Fluid Dynamics

Fluid Dynamics - A collection of 12 lectures

Compiled by Bjørn H Hjertager Department of Mechanical and Structural Engineering and Material Science University of Stavanger Stavanger, Norway

November 2017

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University of Stavanger (UiS)

Course: Lecturer/Instructor:

Fluid Dynamics Professor Bjørn H. Hjertager

Lecture plan Lecture 1: 2 hours Introduction and basic concepts

Exercises: 2 hours

Literature 1) & 2) Chap 1

Lecture 2: 2 hours Properties of fluids

Exercises: 2 hours

1) & 2) Chap 2

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Lecture 3: 2 hours Fluid kinematics 1

Exercises: 2 hours

1) & 2) Chap 4

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Lecture 4: 2 hours Fluid kinematics 2

Exercises: 2 hours

1) & 2) Chap 4

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Lecture 5: 2 hours Differential analysis of fluid flow – 1

Exercises: 2 hours

1) & 2) Chap 9

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Exercises: 2 hours

1) & 2) Chap 9

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Exercises: 2 hours

1) & 2) Chap 9

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Lecture 8: 2 hours Approximate solutions of the Navier– Stokes equation-1

Exercises: 2 hours

1) & 2) Chap 10

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Exercises: 2 hours

1) & 2) Chap 10

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Exercises: 2 hours

1) & 2) Chap 10

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Lecture 11: 2 hours Differential analysis of energy transport Lecture 12: 2 hours Introduction to computational fluid dynamics

Exercises: 2 hours

1) & 3) Chap 6

Exercises: 2 hours

1) & 2) Chap 15

Lecture 1

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Literature 1 2 3

B.H. Hjertager: Fluid Dynamics, 2nd Edition, McGraw-Hill Create, 2015. Y.A. Cengel & J.M. Cimbala: Fluid mechanics: Fundamentals and applications, 3rd Edition in SI units, McGraw-Hill 2014 Y.A. Cengel & A.J. Ghajar: Heat and mass transfer: Fundamentals and applications, 4th Edition, McGraw-Hill, 2011

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Table of contents

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Lecture 1 Introduction and basic concepts

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Lecture 2 Properties of fluids

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Lecture 3 Fluid kinematics 1

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Lecture 4 Fluid kinematics 2

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Lecture 5 Differential analysis of fluid flow 1

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Lecture 8 Approximate solutions of the Navier-Sokes equations 1

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Lecture 11 Differential analysis of energy transport

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Lecture 12 Introduction to computational fluid dynamics

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Chapter 1 INTRODUCTION AND BASIC CONCEPTS Lecture 1 Bjørn H. Hjertager

Based on powerpoints supplied for Fluid Mechanics: Fundamentals and Applications, 2nd Edition Yunus A. Cengel, John M. Cimbala, McGraw-Hill, 2010

Objectives • Understand the basic concepts of Fluid Mechanics. • Recognize the various types of fluid flow problems encountered in practice. • Model engineering problems and solve them in a systematic manner. • Have a working knowledge of accuracy, precision, and significant digits, and recognize the importance of dimensional homogeneity in engineering calculations.

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Outline • • •

Introduction The No-Slip Condition Classification of Fluid Flows 9 9 9 9 9 9 9

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Viscous versus Inviscid Regions of Flow Internal versus External Flow Compressible versus Incompressible Flow Laminar versus Turbulent Flow Natural (or Unforced) versus Forced Flow Steady versus Unsteady Flow One-, Two-, and Three-Dimensional Flows

System and Control Volume Importance of Dimensions and Units Mathematical Modeling of Engineering Problems Problem Solving Technique Engineering Software Packages Accuracy, Precision and Significant Digits 3

1–1 Ŷ INTRODUCTION Mechanics: The oldest physical science that deals with both stationary and moving bodies under the influence of forces. Statics: The branch of mechanics that deals with bodies at rest. Dynamics: The branch that deals with bodies in motion. Fluid mechanics: The science that deals with the behavior of fluids at rest (fluid statics) or in motion (fluid dynamics), and the interaction of fluids with solids or other fluids at the boundaries. Fluid dynamics: Fluid mechanics is also referred to as fluid dynamics by considering fluids at rest as a special case of motion with zero velocity. 8

Fluid mechanics deals with liquids and gases in 4 motion or at rest.

Hydrodynamics: The study of the motion of fluids that can be approximated as incompressible (such as liquids, especially water, and gases at low speeds). Hydraulics: A subcategory of hydrodynamics, which deals with liquid flows in pipes and open channels. Gas dynamics: Deals with the flow of fluids that undergo significant density changes, such as the flow of gases through nozzles at high speeds. Aerodynamics: Deals with the flow of gases (especially air) over bodies such as aircraft, rockets, and automobiles at high or low speeds. Meteorology, oceanography, and hydrology: Deal with naturally occurring flows.

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What Is a Fluid? Fluid: A substance in the liquid or gas phase. A solid can resist an applied shear stress by deforming. A fluid deforms continuously under the influence of a shear stress, no matter how small. In solids, stress is proportional to strain, but in fluids, stress is proportional to strain rate. When a constant shear force is applied, a solid eventually stops deforming at some fixed strain angle, whereas a fluid never stops deforming and approaches a constant rate of strain.

Deformation of a rubber block placed between two parallel plates under the influence of a shear force. The shear stress shown is that on the rubber—an equal but opposite shear stress acts on the upper plate. 9

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Stress: Force per unit area. Normal stress: The normal component of a force acting on a surface per unit area. Shear stress: The tangential component of a force acting on a surface per unit area. Pressure: The normal stress in a fluid at rest. Zero shear stress: A fluid at rest is at a state of zero shear stress. When the walls are removed or a liquid container is tilted, a shear develops as the liquid moves to The normal stress and shear stress at re-establish a horizontal free the surface of a fluid element. For surface. fluids at rest, the shear stress is zero and pressure is the only normal stress. 7

In a liquid, groups of molecules can move relative to each other, but the volume remains relatively constant because of the strong cohesive forces between the molecules. As a result, a liquid takes the shape of the container it is in, and it forms a free surface in a larger container in a gravitational field. A gas expands until it encounters the walls of the container and fills the entire available space. This is because the gas molecules are widely spaced, and the cohesive forces between them are very small. Unlike liquids, a gas in an open container cannot form a free surface.

Unlike a liquid, a gas does not form a free surface, and it expands to fill the entire available space. 10

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Intermolecular bonds are strongest in solids and weakest in gases. Solid: The molecules in a solid are arranged in a pattern that is repeated throughout. Liquid: In liquids molecules can rotate and translate freely. Gas: In the gas phase, the molecules are far apart from each other, and molecular ordering is nonexistent.

The arrangement of atoms in different phases: (a) molecules are at relatively fixed positions in a solid, (b) groups of molecules move about each other in the liquid phase, and (c) individual molecules move about at random 9 in the gas phase.

Gas and vapor are often used as synonymous words. Gas: The vapor phase of a substance is customarily called a gas when it is above the critical temperature. Vapor: Usually implies that the current phase is not far from a state of condensation. Macroscopic or classical approach: Does not require a knowledge of the behavior of individual molecules and provides a direct and easy way to analyze engineering problems. Microscopic or statistical approach: Based on the average behavior of large groups of individual molecules.

On a microscopic scale, pressure is determined by the interaction of individual gas molecules. However, we can measure the pressure on a macroscopic scale with a pressure gage.

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Application Areas of Fluid Mechanics

Fluid dynamics is used extensively in the design of artificial hearts. Shown here is the Penn State Electric Total Artificial Heart.

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1–2 Ŷ THE NO-SLIP CONDITION

A fluid flowing over a stationary surface comes to a complete stop at the surface because of the no-slip condition.

The development of a velocity profile due to the no-slip condition as a fluid flows over a blunt nose.

Flow separation during flow over a curved surface. 13

Boundary layer: The flow region adjacent to the wall in which the viscous effects (and thus the velocity gradients) are significant. 14

1–4 Ŷ CLASSIFICATION OF FLUID FLOWS Viscous versus Inviscid Regions of Flow Viscous flows: Flows in which the frictional effects are significant. Inviscid flow regions: In many flows of practical interest, there are regions (typically regions not close to solid surfaces) where viscous forces are negligibly small compared to inertial or pressure forces.

The flow of an originally uniform fluid stream over a flat plate, and the regions of viscous flow (next to the plate on both sides) and inviscid flow (away from the plate). 15

Internal versus External Flow External flow: The flow of an unbounded fluid over a surface such as a plate, a wire, or a pipe. Internal flow: The flow in a pipe or duct if the fluid is completely bounded by solid surfaces. •

Water flow in a pipe is internal flow, and airflow over a ball is external flow .

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The flow of liquids in a duct is called openchannel flow if the duct is only partially filled with the liquid and there is a free surface.

External flow over a tennis ball, and the turbulent wake region behind. 16

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Compressible versus Incompressible Flow Incompressible flow: If the density of flowing fluid remains nearly constant throughout (e.g., liquid flow). Compressible flow: If the density of fluid changes during flow (e.g., high-speed gas flow) When analyzing rockets, spacecraft, and other systems that involve highspeed gas flows, the flow speed is often expressed by Mach number

Ma = 1 Ma < 1 Ma > 1 Ma >> 1

Sonic flow Subsonic flow Supersonic flow Hypersonic flow 17

Laminar versus Turbulent Flow Laminar flow: The highly ordered fluid motion characterized by smooth layers of fluid. The flow of high-viscosity fluids such as oils at low velocities is typically laminar. Turbulent flow: The highly disordered fluid motion that typically occurs at high velocities and is characterized by velocity fluctuations. The flow of lowviscosity fluids such as air at high velocities is typically turbulent. Transitional flow: A flow that alternates between being laminar and turbulent.

Laminar, transitional, and turbulent flows. 15

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Natural (or Unforced) versus Forced Flow Forced flow: A fluid is forced to flow over a surface or in a pipe by external means such as a pump or a fan. Natural flow: Fluid motion is due to natural means such as the buoyancy effect, which manifests itself as the rise of warmer (and thus lighter) fluid and the fall of cooler (and thus denser) fluid. In this schlieren image of a girl in a swimming suit, the rise of lighter, warmer air adjacent to her body indicates that humans and warm-blooded animals are surrounded by thermal plumes of rising warm air. 19

Steady versus Unsteady Flow •

The term steady implies no change at a point with time.

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The opposite of steady is unsteady.

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The term uniform implies no change with location over a specified region.

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The term periodic refers to the kind of unsteady flow in which the flow oscillates about a steady mean.

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Many devices such as turbines, compressors, boilers, condensers, and heat exchangers operate for long periods of time under the same conditions, and they are classified as steady-flow devices. Oscillating wake of a blunt-based airfoil at Mach number 0.6. Photo (a) is an instantaneous image, while photo (b) is a long-exposure (time-averaged) image. 16

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Comparison of (a) instantaneous snapshot of an unsteady flow, and (b) long exposure picture of the same flow.

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One-, Two-, and Three-Dimensional Flows •

A flow field is best characterized by its velocity distribution.

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A flow is said to be one-, two-, or threedimensional if the flow velocity varies in one, two, or three dimensions, respectively.

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However, the variation of velocity in certain directions can be small relative to the variation in other directions and can be ignored.

Flow over a car antenna is approximately two-dimensional except near the top and bottom of the antenna.

The development of the velocity profile in a circular pipe. V = V(r, z) and thus the flow is two-dimensional in the entrance region, and becomes one-dimensional downstream when the velocity profile fully develops and remains unchanged in 22 the flow direction, V = V(r). 17

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1–5 Ŷ SYSTEM AND CONTROL VOLUME • • •

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System: A quantity of matter or a region in space chosen for study. Surroundings: The mass or region outside the system Boundary: The real or imaginary surface that separates the system from its surroundings. The boundary of a system can be fixed or movable. Systems may be considered to be closed or open. Closed system (Control mass): A fixed amount of mass, and no mass can cross its boundary. 24

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Open system (control volume): A properly selected region in space. • It usually encloses a device that involves mass flow such as a compressor, turbine, or nozzle. • Both mass and energy can cross the boundary of a control volume. • Control surface: The boundaries of a control volume. It can be real or imaginary.

An open system (a control volume) with one inlet and one exit.

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1–6 Ŷ IMPORTANCE OF DIMENSIONS AND UNITS •

Any physical quantity can be characterized by dimensions.

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The magnitudes assigned to the dimensions are called units.

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Some basic dimensions such as mass m, length L, time t, and temperature T are selected as primary or fundamental dimensions, while others such as velocity V, energy E, and volume V are expressed in terms of the primary dimensions and are called secondary dimensions, or derived dimensions.

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Metric SI system: A simple and logical system based on a decimal relationship between the various units.

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English system: It has no apparent systematic numerical base, and various units in this system are related to each other rather arbitrarily. 19

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Some SI Units

Work = Force u Distance 1 J = 1 N·m

The SI unit prefixes are used in all branches of engineering.

The definition of the force units.

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A typical match yields about one one kJ of energy if completely burned. 28

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Dimensional homogeneity All equations must be dimensionally homogeneous.

Unity Conversion Ratios All nonprimary units (secondary units) can be formed by combinations of primary units. Force units, for example, can be expressed as

They can also be expressed more conveniently as unity conversion ratios as:

Unity conversion ratios are identically equal to 1 and are unitless, and thus such ratios (or their inverses) can be inserted conveniently into any calculation to properly convert units. 29

Every unity conversion ratio (as well as its inverse) is exactly equal to one. Shown here are a few commonly used unity conversion ratios.

Always check the units in your calculations. 30

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1–7 Ŷ MATHEMATICAL MODELING OF ENGINEERING PROBLEMS Experimental vs. Analytical Analysis An engineering device or process can be studied either experimentally (testing and taking measurements) or analytically (by analysis or calculations). The experimental approach has the advantage that we deal with the actual physical system, and the desired quantity is determined by measurement, within the limits of experimental error. However, this approach is expensive, time-consuming, and often impractical. The analytical approach (including the numerical approach) has the advantage that it is fast and inexpensive, but the results obtained are subject to the accuracy of the assumptions, approximations, and idealizations made in the analysis.

