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- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Sprache: Englisch
INTRODUCTION TO FLUID DYNAMICS A concise resource that presents a physics-based introduction to fluid dynamics and helps students bridge the gap between mathematical theory and real-world physical properties Introduction to Fluid Dynamics offers a unique physics-based approach to fluid dynamics. Instead of emphasizing specific problem-solving methodologies, this book explains and interprets the physics behind the theory, which helps mathematically-inclined students develop physical intuition while giving more physically-inclined students a better grasp of the underlying mathematics. Real-world examples and end-of-chapter practice problems are included to further enhance student understanding. Written by a highly-qualified author and experienced educator, topics are covered in a progressive manner, enabling maximum reader comprehension from start to finish. Sample topics covered in the book include: * How forces originate in fluids * How to define pressure in a fluid in motion * How to apply conservation laws to deformable substances * How viscous stresses are related to strain rates * How centrifugal forces and viscosity play a role in curved motions and vortex dynamics * How vortices and centrifugal forces are related in external viscous flows * How energy is viscously dissipated in internal viscous flows * How compressibility is related to wave and wave speed Students and instructors in advanced undergraduate or graduate fluid dynamics courses will find immense value in this concise yet comprehensive resource. It enables readers to easily understand complex fluid phenomena, regardless of the academic background they come from.
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Introduction to Fluid Dynamics
Understanding Fundamental Physics
Young J. Moon
Korea University
Seoul, Korea, Rep. of
This edition first published 2022© 2022 John Wiley & Sons, Inc.
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Preface
Fluid dynamics has a long history of more than 400 years. Although it is a classical subject, fundamental physics is neither easy to understand nor simple to convey to students. This book aims to guide undergraduate and beginning graduate students in engineering and science to understand the fundamental physics of fluid dynamics. To inspire students to be curious about the physics, this book begins with questions on the nature of forces; how forces originate and get distributed in the field, how fluids react to these forces with inertia, and how fluid motions are related to hydrostatic and viscous stresses. To stimulate the readers' curiosity, an elementary level of geophysical fluid dynamics, biofluid mechanics, and aeroacoustics are introduced, and some of the problem sets are carefully selected to illustrate flow physics often encountered in nature such as the Great Red Spot on Jupiter, jet streams, flying frogs, dragon lizards (Draco), frogfish, whales, crab nebula in the constellation of Taurus, interstellar bow shock wave (Zeta Ophiuchi).
In this book, the fundamentals of fluid dynamics are covered in Chapters 1, 2, and 3: pressure, macroscopic balance of mass, momentum, and energy, and differential equations of motions. To provide an in-depth physical explanation of the basic concepts, each subject is dealt with the physics of fluid motion and their relevances to the conservation laws of mass, momentum, and energy. Chapter 1 focuses specifically on discussing what is the nature of pressure and how pressure varies with inertia in transient and steady flows. Chapter 2 uses the concept of fluid particles to explain how the laws of conservation of mass, momentum, and energy apply to fluids. Chapter 3 covers the fundamental physics of viscous flow, that is, the deviatoric nature of viscous stresses and strain rates. The Navier–Stokes equations are derived based on the constitutive relation between the viscous stress and the strain rate.
Chapters 4 and 5 discuss the general concepts of physics in curved motion and vortex dynamics, in which the centrifugal force coupled with viscosity governs the dynamics. To discover the complex nature of vortical flows, the Lamb acceleration vector and its divergence and curl are brought to our attention. The divergence and curl of the Lamb vector are of great importance in describing how rotating fluids do viscous shear work or dissipate energy at a rate in a shear-straining field, or how vortices change the strength while being convected, stretched, tilted, or isotropically expanded. Many curved flows in aerodynamics, geophysical fluid dynamics, and biofluid dynamics are introduced from the perspective of vortex dynamics.
In Chapters 6 and 7, the basic physical concepts covered in Chapters 1, 2, 3, 4, and 5 are practiced for various external and internal viscous flows. Chapter 6 discusses the physics of boundary layers, stability and transition, turbulence, separation, drag and lift, aerodynamics, and gliding. Chapter 7 focuses on the hydrodynamic resistance of internal viscous flows with energy supply and frictional losses. Related topics include frictional losses in pipe and duct flow, local losses in systems with flow separation and secondary flow, and valves. Biofluid mechanics share the same interests as internal viscous flows in topics such as blood vessels, branches, and stenosis (or aneurysms), but differs only in scale (e.g. mosquito food canal, fish gill).
