Introduction to Fluid Mechanics, 7th Edition

Introduction to Fluid Mechanics, 7th Edition
Robert W. Fox, Philip J. Pritchard, Alan T. McDonald

Introduction to Fluid Mechanics
Chapter 1 Introduction

Main Topics Definition of a Fluid Basic Equations Methods of Analysis
Dimensions and Units

Definition of a Fluid When a shear stress is applied:
Fluids continuously deform Solids deform or bend

Basic Equations We need forms of the following Conservation of mass
Newton’s second law of motion The principle of angular momentum The first law of thermodynamics The second law of thermodynamics

Methods of Analysis System (or “Closed System”)
Control Volume (or “Open System”)

Dimensions and Units Systems of Dimensions [M], [L], [t], and [T]
[F], [L], [t], and [T] [F],[M], [L], [t], and [T]

Dimensions and Units Systems of Units MLtT FLtT FMLtT SI (kg, m, s, K)
British Gravitational (lbf, ft, s, oR) FMLtT English Engineering (lbf, lbm, ft, s, oR)

Dimensions and Units Systems of Units

Dimensions and Units Preferred Systems of Units SI (kg, m, s, K)
British Gravitational (lbf, ft, s, oR)

Chapter 2 Fundamental Concepts

Main Topics Fluid as a Continuum Velocity Field Stress Field Viscosity
Surface Tension Description and Classification of Fluid Motions

Fluid as a Continuum

Velocity Field

Velocity Field Consider also Steady and Unsteady Flows
1D, 2D, and 3D Flows Timelines, Pathlines, and Streaklines

Stress Field

Viscosity Newtonian Fluids
Most of the common fluids (water, air, oil, etc.) “Linear” fluids

Viscosity Non-Newtonian Fluids
Special fluids (e.g., most biological fluids, toothpaste, some paints, etc.) “Non-linear” fluids

Viscosity Non-Newtonian Fluids

Surface Tension

Description and Classification of Fluid Motions

Chapter 3 Fluid Statics

Main Topics The Basic Equations of Fluid Statics
Pressure Variation in a Static Fluid Hydrostatic Force on Submerged Surfaces Buoyancy

The Basic Equations of Fluid Statics
Body Force

Surface Force

Total Force

Newton’s Second Law

Pressure-Height Relation

Pressure Variation in a Static Fluid
Incompressible Fluid: Manometers

Pressure Variation in a Static Fluid
Compressible Fluid: Ideal Gas Need additional information, e.g., T(z) for atmosphere

Hydrostatic Force on Submerged Surfaces
Plane Submerged Surface

Plane Submerged Surface We can find FR, and y´ and x´, by integrating, or …

Plane Submerged Surface Algebraic Equations – Total Pressure Force

Plane Submerged Surface Algebraic Equations – Net Pressure Force

Curved Submerged Surface

Curved Submerged Surface Horizontal Force = Equivalent Vertical Plane Force Vertical Force = Weight of Fluid Directly Above (+ Free Surface Pressure Force)

Buoyancy

For example, for a hot air balloon (Example 3.8):
Buoyancy For example, for a hot air balloon (Example 3.8):

Chapter 4 Basic Equations in Integral Form for a Control Volume

Main Topics Basic Laws for a System
Relation of System Derivatives to the Control Volume Formulation Conservation of Mass Momentum Equation for Inertial Control Volume Momentum Equation for Inertial Control Volume with Rectilinear Acceleration The Angular Momentum Principle The First Law of Thermodynamics The Second Law of Thermodynamics

Basic Laws for a System Conservation of Mass

Basic Laws for a System Momentum Equation for Inertial Control Volume

Basic Laws for a System The Angular Momentum Principle

Basic Laws for a System The First Law of Thermodynamics

Basic Laws for a System The Second Law of Thermodynamics

Relation of System Derivatives to the Control Volume Formulation
Extensive and Intensive Properties