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Modeling in Engineering Why do we need differential equations? The descriptions of most scientific problems involve equations that relate the changes in some key variables to each other. In the limiting case of infinitesimal or differential changes in variables, we obtain differential equations that provide precise mathematical formulations for the physical principles and laws by representing the rates of change as derivatives. Therefore, differential equations are used to investigate a wide variety of problems in sciences and engineering. Do we always need differential equations? Many problems encountered in practice can be solved without resorting to differential equations and the complications associated with them.

Mathematical modeling of physical problems. 23

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Complex model (very accurate ) vs. Simple model (not-so-accurate) Simplified models are often used in fluid mechanics to obtain approximate solutions to difficult engineering problems. Here, the helicopter's rotor is modeled by a disk, across which is imposed a sudden change in pressure. The helicopter's body is modeled by a simple ellipsoid. This simplified model yields the essential features of the overall air flow field in the vicinity of the ground. The right choice is usually the simplest model that 35 yields satisfactory results.

1–8 Ŷ PROBLEM-SOLVING TECHNIQUE • Step 1: Problem Statement • Step 2: Schematic • Step 3: Assumptions and Approximations • Step 4: Physical Laws • Step 5: Properties • Step 6: Calculations • Step 7: Reasoning, Verification, and Discussion

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A step-by-step approach can greatly simplify problem solving.

The assumptions made while solving an engineering problem must be reasonable and justifiable. 37

Neatness and organization are highly valued by employers. The results obtained from an engineering analysis must be checked for reasonableness. 38

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1–9 Ŷ ENGINEERING SOFTWARE PACKAGES We should always remember that all the computing power and the engineering software packages available today are just tools, and tools have meaning only in the hands of masters. Hand calculators did not eliminate the need to teach our children how to add or subtract, and sophisticated medical software packages did not take the place of medical school training. Neither will engineering software packages replace the traditional engineering education. They will simply cause a shift in emphasis in the courses from mathematics to physics. That is, more time will be spent in the classroom discussing the physical aspects of the problems in greater detail, and less time on the mechanics of solution procedures.

An excellent word-processing program does not make a person a good writer; it simply makes a good writer a more efficient writer. 39

EES (Engineering Equation Solver) (Pronounced as ease): EES is a program that solves systems of linear or nonlinear algebraic or differential equations numerically. It has a large library of built-in thermodynamic property functions as well as mathematical functions. Unlike some software packages, EES does not solve engineering problems; it only solves the equations supplied by the user.

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1–10 Ŷ ACCURACY, PRECISION, AND SIGNIFICANT DIGITS Accuracy error (inaccuracy): The value of one reading minus the true value. In general, accuracy of a set of measurements refers to the closeness of the average reading to the true value. Accuracy is generally associated with repeatable, fixed errors. Precision error: The value of one reading minus the average of readings. In general, precision of a set of measurements refers to the fineness of the resolution and the repeatability of the instrument. Precision is generally associated with unrepeatable, random errors. Significant digits: Digits that are relevant and meaningful. Illustration of accuracy versus precision. Shooter A is more precise, but less accurate, while shooter B is more accurate, but less precise.

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A result with more significant digits than that of given data falsely implies more precision.

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Summary • •

The No-Slip Condition Classification of Fluid Flows 9 9 9 9 9 9 9

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Viscous versus Inviscid Regions of Flow Internal versus External Flow Compressible versus Incompressible Flow Laminar versus Turbulent Flow Natural (or Unforced) versus Forced Flow Steady versus Unsteady Flow One-, Two-, and Three-Dimensional Flows

System and Control Volume Importance of Dimensions and Units Mathematical Modeling of Engineering Problems Problem Solving Technique Engineering Software Packages Accuracy, Precision and Significant Digits

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Chapter 2 PROPERTIES OF FLUIDS Lecture 2 Bjørn H. Hjertager


A drop forms when liquid is forced out of a small tube. The shape of the drop is determined by a balance of pressure, gravity, and surface tension forces. 2

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Objectives • Have a working knowledge of the basic properties of fluids and understand the continuum approximation. • Have a working knowledge of viscosity and the consequences of the frictional effects it causes in fluid flow. • Calculate the capillary rise (or drop) in tubes due to the surface tension effect.

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Outline • Introduction 9 Continuum

• Density and Specific Gravity 9 Density of Ideal Gases

• Vapor Pressure and Cavitation • Energy and Specific Heats • Compressibility and Speed of Sound 9 Coefficient of Compressibility 9 Coefficient of Volume Expansion 9 Speed of Sound and Mach Number

• Viscosity • Surface Tension and Capillary Effect 4

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2–1 Ŷ INTRODUCTION • •

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Property: Any characteristic of a system. Some familiar properties are pressure P, temperature T, volume V, and mass m. Properties are considered to be either intensive or extensive. Intensive properties: Those that are independent of the mass of a system, such as temperature, pressure, and density. Extensive properties: Those whose values depend on the size— or extent—of the system. Specific properties: Extensive properties per unit mass. Criterion to differentiate intensive and extensive properties.

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Continuum •

Matter is made up of atoms that are widely spaced in the gas phase. Yet it is very convenient to disregard the atomic nature of a substance and view it as a continuous, homogeneous matter with no holes, that is, a continuum.

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The continuum idealization allows us to treat properties as point functions and to assume the properties vary continually in space with no jump discontinuities.

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This idealization is valid as long as the size of the system we deal with is large relative to the space between the molecules.

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This is the case in practically all problems.

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In this text we will limit our consideration to substances that can be modeled as a continuum. 33

Despite the relatively large gaps between molecules, a substance can be treated as a continuum because of the very large number of molecules even in an extremely small volume.

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The length scale associated with most flows, such as seagulls in flight, is orders of magnitude larger than the mean free path of the air molecules. Therefore, here, and for all fluid flows considered in this book, the continuum idealization is appropriate. 7

2–2 Ŷ DENSITY AND SPECIFIC GRAVITY Density

Specific volume

Specific gravity: The ratio of the density of a substance to the density of some standard substance at a specified temperature (usually water at 4°C). Specific weight: The weight of a unit volume of a substance.

Density is mass per unit volume; specific volume is volume per unit mass. 8

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Density of Ideal Gases Equation of state: Any equation that relates the pressure, temperature, and density (or specific volume) of a substance. Ideal-gas equation of state: The simplest and best-known equation of state for substances in the gas phase.

Ru: The universal gas constant

The thermodynamic temperature scale in the SI is the Kelvin scale.

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An ideal gas is a hypothetical substance that obeys the relation Pv = RT. The ideal-gas relation closely approximates the P-v-T behavior of real gases at low densities. At low pressures and high temperatures, the density of a gas decreases and the gas behaves like an ideal gas.

Air behaves as an ideal gas, even at very high speeds. In this schlieren image, a bullet traveling at about the speed of sound bursts through both sides of a balloon, forming two expanding shock waves. The turbulent wake of the bullet is also visible.

In the range of practical interest, many familiar gases such as air, nitrogen, oxygen, hydrogen, helium, argon, neon, and krypton and even heavier gases such as carbon dioxide can be treated as ideal gases with negligible error. Dense gases such as water vapor in steam power plants and refrigerant vapor in refrigerators, however, should not be treated as ideal gases since they usually exist at a state near saturation. 10

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2–3 Ŷ VAPOR PRESSURE AND CAVITATION • Saturation temperature Tsat: The temperature at which a pure substance changes phase at a given pressure. • Saturation pressure Psat: The pressure at which a pure substance changes phase at a given temperature. • Vapor pressure (Pv): The pressure exerted by its vapor in phase equilibrium with its liquid at a given temperature. It is identical to the saturation pressure Psat of the liquid (Pv = Psat). • Partial pressure: The pressure of a gas or vapor in a mixture with other gases. For example, atmospheric air is a mixture of dry air and water vapor, and atmospheric pressure is the sum of the partial pressure of dry air and the partial pressure of water vapor. 12

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The vapor pressure (saturation pressure) of a pure substance (e.g., water) is the pressure exerted by its vapor molecules when the system is in phase equilibrium with its liquid molecules at a given temperature.

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There is a possibility of the liquid pressure in liquid-flow systems dropping below the vapor pressure at some locations, and the resulting unplanned vaporization.

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The vapor bubbles (called cavitation bubbles since they form “cavities” in the liquid) collapse as they are swept away from the low-pressure regions, generating highly destructive, extremely high-pressure waves.

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Cavitation damage on a 16-mm by 23-mm aluminum sample tested at 60 m/s for 2.5 h. The sample was located at the cavity collapse region downstream of a cavity generator specifically designed to produce high damage potential.

This phenomenon, which is a common cause for drop in performance and even the erosion of impeller blades, is called cavitation, and it is an important consideration in the design of hydraulic turbines and pumps.

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2–4 Ŷ ENERGY AND SPECIFIC HEATS •

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Energy can exist in numerous forms such as thermal, mechanical, kinetic, potential, electric, magnetic, chemical, and nuclear, and their sum constitutes the total energy, E of a system. Thermodynamics deals only with the change of the total energy. Macroscopic forms of energy: Those a system possesses as a whole with respect to some outside reference frame, such as kinetic and potential energies. Microscopic forms of energy: Those related to the molecular structure of a system and the degree of the molecular activity. Internal energy, U: The sum of all the microscopic forms of energy. Kinetic energy, KE: The energy that a system possesses as a result of its motion relative to some reference frame. Potential energy, PE: The energy that a system possesses as a result The macroscopic energy of an of its elevation in a gravitational field. object changes with velocity and

elevation. 38

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Enthalpy Energy of a flowing fluid

P/U is the flow energy, also called the flow work, which is the energy per unit mass needed to move the fluid and maintain flow. for a P = const. process For a T = const. process

The internal energy u represents the microscopic energy of a nonflowing fluid per unit mass, whereas enthalpy h represents the microscopic energy of a flowing fluid per unit mass.

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Specific Heats Specific heat at constant volume, cv: The energy required to raise the temperature of the unit mass of a substance by one degree as the volume is maintained constant. Specific heat at constant pressure, cp: The energy required to raise the temperature of the unit mass of a substance by one degree as the pressure is maintained constant.

Specific heat is the energy required to raise the temperature of a unit mass of a substance by one degree in a specified way.

Constantvolume and constantpressure specific heats cv and cp (values are for helium gas). 18

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2–5 Ŷ COMPRESSIBILITY AND SPEED OF SOUND Coefficient of Compressibility We know from experience that the volume (or density) of a fluid changes with a change in its temperature or pressure. Fluids usually expand as they are heated or depressurized and contract as they are cooled or pressurized. But the amount of volume change is different for different fluids, and we need to define properties that relate volume changes to the changes in pressure and temperature. Two such properties are: the bulk modulus of elasticity N the coefficient of volume expansion E.

Fluids, like solids, compress when the applied pressure is 19 increased from P1 to P2.

Coefficient of compressibility (also called the bulk modulus of compressibility or bulk modulus of elasticity) for fluids

The coefficient of compressibility represents the change in pressure corresponding to a fractional change in volume or density of the fluid while the temperature remains constant. What is the coefficient of compressibility of a truly incompressible substance (v = constant)? A large value of N indicates that a large change in pressure is needed to cause a small fractional change in volume, and thus a fluid with a large N is essentially incompressible. This is typical for liquids, and explains why liquids are usually considered to be incompressible.

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Water hammer: Characterized by a sound that resembles the sound produced when a pipe is “hammered.” This occurs when a liquid in a piping network encounters an abrupt flow restriction (such as a closing valve) and is locally compressed. The acoustic waves that are produced strike the pipe surfaces, bends, and valves as they propagate and reflect along the pipe, causing the pipe to vibrate and produce the familiar sound. Water hammering can be quite destructive, leading to leaks or even structural damage. The effect can be suppressed with a water hammer arrestor.

Water hammer arrestors: (a) A large surge tower built to protect the pipeline against water hammer damage. (b) Much smaller arrestors used for supplying 21 water to a household washing machine.

The coefficient of compressibility of an ideal gas is equal to its absolute pressure, and the coefficient of compressibility of the gas increases with increasing pressure.

The percent increase of density of an ideal gas during isothermal compression is equal to the percent increase in pressure. Isothermal compressibility: The inverse of the coefficient of compressibility. The isothermal compressibility of a fluid represents the fractional change in volume or density corresponding to a unit change in pressure.

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Coefficient of Volume Expansion The density of a fluid depends more strongly on temperature than it does on pressure. The variation of density with temperature is responsible for numerous natural phenomena such as winds, currents in oceans, rise of plumes in chimneys, the operation of hot-air balloons, heat transfer by natural convection, and even the rise of hot air and thus the phrase “heat rises”. To quantify these effects, we need a property that represents the variation of the density of a fluid with temperature at constant pressure.

Natural convection over a woman’s hand.

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The coefficient of volume expansion (or volume expansivity): The variation of the density of a fluid with temperature at constant pressure.

A large value of E for a fluid means a large change in density with temperature, and the product E 'T represents the fraction of volume change of a fluid that corresponds to a temperature change of ' T at constant pressure. The volume expansion coefficient of an ideal gas (P = URT ) at a temperature T is equivalent to the inverse of the temperature: The coefficient of volume expansion is a measure of the change in volume of a substance with 24 temperature at constant pressure. 42

In the study of natural convection currents, the condition of the main fluid body that surrounds the finite hot or cold regions is indicated by the subscript “infinity” to serve as a reminder that this is the value at a distance where the presence of the hot or cold region is not felt. In such cases, the volume expansion coefficient can be expressed approximately as

The combined effects of pressure and temperature changes on the volume change of a fluid can be determined by taking the specific volume to be a function of T and P.

The fractional change in volume (or density) due to changes in pressure and temperature can be expressed approximately as

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The variation of the coefficient of volume expansion of water with temperature in the range of 20°C to 50°C.

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Speed of Sound and Mach Number Speed of sound (sonic speed): The speed at which an infinitesimally small pressure wave travels through a medium. Control volume moving with the small pressure wave along a duct.

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Propagation of a small pressure wave along a duct.

Speed of Sound and Mach Number 2 Speed of sound (sonic speed):.

For any fluid

For an ideal gas The speed of sound changes with temperature and varies with the fluid. 30

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Mach number Ma: The ratio of the actual speed of the fluid (or an object in still fluid) to the speed of sound in the same fluid at the same state.

The Mach number depends on the speed of sound, which depends on the state of the fluid.

The Mach number can be different at different temperatures even if the flight speed is the same.