Lastly, Chapter 8 covers the compressible flow physics of sound waves, Mach waves, shock waves, and isentropic flows in rocket nozzles. The fundamental physics of each topic is based on the speed of sound, the Doppler effect, and the volumetric dilatational rate, that is, the rate of change of volume associated with the compression or expansion work done on a fluid at a rate. The physics of compressible flows can also be observed in our planetary systems. Waves in plasma, similar to waves in fluids, are brought for discussion on the relationship between waves and forces.
This book aims to introduce students to a simple yet unique methodology by which to learn fluid dynamics through an understanding of fundamental physics, and we will all do our best to achieve it.
SeoulJanuary 2022
Young J. Moon
Acknowledgments
In preparation of this book, various textbooks and materials are referenced. Among those, I would like to express my special appreciation to An Introduction to Fluid Dynamics by G.K. Batchelor, An Informal Introduction to Theoretical Fluid Mechanics by J. Lighthill, and Illustrated Experiments in Fluid Mechamics by National Committee for Fluid Mechanics Films (prefaced by A.H. Shapiro). I am also grateful to the undergraduate and graduate students of the School of Mechanical Engineering and also to the graduate students in the Computational Fluid Dynamics and Acoustics Laboratory, Korea University, Seoul, where I have lectured fluid dynamics for the past 30 years. I thank Minsung Kim for his assistance in preparing this manuscript. Lastly, I would like to extend my gratitude to my dear wife, Inmee, and my daughters, Min and Hee, for their constant support and advices.
About the Companion Website
This book is accompanied by a companion website.
www.wiley.com/go/Moon/IntroductiontoFluidDynamics
This website includes:
Instructor Manual
1Pressure
1.1 Microscopic Physical Properties
An element of a fluid or solid may be viewed as a continuous space with homogeneously distributed molecules. The differences between fluids and solids lay in the molecular spacings and intermolecular forces. In a solid phase, molecules are closely and orderly arranged, having a cohesive structure. Fluids have material properties that differ vastly from those of solids. Gas molecules are spaced far apart and disorderly arranged, and the cohesiveness among the molecules is very weak. In contrast, molecules in the liquid phase have partially ordered arrangements and a cohesiveness that lies in between those of solid and gas molecules, thus exhibiting mixed properties of both states of matter. Furthermore, liquid molecules are often grouped, which constantly break and reform when liquid elements are in relative motions.1 Many physical properties of liquids can be explained from this unique feature.
1.1.1 Continuum
Fluids are made up of billions and billions of molecules. For example, air at sea-level ( and ) contains molecules in a cube of dimensions , i.e. a volume of . If our scale of interest is larger than the distance between molecules (e.g. roughly 10 ), fluid properties such as density, pressure, temperature can be described by continuous functions of space and time; we then can say the medium is a continuum. With the continuum assumption, physical quantities can be defined at a point , representing an averaged value of the molecules surrounding that point.2
This continuum hypothesis is valid for a large-scale flow over an aircraft wing (e.g. chord length: 5 and span: 15 ). It even works for a small-scale flow over a mosquito wing (e.g. chord length: and span: ), since there exists roughly 0.1 million air molecules across the mosquito's wing chord. However, the continuum model fails for air at very high altitudes because the molecular spacings become too large (e.g. rarefied gas).
The validity of the continuum hypothesis is determined by the Knudsen number
where is the mean free path of the molecules and is the characteristic length. For example, = at the standard atmospheric condition. However, = at an elevation of , = at , and = at . If is less than 0.01, the medium can be assumed as a continuum.
1.2 Forces
In fluids, forces originate either from within fluid volumes or at external boundaries. In response to the forces, fluids move3 with inertia; thus, forces are distributed in the flow field, changing the momentum of the fluid at a rate. It is to be noted that this interaction occurs under the laws of conservation of mass, momentum, and energy.