Reynolds Transport Theorem

Interpreting the Scalar Product

Conservation of Mass Basic Law, and Transport Theorem

Conservation of Mass

Conservation of Mass Incompressible Fluids Steady, Compressible Flow

Momentum Equation for Inertial Control Volume
Basic Law, and Transport Theorem

Special Case: Bernoulli Equation Steady Flow No Friction Flow Along a Streamline Incompressible Flow

Special Case: Control Volume Moving with Constant Velocity

Momentum Equation for Inertial Control Volume with Rectilinear Acceleration

The Angular Momentum Principle

The First Law of Thermodynamics

The First Law of Thermodynamics
Work Involves Shaft Work Work by Shear Stresses at the Control Surface Other Work

The Second Law of Thermodynamics

Chapter 5 Introduction to Differential Analysis of Fluid Motion

Main Topics Conservation of Mass
Stream Function for Two-Dimensional Incompressible Flow Motion of a Fluid Particle (Kinematics) Momentum Equation

Conservation of Mass Basic Law for a System

Conservation of Mass Rectangular Coordinate System

Conservation of Mass Rectangular Coordinate System
“Continuity Equation”

Conservation of Mass Rectangular Coordinate System “Del” Operator

Conservation of Mass Rectangular Coordinate System

Conservation of Mass Rectangular Coordinate System
Incompressible Fluid: Steady Flow:

Conservation of Mass Cylindrical Coordinate System

Conservation of Mass Cylindrical Coordinate System “Del” Operator

Conservation of Mass Cylindrical Coordinate System

Conservation of Mass Cylindrical Coordinate System
Incompressible Fluid: Steady Flow:

Stream Function for Two-Dimensional Incompressible Flow
Two-Dimensional Flow Stream Function y

Stream Function for Two-Dimensional Incompressible Flow
Cylindrical Coordinates Stream Function y(r,q)

Motion of a Fluid Particle (Kinematics)
Fluid Translation: Acceleration of a Fluid Particle in a Velocity Field Fluid Rotation Fluid Deformation Angular Deformation Linear Deformation

Fluid Translation: Acceleration of a Fluid Particle in a Velocity Field

Fluid Translation: Acceleration of a Fluid Particle in a Velocity Field (Cylindrical)

Fluid Rotation

Fluid Deformation: Angular Deformation

Fluid Deformation: Linear Deformation

Momentum Equation Newton’s Second Law

Momentum Equation Forces Acting on a Fluid Particle

Momentum Equation Differential Momentum Equation

Momentum Equation Newtonian Fluid: Navier-Stokes Equations

Momentum Equation Special Case: Euler’s Equation

Computational Fluid Dynamics
Some Applications

Computational Fluid Dynamics
Discretization

Chapter 6 Incompressible Inviscid Flow

Main Topics Momentum Equation for Frictionless Flow: Euler’s Equation
Euler’s Equation in Streamline Coordinates Bernoulli Equation – Integration of Euler’s Equation Along a Streamline for Steady Flow The Bernoulli Equation Interpreted as an Energy Equation Energy Grade Line and Hydraulic Grade Line

Momentum Equation for Frictionless Flow: Euler’s Equation
Continuity

Rectangular Coordinates

Cylindrical Coordinates

Euler’s Equation in Streamline Coordinates
Along a Streamline (Steady Flow, ignoring body forces) Normal to the Streamline (Steady Flow, ignoring body forces)

Bernoulli Equation – Integration of Euler’s Equation Along a Streamline for Steady Flow
Euler’s Equation in Streamline Coordinates (assuming Steady Flow)

Integration Along s Coordinate

No Friction Flow Along a Streamline Incompressible Flow

Static, Stagnation, and Dynamic Pressures (Ignore Gravity) Stagnation Static Dynamic

The Bernoulli Equation Interpreted as an Energy Equation

Basic Equation No Shaft Work No Shear Force Work No Other Work Steady Flow Uniform Flow and Properties

Hence Assumption 6: Incompressible Assumption 7:

No Shaft Work No Shear Force Work No Other Work Steady Flow Uniform Flow and Properties Incompressible Flow u2 – u1 – dQ/dm = 0