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2–6 Ŷ VISCOSITY Viscosity: A property that represents the internal resistance of a fluid to motion or the “fluidity”. Drag force: The force a flowing fluid exerts on a body in the flow direction. The magnitude of this force depends, in part, on viscosity The viscosity of a fluid is a measure of its “resistance to deformation.” Viscosity is due to the internal frictional force that develops between different layers of fluids as they are forced to move relative to each other.

A fluid moving relative to a body exerts a drag force on the body, partly because of friction caused by viscosity.

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Newtonian fluids: Fluids for which the rate of deformation is proportional to the shear stress.

Shear stress The behavior of a fluid in laminar flow between two parallel plates when the upper plate moves with a constant velocity.

Shear force

P coefficient of viscosity Dynamic (absolute) viscosity kg/m s or N s/m2 or Pa s 1 poise = 0.1 Pa s 34 47

The rate of deformation (velocity gradient) of a Newtonian fluid is proportional to shear stress, and the constant of proportionality is the viscosity.

Variation of shear stress with the rate of deformation for Newtonian and non-Newtonian fluids (the slope of a curve at a point is the apparent viscosity of the fluid at that point).

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Kinematic viscosity m2/s or stoke 1 stoke = 1 cm2/s For liquids, both the dynamic and kinematic viscosities are practically independent of pressure, and any small variation with pressure is usually disregarded, except at extremely high pressures. For gases, this is also the case for dynamic viscosity (at low to moderate pressures), but not for kinematic viscosity since the density of a gas is proportional to its pressure.

Dynamic viscosity, in general, does not depend on pressure, but kinematic viscosity does.

For gases: For liquids

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The viscosity of a fluid is directly related to the pumping power needed to transport a fluid in a pipe or to move a body through a fluid. Viscosity is caused by the cohesive forces between the molecules in liquids and by the molecular collisions in gases, and it varies greatly with temperature. In a liquid, the molecules possess more energy at higher temperatures, and they can oppose the large cohesive intermolecular forces more strongly. As a result, the energized liquid molecules can move more freely.

The viscosity of liquids decreases and the viscosity of gases increases with temperature.

In a gas, the intermolecular forces are negligible, and the gas molecules at high temperatures move randomly at higher velocities. This results in more molecular collisions per unit volume per unit time and therefore in greater resistance to flow. 37

The variation of dynamic (absolute) viscosity of common fluids with temperature at 1 atm (1 Ns/m2 = 1kg/ms )

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L

length of the cylinder number of revolutions per unit time

This equation can be used to calculate the viscosity of a fluid by measuring torque at a specified angular velocity. Therefore, two concentric cylinders can be used as a viscometer, a device that measures viscosity.

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2–7 Ŷ SURFACE TENSION AND CAPILLARY EFFECT •

Liquid droplets behave like small balloons filled with the liquid on a solid surface, and the surface of the liquid acts like a stretched elastic membrane under tension.

•

The pulling force that causes this tension acts parallel to the surface and is due to the attractive forces between the molecules of the liquid.

•

The magnitude of this force per unit length is called surface tension (or coefficient of surface tension) and is usually expressed in the unit N/m.

•

This effect is also called surface energy [per unit area] and is expressed in the equivalent unit of N m/m2.

41

Attractive forces acting on a liquid molecule at the surface and deep inside the liquid. Stretching a liquid film with a Ushaped wire, and the forces acting on the movable wire of length b.

Surface tension: The work done per unit increase in the surface area of the liquid. 51

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The free-body diagram of half a droplet or air bubble and half a soap bubble.

43

Capillary effect: The rise or fall of a liquid in a small-diameter tube inserted into the liquid. Capillaries: Such narrow tubes or confined flow channels. The capillary effect is partially responsible for the rise of water to the top of tall trees. Meniscus: The curved free surface of a liquid in a capillary tube.

Capillary Effect

The strength of the capillary effect is quantified by the contact (or wetting) angle, defined as the angle that the tangent to the liquid surface makes with the solid surface at the point of contact.

The meniscus of colored water in a 4-mm-inner-diameter glass tube. Note that the edge of the meniscus meets the wall of the capillary tube at a very small contact angle. 44

The contact angle for wetting and nonwetting fluids. 52

The capillary rise of water and the capillary fall of mercury in a smalldiameter glass tube.

The forces acting on a liquid column that has risen in a tube due to the capillary effect.

¾ Capillary rise is inversely proportional to the radius of the tube and density of the liquid.

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Summary • Introduction 9 Continuum

• Density and Specific Gravity 9 Density of Ideal Gases

• Vapor Pressure and Cavitation • Energy and Specific Heats • Compressibility and Speed of Sound 9 Coefficient of Compressibility 9 Coefficient of Volume Expansion 9 Speed of Sound and Mach Number

• Viscosity • Surface Tension and Capillary Effect 49

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Chapter 4

FLUID KINEMATICS-1 Lecture 3 Bjørn H. Hjertager


Satellite image of a hurricane near the Florida coast; water droplets move with the air, enabling us to visualize the counterclockwise swirling motion. However, the major portion of the hurricane is actually irrotational, while only the core (the eye of the storm) is rotational. 57

2

Objectives • Understand the role of the material derivative in transforming between Lagrangian and Eulerian descriptions • Distinguish between various types of flow visualizations and methods of plotting the characteristics of a fluid flow

3

Outline •

Lagrangian and Eulerian Descriptions 9 Acceleration Field 9 Material Derivative

•

Flow Patterns and Flow Visualization 9 9 9 9

•

Streamlines and Streamtubes, Pathlines, Streaklines, Timelines Refractive Flow Visualization Techniques Surface Flow Visualization Techniques

Plots of Fluid Flow Data 9 Vector Plots, Contour Plots

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4–1 Ŷ LAGRANGIAN AND EULERIAN DESCRIPTIONS Kinematics: The study of motion. Fluid kinematics: The study of how fluids flow and how to describe fluid motion. There are two distinct ways to describe motion: Lagrangian and Eulerian Lagrangian description: To follow the path of individual objects. This method requires us to track the position and velocity of each individual fluid parcel (fluid particle) and take to be a parcel of fixed identity.

With a small number of objects, such as billiard balls on a pool table, individual objects can be tracked.

• • • •

•

In the Lagrangian description, one must keep track of the position and velocity of individual particles.

5

A more common method is Eulerian description of fluid motion. In the Eulerian description of fluid flow, a finite volume called a flow domain or control volume is defined, through which fluid flows in and out. Instead of tracking individual fluid particles, we define field variables, functions of space and time, within the control volume. The field variable at a particular location at a particular time is the value of the variable for whichever fluid particle happens to occupy that location at that time. For example, the pressure field is a scalar field variable. We define the velocity field as a vector field variable.

Collectively, these (and other) field variables define the flow field. The velocity field can be expanded in Cartesian coordinates as

6

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In the Eulerian description, one defines field variables, such as the pressure field and the velocity field, at any location and instant in time.

•

In the Eulerian description we don’t really care what happens to individual fluid particles; rather we are concerned with the pressure, velocity, acceleration, etc., of whichever fluid particle happens to be at the location of interest at the time of interest.

•

While there are many occasions in which the Lagrangian description is useful, the Eulerian description is often more convenient for fluid mechanics applications.

•

Experimental measurements are generally more suited to the Eulerian description. 7

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A Steady Two-Dimensional Velocity Field

Flow field near the bell mouth inlet of a hydroelectric dam; a portion of the velocity field of Example 4-1 may be used as a first-order approximation of this physical flow field. Velocity vectors for the velocity field of Example 4–1. The scale is shown by the top arrow, and the solid black curves represent the approximate shapes of some streamlines, based on the calculated velocity vectors. The stagnation point is indicated by the circle. The shaded region represents a portion of the flow field that can approximate flow into an inlet.

9

Acceleration Field The equations of motion for fluid flow (such as Newton’s second law) are written for a fluid particle, which we also call a material particle. If we were to follow a particular fluid particle as it moves around in the flow, we would be employing the Lagrangian description, and the equations of motion would be directly applicable. For example, we would define the particle’s location in space in terms of a material position vector (xparticle(t), yparticle(t), zparticle(t)).

Newton’s second law applied to a fluid particle; the acceleration vector (purple arrow) is in the same direction as the force vector (green arrow), but the velocity vector (blue arrow) may act in a different direction.

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Local acceleration

Advective (convective) acceleration

11

The components of the acceleration vector in cartesian coordinates:

When following a fluid particle, the xcomponent of velocity, u, is defined as dxparticle/dt. Similarly, v=dyparticle/dt and w=dzparticle/dt. Movement is shown here only in two dimensions for simplicity. Flow of water through the nozzle of a garden hose illustrates that fluid particles may accelerate, even in a steady flow. In this example, the exit speed of the water is much higher than the water speed in the hose, implying that fluid particles have accelerated 12 even though the flow is steady. 62

Material Derivative The total derivative operator d/dt in this equation is given a special name, the material derivative; it is assigned a special notation, D/Dt, in order to emphasize that it is formed by following a fluid particle as it moves through the flow field. Other names for the material derivative include total, particle, Lagrangian, Eulerian, and substantial derivative.

The material derivative D/Dt is defined by following a fluid particle as it moves throughout the flow field. In this illustration, the fluid particle is accelerating to the right as it moves up and to the right.

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Material Acceleration of a Steady Velocity Field

Acceleration vectors for the velocity field of Examples 4– 1 and 4–3. The scale is shown by the top arrow, and the solid black curves represent the approximate shapes of some streamlines, based on the calculated velocity vectors. The stagnation point is indicated by the color circle.

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4–2 Ŷ FLOW PATTERNS AND FLOW VISUALIZATION •

Flow visualization: The visual examination of flow field features.

•

While quantitative study of fluid dynamics requires advanced mathematics, much can be learned from flow visualization.

•

Flow visualization is useful not only in physical experiments but in numerical solutions as well [computational fluid dynamics (CFD)]. In fact, the very first thing an engineer using CFD does after obtaining a numerical solution is simulate some form of flow visualization.

•

Spinning baseball. The late F. N. M. Brown devoted many years to developing and using smoke visualization in wind tunnels at the University of Notre Dame. Here the flow speed is about 23 m/s and the ball is rotated at 630 rpm.

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Streamlines and Streamtubes Streamline: A curve that is everywhere tangent to the instantaneous local velocity vector. Streamlines are useful as indicators of the instantaneous direction of fluid motion throughout the flow field. For example, regions of recirculating flow and separation of a fluid off of a solid wall are easily identified by the streamline pattern. Streamlines cannot be directly observed experimentally except in steady flow fields.

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Streamlines for a steady, incompressible, two-dimensional velocity field

Streamlines (solid black curves) for the velocity field of Example 4–4; velocity vectors (color arrows) are superimposed for comparison. The agreement is excellent in the sense that the velocity vectors point everywhere tangent to the streamlines. Note that speed cannot be determined directly from the streamlines alone. 21

A streamtube consists of a bundle of streamlines much like a communications cable consists of a bundle of fiber-optic cables. Since streamlines are everywhere parallel to the local velocity, fluid cannot cross a streamline by definition. Fluid within a streamtube must remain there and cannot cross the boundary of the streamtube. Both streamlines and streamtubes are instantaneous quantities, defined at a particular instant in time according to the velocity field at that instant.

A streamtube consists of a bundle of individual streamlines.

In an incompressible flow field, a streamtube (a) decreases in diameter as the flow accelerates or converges and (b) increases in 22 diameter as the flow decelerates or diverges. 67

Pathlines •

Pathline: The actual path traveled by an individual fluid particle over some time period.

•

A pathline is a Lagrangian concept in that we simply follow the path of an individual fluid particle as it moves around in the flow field.

•

A pathline is formed by following the actual path of a fluid particle. Thus, a pathline is the same as the fluid particle’s material Pathlines produced by white tracer particles suspended position vector (xparticle(t), yparticle(t), zparticle(t)) traced out in water and captured by time-exposure photography; over some finite time interval. as waves pass horizontally, each particle moves in an elliptical path during one wave period.

23

Particle image velocimetry (PIV): A modern experimental technique that utilizes short segments of particle pathlines to measure the velocity field over an entire plane in a flow. Recent advances also extend the technique to three dimensions. In PIV, tiny tracer particles are suspended in the fluid. However, the flow is illuminated by two flashes of light (usually a light sheet from a laser) to produce two bright spots (recorded by a camera) for each moving particle. Then, both the magnitude and direction of the velocity vector at each particle location can be inferred, assuming that the tracer particles are small enough that they move with the fluid.

Stereo PIV measurements of the wing tip vortex in the wake of a NACA-66 airfoil at angle of attack. Color contours denote the local vorticity, normalized by the minimum value, as indicated in the color map. Vectors denote fluid motion in the plane of measurement. The black line denotes the location of the upstream wing trailling edge. Coordinates are normalized by the airfoil chord, and the origin is the wing root.

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Streaklines Streakline: The locus of fluid particles that have passed sequentially through a prescribed point in the flow. Streaklines are the most common flow pattern generated in a physical experiment. If you insert a small tube into a flow and introduce a continuous stream of tracer fluid (dye in a water flow or smoke in an air flow), the observed pattern is a streakline.

A streakline is formed by continuous introduction of dye or smoke from a point in the flow. Labeled tracer particles (1 through 8) were introduced sequentially. 26

69

Streaklines produced by colored fluid introduced upstream; since the flow is steady, these streaklines are the same as streamlines and pathlines. •

Streaklines, streamlines, and pathlines are identical in steady flow but they can be quite different in unsteady flow.

•

The main difference is that a streamline represents an instantaneous flow pattern at a given instant in time, while a streakline and a pathline are flow patterns that have some age and thus a time history associated with them.

•

A streakline is an instantaneous snapshot of a time-integrated flow pattern.

•

A pathline, on the other hand, is the time-exposed flow path of an individual particle over some time period.

27

In the figure, streaklines are introduced from a smoke wire located just downstream of a circular cylinder of diameter D aligned normal to the plane of view. When multiple streaklines are introduced along a line, as in the figure, we refer to this as a rake of streaklines. The Reynolds number of the flow is Re = 93.

Smoke streaklines introduced by a smoke wire at two different locations in the wake of a circular cylinder: (a) smoke wire just downstream of the cylinder and (b) smoke wire located at x/D = 150. The time-integrative nature of streaklines 28 is clearly seen by comparing the two photographs. 70

Because of unsteady vortices shed in an alternating pattern from the cylinder, the smoke collects into a clearly defined periodic pattern called a Kármán vortex street. A similar pattern can be seen at much larger scale in the air flow in the wake of an island.

