Open Outlet
5.3.5 Helmholtz Resonator
5.3.6 Transmission Loss of Resonators
5.3.7 Interferential Resonator
5.3.7.1 Wave Transfer Matrix of Parallel Two-Ports
5.3.7.2 Wave Transfer Matrix of the HQ Tube
5.3.7.3 Herschel-Quincke Tube Resonator
5.3.8 Straight-Through Resonator
5.4 Expansion Chambers
5.4.1 Through-Flow Expansion Chambers
5.4.2 Transmission Loss of Expansion Chambers
5.4.3 Pure Expansion Chambers
5.4.4 The Strongest Pure Expansion Chamber
5.4.5 Tuning Inlet and Outlet Duct Extensions
5.4.6 Effect of Inlet and Outlet Duct Configurations
5.4.7 Division of a Pure Expansion Chamber
5.4.8 Low-Frequency Response of Chambers
5.4.9 Chambers with a Perforated Duct Bridge
5.4.10 Packed Chambers
5.5 Reciprocal Two-Ports
5.6 Some Practical Issues
5.6.1 Irregular Geometry
5.6.2 Variable Mean Flow Conditions
5.6.3 Multiple Outlet Ducts
5.6.4 Flow Excited Resonators and Chambers
5.7 Flow Rate and Back-Pressure Calculation
5.7.1 Calculation of Mean Temperature Drop
5.8 Shell Noise
References
6 Multi-Modal Sound Propagation in Ducts
6.1 Introduction
6.2 Uniform Ducts with Axial Mean Flow
6.3 Boundary Condition on Impermeable Walls
6.3.1 No-Slip Model
6.3.2 Partial-Slip Model
6.3.3 Ingard-Myers Model
6.3.4 Modified Ingard-Myers Models
6.4 Wave Transmission in a Uniform Duct with Uniform Mean Flow
6.4.1 General Solution of the Convected Wave Equation
6.4.2 Modal Wave Transfer Matrix
6.5 Hard-Walled Ducts with Uniform Mean Flow
6.5.1 Eigenvalues and the Orthogonality of Eigenfunctions
6.5.2 Propagating and Evanescent Modes
6.5.3 Modal Propagation Angles
6.5.4 Transverse Modes of Common Duct Sections
6.5.4.1 Rectangular Ducts
6.5.4.2 Hollow Circular Ducts
6.5.4.3 Annular Circular Ducts
6.5.4.4 Spinning Modes
6.5.5 Numerical Determination of Transverse Duct Modes
6.5.6 Time-Averaged Acoustic Power
6.6 Hard-Walled Uniform Ducts Packed with Porous Material
6.7 Lined Uniform Ducts with Uniform Mean Flow
6.7.1 Dispersion Equation for Uniformly Lined Circular Ducts
6.7.1.1 Hard-Liner Solution for Hollow Ducts
6.7.1.2 Iterative Graphical Solution
6.7.2 Dispersion Equations for Uniformly Lined Rectangular Ducts
6.7.3 Discussion of Transverse Modes
6.7.3.1 Surface Modes
6.7.3.2 Orthogonality of Modes
6.7.4 Liner Optimization
6.7.5 Multi-Modal Attenuation Characteristics
6.7.6 Non-Uniformly Lined Ducts
6.8 Uniform Ducts with Sheared Mean Flow
6.8.1 Solution of the Pridmore-Brown Equation
6.8.2 Effect of the Mean Boundary Layer Thickness
6.9 Ducts with Axially Non-Uniform Cross-Sectional Area
6.10 Circularly Curved Ducts
6.10.1 Rectangular Ducts
6.10.2 Numerical Determination of Angular Wavenumbers
6.10.3 Fundamental Mode Approximation
6.11 Uniform Ducts with Mean Swirl
6.12 Ducts with Mean Temperature Gradient
6.12.1 Ducts without Mean Flow
6.12.2 Effect of Mean Flow
References
7 Transmission of Wave Modes in Coupled Ducts
7.1 Introduction
7.2 Weak Form of the Convected Wave Equation
7.3 Ducts with Identical Sections
7.4 Sudden Area Changes
7.4.1 Open Sudden Expansion
7.4.1.1 Closed Through-Flow Expansion
7.4.1.2 Closed Flow-Reversing Expansion
7.4.2 Sudden Area Contraction
7.4.3 Open Area Change with Multiple Ducts
7.5 Perforated Baffles
7.6 Cavity Coupled with Multiple Ducts
7.6.1 Closed Cavity Modes
7.6.2 The Green Function of the Cavity
7.6.3 Coupling the Cavity with Ducts
7.7 Coupled Perforated Ducts
7.7.1 Acoustic Field in a Duct with a Single Aperture
7.7.2 Wave Transfer Across a Row of Apertures
7.7.3 Dissipative Silencers
7.7.4 Lined Ducts
7.8 Contracted Models of Silencers
7.8.1 Expansion Chamber with Offset Inlet and Outlet Ducts
7.8.2 Expansion Chamber with Double Outlet
7.8.3 Flow-Reversing Chamber
7.8.4 Through-Flow Resonator and Muffler
7.8.5 Three-Pass Muffler
References
8 Effects of Viscosity and Thermal Conductivity
8.1 Introduction
8.2 Convected Wave Equation for a Viscothermal Fluid
8.3 Low Reduced Frequency Theory
8.3.1 Circular Hollow Ducts
8.3.1.1 Hard-Walled Ducts
8.3.1.2 Wide-Duct Approximation
8.3.1.3 Effect of Parabolic Mean Flow Velocity Profile
8.3.1.4 Effect of Turbulent Boundary Layer
8.3.2 Circular Annular Ducts
8.3.3 Rectangular Ducts
8.4 Time-Averaged Acoustic Power
8.5 Sudden Area Changes and Junctions