Energy Grade Line and Hydraulic Grade Line
Energy Equation

Energy Grade Line (EGL) Hydraulic Grade Line (HGL)

Irrotational Flow Irrotationality Condition

Irrotational Flow Velocity Potential

Irrotational Flow Velocity Potential automatically satisfies Irrotationality Condition

Irrotational Flow 2D Incompressible, Irrotational Flow

Irrotational Flow Elementary Plane Flows

Irrotational Flow Superposition

Chapter 7 Dimensional Analysis and Similitude

Main Topics Nondimensionalizing the Basic Differential Equations
Nature of Dimensional Analysis Buckingham Pi Theorem Significant Dimensionless Groups in Fluid Mechanics Flow Similarity and Model Studies

Nondimensionalizing the Basic Differential Equations
Example: Steady Incompressible Two-dimensional Newtonian Fluid

Nondimensionalizing the Basic Differential Equations

Nature of Dimensional Analysis
Example: Drag on a Sphere Drag depends on FOUR parameters: sphere size (D); speed (V); fluid density (r); fluid viscosity (m) Difficult to know how to set up experiments to determine dependencies Difficult to know how to present results (four graphs?)

Example: Drag on a Sphere Only one dependent and one independent variable Easy to set up experiments to determine dependency Easy to present results (one graph)

Buckingham Pi Theorem Step 1:
List all the dimensional parameters involved Let n be the number of parameters Example: For drag on a sphere, F, V, D, r, m, and n = 5

Buckingham Pi Theorem Step 2
Select a set of fundamental (primary) dimensions For example MLt, or FLt Example: For drag on a sphere choose MLt

List the dimensions of all parameters in terms of primary dimensions Let r be the number of primary dimensions Example: For drag on a sphere r = 3

Select a set of r dimensional parameters that includes all the primary dimensions Example: For drag on a sphere (m = r = 3) select r, V, D

Set up dimensional equations, combining the parameters selected in Step 4 with each of the other parameters in turn, to form dimensionless groups There will be n – m equations Example: For drag on a sphere

Buckingham Pi Theorem Step 5 (Continued) Example: For drag on a sphere

Check to see that each group obtained is dimensionless Example: For drag on a sphere

Significant Dimensionless Groups in Fluid Mechanics
Reynolds Number Mach Number

Froude Number Weber Number

Euler Number Cavitation Number

Flow Similarity and Model Studies
Geometric Similarity Model and prototype have same shape Linear dimensions on model and prototype correspond within constant scale factor Kinematic Similarity Velocities at corresponding points on model and prototype differ only by a constant scale factor Dynamic Similarity Forces on model and prototype differ only by a constant scale factor

Example: Drag on a Sphere

Example: Drag on a Sphere For dynamic similarity … … then …

Incomplete Similarity Sometimes (e.g., in aerodynamics) complete similarity cannot be obtained, but phenomena may still be successfully modelled

Scaling with Multiple Dependent Parameters Example: Centrifugal Pump Pump Head Pump Power

Scaling with Multiple Dependent Parameters Example: Centrifugal Pump Head Coefficient Power Coefficient

Scaling with Multiple Dependent Parameters Example: Centrifugal Pump (Negligible Viscous Effects) If … … then …

Scaling with Multiple Dependent Parameters Example: Centrifugal Pump Specific Speed

Chapter 8 Internal Incompressible Viscous Flow

Main Topics Entrance Region
Fully Developed Laminar Flow Between Infinite Parallel Plates Fully Developed Laminar Flow in a Pipe Turbulent Velocity Profiles in Fully Developed Pipe Flow Energy Considerations in Pipe Flow Calculation of Head Loss Solution of Pipe Flow Problems Flow Measurement

Entrance Region

Fully Developed Laminar Flow Between Infinite Parallel Plates
Both Plates Stationary

Both Plates Stationary Transformation of Coordinates

Both Plates Stationary Shear Stress Distribution Volume Flow Rate

Both Plates Stationary Flow Rate as a Function of Pressure Drop Average and Maximum Velocities