Kármán vortices visible in the clouds in the wake of Alexander Selkirk Island in the southern Pacific Ocean.

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Comparison of Flow Patterns in an Unsteady Flow

An unsteady, incompressible, two-dimensional velocity field

Streamlines, pathlines, and streaklines for the oscillating velocity field of Example 4–5. The streaklines and pathlines are wavy because of their integrated time history, but the streamlines are not wavy since they represent an instantaneous snapshot of the velocity field.

31

Timelines Timeline: A set of adjacent fluid particles that were marked at the same (earlier) instant in time. Timelines are particularly useful in situations where the uniformity of a flow (or lack thereof) is to be examined.

Timelines are formed by marking a line of fluid particles, and then watching that line move (and deform) through the flow field; timelines are shown at t = 0, t1, t2, and t3. Timelines produced by a hydrogen bubble wire are used to visualize the boundary layer velocity profile shape. Flow is from left to right, and the hydrogen bubble wire is located to the left of the field of view. Bubbles near the wall reveal a flow instability that leads to turbulence. 72

32

Refractive Flow Visualization Techniques It is based on the refractive property of light waves. The speed of light through one material may differ somewhat from that in another material, or even in the same material if its density changes. As light travels through one fluid into a fluid with a different index of refraction, the light rays bend (they are refracted). Two primary flow visualization techniques that utilize the fact that the index of refraction in air (or other gases) varies with density: the shadowgraph technique and the schlieren technique. Interferometry is a visualization technique that utilizes the related phase change of light as it passes through air of varying densities as the basis for flow visualization. These techniques are useful for flow visualization in flow fields where density changes from one location in the flow to another, such as such as natural convection flows (temperature differences cause the density variations), mixing flows (fluid species cause the density variations), and supersonic flows (shock waves and expansion waves cause the density variations). 33

Unlike flow visualizations involving streaklines, pathlines, and timelines, the shadowgraph and schlieren methods do not require injection of a visible tracer (smoke or dye). Rather, density differences and the refractive property of light provide the necessary means for visualizing regions of activity in the flow field, allowing us to “see the invisible.” The image (a shadowgram) produced by the shadowgraph method is formed when the refracted rays of light rearrange the shadow cast onto a viewing screen or camera focal plane, causing bright or dark patterns to appear in the shadow. The dark patterns indicate the location where the refracted rays originate, while the bright patterns mark where these rays end up, and can be misleading. As a result, the dark regions are less distorted than the bright regions and are more useful in the interpretation of the shadowgram. 73

Shadowgram of a 14.3 mm sphere in free flight through air at Ma 3.0. A shock wave is clearly visible in the shadow as a dark band that curves around the sphere and is called a bow wave (see Chap. 12). 34

A shadowgram is not a true optical image; it is, after all, merely a shadow. A schlieren image, involves lenses (or mirrors) and a knife edge or other cutoff device to block the refracted light and is a true focused optical image. Schlieren imaging is more complicated to set up than is shadowgraphy but has a number of advantages. A schlieren image does not suffer from optical distortion by the refracted light rays. Schlieren imaging is also more sensitive to weak density gradients such as those caused by natural convection or by gradual phenomena like expansion fans in supersonic flow. Color schlieren imaging techniques have also been developed. One can adjust more components in a schlieren setup.

Schlieren image of natural convection due to a barbeque grill. 35

Surface Flow Visualization Techniques •

The direction of fluid flow immediately above a solid surface can be visualized with tufts—short, flexible strings glued to the surface at one end that point in the flow direction.

•

Tufts are especially useful for locating regions of flow separation, where the flow direction suddenly reverses.

•

A technique called surface oil visualization can be used for the same purpose—oil placed on the surface forms streaks called friction lines that indicate the direction of flow.

•

If it rains lightly when your car is dirty (especially in the winter when salt is on the roads), you may have noticed streaks along the hood and sides of the car, or even on the windshield.

•

This is similar to what is observed with surface oil visualization.

•

Lastly, there are pressure-sensitive and temperature-sensitive paints that enable researchers to observe the pressure or temperature distribution along solid surfaces. 36

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4–3 Ŷ PLOTS OF FLUID FLOW DATA Regardless of how the results are obtained (analytically, experimentally, or computationally), it is usually necessary to plot flow data in ways that enable the reader to get a feel for how the flow properties vary in time and/or space. You are already familiar with time plots, which are especially useful in turbulent flows (e.g., a velocity component plotted as a function of time), and xy-plots (e.g., pressure as a function of radius). In this section, we discuss three additional types of plots that are useful in fluid mechanics— profile plots, vector plots, and contour plots.

37

Profile Plots A profile plot indicates how the value of a scalar property varies along some desired direction in the flow field. In fluid mechanics, profile plots of any scalar variable (pressure, temperature, density, etc.) can be created, but the most common one used in this book is the velocity profile plot. Since velocity is a vector quantity, we usually plot either the magnitude of velocity or one of the components of the velocity vector as a function of distance in some desired direction. Profile plots of the horizontal component of velocity as a function of vertical distance; flow in the boundary layer growing along a horizontal flat plate: (a) standard profile plot and (b) profile plot with arrows. 75

38

Vector Plots A vector plot is an array of arrows indicating the magnitude and direction of a vector property at an instant in time. Streamlines indicate the direction of the instantaneous velocity field, they do not directly indicate the magnitude of the velocity (i.e., the speed). A useful flow pattern for both experimental and computational fluid flows is thus the vector plot, which consists of an array of arrows that indicate both magnitude and direction of an instantaneous vector property. Vector plots can also be generated from experimentally obtained data (e.g., from PIV measurements) or numerically from CFD calculations. Fig. 4-4: Velocity vector plot Fig. 4-14: Acceleration vector plot. Both generated analytically.

Results of CFD calculations of a twodimensional flow field consisting of free-stream flow impinging on a block of rectangular cross section. (a) streamlines, (b) velocity vector plot of the upper half of the flow, and (c) velocity vector plot, close-up view revealing more details in the separated flow region.

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A contour plot shows curves of constant values of a scalar property (or magnitude of a vector property) at an instant in time.

Contour Plots

Contour plots (also called isocontour plots) are generated of pressure, temperature, velocity magnitude, species concentration, properties of turbulence, etc. A contour plot can quickly reveal regions of high (or low) values of the flow property being studied. A contour plot may consist simply of curves indicating various levels of the property; this is called a contour line plot. Alternatively, the contours can be filled in with either colors or shades of gray; this is called a filled contour plot. Contour plots of the pressure field due to flow impinging on a block, as produced by CFD calculations; only the upper half is shown due to symmetry; (a) filled color scale contour plot and (b) contour line plot where pressure values are displayed in units of Pa gage pressure.

41

Summary •

Lagrangian and Eulerian Descriptions 9 Acceleration Field 9 Material Derivative

•

Flow Patterns and Flow Visualization 9 9 9 9

•

Streamlines and Streamtubes, Pathlines, Streaklines, Timelines Refractive Flow Visualization Techniques Surface Flow Visualization Techniques

Plots of Fluid Flow Data 9 Vector Plots, Contour Plots

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Chapter 4

FLUID KINEMATICS-2 Lecture 4 Bjørn H. Hjertager


Objectives • Appreciate the many ways that fluids move and deform • Distinguish between rotational and irrotational regions of flow based on the flow property vorticity • Understand the usefulness of the Reynolds transport theorem

2

79

Outline •

Other Kinematic Descriptions 9 Types of Motion or Deformation of Fluid Elements

•

Vorticity and Rotationality 9 Comparison of Two Circular Flows

•

The Reynolds Transport Theorem 9 Alternate Derivation of the Reynolds Transport Theorem 9 Relationship between Material Derivative and RTT

3

4–4 Ŷ OTHER KINEMATIC DESCRIPTIONS Types of Motion or Deformation of Fluid Elements In fluid mechanics, an element may undergo four fundamental types of motion or deformation: (a) translation, (b) rotation, (c) linear strain (also called extensional strain), and (d) shear strain. All four types of motion or deformation usually occur simultaneously. It is preferable in fluid dynamics to describe the motion and deformation of fluid elements in terms of rates such as velocity (rate of translation), angular velocity (rate of rotation), linear strain rate (rate of linear strain), and shear strain rate (rate of shear strain). In order for these deformation rates to be useful in the calculation of fluid flows, we must express them in terms of velocity and derivatives of velocity. 80

Fundamental types of fluid element motion or deformation: (a) translation, (b) rotation, (c) linear strain, 4 and (d) shear strain.

A vector is required in order to fully describe the rate of translation in three dimensions. The rate of translation vector is described mathematically as the velocity vector.

Rate of rotation (angular velocity) at a point: The average rotation rate of two initially perpendicular lines that intersect at that point. Rate of rotation of fluid element about point P

For a fluid element that translates and deforms as sketched, the rate of rotation at point P is defined as the average rotation rate of two initially perpendicular lines (lines a and b).

5

The first approximation is due to the fact that as the size of the fluid element shrinks to a point, dx o 0, and at the same time dt o 0. Thus, the second term in the denominator is second-order compared to the firstorder term dx and can be neglected. The second approximation is because as dt o 0 angle D is very small, and tanD o D. 6

81

The rate of rotation vector is equal to the angular velocity vector.

7

Linear strain rate: The rate of increase in length per unit length. Mathematically, the linear strain rate of a fluid element depends on the initial orientation or direction of the line segment upon which we measure the linear strain.

8

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Using the lengths marked in the figure, the linear strain rate in the xa-direction is

9

Volumetric strain rate or bulk strain rate: The rate of increase of volume of a fluid element per unit volume. This kinematic property is defined as positive when the volume increases. Another synonym of volumetric strain rate is also called rate of volumetric dilatation, (the iris of your eye dilates (enlarges) when exposed to dim light). The volumetric strain rate is the sum of the linear strain rates in three mutually orthogonal directions.

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The volumetric strain rate is zero in an incompressible flow.

Air being compressed by a piston in a cylinder; the volume of a fluid element in the cylinder decreases, corresponding to a negative rate of volumetric dilatation.

11

Shear strain rate at a point: Half of the rate of decrease of the angle between two initially perpendicular lines that intersect at the point. Shear strain rate, initially perpendicular lines in the x- and y-directions:

For a fluid element that translates and deforms as sketched, the shear strain rate at point P is defined as half of the rate of decrease of the angle between two initially perpendicular lines (lines a and b).

12

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Shear strain rate in Cartesian coordinates:

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Figure shows a general (although two-dimensional) situation in a compressible fluid flow in which all possible motions and deformations are present simultaneously. In particular, there is translation, rotation, linear strain, and shear strain. Because of the compressible nature of the fluid flow, there is also volumetric strain (dilatation). You should now have a better appreciation of the inherent complexity of fluid dynamics, and the mathematical sophistication required to fully describe fluid motion.

A fluid element illustrating translation, rotation, linear strain, shear strain, and volumetric strain. 15

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4–5 Ŷ VORTICITY AND ROTATIONALITY Another kinematic property of great importance to the analysis of fluid flows is the vorticity vector, defined mathematically as the curl of the velocity vector

Vorticity is equal to twice the angular velocity of a fluid particle

The direction of a vector cross product is determined by the righthand rule. 87

The vorticity vector is equal to twice the angular velocity vector of a rotating fluid particle.

18

•

If the vorticity at a point in a flow field is nonzero, the fluid particle that happens to occupy that point in space is rotating; the flow in that region is called rotational.

•

Likewise, if the vorticity in a region of the flow is zero (or negligibly small), fluid particles there are not rotating; the flow in that region is called irrotational.

•

Physically, fluid particles in a rotational region of flow rotate end over end as they move along in the flow.

The difference between rotational and irrotational flow: fluid elements in a rotational region of the flow rotate, but those in an irrotational region of the flow do not.

19

For a two-dimensional flow in the xy-plane, the vorticity vector always points in the z- or z-direction. In this illustration, the flag-shaped fluid particle rotates in the counterclockwise direction as it moves in the xy-plane; its vorticity points in the positive z-direction as shown. 88

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Determination of Rotationality in a Two-Dimensional Flow steady, incompressible, twodimensional velocity field:

Vorticity:

Deformation of an initially square fluid parcel subjected to the velocity field of Example 4–8 for a time period of 0.25 s and 0.50 s. Several streamlines are also plotted in the first quadrant. It is clear that this flow is rotational.

23

For a two-dimensional flow in the rTplane, the vorticity vector always points in the z (or z) direction. In this illustration, the flag-shaped fluid particle rotates in the clockwise direction as it moves in the ru-plane; its vorticity points in the z-direction as shown. 24

90

Comparison of Two Circular Flows Streamlines and velocity profiles for (a) flow A, solid-body rotation and (b) flow B, a line vortex. Flow A is rotational, but flow B is irrotational everywhere except at the origin.

25

A simple analogy can be made between flow A and a merry-goround or roundabout, and flow B and a Ferris wheel. As children revolve around a roundabout, they also rotate at the same angular velocity as that of the ride itself. This is analogous to a rotational flow. In contrast, children on a Ferris wheel always remain oriented in an upright position as they trace out their circular path. This is analogous to an irrotational flow.

A simple analogy: (a) rotational circular flow is analogous to a roundabout, while (b) irrotational circular flow is analogous to a Ferris wheel. 91

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4–6 Ŷ THE REYNOLDS TRANSPORT THEOREM Two methods of analyzing the spraying of deodorant from a spray can: (a) We follow the fluid as it moves and deforms. This is the system approach—no mass crosses the boundary, and the total mass of the system remains fixed. (b) We consider a fixed interior volume of the can. This is the control volume approach— mass crosses the boundary.

The relationship between the time rates of change of an extensive property for a system and for a control volume is expressed by the Reynolds transport theorem (RTT).

The Reynolds transport theorem (RTT) provides a link between the system approach and the control volume approach. 92

28

A moving system (hatched region) and a fixed control volume (shaded region) in a diverging portion of a flow field at times t and t+'t. The upper and lower bounds are streamlines of the flow.

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The time rate of change of the property B of the system is equal to the time rate of change of B of the control volume plus the net flux of B out of the control volume by mass crossing the control surface. This equation applies at any instant in time, where it is assumed that the system and the control volume occupy the same space at that particular instant in time.

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Reynolds transport theorem applied to a control volume moving at constant velocity.

Relative velocity crossing a control surface is found by vector addition of the absolute velocity of the fluid and the negative of the local velocity of the control surface.