1.2.1 Surface Forces
Surface forces act across a surface element via direct contacts of fluid molecules, and these forces defined by per unit area are called stresses. There are two different types of stresses: pressure (or hydrostatic) and viscous stresses. Pressure represents a measure of repulsiveness of molecules against compressive forces. It acts in the direction normal and inward to the surface of a unit area, and the magnitude is direction-independent (i.e. isotropic).
Meanwhile, viscous stress represents a measure of resistance of molecules when a fluid element is strained (or deformed) at a rate against frictions. Thus, the magnitude is proportional to the viscosity4 and the straining rate of the fluid. Due to the nature of frictional intermolecular interactions, a viscous stress vector acts on the surface at an angle, which depends on the orientation of the surface.5 It is a direction-dependent quantity (i.e. nonisotropic).
It is worth to note that the magnitude of the surface force is proportional to the contact surface area of the fluid. For example, one way to increase the lift of an aircraft is to simply increase the surface area, i.e. a planform area of the aircraft such as wings, fuselage. On the other hand, engineers also make efforts to reduce the increased frictional drag, using surface controls such as riblets, re-laminarization of the boundary layer.6
1.2.2 Volumetric Body Forces
In the fields of gravity, electricity, and magnetism, a force acts through the volume of a body that carries mass or electric charge. This is called volumetric body force. The most common volumetric body forces are the gravitational body force and the Lorentz force. The centrifugal and Coriolis forces are also volumetric body forces, but these are fictitious; they are only concerned when the equation of motion is transformed from an inertial reference frame to a rotating reference frame. The Coriolis force is particularly important in geophysical fluid dynamics because it is a nonconservative body force that produces rotation; in this case, the line of action of the net force does not go through the center of mass of the fluid particles.7
1.3 Pressure
In a volume of space, fluid molecules in random oscillations retain a certain level of repulsiveness. If a small piece of solid is immersed in a fluid, the fluid molecules exert normal, compressive forces to the surface of the solid, and as the body's volume shrinks to a point, the normal force per unit area (or normal stress) becomes independent of direction. This omnidirectional and compressive forces per unit area is called pressure.
Gases are compressible fluids that can change volume due to normal force or heat. According to Boyle's law, if a gas is slowly compressed by normal force, pressure is inversely proportional to gas volume at a constant temperature
where , , and are the pressure, density, and volume, respectively. If the gas is slowly heated up with a fixed pressure , Charles' law states:
With Boyle's and Charles' laws, we can now obtain the ideal gas law:
where is the universal gas constant. The ideal gas law shows that pressure of a gas can be defined by two state variables in thermodynamic equilibrium, e.g. density and temperature.
On the contrary, liquid pressure cannot be defined by the equation of state with two thermodynamic variables. As shown in the – diagram (Figure 1.1), liquid volume can hardly be changed by compressive forces, but its pressure can easily be changed to suit the local force equilibrium. If an external normal force is applied to compress a liquid, an electrostatic repulsion immediately acts as a restoring force to resist the external deformation.8 In a cup of water, for example the volume of water remains the same, but the pressure changes vertically due to its own weight. However, the volume of the liquid more easily changes with heat, as a result, evaporation results in a large normal compressive force.
Figure 1.1– diagram and isothermal lines of liquid and vapor
Example 1.1 Physical illustration of pressure
A bullet is fired into two paint cans, one of which is full of liquid and the other full of air (YouTube, W. Lewin, MIT, Classical Physics, Lec-27) [3]. Which one will explode and which one will not?
This experiment demonstrates three physical principles:
The compressibility of the fluid: nearly incompressible water vs. compressible air.
Squeezing in the bullet into the paint can containing water causes the pressure inside the can to substantially increase (
Figure 1.1
). As a result, the paint can must explode because of the pressure force. The can containing air will not explode because the air locally changes the volume with an increase of pressure, forming a shock wave.
The force exerted by the bullet is transmitted through the fluid by pressure. Pascal's law states that a change in pressure at any point in an enclosed fluid at rest is transmitted undiminished to all points in the fluid.