8.6 Coupled Narrow Ducts with Porous Walls
References
9 Reflection and Radiation at Open Duct Terminations
9.1 Introduction
9.2 Reflection Matrix and End-Correction
9.3 Flanged and Unflanged Open Terminations without Mean Flow
9.3.1 Exterior Surface Helmholtz Equation
9.3.2 Flanged Open End
9.3.2.1 Circular Ducts
9.3.3 Unflanged Open End
9.4 Reflection Matrix at an Unflanged Open End with Mean Flow
9.4.1 The Exhaust Problem
9.4.2 Circular Duct
9.4.2.1 Plane-Wave Reflection Coefficient
9.4.2.2 Reflection of Higher-Order Incident Modes
9.4.3 Reflection at Flow Intakes
9.4.3.1 Plane-Wave Reflection Coefficient
9.5 Acoustic Radiation from Open Ends of Ducts
9.5.1 Modal Radiation Transfer Function
9.5.2 Radiated Acoustic Power
9.5.3 Flanged Open End without Mean Flow
9.5.3.1 Circular Ducts
9.5.3.2 Rectangular Ducts
9.5.4 Unflanged Circular Open End without Mean Flow
9.5.5 Radiation from Unflanged Circular Open End with Mean Flow
9.5.6 Simple-Source Approximation
9.5.6.1 Effect of Vorticity
9.5.7 Power Source Model
9.5.8 Effect of Reflecting Surfaces
References
10 Modeling of Ducted Acoustic Sources
10.1 Introduction
10.2 One-Port Sources Characterized by Unsteady Mass Injection
10.3 Moving the Active Plane of One-Port Sources
10.4 Two-Port Sources Characterized by Fluctuating Force Application
10.4.1 Flow Noise
10.5 Two-Port Sources Characterized by Ducted Combustion
10.5.1 Combustion Oscillations and Instability
10.6 Moving Source Planes of Two-Port Sources
10.7 Ducted Loudspeakers
References
11 Radiated Sound Pressure Prediction
11.1 Introduction
11.2 Calculation of Sound Pressure Field of Ducted Sources
11.2.1 Ducted One-Port Sources
11.2.1.1 One-Dimensional Sources
11.2.2 Ducted Two-Port Sources
11.2.3 Multiple Radiating Outlets
11.3 Analysis of Sound Pressure
11.4 Insertion Loss
11.4.1 Noise Reduction
11.4.2 Attenuation
11.5 Multi-Modal Transmission Loss Calculations
11.6 In-Duct Sources Characterized by Acoustic Power
11.6.1 The ASHRAE Method
References
12 Measurement Methods
12.1 Introduction
12.2 Measurement of In-Duct Acoustic Field
12.2.1 Multi-Modal Wave Field Decomposition
12.2.2 The Two-Microphone Method
12.2.2.1 Calibration of Microphones
12.2.2.2 Signal Enhancement
12.2.3 Measurement of the Plane-Wave Reflection Coefficient
12.2.4 Measurement of Wavenumbers
12.3 Measurement of Passive Acoustic Two-Ports
12.3.1 Basics of the Four Microphone Method
12.3.2 Measurement of Attenuation
12.3.3 Measurement of Transmission Loss
12.3.4 Measurement of the Wave Transfer Matrix
12.4 Measurement of One-Port Source Characteristics
12.4.1 The Two-Load Method
12.4.1.1 Implementation with Non-Calibrated Loads
12.4.1.2 Implementation with Calibrated Loads
12.4.2 Geometrical Interpretation of the Two-Load Method
12.4.3 The Apollonian Circle of Two Loads
12.4.3.1 Upper and Lower Bounds for Source Pressure Strength
12.4.4 Calculation Bounds to Sound Pressure
12.4.5 The Three-Load Method
12.4.6 Over-Determined Methods
12.4.6.1 Over-Determined Two-Load Method
12.4.6.2 Over-Determined Three-Load Method
12.4.7 The Fuzzy Two-Load Method
12.4.8 The Explicit N-Load Method
12.5 Measurement of Two-Port Source Characteristics
References
13 System Search and Optimization
13.1 Introduction
13.2 Direct Random Search
13.3 Interval Analysis
13.4 The Inverse Method
13.4.1 Acoustic Path Space
13.4.2 Acoustic Path Space on the Attenuation Plane
13.4.3 Signature of Acoustic Paths
13.4.4 System Search in Acoustic Path Space
13.4.5 Acoustic Path Spaces for Different Targets
13.4.5.1 Noise Reduction
13.4.5.2 Insertion Loss
References
Appendix A Basic Equations of Fluid Motion
A.1 Integral Forms of Conservation Laws
A.1.1 Conservation of Mass
A.1.2 Conservation of Momentum
A.1.3 Conservation of Energy
A.2 State Equations and the Speed of Sound
A.3 Equations of Motion of Ideal Fluids
A.3.1 Continuity Equation
A.3.2 Momentum Equation
A.3.3 Energy Equation
A.4 Equation of Motion of Newtonian Fluids
A.4.1 Momentum Equation
A.4.2 Energy Equation
References
Appendix B Acoustic Properties of Rigid-Frame Fibrous Materials
References
Appendix C Impedance of Compact Apertures
C.1 Empirical and Semi-Empirical Models
C.2 Theoretical Models
References
Index