33

An example control volume in which there is one well-defined inlet (1) and two well-defined outlets (2 and 3). In such cases, the control surface integral in the RTT can be more conveniently written in terms of the average values of fluid properties crossing each inlet and outlet. 34 95

Alternate Derivation of the Reynolds Transport Theorem

A more elegant mathematical derivation of the Reynolds transport theorem is possible through use of the Leibniz theorem The Leibniz theorem takes into account the change of limits a(t) and b(t) with respect to time, as well as the unsteady changes of integrand G(x, t) with time.

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The three-dimensional Leibniz theorem is required when calculating the time derivative of a volume integral for which the volume itself moves and/or deforms with time. It turns out that the three-dimensional form of the Leibniz theorem can be used in an alternative derivation of the Reynolds transport theorem. 97

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The material volume (system) and control volume occupy the same space at time t (the blue shaded area), but move and deform differently. At a later time they are not coincident.

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Relationship between Material Derivative and RTT While the Reynolds transport theorem deals with finite-size control volumes and the material derivative deals with infinitesimal fluid particles, the same fundamental physical interpretation applies to both.

The Reynolds transport theorem for finite volumes (integral analysis) is analogous to the material derivative for infinitesimal volumes (differential analysis). In both cases, we transform from a Lagrangian or system viewpoint to an Eulerian or control volume viewpoint.

Just as the material derivative can be applied to any fluid property, scalar or vector, the Reynolds transport theorem can be applied to any scalar or vector property as well.

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Summary •

Other Kinematic Descriptions 9 Types of Motion or Deformation of Fluid Elements

•

Vorticity and Rotationality 9 Comparison of Two Circular Flows

•

The Reynolds Transport Theorem 9 Alternate Derivation of the Reynolds Transport Theorem 9 Relationship between Material Derivative and RTT

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100

Chapter 9 DIFFERENTIAL ANALYSIS OF FLUID FLOW – 1 Lecture 5 Bjørn H. Hjertager


The fundamental differential equations of fluid motion are derived in this chapter, and we show how to solve them analytically for some simple flows. More complicated flows, such as the air flow induced by a tornado shown here, cannot be solved exactly. 101

2

Objectives

• Understand how the differential equation of conservation of mass is derived and applied • Calculate the stream function and plot streamlines for a known velocity field

3

Outline • •

Introduction Conservation of mass-The continuity equation 9 9 9 9 9

•

Derivation Using the Divergence Theorem Derivation Using an Infinitesimal Control Volume Alternative Form of the Continuity Equation Continuity Equation in Cylindrical Coordinates Special Cases of the Continuity Equation

The stream function 9 The Stream Function in Cartesian Coordinates 9 The Stream Function in Cylindrical Coordinates 9 The Compressible Stream Function

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9–1 Ŷ INTRODUCTION The control volume technique is useful when we are interested in the overall features of a flow, such as mass flow rate into and out of the control volume or net forces applied to bodies. Differential analysis, on the other hand, involves application of differential equations of fluid motion to any and every point in the flow field over a region called the flow domain. Boundary conditions for the variables must be specified at all boundaries of the flow domain, including inlets, outlets, and walls. If the flow is unsteady, we must march our solution along in time as the flow field changes.

(a) In control volume analysis, the interior of the control volume is treated like a black box, but (b) in differential analysis, all the details of the flow are solved at every point within the flow domain.

5

9–2 Ŷ CONSERVATION OF MASS—THE CONTINUITY EQUATION

Bsys =m;b=m/m=1

The net rate of change of mass within the control volume is equal to the rate at which mass flows into the control volume minus the rate at which mass flows out of the control volume. To derive a differential conservation equation, we imagine shrinking a control volume to infinitesimal size. 103

6

Derivation Using the Divergence Theorem The quickest and most straightforward way to derive the differential form of conservation of mass is to apply the divergence theorem (Gauss’s theorem).

This equation is the compressible form of the continuity equation since we have not assumed incompressible flow. It is valid at any point in the flow domain.

7

Derivation Using an Infinitesimal Control Volume At locations away from the center of the box, we use a Taylor series expansion about the center of the box.

A small box-shaped control volume centered at point P is used for derivation of the differential equation for conservation of mass in Cartesian coordinates; the red dots indicate the center of each face. 8

104

The mass flow rate through a surface is equal to UVnA.

The inflow or outflow of mass through each face of the differential control volume; the red dots indicate the center of each face.

9

The divergence operation in Cartesian and cylindrical coordinates.

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10

Fuel and air being compressed by a piston in a cylinder of an internal combustion engine.

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Nondimensional density as a function of nondimensional time for Example 9–1.

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Alternative Form of the Continuity Equation

As a material element moves through a flow field, its density changes according to Eq. 9–10.

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Continuity Equation in Cylindrical Coordinates

Velocity components and unit vectors in cylindrical coordinates: (a) twodimensional flow in the xy- or rT-plane, (b) three-dimensional flow.

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Special Cases of the Continuity Equation Special Case 1: Steady Compressible Flow

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Special Case 2: Incompressible Flow

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Converging duct, designed for a high-speed wind tunnel (not to scale). 18

109

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Streamlines for the converging duct of Example 9–2. 110

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Streamlines and velocity profiles for (a) a line vortex flow and (b) a spiraling line vortex/sink flow. 112

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Discussion The final result is general—not limited to Cartesian coordinates. It applies to unsteady as well as steady flows.

(a) In an incompressible flow field, fluid elements may translate, distort, and rotate, but they do not grow or shrink in volume; (b) in a compressible flow field, fluid elements may grow or shrink in volume as they translate, distort, 26 and rotate. 113

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9–3 Ŷ THE STREAM FUNCTION The Stream Function in Cartesian Coordinates

Incompressible, two-dimensional stream function in Cartesian coordinates: stream function \

There are several definitions of the stream function, depending on the type of flow under consideration as well as the coordinate system being used. 114

28

Curves of constant stream function represent streamlines of the flow.

Curves of constant \ are streamlines of the flow. 29

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Streamlines for the velocity field of Example 9–8; the value of constant \ is indicated for each streamline, and velocity vectors are shown at four locations.

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Streamlines for the velocity field of Example 9–9; the value of constant \ is indicated for each streamline. 34

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The difference in the value of \ from one streamline to another is equal to the volume flow rate per unit width between the two streamlines.

(a) Control volume bounded by streamlines \1 and \2 and slices A and B in the xy-plane; (b) magnified view of the region around infinitesimal length ds. 35

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The value of \ increases to the left of the direction of flow in the xy-plane.

Illustration of the “leftside convention.” In the xy-plane, the value of the stream function always increases to the left of the flow direction. In the figure, the stream function increases to the left of the flow direction, regardless of how much the flow twists and turns. When the streamlines are far apart (lower right of figure), the magnitude of velocity (the fluid speed) in that vicinity is small relative to the speed in locations where the streamlines are close together (middle region). This is because as the streamlines converge, the cross-sectional area between them decreases, and the velocity must increase to maintain the flow rate between the streamlines.

37

Streaklines produced by Hele–Shaw flow over an inclined plate. The streaklines model streamlines of potential flow (Chap. 10) over a two-dimensional inclined plate of the same cross-sectional shape.

119

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Streamlines for freestream flow along a wall with a narrow suction slot; streamline values are shown in units of m2/s; the thick streamline is the dividing streamline. The direction of the velocity vector at point A is determined by the left-side 39 convention.

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The Stream Function in Cylindrical Coordinates

Flow over an axisymmetric body in cylindrical coordinates with rotational symmetry about the z-axis; neither the geometry nor the velocity field depend on T, and uT = 0.

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Streamlines for the velocity field of Example 9–12, with K = 10 m2/s and C = 0; the value of constant \ is indicated for several streamlines.

43

The Compressible Stream Function

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Summary • •

Introduction Conservation of mass-The continuity equation 9 9 9 9 9

•

Derivation Using the Divergence Theorem Derivation Using an Infinitesimal Control Volume Alternative Form of the Continuity Equation Continuity Equation in Cylindrical Coordinates Special Cases of the Continuity Equation

The stream function 9 The Stream Function in Cartesian Coordinates 9 The Stream Function in Cylindrical Coordinates 9 The Compressible Stream Function

45

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124



Objectives

• Understand how the differential equation of conservation of linear momentum is derived and applied • Calculate the pressure field for a known velocity field

2

125

Outline •

The differential linear momentum equation-Cauchy’s equation 9 9 9 9

Derivation Using the Divergence Theorem Derivation Using an Infinitesimal Control Volume Alternative Form of Cauchy’s Equation Derivation Using Newton’s Second Law

• The Navier-Stokes equation 9 Introduction 9 Newtonian versus Non-Newtonian Fluids 9 Derivation of the Navier–Stokes Equation for Incompressible, Isothermal Flow 9 Continuity and Navier–Stokes Equations in Cartesian Coordinates 9 Continuity and Navier–Stokes Equations in Cylindrical Coordinates 9 Calculation of the Pressure Field for a Known Velocity Field 3

THE LINEAR MOMENTUM EQUATION

Newton’s second law can be stated as The sum of all external forces acting on a system is equal to the time rate of change of linear momentum of the system. This statement is valid for a coordinate system that is at rest or moves with a constant velocity, called an inertial coordinate system or inertial reference frame.4 126

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Ŷ FORCES ACTING ON A CONTROL VOLUME The forces acting on a control volume consist of Body forces that act throughout the entire body of the control volume (such as gravity, electric, and magnetic forces) and Surface forces that act on the control surface (such as pressure and viscous forces and reaction forces at points of contact). Only external forces are considered in the analysis.

Total force acting on control volume:

The total force acting on a control volume is composed of body forces and surface forces; body force is shown on a differential volume element, and surface force is shown on a differential surface element. 7

The most common body force is that of gravity, which exerts a downward force on every differential element of the control volume.

The gravitational force acting on a differential volume element of fluid is equal to its weight; the axes have been rotated so that the gravity vector acts downward in the negative z-direction.

Surface forces are not as simple to analyze since they consist of both normal and tangential components. Normal stresses are composed of pressure (which always acts inwardly normal) and viscous stresses. Shear stresses are composed entirely of viscous stresses.

8

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Surface force acting on a differential surface element: j – coordinate direction of stress i – coordinate direction of the area on which the stress is acting

Total surface force acting on control surface:

Total force:

When coordinate axes are rotated (a) to (b), the components of the surface force change, even though the force itself remains the same; only two dimensions are shown here. 9

Momentum-Flux Correction Factor, E The velocity across most inlets and outlets is not uniform. The control surface integral of Eq. 6–17 may be converted into algebraic form using a dimensionless correction factor E, called the momentum-flux correction factor.

E is always greater than or equal to 1. E is close to 1 for turbulent flow and not very close to 1 for fully developed laminar flow.

10

129

For turbulent flow ȕ may have an insignificant effect at inlets and outlets, but for laminar flow ȕ may be important and should not be neglected. It is wise to include ȕ in all momentum control volume problems. 11

9–4 Ŷ THE DIFFERENTIAL LINEAR MOMENTUM EQUATION—CAUCHY’S EQUATION

Positive components of the stress tensor in Cartesian coordinates on the positive (right, top, and front) faces of an infinitesimal rectangular control volume. The dots indicate the center of each face. Positive components on the negative (left, bottom, and back) faces are in the opposite direction of those shown here. 12

130

Derivation Using the Divergence Theorem

An extended form of the divergence theorem is useful not only for vectors, but also for tensors. In the equation, Gij is a second-order tensor, V is a volume, and A is the surface area that encloses and defines the volume.

Cauchy’s equation is a differential form of the linear momentum equation. It applies 13 to any type of fluid.

Derivation Using an Infinitesimal Control Volume (A)

Inflow and outflow of the x-component of linear momentum through each face of an infinitesimal control volume; the dots indicate the center of each face. 14

131

The gravity vector is not necessarily aligned with any particular axis, in general, and there are three components of the body force acting on an infinitesimal fluid element.

15

Sketch illustrating the surface forces acting in the x-direction ( j – index) due to the appropriate stress tensor component on each face (i –index) of the differential control volume; the dots indicate the center of each face.

Put all contributions ( colored ) into (A), devide with dx dy dz, rearrange and get:

16

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Alternative Form of Cauchy’s Equation

____________

0 because of continuity

18

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Derivation Using Newton’s Second Law

If the differential fluid element is a material element, it moves with the flow and Newton’s second law applies directly.

19

9–5 Ŷ THE NAVIER–STOKES EQUATION Introduction Wij, called the viscous stress tensor or the deviatoric stress tensor

Mechanical pressure is the mean normal stress acting inwardly on a fluid element. For fluids at rest, the only stress on a fluid element is the hydrostatic pressure, which always acts inward and normal to any surface. 134

20

Newtonian versus Non-Newtonian Fluids Rheology: The study of the deformation of flowing fluids. Newtonian fluids: Fluids for which the shear stress is linearly proportional to the shear strain rate. Non-Newtonian fluids: Fluids for which the shear stress is not linearly related to the shear strain rate. Viscoelastic: A fluid that returns (either fully or partially) to its original shape after the applied stress is released. Rheological behavior of fluids—shear stress as a function of shear strain rate. In some fluids a finite stress called the yield stress is required before the fluid begins to flow at all; such fluids are called Bingham plastic fluids.

Some non-Newtonian fluids are called shear thinning fluids or pseudoplastic fluids, because the more the fluid is sheared, the less viscous it becomes. Plastic fluids are those in which the shear thinning effect is extreme.

21

Shear thickening fluids or dilatant fluids: The more the fluid is sheared, the more viscous it becomes.

When an engineer falls into quicksand (a dilatant fluid), the faster he tries to move, the more viscous the fluid becomes. 22

135

Derivation of the Navier–Stokes Equation for Incompressible, Isothermal Flow

The incompressible flow approximation implies constant density, and the isothermal approximation implies constant viscosity.

23

Strain rate tensor

24

136

The Laplacian operator, shown here in both Cartesian and cylindrical coordinates, appears in the viscous term of the incompressible Navier– Stokes equation.

25

The Navier–Stokes equation is an unsteady, nonlinear, secondorder, partial differential equation. Equation 9–60 has four unknowns (three velocity components and pressure), yet it represents only three equations (three components since it is a vector equation). The Navier–Stokes equation is the cornerstone of fluid mechanics.

Obviously we need another equation to make the problem solvable. The fourth equation is the incompressible continuity equation (Eq. 9– 16).

