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)

Introduction to Fluid Mechanics 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

Introduction to Fluid Mechanics 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

The Basic Equations of Fluid Statics Surface Force

The Basic Equations of Fluid Statics Surface Force

The Basic Equations of Fluid Statics Surface Force

The Basic Equations of Fluid Statics Total Force

The Basic Equations of Fluid Statics Newton’s Second Law

The Basic Equations of Fluid Statics 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

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

Hydrostatic Force on Submerged Surfaces Plane Submerged Surface Algebraic Equations – Total Pressure Force

Hydrostatic Force on Submerged Surfaces Plane Submerged Surface Algebraic Equations – Net Pressure Force

Hydrostatic Force on Submerged Surfaces Curved Submerged Surface

Hydrostatic Force on Submerged Surfaces 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):

Introduction to Fluid Mechanics 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

Relation of System Derivatives to the Control Volume Formulation Reynolds Transport Theorem

Relation of System Derivatives to the Control Volume Formulation 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

Momentum Equation for Inertial Control Volume

Momentum Equation for Inertial Control Volume Special Case: Bernoulli Equation Steady Flow No Friction Flow Along a Streamline Incompressible Flow

Momentum Equation for Inertial Control Volume Special Case: Control Volume Moving with Constant Velocity

Momentum Equation for Inertial Control Volume with Rectilinear Acceleration

The Angular Momentum Principle Basic Law, and Transport Theorem

The Angular Momentum Principle

The First Law of Thermodynamics Basic Law, and Transport Theorem

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

The Second Law of Thermodynamics Basic Law, and Transport Theorem

The Second Law of Thermodynamics

Introduction to Fluid Mechanics 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

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

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

Motion of a Fluid Particle (Kinematics)

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

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

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

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

Motion of a Fluid Particle (Kinematics) Fluid Rotation

Motion of a Fluid Particle (Kinematics) Fluid Rotation

Motion of a Fluid Particle (Kinematics) Fluid Rotation

Motion of a Fluid Particle (Kinematics) Fluid Deformation: Angular Deformation

Motion of a Fluid Particle (Kinematics) Fluid Deformation: Angular Deformation

Motion of a Fluid Particle (Kinematics) Fluid Deformation: Linear Deformation

Momentum Equation Newton’s Second Law

Momentum Equation Forces Acting on a Fluid Particle

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

Introduction to Fluid Mechanics 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

Momentum Equation for Frictionless Flow: Euler’s Equation Rectangular Coordinates

Momentum Equation for Frictionless Flow: Euler’s Equation 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)

Bernoulli Equation – Integration of Euler’s Equation Along a Streamline for Steady Flow Integration Along s Coordinate

Bernoulli Equation – Integration of Euler’s Equation Along a Streamline for Steady Flow No Friction Flow Along a Streamline Incompressible Flow

Bernoulli Equation – Integration of Euler’s Equation Along a Streamline for Steady Flow Static, Stagnation, and Dynamic Pressures (Ignore Gravity) Stagnation Static Dynamic

The Bernoulli Equation Interpreted as an Energy Equation

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

The Bernoulli Equation Interpreted as an Energy Equation Hence Assumption 6: Incompressible Assumption 7:

The Bernoulli Equation Interpreted as an Energy Equation 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 and Hydraulic Grade Line Energy Grade Line (EGL) Hydraulic Grade Line (HGL)

Energy Grade Line and Hydraulic Grade Line

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

Introduction to Fluid Mechanics 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

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?)

Nature of Dimensional Analysis 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)

Nature of Dimensional Analysis

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

Buckingham Pi Theorem Step 3 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

Buckingham Pi Theorem Step 4 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

Buckingham Pi Theorem Step 5 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

Buckingham Pi Theorem Step 6 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

Significant Dimensionless Groups in Fluid Mechanics Froude Number Weber Number

Significant Dimensionless Groups in Fluid Mechanics 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

Flow Similarity and Model Studies Example: Drag on a Sphere

Flow Similarity and Model Studies Example: Drag on a Sphere For dynamic similarity … … then …

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

Flow Similarity and Model Studies Scaling with Multiple Dependent Parameters Example: Centrifugal Pump Pump Head Pump Power

Flow Similarity and Model Studies Scaling with Multiple Dependent Parameters Example: Centrifugal Pump Head Coefficient Power Coefficient