1.4 Pressure in a Fluid at Rest
1.4.1 Pascal's Experiment
The experiment conducted by Pascal in 1646 presents many interesting truths about pressure. The water being poured through a long cylindrical vertical tube on top of a barrel applies weight so that pressure acts normally to every surface element of the water as well as to the inner wall of the barrel. At the same height, the water is expelled through the holes with equal strength since it acts in all directions with the same magnitude. In fact, the pressure in the barrel does not change with the diameter of the vertical tube because the water weight augmented by increasing the diameter is to be distributed over the increased surface area.
1.4.2 Hydrostatic Pressure
When fluids are at rest in the gravitational field, there is no other external force acting on the fluid except its own weight. In this case, the action force, i.e. gravitational body force proportional to the fluid volume and density, is locally in static force equilibrium with the reaction force from the other side that supports the fluid above. As a result, the fluid is in the state of compression by two forces in action and reaction, and the pressure (or hydrostatic stress) at this point is defined per unit area with these two normal forces in local static equilibrium (Figure 1.2).
Figure 1.2 Pressure (or hydrostatic stress) determined by two forces in action and reaction; the top and bottom boundaries are exposed to the ambient pressure
Figure 1.3 Hydrostatic stresses acting on the surfaces of an infinitesimal fluid element
For an infinitesimal element of fluid, for example a static force balance can be set in the vertical direction as follows:
where is the pressure difference, the density is a constant for incompressible fluid, and is the gravitational acceleration (Figure 1.3).
By integrating Eq. (1.5) from 0 to , we can obtain a pressure distribution:
where is the pressure at the reference point, and is called the hydrostatic pressure. Equation (1.5) also proves that as the volume of the fluid element approaches zero, must approach zero since the area of the element is one order larger.
1.4.3 Earth Surrounded by Fluids
On a grand scale, all creatures living on the Earth are surrounded by two fluids: air and water. As opposed to outer space where no matter exists, the Earth's atmosphere is filled with air molecules. At sea level, the atmospheric pressure is 101,000 Pascals (or denoted by ) because air molecules occupy the space from the troposphere (0–12 ) to the thermosphere (85–110 ).9 Note that the atmospheric pressure can be obtained by integrating Eq. (1.5) with the equation of state, expressing the density as a function of pressure and temperature; the temperature distribution in the atmosphere is usually known as a function of height. Meanwhile, the pressure in the oceans and rivers varies more significantly with depth because the density of water is approximately 1000 times greater than the density of air.
Example 1.2 Magic cup?
Let us suppose we have water in a cup with a radius of and a height of . When the cup is held upside down with a paper plate placed at the bottom (Figure 1.4), the weight of water lowers the pressure of the air trapped inside the cup (e.g. height is ). The change of air volume may not be noticeable, but the trapped air expands by a fraction of its volume. According to Boyle's law, this very small change of air volume lowers the air pressure and creates a suction force to hold the water.
Numerics:
Cup cross-sectional area () =
Water weight ():
Pressure change () = =
Boyle's law:
;
Thus, , and with , .
Density decrease of air () =
1.4.4 Buoyant Force
Buoyancy is another form of gravitational effect. It is created by the hydrostatic pressure acting on the surface of an immersed matter of a different density, where the hydrostatic pressure results from the gravitational body force of the fluid. If a solid body of density is immersed in a fluid of density , the hydrostatic pressure acting on the solid surface varies with depth (Figure 1.5). If the hydrostatic pressure is integrated over the solid body, the forces in the horizontal direction are all cancelled off, resulting in no net force. In the vertical direction, however, there exists a net pressure force acting in the positive direction. This buoyant force is written as follows
where denotes a segmented vertical column of the solid body of height , an incremental area projected to the horizontal plane, an incremental volume of the solid, and the mass of the fluid for the volume occupied by the solid. Equation (1.7) shows that the buoyant force is equivalent to the gravitational body force of the fluid that occupies the volume of the solid.
Figure 1.4 Suction (or negative) pressure inside the cup
Figure 1.5 Hydrostatic stress vectors on the surface of a solid body immersed in a fluid; densities of solid and fluid denoted by and
The net upward force acting on the submerged body is the difference between the buoyant force and the weight of the body:
It is clearly shown in Eq. (1.8) that the buoyant effect occurs due to the density difference between the solid body and the fluid. If the solid were replaced by a fluid of the same density , there would be no net vertical force acting on the fluid replacing the space of solid. If , then , and the force magnitude is proportional to the density difference and volume. There are many examples of submergence of matter in different phases, e.g. a piece of wood or submarine in water, a hot air balloon in the air, the chimney (or stack) effect.