137

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Continuity and Navier–Stokes Equations in Cartesian Coordinates

27

Continuity and Navier–Stokes Equations in Cylindrical Coordinates

28

138

An alternative form for the first two viscous terms in the r- and Tcomponents of the Navier–Stokes equation.

29

9–6 Ŷ DIFFERENTIAL ANALYSIS OF FLUID FLOW PROBLEMS There are two types of problems for which the differential equations (continuity and Navier–Stokes) are useful: • Calculating the pressure field for a known velocity field (This lecture) • Calculating both the velocity and pressure fields for a flow of known geometry and known boundary conditions (Next lecture)

A general three-dimensional but incompressible flow field with constant properties requires four equations to solve for four unknowns. 139

30

Calculation of the Pressure Field for a Known Velocity Field •

The first set of examples involves calculation of the pressure field for a known velocity field.

•

Since pressure does not appear in the continuity equation, we can theoretically generate a velocity field based solely on conservation of mass.

•

However, since velocity appears in both the continuity equation and the Navier–Stokes equation, these two equations are coupled.

•

In addition, pressure appears in all three components of the Navier–Stokes equation, and thus the velocity and pressure fields are also coupled.

•

This intimate coupling between velocity and pressure enables us to calculate the pressure field for a known velocity field. 31

32

140

33

Fig 9-46 For a two-dimensional flow field in the xy-plane, crossdifferentiation reveals whether pressure P is a smooth function. 34

141

35

The velocity field in an incompressible flow is not affected by the absolute magnitude of pressure, but only by pressure differences. Since pressure appears only as a gradient in the incompressible Navier–Stokes equation, the absolute magnitude of pressure is not relevant—only pressure differences matter.

Filled pressure contour plot, velocity vector plot, and streamlines for downward flow of air through a channel with blockage: (a) case 1; (b) case 2—identical to case 1, except P is everywhere increased by 500 Pa. On the colour-scale contour plots, blue is low pressure and red is high pressure. 142

36

Will be discussed in the spring course MSK600 CFD

Fig 9-49 Streamlines and velocity profiles for a line vortex. 143

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40

144

Fig 9-50 For a two-dimensional flow field in the rT-plane, crossdifferentiation reveals whether pressure P is a smooth function.

41

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Fig 9-51.The two-dimensional line vortex is a simple approximation of a tornado; the lowest pressure is at the center of the vortex.

43

Summary •

The differential linear momentum equation-Cauchy’s equation 9 9 9 9

Derivation Using the Divergence Theorem Derivation Using an Infinitesimal Control Volume Alternative Form of Cauchy’s Equation Derivation Using Newton’s Second Law

• The Navier-Stokes equation 9 Introduction 9 Newtonian versus Non-Newtonian Fluids 9 Derivation of the Navier–Stokes Equation for Incompressible, Isothermal Flow 9 Continuity and Navier–Stokes Equations in Cartesian Coordinates 9 Continuity and Navier–Stokes Equations in Cylindrical Coordinates 9 Calculation of the Pressure Field for a Known Velocity Field 44

146



Objectives • Obtain analytical solutions of the equations of motion for simple flow fields

2

147

Outline • Differential analysis of fluid flow problems 9 Exact Solutions of the Continuity and Navier– Stokes Equations • Fully developed Couette flow • Vertical wall oil film • Fully developed flow in a round pipe - Poiseuille flow • Sudden motion of an infinite flat plate

3

9–6 Ŷ DIFFERENTIAL ANALYSIS OF FLUID FLOW PROBLEMS There are two types of problems for which the differential equations (continuity and Navier–Stokes) are useful: • Calculating the pressure field for a known velocity field (Previous lecture) • Calculating both the velocity and pressure fields for a flow of known geometry and known boundary conditions (This lecture)

A general three-dimensional but incompressible flow field with constant properties requires four equations to solve for four unknowns. 148

4

Continuity and Navier–Stokes Equations in Cartesian Coordinates

5


6

149

Exact Solutions of the Continuity and Navier–Stokes Equations

Procedure for solving the incompressible continuity and Navier–Stokes equations.

Boundary Conditions

A piston moving at speed VP in a cylinder. A thin film of oil is sheared between the piston and the cylinder; a magnified view of the oil film is shown. The no-slip boundary condition requires that the velocity of fluid adjacent to a wall equal that of the wall. 7

At an interface between two fluids, the velocity of the two fluids must be equal. In addition, the shear stress parallel to the interface must be the same in both fluids.

Along a horizontal free surface of water and air, the water and air velocities must be equal and the shear stresses must match. However, since Pair

Pressure may change along a boundary layer (x-direction), but the change in pressure across a boundary layer (y-direction) is negligible.

Highly magnified view of the boundary layer along the surface of a body, showing that velocity component v is much smaller than u.

213

10

The pressure in the irrotational region of flow outside of a boundary layer can be measured by static pressure taps in the surface of the wall. Two such pressure taps are sketched.

(10-67)

11

ࢾ̱

࢞ ࢂ࢞ൗ ࣏

Outer flow speed parallel to the wall is U(x) and is obtained from the outer flow pressure, P(x). This speed appears in the x-component of the boundary layer momentum equation, Eq. 10–70. 12

214

The boundary layer equation set is parabolic, so boundary conditions need to be specified on only three sides of the flow domain. 13

The Boundary Layer Procedure

Fig 10-93 Summary of the boundary layer procedure for steady, incompressible, two-dimensional boundary layers in the xy-plane. 14

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K

y U

Q x

d2 f ' d f' f 0 dK 2 dK df df 0, 1 dK K 0 dK K of 2

17

ĺ

ĺ

ĺ

18

217

K

y U

Q x wu wy

u Uf ' Ww

P

du dy

PU y 0

Ww

0.332 PU

Ww

0.332 UU 2

Ww

0.332

UU

U

U df ' Q x dK

U df ' Q x dK K

UU Px P UUx

0

0.332 UU 2 0.332

UU P 2 U 2U 2P x UU 2 UUx

P

2

Re x

19

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Displacement Thickness

Displacement thickness defined by a streamline outside of the boundary layer. Boundary layer thickness is exaggerated.

For a laminar flat plate boundary layer, the displacement thickness is roughly one-third of the 99 percent boundary layer thickness. 21

The boundary layer affects the irrotational outer flow in such a way that the wall appears to take the shape of the displacement thickness. The apparent U(x) differs from the original approximation because of the “thicker” wall.

Displacement thickness is the imaginary increase in thickness of the wall, as seen by the outer flow, due to the effect of the growing boundary layer.

The effect of boundary layer growth on flow entering a two-dimensional channel: the irrotational flow between the top and bottom boundary layers accelerates as indicated by (a) actual velocity profiles, and (b) change in apparent core flow due to the displacement thickness of the boundary layer (boundary layers greatly exaggerated for clarity).

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Momentum Thickness

A control volume is defined by the thick red dashed line, bounded above by a streamline outside of the boundary layer, and bounded below by the flat plate; FD, x is the viscous force of the plate acting on the control volume.

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Turbulent Flat Plate Boundary Layer Illustration of the unsteadiness of a turbulent boundary layer; the thin, wavy black lines are instantaneous profiles, and the thick blue line is a long time-averaged profile.

All turbulent expressions discussed here represent time-averaged values. One common empirical approximation for the timeaveraged velocity profile of a turbulent flat plate boundary layer is the one seventh-power law

Fig-10-113 Comparison of laminar and turbulent flat plate boundary layer profiles, non-dimensionalized by boundary layer thickness.

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Another common approximation is the log law, a semi-empirical expression that turns out to be valid not only for flat plate boundary layers but also for fully developed turbulent pipe flow velocity profiles. The log law turns out to be applicable for nearly all wall-bounded turbulent boundary layers, not just flow over a flat plate. The log law is commonly expressed in variables nondimensionalized by a characteristic velocity called the friction velocity u*.

A clever expression that is valid all the way to the wall is called Spalding’s law of the wall,

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Boundary Layers with Pressure Gradients When the flow in the inviscid and/or irrotational outer flow region (outside of the boundary layer) accelerates, U(x) increases and P(x) decreases. We refer to this as a favorable pressure gradient. It is favorable or desirable because the boundary layer in such an accelerating flow is usually thin, hugs closely to the wall, and therefore is not likely to separate from the wall.

Boundary layers with nonzero pressure gradients occur in both external flows and internal flows: (a) boundary layer developing along the fuselage of an airplane and into the wake, and (b) boundary layer growing on the wall of a diffuser (boundary layer thickness exaggerated in both cases).

When the outer flow decelerates, U(x) decreases, P(x) increases, and we have an unfavorable or adverse pressure gradient. As its name implies, this condition is not desirable because the boundary layer is usually thicker, does not hug closely to the wall, and is much more likely to separate 36 from the wall. 226

The boundary layer along a body immersed in a free stream is typically exposed to a favorable pressure gradient in the front portion of the body and an adverse pressure gradient in the rear portion of the body.

The closed streamline indicates a region of recirculating flow called a separation bubble.

Examples of boundary layer separation in regions of adverse pressure gradient: (a) an airplane wing at a moderate angle of attack, (b) the same wing at a high angle of attack (a stalled wing), and (c) a wide-angle diffuser in which the boundary layer cannot remain attached and separates on one side.

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CFD calculations of flow over a two dimensional bump: (a) solution of the Euler equation with outer flow streamlines plotted (no flow separation), (b) laminar flow solution showing flow separation on the downstream side of the bump,

39

CFD calculations of flow over a twodimensional bump: (c) close-up view of streamlines near the separation point, and (d) close-up view of velocity vectors, same view as (c). The dashed line is a dividing streamline – fluid below this streamline is “trapped” in the recirculating separation bubble.

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Flow visualization comparison of laminar and turbulent boundary layers in an adverse pressure gradient; flow is from left to right. (a) The laminar boundary layer separates at the corner, but (b) the turbulent one does not. Photographs taken by M. R. Head in 1982 as visualized with titanium tetrachloride. 41

CFD calculation of turbulent flow over the same bump as that of Fig. 10– 124. Compared to the laminar result of Fig. 10–124b, the turbulent boundary layer is more resistant to flow separation and does not separate in the adverse pressure gradient region in the rear portion of the bump. The turbulent boundary layer remains attached (no flow separation), in contrast to the laminar boundary layer that separates off the rear portion of the bump. In the turbulent case, the outer flow Euler solution is a reasonable approximation over the entire bump since there is no flow separation and since the boundary layer remains very thin. 229

42

The Momentum Integral Technique for Boundary Layers In many practical engineering applications, we do not need to know all the details inside the boundary layer; rather we seek reasonable estimates of gross features of the boundary layer such as boundary layer thickness and skin friction coefficient. The momentum integral technique utilizes a control volume approach to obtain such quantitative approximations of boundary layer properties along surfaces with zero or nonzero pressure gradients. It is valid for both laminar and turbulent boundary layers.

Control volume (thick red dashed black line) used in derivation of the momentum integral equation.

43

Mass flow balance on the control volume of Fig. 10–127.

44

230

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231

Integration of a known (or assumed) velocity profile is required when using the Kármán integral equation.

47

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Summary •

THE BOUNDARY LAYER APPROXIMATION 9 9 9 9 9 9 9 9

Basics of Boundary Layers The Boundary Layer Equations The Boundary Layer Procedure Displacement Thickness Momentum Thickness Turbulent Flat Plate Boundary Layer Boundary Layers with Pressure Gradients The Momentum Integral Technique for Boundary Layers

50

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Chapter6 FUNDAMENTALS OF CONVECTION DIFFERENTIAL ENERGY EQUATION

Lecture 11 BjørnH.Hjertager

Based on powerpoints supplied for Heat and Mass Transfer: Fundamentals & Applications Fourth Edition, Yunus A. Cengel, Afshin J. Ghajar, McGraw-Hill, 2011

Objectives •

Understand the physical mechanism of convection and its classification

•

Visualize the development of velocity and thermal boundary layers during flow over surfaces

•

Gain a working knowledge of the dimensionless Reynolds, Prandtl, and Nusselt numbers

•

Derive the differential equations that govern convection on the basis of mass, momentum, and energy balances, and solve these equations for some simple cases such as laminar flow over a flat plate

•

Non-dimensionalize the convection equations and obtain the functional forms of friction and heat transfer coefficients

•

Use analogies between momentum and heat transfer, and determine heat transfer coefficient from knowledge of friction coefficient

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Outline •

PhysicalMechanismofConvection – Nusselt Number

•

VelocityBoundaryLayer – Wallshearstress

•

ThermalBoundaryLayer – Prandtl Number

•

DerivationofDifferentialEnergyEquations – Mechanical and – Thermal

• SolutionsofConvectionEquationsforaFlatPlate • NonͲdimensionalized ConvectionEquationsandSimilarity • FunctionalFormsofFrictionandConvectionCoefficients • AnalogiesBetweenMomentumandHeatTransfer

3

PHYSICAL MECHANISM OF CONVECTION Conduction and convection both require the presence of a material medium but convection requires fluid motion. Convection involves fluid motion as well as heat conduction. Heat transfer through a solid is always by conduction. Heat transfer through a fluid is by convection in the presence of bulk fluid motion and by conduction in the absence of it. Therefore, conduction in a fluid can be viewed as the limiting case of convection, corresponding to the case of quiescent fluid. 4

236

The fluid motion enhances heat transfer, since it brings warmer and cooler chunks of fluid into contact, initiating higher rates of conduction at a greater number of sites in a fluid. The rate of heat transfer through a fluid is much higher by convection than it is by conduction. In fact, the higher the fluid velocity, the higher the rate of heat transfer.

Heat transfer through a fluid sandwiched between two parallel plates.

5

Convection heat transfer strongly depends on the fluid properties dynamic viscosity, thermal conductivity, density, and specific heat, as well as the fluid velocity. It also depends on the geometry and the roughness of the solid surface, in addition to the type of fluid flow (such as being streamlined or turbulent). Newton’s law of cooling

Convection heat transfer coefficient, h: The rate of heat transfer between a solid surface and a fluid per unit surface area per unit temperature difference. 6

237

No-slip condition: A fluid in direct contact with a solid “sticks” to the surface due to viscous effects, and there is no slip. Boundary layer: The flow region adjacent to the wall in which the viscous effects (and thus the velocity gradients) are significant. The fluid property responsible for the no-slip condition and the development of the boundary layer is viscosity.