Flow Similarity and Model Studies Scaling with Multiple Dependent Parameters Example: Centrifugal Pump (Negligible Viscous Effects) If … … then …

Flow Similarity and Model Studies Scaling with Multiple Dependent Parameters Example: Centrifugal Pump Specific Speed

Introduction to Fluid Mechanics 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

Fully Developed Laminar Flow Between Infinite Parallel Plates Both Plates Stationary Transformation of Coordinates

Fully Developed Laminar Flow Between Infinite Parallel Plates Both Plates Stationary Shear Stress Distribution Volume Flow Rate

Fully Developed Laminar Flow Between Infinite Parallel Plates Both Plates Stationary Flow Rate as a Function of Pressure Drop Average and Maximum Velocities

Fully Developed Laminar Flow Between Infinite Parallel Plates Upper Plate Moving with Constant Speed, U

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

Fully Developed Laminar Flow in a Pipe Volume Flow Rate Flow Rate as a Function of Pressure Drop

Fully Developed Laminar Flow in a Pipe Average Velocity Maximum Velocity

Turbulent Velocity Profiles in Fully Developed Pipe Flow

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

Calculation of Head Loss Laminar Friction Factor Turbulent Friction Factor

Calculation of Head Loss

Calculation of Head Loss Minor Losses Examples: Inlets and Exits; Enlargements and Contractions; Pipe Bends; Valves and Fittings

Calculation of Head Loss Minor Loss: Loss Coefficient, K Minor Loss: Equivalent Length, Le

Calculation of Head Loss Pumps, Fans, and Blowers

Calculation of Head Loss Noncircular Ducts Example: Rectangular Duct

Solution of Pipe Flow Problems Energy Equation

Solution of Pipe Flow Problems Major Losses

Solution of Pipe Flow Problems Minor Losses

Solution of Pipe Flow Problems Single Path Find Dp for a given L, D, and Q Use energy equation directly Find L for a given Dp, D, and Q

Solution of Pipe Flow Problems 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

Solution of Pipe Flow Problems Multiple-Path Systems Example:

Solution of Pipe Flow Problems 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

Introduction to Fluid Mechanics 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

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

Laminar Flat-Plate Boundary Layer: Exact Solution Boundary Conditions

Laminar Flat-Plate Boundary Layer: Exact Solution Equations are Coupled, Nonlinear, Partial Differential Equations Blasius Solution: Transform to single, higher-order, nonlinear, ordinary differential equation

Laminar Flat-Plate Boundary Layer: Exact Solution 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

Use of the Momentum Equation for Flow with Zero Pressure Gradient Laminar Flow Example: Assume a Polynomial Velocity Profile (Item 2) The wall shear stress tw is then (Item 3)

Use of the Momentum Equation for Flow with Zero Pressure Gradient Laminar Flow Results (Polynomial Velocity Profile) Compare to Exact (Blasius) results!

Use of the Momentum Equation for Flow with Zero Pressure Gradient Turbulent Flow Example: 1/7-Power Law Profile (Item 2)

Use of the Momentum Equation for Flow with Zero Pressure Gradient 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 662-215

Lift Induced Drag

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

Lift Induced Drag (Continued)

Introduction to Fluid Mechanics 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

Introduction and Classification of Fluid Machines Machines for Doing Work on a Fluid Pumps Fans Blowers Compressors

Introduction and Classification of Fluid Machines Machines for Doing Work on a Fluid

Introduction and Classification of Fluid Machines Machines for Doing Work on a Fluid

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

Introduction and Classification of Fluid Machines Machines for Extracting Work (Power) from a Fluid

Turbomachinery Analysis The Angular Momentum Principle Apply to Control Volume:

Turbomachinery Analysis Euler Turbomachine Equation Mechanical Power: Theoretical Head:

Turbomachinery Analysis Velocity Diagrams

Turbomachinery Analysis 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

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

Turbomachinery Analysis Example: Idealized Centrifugal Pump (Continued)

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

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

Performance Characteristics Machines for Doing Work on a Fluid

Performance Characteristics Machines for Extracting Work (Power) from a Fluid

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

Performance Characteristics Dimensional Analysis and Specific Speed Torque Coefficient:

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

Performance Characteristics Dimensional Analysis and Specific Speed

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

Applications to Fluid Systems Machines for Doing Work on a Fluid

Applications to Fluid Systems Machines for Doing Work on a Fluid Pump Wear

Applications to Fluid Systems Machines for Doing Work on a Fluid Pumps in Series

Applications to Fluid Systems Machines for Doing Work on a Fluid Pumps in Parallel

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

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

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

Applications to Fluid Systems Machines for Doing Work on a Fluid Positive-Displacement Pumps

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

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

Introduction to Fluid Mechanics 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

Specific Energy, Momentum Equation, and Specific Force Froude Number Criterion Froude Number Criterion (Rectangular Channel)

Specific Energy, Momentum Equation, and Specific Force

Specific Energy, Momentum Equation, and Specific Force Momentum Equation (Steady State)

Specific Energy, Momentum Equation, and Specific Force

Specific Energy, Momentum Equation, and Specific Force 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)

Discharge Measurement Using Weirs Suppressed Rectangular Weir Contracted Rectangular Weir Effective Length

Discharge Measurement Using Weirs Triangular Weir

Discharge Measurement Using Weirs Broad-Crested Weir

Introduction to Fluid Mechanics 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

Review of Thermodynamics Specific Heat Formulas

Review of Thermodynamics Internal Energy and Enthalpy

Review of Thermodynamics Entropy

Review of Thermodynamics The Second Law of Thermodynamics

Review of Thermodynamics Isentropic (Reversible Adiabatic) Processes

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

Propagation of Sound Waves Types of Flow – The Mach Cone

Propagation of Sound Waves 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

Introduction to Fluid Mechanics 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

Basic Equations for One-Dimensional Compressible Flow Continuity Momentum

Basic Equations for One-Dimensional Compressible Flow First Law of Thermodynamics Second Law of Thermodynamics

Basic Equations for One-Dimensional Compressible Flow Equation of State Property Relations

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

Isentropic Flow of an Ideal Gas – Area Variation

Isentropic Flow of an Ideal Gas – Area Variation Subsonic, Supersonic, and Sonic Flows

Isentropic Flow of an Ideal Gas – Area Variation Reference Stagnation and Critical Conditions for Isentropic Flow

Isentropic Flow of an Ideal Gas – Area Variation Property Relations

Isentropic Flow of an Ideal Gas – Area Variation Isentropic Flow in a Converging Nozzle

Isentropic Flow of an Ideal Gas – Area Variation Isentropic Flow in a Converging Nozzle

Isentropic Flow of an Ideal Gas – Area Variation Isentropic Flow in a Converging-Diverging Nozzle

Isentropic Flow of an Ideal Gas – Area Variation Isentropic Flow in a Converging-Diverging Nozzle

Flow in a Constant-Area Duct with Friction Control Volume

Flow in a Constant-Area Duct with Friction Basic Equations for Adiabatic Flow

Flow in a Constant-Area Duct with Friction Adiabatic Flow: The Fanno Line

Flow in a Constant-Area Duct with Friction Fanno-Line Flow Functions for One-Dimensional Flow of an Ideal Gas

Flow in a Constant-Area Duct with Friction Fanno-Line Relations

Flow in a Constant-Area Duct with Friction Fanno-Line Relations (Continued)

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

Frictionless Flow in a Constant-Area Duct with Heat Exchange Basic Equations for Flow with Heat Exchange

Frictionless Flow in a Constant-Area Duct with Heat Exchange Heat Exchange: The Rayleigh Line

Frictionless Flow in a Constant-Area Duct with Heat Exchange 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

Oblique Shocks and Expansion Waves Mach Angle and Oblique Shock Angle

Oblique Shocks and Expansion Waves Oblique Shock: Control Volume

Oblique Shocks and Expansion Waves Oblique Shock: Useful Formulas

Oblique Shocks and Expansion Waves Oblique Shock Relations

Oblique Shocks and Expansion Waves Oblique Shock Relations (Continued)

Oblique Shocks and Expansion Waves Oblique Shock: Deflection Angle

Oblique Shocks and Expansion Waves Oblique Shock: Deflection Angle

Oblique Shocks and Expansion Waves Expansion and Compression Waves

Oblique Shocks and Expansion Waves Expansion Wave: Control Volume

Oblique Shocks and Expansion Waves Expansion Wave: Prandtl-Meyer Expansion Function

Oblique Shocks and Expansion Waves Expansion Wave: Isentropic Relations