Upper Plate Moving with Constant Speed, U

Fully Developed Laminar Flow in a Pipe
Velocity Distribution Shear Stress Distribution

Volume Flow Rate Flow Rate as a Function of Pressure Drop

Average Velocity Maximum Velocity

Turbulent Velocity Profiles in Fully Developed Pipe Flow

Energy Considerations in Pipe Flow
Energy Equation

Energy Considerations in Pipe Flow
Head Loss

Calculation of Head Loss
Major Losses: Friction Factor

Laminar Friction Factor Turbulent Friction Factor

Minor Losses Examples: Inlets and Exits; Enlargements and Contractions; Pipe Bends; Valves and Fittings

Minor Loss: Loss Coefficient, K Minor Loss: Equivalent Length, Le

Pumps, Fans, and Blowers

Noncircular Ducts Example: Rectangular Duct

Solution of Pipe Flow Problems
Energy Equation

Major Losses

Minor Losses

Single Path Find Dp for a given L, D, and Q Use energy equation directly Find L for a given Dp, D, and Q

Single Path (Continued) Find Q for a given Dp, L, and D Manually iterate energy equation and friction factor formula to find V (or Q), or Directly solve, simultaneously, energy equation and friction factor formula using (for example) Excel Find D for a given Dp, L, and Q Manually iterate energy equation and friction factor formula to find D, or

Multiple-Path Systems Example:

Multiple-Path Systems Solve each branch as for single path Two additional rules The net flow out of any node (junction) is zero Each node has a unique pressure head (HGL) To complete solution of problem Manually iterate energy equation and friction factor for each branch to satisfy all constraints, or Directly solve, simultaneously, complete set of equations using (for example) Excel

Flow Measurement Direct Methods
Examples: Accumulation in a Container; Positive Displacement Flowmeter Restriction Flow Meters for Internal Flows Examples: Orifice Plate; Flow Nozzle; Venturi; Laminar Flow Element

Flow Measurement Linear Flow Meters
Examples: Float Meter (Rotameter); Turbine; Vortex; Electromagnetic; Magnetic; Ultrasonic

Flow Measurement Traversing Methods
Examples: Pitot (or Pitot Static) Tube; Laser Doppler Anemometer

Chapter 9 External Incompressible Viscous Flow

Main Topics The Boundary-Layer Concept Boundary-Layer Thicknesses
Laminar Flat-Plate Boundary Layer: Exact Solution Momentum Integral Equation Use of the Momentum Equation for Flow with Zero Pressure Gradient Pressure Gradients in Boundary-Layer Flow Drag Lift

The Boundary-Layer Concept

Boundary Layer Thicknesses

Boundary Layer Thicknesses
Disturbance Thickness, d Displacement Thickness, d* Momentum Thickness, q

Laminar Flat-Plate Boundary Layer: Exact Solution
Governing Equations

Boundary Conditions

Equations are Coupled, Nonlinear, Partial Differential Equations Blasius Solution: Transform to single, higher-order, nonlinear, ordinary differential equation

Results of Numerical Analysis

Momentum Integral Equation
Provides Approximate Alternative to Exact (Blasius) Solution

Momentum Integral Equation
Equation is used to estimate the boundary-layer thickness as a function of x: Obtain a first approximation to the freestream velocity distribution, U(x). The pressure in the boundary layer is related to the freestream velocity, U(x), using the Bernoulli equation Assume a reasonable velocity-profile shape inside the boundary layer Derive an expression for tw using the results obtained from item 2

Use of the Momentum Equation for Flow with Zero Pressure Gradient
Simplify Momentum Integral Equation (Item 1) The Momentum Integral Equation becomes

Laminar Flow Example: Assume a Polynomial Velocity Profile (Item 2) The wall shear stress tw is then (Item 3)

Laminar Flow Results (Polynomial Velocity Profile) Compare to Exact (Blasius) results!