The development of a velocity profile due to the no-slip condition as a fluid flows over a blunt nose.

A fluid flowing over a stationary surface comes to a complete stop at the surface because of the no-slip condition. 7

An implication of the no-slip condition is that heat transfer from the solid surface to the fluid layer adjacent to the surface is by pure conduction, since the fluid layer is motionless, and can expressed by the Fourier’s law as

The determination of the convection heat transfer coefficient when the temperature distribution within the fluid is known

The convection heat transfer coefficient, in general, varies along the flow (or x-) direction. The average or mean convection heat transfer coefficient for a surface in such cases is determined by properly averaging the local convection heat transfer coefficients over the entire surface area As or length L as

8

238

Nusselt Number In convection studies, it is common practice to nondimensionalize the governing equations and combine the variables, which group together into dimensionless numbers in order to reduce the number of total variables. Nusselt number: Dimensionless convection heat transfer coefficient

Lc characteristic length

Heat transfer through a fluid layer of thickness L and temperature difference 'T.

The Nusselt number represents the enhancement of heat transfer through a fluid layer as a result of convection relative to conduction across the same fluid layer. The larger the Nusselt number, the more effective the convection. A Nusselt number of Nu = 1 for a fluid layer represents heat transfer across the layer by pure conduction. 9

Convection in daily life •

We turn on the fan on hot summer days to help our body cool more effectively. The higher the fan speed, the better we feel.

•

We stir our soup and blow on a hot slice of pizza to make them cool faster.

•

The air on windy winter days feels much colder than it actually is.

•

The simplest solution to heating problems in electronics packaging is to use a large enough fan.

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240

VELOCITY BOUNDARY LAYER Velocity boundary layer: The region of the flow above

the plate bounded by G in which the effects of the viscous shearing forces caused by fluid viscosity are felt. The boundary layer thickness, G, is typically defined as the distance y from the surface at which u = 0.99V. The hypothetical line of u = 0.99V divides the flow over a plate into two regions: Boundary layer region: The viscous effects and the velocity changes are significant. Irrotational flow region: The frictional effects are negligible and the velocity remains essentially constant.

13

THERMAL BOUNDARY LAYER A thermal boundary layer develops when a fluid at a specified temperature flows over a surface that is at a different temperature. Thermal boundary layer: The flow region over the surface in which the temperature variation in the direction normal to the surface is significant. The thickness of the thermal boundary layer Gt at any location along the surface is defined as the distance from the surface at which the temperature difference T í Ts equals 0.99(Tfí Ts). The thickness of the thermal boundary layer increases in the flow direction, since the effects of heat transfer are felt at greater distances from the surface further down stream.

Thermal boundary layer on a flat plate (the fluid is hotter than the plate surface). 241

The shape of the temperature profile in the thermal boundary layer dictates the convection heat transfer between a solid surface and the fluid flowing over it. 14

Prandtl Number The relative thickness of the velocity and the thermal boundary layers is best described by the dimensionless parameter Prandtl number

The Prandtl numbers of gases are about 1, which indicates that both momentum and heat dissipate through the fluid at about the same rate. Heat diffuses very quickly in liquid metals (Pr > 1) relative to momentum. Consequently the thermal boundary layer is much thicker for liquid metals and much thinner for oils relative to the velocity boundary layer.

15

Reynolds Number

At large Reynolds numbers, the inertial forces, which are proportional to the fluid density and the square of the fluid velocity, are large relative to the viscous forces, and thus the viscous forces cannot prevent the random and rapid fluctuations of the fluid (turbulent). At small or moderate Reynolds numbers, the viscous forces are large enough to suppress these fluctuations and to keep the fluid “in line” (laminar).

The transition from laminar to turbulent flow depends on the geometry, surface roughness, flow velocity, surface temperature, and type of fluid. The flow regime depends mainly on the ratio of inertia forces to viscous forces (Reynolds number).

Critical Reynolds number, Recr: The Reynolds number at which the flow becomes turbulent. The value of the critical Reynolds number is different for different geometries and flow conditions. The Reynolds number can be viewed as the ratio of inertial forces to viscous forces acting on a fluid element. 242

16

Differential equation of energy change • Mechanical energy equation • Thermal energy equation (1st law of Thermodynamics)

17

Mechanical energy equation • Conservation of kinetic energy (ȡ v2 /2) The momentum equation is the Newtons 2. law for flowing system: d mv v = i u j v k w = i u1 j u2 k u3 dt The dot product becomes:

K=

K v =v

d mv dt

v 2 = u 2 v 2 w2

d §1 2· ¨ mv ¸ dt © 2 ¹ u12 u22 u32

ui ui = ui2

Tensor notation

The momentum balance : w U v U v v w t

d mv dt

p W ij U g

K

Volume

Volume

243

18

The momentum balance in tensor notation: w w U ui U u jui wt wx j

d mv dt

wp wW ji U gi wxi wx j

K

Volume

Volume

Taking the dot product means multiplying with ui and use ui dui

d(1 / 2 ui2 )

The results becomes: w §1 2· w ¨ U ui ¸ wt © 2 ¹ wx j

wW ji wp ui ui U ui gi wxi wx j

§1 2· u u U j i ¸ ¨ ©2 ¹

19

Apply the following change for the two terms on the rigth hand side: w pui wxi w W ji ui wx j

ui

wu wp p i wxi wxi

ui

wW ji wx j

W ji

wui wx j

ui

wu w pui p i wxi wxi

wp wxi

ui

wW ji wx j

wu w W ji ui W ji i wx j wx j

Insert and get

20

244

Insert and get the mechanical energy equation in differential form: w §1 2· w §1 2· ¨ U ui ¸ ¨ U u jui ¸ wt © 2 ¹ wx j © 2 ¹

Transient term

Convection term

w pui wxi

§ wui · ¨p ¸ wxi ¹ ©

Work done by the pressure

Reversible pressure x volume work

w W ji ui wx j

Viscous work

§ wui · ¨¨W ji ¸¸ x w j ©

¹ Irreversible viscous work that goes to heat

U ui gi N Work against the gravity

This energy equation will be used later for the thermal energy equation. 21

Derivation of differential equation of energy change • The 1. law of thermodynamics is the conservation of energy equation which we can write as

(A)

• We will make a balance on an element (CV) as we did for the continuity and momentum equations in Lectures 5 and 6 • We shall consider the two types of energy – Internal energy e [J/kg] – Kinetic energy ȡ ui2 /2 [J/kg]

• Note that the energy equation is a scalar not vector equation! 245

22

• If we look at the first terms in Equation (A) we note that energy may enter and leave the CV by two mechanisms: – Convection ĺ bulk transport • (ȡ uj) x Area x ( e + ui2 /2 ) [J/s] • Here (ȡ uj) is the mass flux in j-direction that brings energy into or out of the CV

– Molecular transport or conductionĺ due to gradients in the temperature field ĺ heat flux • qi x Area [J/s] • Here index i indicate heat flux component transported in the idirection

• We will start with the Convection contribution to the energy equation and follow with Molecular contribution 23

• Convective energy transport ª U v ¬

e+ 1

2 u i2 º¼

ª U u e + 1 2 u i2 º ¬ ¼

y 'y

ª U w e + 1 2 u i2 º ¬ ¼

ª U u e + 1 2 u i2 º ¬ ¼

x

ª U w e + 1 2 u ¬

ª Rate of º « energy in » ¬ ¼

^¬ª Uu e+1 2 u º¼ 2 i

^

x

z 'z

2 i

º¼

ª U v e + 1 2 u i2 º ¬ ¼

x 'x

y

z

ª Rate of º « energy out » : ¬ ¼

ª¬ U u e+1 2 ui2 º¼

x 'x

` 'y'z

`

¬ª U v e+1 2 ui2 ¼º ¬ª U v e+1 2 ui2 ¼º 'x'z y y 'y

^

`

ª¬ U w e+1 2 ui2 º¼ ª¬ U w e+1 2 ui2 º¼ 'x'y z z 'z 246

(B)

24

• Molecular energy (heat) transport qy

qz

y 'y

qx

z 'z

qx

x

x 'x

qx is the heat flux in x-direction

qy

qz

y

z

ª Rate of º ª Rate of º «energy (heat) in » « energy (heat) out » : ¬ ¼ ¬ ¼

^q

x x

^

qx

x 'x

` 'y'z

qy qy

y 'y

^qz z qz

z 'z

y

` 'x'z

(C)

` 'x'y

25

• Work done by system on surroundings • There are three parts of work that are done – Work by the gravity – Work done by the surface forces: • Work done by pressure • Work done by the viscous forces (shear stresses)

26

247

• Pressure and gravity work

pu

g xu

pu

x

x 'x

ª Work done by º « pressure forces » : « » «¬on surroundings »¼ º 'x'z ª pw pw ª pu pu º 'y 'z ª pv pv º 'y 'x z z x ' ¬ ¼ x 'x ¼ ¬ x «¬ y y 'y » ¼ ª Work done by º «gravity forces » : ªug vg wg º U'x'y 'z y z¼ « » ¬ x «¬on surroundings »¼

(D)

(E) 27

• Work from viscous forces (stresses) W z

ª Work done by º « viscous forces » : y « » «¬ on surroundings »¼

W xy u

xz

u

z 'z

y 'y

W xx u

W xy u

x

W xz u

W xx u

x 'x

y

z

x

^

`

'y 'z ª¬W xx u W yx v W zx wº¼ ª¬W xx u W yx v W zx wº¼ x 'x x

^

`

'x'z ª¬W xy u W yy v W zy wº¼ ª¬W xy u W yy v W zy wº¼ y 'y y ° ª º ª º ½° 'x'y ® «W xz u W yz v W zz w» «W xz u W yz v W zz w» ¾ N »¼ z 'z «¬ N »¼ z ¿° °¯ «¬ 248

(F) 28

• Accumulation (G) ª Accumulation of º w § ª 1 2º· 'x'y 'z ¨ U «e ui » ¸ «energy »: wt © ¬ 2 ¼ ¹ ¬ ¼ Insert contributions (B) to (G) into the energy balance (A);

devide with 'x'y 'z; and get: Accumulation of energy: w § ª 1 2º· ¨ U e ui ¸ wt © «¬ 2 »¼ ¹

29

Convective energy transport: ª U u e+1 2 ui2 º ª U u e+1 2 ui2 º ¬ ¼x ¬ ¼ x 'x 'x

ª U v e+1 2 ui2 º ª U v e+1 2 ui2 º ¬ ¼y ¬ ¼ y 'y 'y

ª U w e+1 2 ui2 º ª U w e+1 2 ui2 º ¬ ¼z ¬ ¼ z 'z 'z Heat transport:

qx x qx

x 'x

qy qy

'x Pressure work:

pu x pu 'x

x 'x

y

y 'y

'y pv y pv

y 'y

'y

qz z q z

z 'z

'z

pw z pw 'z

z 'z

30

249

Stress work ª¬W xx u W yx v W zx wº¼ ª¬W xx u W yx v W zx wº¼ x 'x x 'x ª¬W xy u W yy v W zy wº¼ ª¬W xy u W yy v W zy wº¼ y 'y y 'y ª¬W xz u W yz v W zz wº¼ ª¬W xz u W yz v W zz wº¼ z 'z z 'z Gravity work:

U ª¬u g x v g y w g z º¼

31

Let 'x, ǻy and ǻz o 0 using lim

'x o 0

f x 'x f x wf o and by 'x wx

using tensor notation we get: w § ª 1 2º· ¨ U «e ui » ¸ wt © ¬ 2 ¼ ¹

§ ª 1 2 º · wqi u U j ¨ «e 2 ui » ¸ wx ¬ ¼¹ © i w w pui W ij ui U ui gi wxi wx j

w wx j

32

250

• Final form

w § ª 1 2º· w § ª 1 2º· e u u U U i »¸ ¨ « ¨ j «e ui » ¸ wt © ¬ 2 ¼ ¹ wx j © ¬ 2 ¼¹

Accumulation of energy; transient term

Convection of energy: bulk transport

wq i wx Ni

Heat transferred; molecular energy transfer

w w U ui gi pui W ij ui N wx wx j Work done by i

the gravity

Work done by pressure Work done by the viscous forces

33

It is normal to subtract the mechanical energy equation from the thermal: w §1 2· w §1 2· u u u U U i ¸ j i ¸ ¨ ¨ wt © 2 ¹ wx j © 2 ¹

§ wu · w pui ¨ p i ¸ wxi wxi ¹ ©

§ wu · w W ji ui ¨W ji i ¸ U ui gi + ¨ wx ¸ wx j j ¹ © wqi w § ª 1 2º· w § ª 1 2º· U U e u u e u i i »¸ ¨ ¸ ¨ j« wt © «¬ 2 »¼ ¹ wx j © wxi ¬ 2 ¼¹

(A)

(B)

w w pui W ij ui U ui gi wxi wx j

The result (B) - (A) is: w w Ue Uu j e wt wx j

wqi wxi

wu p i wx i Reversible pressure x volume work

251

wui wx j

W ji

Irreversible work; always > 0

34

Alternative forms : It is also normal to introduce the enthalpy: e p ; e

h

h p

U

Insert and get:

w§ p · w ¨U h U ¸ wt © ¹ wxi

U

§ Uu h p · ¨ i U ¸¹ ©

wqi wu wu p i W ji i wxi wxi wx j

Rearrange and get: p

w w U h U ui h wt wxi

wui wp ui wxi wxi

wq wp w wu wu i ui p p i W ji i wxi wx j wxi wt wxi

Final equation: w w U h U ui h wt wxi

wqi wp wp ui wxi wt wx i

wu W ji i wx j

Reversible Irreversible work; volume x pressure always > 0 work

35

Alternative forms : It is also normal to introduce the temperature: h

³

T

Tref

c p dT

c pm (T Tref )