Example 1.3 Solar chimney
EnviroMission, an Australian public company, proposed to build the first large-scale solar updraft tower (about twice as tall as the Empire State Building) in La Paz County, Arizona. This towering hot air chimney is designed to produce 200 MW electrical energy with turbines (Figure 1.6).
Figure 1.6 Solar chimney; two Empire State Buildings high
1.5 Pressure in a Fluid in Motion
We have shown so far that the pressure in a fluid at rest is determined by the local balance of action and reaction forces: gravitational body force vs. reaction force from the ground. The next question is how pressure can be defined in a fluid in motion. The answer to this question is not that simple, especially when the fluid is viscously strained at a rate. In viscous flows, viscous stresses are directly related to the strain rates through viscosity under Newton's law of viscosity. Therefore, the mean normal stress (or mechanical pressure) at a point may or may not be the same as the pressure. Nevertheless, we assume Stokes' hypothesis that mechanical pressure would be the same as pressure. Once Stokes' hypothesis is assumed, the pressure of a fluid in motion can be defined in the same manner as it is defined in a fluid at rest.10
In this section, we are going to demonstrate with a simple syringe-like device (Figure 1.7) that the pressure of an incompressible fluid (e.g. water) inside the device is determined by the local balance of action and reaction forces. Here, the action force is the normal force originated by the piston, and the reaction force can be the inertial resistance force of the accelerating fluid or the frictional resistance force of the shear-straining fluid. We assume that the syringe-like device is immersed in water and fixed to the ground. We also neglect any inertial effect of water being sucked into the device inlet, for the sake of simplicity.
1.5.1 Transient Inertia Force
1.5.1.1 Slug Motion
Let us suppose we have a straight tube and a force is applied to the piston to accelerate the water at rest (Figure 1.7). Then, the water in the tube is forced to be in the state of compression by forces in action and reaction: the force applied to the piston vs. the inertial resistance force of the water. In this case, the magnitude of the inertial resistance force depends not only on the local time acceleration of the piston but also on the amount of mass of water being accelerated in time. Therefore, the pressure of the water inside the tube linearly decreases with distance and its gradient depends on the acceleration of the piston.
Figure 1.7 Pressure changes with different reaction forces: (i) transient inertia force, (ii) convective inertia force, and (iii) frictional resistance force
To quantify the pressure inside the tube, an equation of motion can be set for a water element. The inertial resistance force produced by transient acceleration of the water element is balanced with a local net pressure force:
where is the incremental pressure difference over and is the cross-sectional area of the water element. The inertial resistance force of the water element can be written as follows:
where is the water density, is the piston speed (which is generally time-dependent), and is the incremental distance in the streamwise direction. Note that and are the normal stresses determined by the local balance of action and reaction forces.
Equation (1.9) can be written per unit volume as follows:
and the pressure distribution inside the tube can be obtained by integrating Eq. (1.11) with the boundary condition at the exit, 11
where is the tube length.
Equation (1.12) shows that the pressure inside the tube linearly decreases with distance because the inertial resistance force is linearly proportional to the mass being accelerated in time, and the slope of pressure distribution is proportional to the magnitude of the acceleration of the piston itself. Equation (1.12) is only valid if .
[Notes] If the piston is suddenly pulled in the opposite direction, i.e. , the pressure distribution inside the tube is obtained as follows:
where the pressure at the tube exit is lower than , since the water outside is spatially accelerated into the tube. From the conservation law of energy (or Bernoulli's principle), the pressure at the exit can be estimated as follows:
if we neglect the local transient acceleration of the water outside and any possible flow separations near the tube exit. Equation (1.13) shows that the pressure along the tube linearly decreases from at the tube exit to at the head of the piston.
Example 1.4 Oscillation of water in a U-tube
The water in a U-tube oscillates with the same velocity as a slug flow (Figure 1.8) [4]. Find the instantaneous pressure distribution inside the tube.