Turbulent Flow Example: 1/7-Power Law Profile (Item 2)

Turbulent Flow Results (1/7-Power Law Profile)

Pressure Gradients in Boundary-Layer Flow

Drag Drag Coefficient with or

Drag Pure Friction Drag: Flat Plate Parallel to the Flow
Pure Pressure Drag: Flat Plate Perpendicular to the Flow Friction and Pressure Drag: Flow over a Sphere and Cylinder Streamlining

Drag Flow over a Flat Plate Parallel to the Flow: Friction Drag
Boundary Layer can be 100% laminar, partly laminar and partly turbulent, or essentially 100% turbulent; hence several different drag coefficients are available

Drag Flow over a Flat Plate Parallel to the Flow: Friction Drag (Continued) Laminar BL: Turbulent BL: … plus others for transitional flow

Drag Flow over a Flat Plate Perpendicular to the Flow: Pressure Drag
Drag coefficients are usually obtained empirically

Drag Flow over a Flat Plate Perpendicular to the Flow: Pressure Drag (Continued)

Drag Flow over a Sphere and Cylinder: Friction and Pressure Drag

Drag Flow over a Sphere and Cylinder: Friction and Pressure Drag (Continued)

Streamlining Used to Reduce Wake and hence Pressure Drag

Lift Mostly applies to Airfoils Note: Based on planform area Ap

Lift Examples: NACA 23015; NACA

Lift Induced Drag

Lift Induced Drag (Continued) Reduction in Effective Angle of Attack:
Finite Wing Drag Coefficient:

Lift Induced Drag (Continued)

Chapter 10 Fluid Machinery

Main Topics Introduction and Classification of Fluid Machines
Turbomachinery Analysis Performance Characteristics Applications to Fuid Systems

Introduction and Classification of Fluid Machines
Positive Displacement Turbomachines Radial-Flow (Centrifugal) Axial-Flow Mixed-Flow

Machines for Doing Work on a Fluid Pumps Fans Blowers Compressors

Machines for Doing Work on a Fluid

Machines for Extracting Work (Power) from a Fluid Hydraulic Turbines (Impulse and Reaction) Gas Turbines (Impulse and Reaction)

Machines for Extracting Work (Power) from a Fluid

Turbomachinery Analysis
The Angular Momentum Principle Apply to Control Volume:

Euler Turbomachine Equation Mechanical Power: Theoretical Head:

Velocity Diagrams

Example: Idealized Centrifugal Pump Negligible torque due to surface forces (viscous and pressure). Inlet and exit flow tangent to blades. Uniform flow at inlet and exit. Zero inlet tangential velocity

Example: Idealized Centrifugal Pump (Continued) Head Equation: Shutoff Head:

Example: Idealized Centrifugal Pump (Continued)

Machines for Doing Work on a Fluid Hydraulic Power: Pump Efficiency:

Machines for Extracting Work (Power) from a Fluid Hydraulic Power: Turbine Efficiency:

Performance Characteristics

Machines for Extracting Work (Power) from a Fluid

Dimensional Analysis and Specific Speed Flow Coefficient: Head Coefficient: Power Coefficient:

Dimensional Analysis and Specific Speed Torque Coefficient:

Dimensional Analysis and Specific Speed Specific Speed: Specific Speed (Customary Units): Specific Speed (Customary Units):

Dimensional Analysis and Specific Speed

Similarity Rules For Dynamic Similarity: … and … … so …

Applications to Fluid Systems

Machines for Doing Work on a Fluid Pump Wear

Machines for Doing Work on a Fluid Pumps in Series

Machines for Doing Work on a Fluid Pumps in Parallel

Machines for Doing Work on a Fluid Fans, Blowers, and Compressors

Machines for Doing Work on a Fluid Positive-Displacement Pumps

Machines for Doing Work on a Fluid Propellers Speed of Advance Coefficient: Thrust, Torque, Power Coefficients, and Propeller Efficiency

Machines for Extracting Work (Power) from a Fluid Hydraulic Turbines Wind-Power Machines