Insert and get: Dp Dt

w w U c pmT U ui c pmT wt wxi

wq wp wu wp i ui W ji i wxi wt wxi wx j

36

252

Fouriers law and Newtons law of viscosity are inserted: qi

k

wT ; W ji wxi

§ wui wu j · 2 wu G ij P i ; G ij ¸ ¨ wx ¸ wxi © j wxi ¹ 3

1 for i j ® ¯0 for i z j

P¨

Result is: w w U c pmT U ui c pmT wt wxi

w wxi

§ wT ¨k © wxi

· Dp P ) conservative form ¸ Dt ¹

Here ) is the socalled dissipation function expressed as: ª§ wu · 2 § wv · 2 § ww ·2 º ) 2 «¨ ¸ ¨ ¸ ¨ ¸ » «¬© wx ¹ © wy ¹ © wz ¹ »¼ ª§ wu wv · 2 § wv ww · 2 § ww wu · 2 º + «¨ ¸ ¨ ¸ » ¸ ¨ «¬© wy wx ¹ © wz wy ¹ © wx wz ¹ »¼ ) always ! 0

w w U c pmT U ui c pmT wt wxi

37

w § wT · Dp P ) ¨k ¸ wxi © wxi ¹ Dt

Differentiate the left side: § wU w · w w c pmT ¨ U ui ¸ U c pmT U ui c pmT wt wxi wt wxi ©

¹ { 0 from continuity equation

=

U U

w w c pmT U ui c pmT wt wxi D c pmT Dt

w § wT ¨k wxi © wxi

w wxi

§ wT ¨k © wxi

· Dp P ) ¸ Dt ¹

w wxi

§ wT ¨k © wxi

· Dp P ) ¸ Dt ¹

· Dp P ) ¸ Dt ¹

non-conservative form

38

253

Further assumptions c pm k p are assumed constant; devide by U c pm and get: wT wT ui wt wxi

k w 2T P ) U c pm wxi wxi U c pm N D

For small velocities ) | 0 Solid material u i wT wt

0

w 2T wT written out D wxi wxi wt

§ w 2T w 2T w 2T · D¨ 2 2 2 ¸ wy wz ¹ © wx

or wT wt

D 2T Heat conduction equation 39

Summary of mass, momentum and energy conservation equations for constant P , cpm , k and ȡ : Mass: wui wxi

0

Momentum (Navier Stokes): wu wu U i Uu j i wt wx j

w 2 ui wp P U gi wxi wxi wxi

Energy: wT wT Uu j U wt wx j

k w 2T P ) c pm wxi wxi c pm 40

254

41

42

255

43

44

256

SOLUTIONS OF CONVECTION EQUATIONS FOR A FLAT PLATE

45

46

257

47

48

258

The Energy Equation

49

50

259

51

NONDIMENSIONALIZED CONVECTION EQUATIONS AND SIMILARITY

52

260

53

FUNCTIONAL FORMS OF FRICTION AND CONVECTION COEFFICIENTS

54

261

55

56

262

ANALOGIES BETWEEN MOMENTUM AND HEAT TRANSFER

57

58

263

59

60

264

61

Some important results from convection equations The velocity boundary layer thickness

The local skin friction coefficient

Local Nusselt number The thermal boundary layer thickness

Reynolds analogy Modified Reynolds analogy or Chilton-Colburn analogy 62

265

Summary •

PhysicalMechanismofConvection – NusseltNumber

•

VelocityBoundaryLayer – Wallshearstress

•

ThermalBoundaryLayer – PrandtlNumber

•

DerivationofDifferentialEnergyEquations – MechanicalandThermal

• SolutionsofConvectionEquationsforaFlatPlate • NonͲdimensionalizedConvectionEquationsandSimilarity • FunctionalFormsofFrictionandConvectionCoefficients • AnalogiesBetweenMomentumandHeatTransfer

63

Dimensional analysis of the energy equation (optional material) Energy equation non-conservative form; c pm , P and k constant

U c pm

wT wT U ui c pm wt wxi

w 2T wp wp +ui P ) k wxi wxi wt wxi

(A)

Dissipationfunction )

ª§ wu · 2 § wv · 2 § ww · 2 º 2 «¨ ¸ ¨ ¸ ¨ ¸ » «¬© wx ¹ © wy ¹ © wz ¹ »¼ ª§ wu wv · 2 § wv ww ·2 § ww wu · 2 º + «¨ ¸ ¨ ¸ » ¸ ¨ «¬© wy wx ¹ © wz wy ¹ © wx wz ¹ »¼ § wu · § wu wu j · § wui wu j · 2¨ i ¸ ¨ i ¸¸ ¨¨ ¸¸ ¨ w x w x w x w x w x i ¹© j i ¹ © i¹ © j 2

)

266

64

Characterstic quantities: L; u c ; 'T

Tc

Normalised variables ui*

ui uc

ui

t*

tuc L

t

T*

T Tc

T

)*

) § uc · ¨L¸ © ¹

2

ui* uc ; t*

L ; uc

xi L

xi

xi* L;

p U uc2

p

p* U uc2 ;

xi* p*

T *Tc

)

§u · )* ¨ c ¸ ©L¹

2

65

Insert into (A):

U c pm

wT *Tc wT *Tc * U ui uc c pm * wxi L * L wt uc

w 2T *Tc k * wxi Li wxi* L

wp* U uc2 * wp* U uc2 * § uc · ) +u i v c P ¨L¸ L wxi* L © ¹ wt * uc * wT * * wT ui * wt * wxi

2

L U c pm ucTc

w 2T * U c pm uc L wxi*wxi* k

uc2 wp* uc2 * wp* P uc u + )* i * c pmTc wt c pmTc wxi U c pmTc L k

U c pm uc L

P

k ; U uc L P c pm NN 1

Re

1

P uc U c pm c pmTc L

P

267

P uc2

U vc L c pm P N kT N N c 1

Pr

k

Re

1

Br

Pr

66

Insert dimensionless numbers : * wT * * wT ui * wt * wxi

1 w 2T * Re Pr wxi*wxi* uc2 c pmTc

Re

U uc L ; P

Pr

P c pm k

§ wp* * wp* · Br )* +u ¨ * ¸ i wxi ¹ Re Pr © wt ;

Brinkman number Br reads:

Br

P uc2 kTc

uc2 P 2 L kTc L2

viscous dissipation conduction heat transfer 67

Introduce the Eckert number, Ec: Ec

uc2 c pmTc

kinetic energy thermal energy

Insert and get final dimensionless equation: * wT * * wT ui * wt * wxi

w 2T * 1 Re Pr wxi*wxi* § wp* * wp* · Br Ec ¨ * +u i )* ¸ wxi ¹ Re Pr © wt 68

268

Chapter 15 INTRODUCTION TO COMPUTATIONAL FLUID DYNAMICS Lecture 12 Bjørn H. Hjertager


Flow over a male swimmer simulated using the ANSYS® FLUENT® CFD code. The image shows simulated oil flow lines along the surface of the body. Flow separation in the region of the neck is visible. 269

2

Objectives • Understand the importance of a high-quality, good resolution mesh • Apply appropriate boundary conditions to computational domains • Understand how to apply CFD to basic engineering problems and how to determine whether the output is physically meaningful • Realize that you need much further study and practice to use CFD successfully (There will be a 5 ECTS CFD course (MSK 600) available in the spring semester)

3

Outline •

•

•

•

INTRODUCTION AND FUNDAMENTALS 9 Motivation 9 Equations of Motion 9 Solution Procedure 9 Grid Generation and Grid Independence 9 Boundary Conditions 9 Practice Makes Perfect LAMINAR CFD CALCULATIONS 9 Pipe Flow Entrance Region at Re = 500 9 Flow around a Circular Cylinder at Re = 150 TURBULENT CFD CALCULATIONS 9 Flow around a Circular Cylinder at Re = 10,000 9 Flow around a Circular Cylinder at Re = 107 CFD WITH HEAT TRANSFER 9 Temperature Rise through a Cross-Flow Heat Exchanger 9 Cooling of an Array of Integrated Circuit Chips 4

270

15–1 Ŷ INTRODUCTION AND FUNDAMENTALS Motivation Modern engineers apply both experimental and CFD analyses, and the two complement each other. For example, engineers may obtain global properties, such as lift, drag, pressure drop, or power, experimentally, but use CFD to obtain details about the flow field, such as shear stresses, velocity and pressure profiles, and flow streamlines. In addition, experimental data are often used to validate CFD solutions by matching the computationally and experimentally determined global quantities. CFD is then employed to shorten the design cycle through carefully controlled parametric studies, thereby reducing the required amount of experimental testing.

5

Equations of Motion Continuity equation 15-1 Navier–Stokes equation 15-2

6

271

Solution Procedure 1. A computational domain is chosen, and a grid (also called a mesh) is generated; the domain is divided into many small elements called cells. 2. Boundary conditions are specified on each edge of the computational domain (2-D flows) or on each face of the domain (3-D flows). 3. The type of fluid (water, air, gasoline, etc.) is specified, along with fluid properties (temperature, density, viscosity, etc.). 4. Numerical parameters and solution algorithms are selected. 5. Starting values for all flow field variables are specified for each cell. 6. Beginning with the initial guesses, discretized forms of Eqs. 15–1 and 15–2 are solved iteratively, usually at the center of each cell. 7. Once the solution has converged, flow field variables such as velocity and pressure are plotted and analyzed graphically. 8. Global properties of the flow field, such as pressure drop, and integral properties, such as forces (lift and drag) and moments acting on a body, are calculated from the converged solution

7

A quality grid is essential to a quality CFD simulation. 272

8

9

Grid Generation and Grid Independence

Fewer cells in structured than unstuctured grids: 1) 32; a) 76; b) 38 cells 273

10

11

12

274

13

Boundary Conditions Appropriate boundary conditions are required in order to obtain an accurate CFD solution.

14

275

Wall Boundary Conditions The simplest boundary condition is that of a wall. Since fluid cannot pass through a wall, the normal component of velocity is set to zero relative to the wall along a face on which the wall boundary condition is prescribed. In addition, because of the no-slip condition, we usually set the tangential component of velocity at a stationary wall to zero as well.

15

Inflow/Outflow Boundary Conditions There are several options at the boundaries through which fluid enters the computational domain (inflow) or leaves the domain (outflow). They are generally categorized as either velocity-specified conditions or pressurespecified conditions. At a velocity inlet, we specify the velocity of the incoming flow along the inlet face. If energy and/or turbulence equations are being solved, the temperature and/or turbulence properties of the incoming flow need to be specified as well.

16

276

17

Miscellaneous Boundary Conditions Some boundaries of a computational domain are neither walls nor inlets or outlets, but rather enforce some kind of symmetry or periodicity. For example, the periodic boundary condition is useful when the geometry involves repetition. Periodic boundary conditions must be specified as either translational (periodicity applied to two parallel faces, or rotational (periodicity applied to two radially oriented faces).

18

277

19

Internal Boundary Conditions The final classification of boundary conditions is imposed on faces or edges that do not define a boundary of the computational domain, but rather exist inside the domain. When an interior boundary condition is specified on a face, flow crosses through the face without any user-forced changes, just as it would cross from one interior cell to another. This boundary condition is necessary for situations in which the computational domain is divided into separate blocks or zones, and enables communication between blocks.

20

278

21

15–2 Ŷ LAMINAR CFD CALCULATIONS Computational fluid dynamics does an excellent job at computing incompressible, steady or unsteady, laminar flow, provided that the grid is well resolved and the boundary conditions are properly specified. We show several simple examples of laminar flow solutions, paying particular attention to grid resolution and appropriate application of boundary conditions. In all examples in this section, the flows are incompressible and two-dimensional (or axisymmetric).

22

279

Pipe Flow Entrance Region at Re = 500

23

24

280

25

Flow around a Circular Cylinder at Re 150

26

281

27

28

282

29

30

283

31

32

284

33

15–3 Ŷ TURBULENT CFD CALCULATIONS CFD simulations of turbulent flow are much more difficult than those of laminar flow, even for cases in which the flow field is steady in the mean (statisticians refer to this condition as stationary). The reason is that the finer features of the turbulent flow field are always unsteady and three-dimensional—random, swirling, vortical structures called turbulent eddies of all orientations arise in a turbulent flow. Some CFD calculations use a technique called direct numerical simulation (DNS), in which an attempt is made to resolve the unsteady motion of all the scales of the turbulent flow.

34

285

When using a turbulence model, the steady Navier–Stokes equation is replaced by what is called the Reynolds-averaged Navier–Stokes (RANS) equation, shown here for steady (stationary), incompressible, turbulent flow,

Specific Reynolds stress tensor

35

Many RANS turbulence models are available • • • •

Algebraic One-equation Two-equation Reynolds stress

The most popular are the two-equation models • •

k-İ k-Ȧ

These models calculates the Reynolds stresses by introducing the socalled turbulent viscosity which can be determined by the two variables (k) and (eg. İ or Ȧ) These two variables are determined by modelled transport equations deduced from the Navier-Stokes equations

36

286

37

Flow around a Circular Cylinder at Re = 10,000

a) CD =0.647 Į=140 deg b) CD =0.742 Į=104 deg c) CD =0.753 Į=102 deg Experiments CD =1.15 Į=82 deg 38

287

39

Flow around a Circular Cylinder at Re = 107

40

288

41

15–4 Ŷ CFD WITH HEAT TRANSFER By coupling the differential form of the energy equation with the equations of fluid motion, we can use a computational fluid dynamics code to calculate properties associated with heat transfer (e.g., temperature distributions or rate of heat transfer from a solid surface to a fluid). Since the energy equation is a scalar equation, only one extra transport equation (typically for either temperature or enthalpy) is required, and the computational expense (CPU time and RAM requirements) is not increased significantly. Heat transfer capability is built into most commercially available CFD codes, since many practical problems in engineering involve both fluid flow and heat transfer. As mentioned previously, additional boundary conditions related to heat transfer need to be specified. 289

42

43

Temperature Rise through a Cross-Flow Heat Exchanger

44

290

45

46

291

Cooling of an Array of Integrated Circuit Chips

47

Chip surface limits: • Average maximum surface temperature 150 deg C • Maximum surface temperature 180 deg C

48

292

49

50

293

51

Summary •

•

•

•

INTRODUCTION AND FUNDAMENTALS 9 Motivation 9 Equations of Motion 9 Solution Procedure 9 Grid Generation and Grid Independence 9 Boundary Conditions 9 Practice Makes Perfect LAMINAR CFD CALCULATIONS 9 Pipe Flow Entrance Region at Re = 500 9 Flow around a Circular Cylinder at Re = 150 TURBULENT CFD CALCULATIONS 9 Flow around a Circular Cylinder at Re = 10,000 9 Flow around a Circular Cylinder at Re = 107 CFD WITH HEAT TRANSFER 9 Temperature Rise through a Cross-Flow Heat Exchanger 9 Cooling of an Array of Integrated Circuit Chips

52

294

Fluid Dynamics

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