Chapter 11 Open-Channel Flow

Main Topics Steady Uniform Flow
Specific Energy, Momentum Equation, and Specific Force Steady, Gradually Varied Flow Rapidly Varied Flow Discharge Measurement

Steady Uniform Flow Control Volume

Steady Uniform Flow Chezy Equation Chezy Coefficient

Steady Uniform Flow Manning Equation (SI)
Manning Equation (US Customary)

Steady Uniform Flow Manning Roughness Coefficients

Specific Energy, Momentum Equation, and Specific Force

Froude Number Criterion Froude Number Criterion (Rectangular Channel)

Momentum Equation (Steady State)

Critical Flow Examples

Steady, Gradually Varied Flow
Differential Equation

Rapidly Varied Flow Example: Hydraulic Jump

Rapidly Varied Flow Hydraulic Jump (Continued)

Discharge Measurement Using Weirs
Example (Sharp Crested Weir)

Suppressed Rectangular Weir Contracted Rectangular Weir Effective Length

Triangular Weir

Broad-Crested Weir

Chapter 12 Introduction to Compressible Flow

Main Topics Review of Thermodynamics Propagation of Sound Waves
Reference State: Local Isentropic Stagnation Conditions Critical Conditions

Review of Thermodynamics
Ideal Gas

Specific Heat Formulas

Internal Energy and Enthalpy

Entropy

The Second Law of Thermodynamics

Isentropic (Reversible Adiabatic) Processes

Propagation of Sound Waves
Speed of Sound Solids and Liquids: Ideal Gas:

Types of Flow – The Mach Cone

Types of Flow – The Mach Cone (Continued) Mach Angle:

Reference State: Local Isentropic Stagnation Conditions

Reference State: Local Isentropic Stagnation Conditions
Computing Equations

Critical Conditions Computing Equations

Chapter 13 Compressible Flow

Main Topics Basic Equations for One-Dimensional Compressible Flow
Isentropic Flow of an Ideal Gas – Area Variation Flow in a Constant Area Duct with Friction Frictionless Flow in a Constant-Area Duct with Heat Exchange Normal Shocks Supersonic Channel Flow with Shocks Oblique Shocks and Expansion Waves

Basic Equations for One-Dimensional Compressible Flow
Control Volume

Continuity Momentum

First Law of Thermodynamics Second Law of Thermodynamics

Equation of State Property Relations

Isentropic Flow of an Ideal Gas – Area Variation
Basic Equations for Isentropic Flow

Subsonic, Supersonic, and Sonic Flows

Reference Stagnation and Critical Conditions for Isentropic Flow

Property Relations

Isentropic Flow in a Converging Nozzle

Isentropic Flow in a Converging-Diverging Nozzle

Flow in a Constant-Area Duct with Friction
Control Volume

Basic Equations for Adiabatic Flow

Adiabatic Flow: The Fanno Line

Fanno-Line Flow Functions for One-Dimensional Flow of an Ideal Gas

Fanno-Line Relations

Fanno-Line Relations (Continued)

Frictionless Flow in a Constant-Area Duct with Heat Exchange
Control Volume

Basic Equations for Flow with Heat Exchange

Heat Exchange: The Rayleigh Line

Rayleigh-Line Relations

Normal Shocks Control Volume

Normal Shocks Basic Equations for a Normal Shock

Normal Shocks Intersection of Fanno & Rayleigh Lines

Normal Shocks Normal Shock Relations

Normal Shocks Normal Shock Relations (Continued)

Supersonic Channel Flow with Shocks
Flow in a Converging-Diverging Nozzle

Oblique Shocks and Expansion Waves
Typical Application

Mach Angle and Oblique Shock Angle

Oblique Shock: Control Volume

Oblique Shock: Useful Formulas

Oblique Shock Relations

Oblique Shock Relations (Continued)

Oblique Shock: Deflection Angle

Expansion and Compression Waves

Expansion Wave: Control Volume

Expansion Wave: Prandtl-Meyer Expansion Function

Expansion Wave: Isentropic Relations

Introduction to Fluid Mechanics, 7th Edition

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