An equation of motion can be set for an infinitesimal slice of control volume in the U-tube. In this case, the gravitational body and pressure forces are balanced with an unsteady inertial force:
where is the vertical coordinate originating from the bottom and is the vertical velocity.
By integrating Eq. (1.15) from to the free surface of the water, we can obtain the pressure inside the U-tube:
As shown in Figure 1.9, pressure linearly increases from the free surface () to the bottom () with a slope of . Note that the slope changes in time with the sign and magnitude of , while the water column is accelerating (or decelerating).
For the sake of simplicity, we assume that two vertical tubes are connected by a horizontal junction, which at both ends will have two different pressures. Due to the transient inertial effect of the fluid, pressure varies linearly across this horizontal tube:
where is the pressure at the right end of the junction, is the length, is the horizontal velocity, , and . The two pressures will become the same when the two water columns level even. Note also that the gradient discontinuity of pressure at the two end points will be rounded off by the centrifugal effect, which is not included in the present model.
Figure 1.8 Oscillation of water in a U-tube
Figure 1.9 Instantaneous pressure distribution in a U-tube; thin solid (: highest), dashed (), and thick solid (: highest); filled circles (left-end junction) and hollow circles (right-end junction)
Figure 1.10 Pressure changes in the nozzle (a) and diffuser (b) with changes of fluid momentums in space; a thrust force in reaction to (nozzle) and a drag force in reaction to (diffuser)
1.5.2 Convective Inertia Force
1.5.2.1 Nozzle
If a nozzle is attached to the tube exit, a force is required to move the piston at a constant speed (Figure 1.7). In this case, the water in the nozzle is forced to be in the state of compression by forces in action and reaction: the force applied to the piston vs. the inertial resistance force of the accelerating water in the nozzle.
In contrast to the transient inertial force in the previous case, the inertial resistance force is the reaction force of the fluid being accelerated in the nozzle. If we look at the water element in the nozzle, it is laterally being contracted and longitudinally being stretched (i.e. linearly strained) at a rate while it moves downstream. Over an infinitesimally small control volume fixed in space, the water is convectively accelerated in the streamwise direction,12 and in reaction to this acceleration, the water upstream is forced backward (i.e. a principle of thrust force).
The magnitude of this inertial resistance force depends not only on the convective acceleration of the water, which is the product of the local velocity and the straining rate of the fluid , but also on the amount of mass of water being accelerated in the nozzle. Therefore, the pressure in the nozzle is increased in the upstream direction because the amount of volume-occupying mass to be accelerated is accumulated in that direction (Figure 1.10a).
At this point, special attention should be paid to the definition of the pressure of a fluid in straining (or deforming) motion. When the water in the nozzle is linearly strained at a rate in the streamwise and transverse directions, normal viscous stresses are exerted in the direction normal to the surfaces of the water element. However, they do not build any energy density to be added to the pressure (or hydrostatic stress) since the sum of the strain rates in the , , and directions is zero in incompressible fluids; hence, mechanical pressure is equal to pressure.13
To quantify the pressure distribution in the tube and the nozzle, an equation of motion can be set for a water element. The inertial resistance force associated with convective acceleration of the fluid is balanced with the net pressure force:
where is the incremental pressure difference over , and and are the normal stresses determined by two local forces in action and reaction. The inertial resistance force of the water element is defined as follows:
where represents the mass of the water element and is the convective acceleration.
Equation (1.18) can be written per unit volume as follows:
and the pressure distribution in the nozzle can be obtained by integrating Eq. (1.20) from to the nozzle exit:
where is the velocity at the nozzle exit. It is shown that is greater than due to the increase of kinetic energy (per unit volume) between two stations at and the nozzle exit (i.e. production of thrust force). The pressure difference is also linearly proportional to the density, meaning that the pressure change is related to the mass in convective acceleration in the nozzle. Equation (1.21) is the well-known Bernoulli's principle.
By conservation of mass, , the pressure at (denoting a nozzle entrance) is expressed as follows:
where is the contraction-ratio of the nozzle.
1.5.2.2 Diffuser
In diffusers, a fluid is forced to be in the state of compression by a local balance of forces in action and reaction: an inertia force of the fluid upstream vs. a reaction force from downstream. The reaction force can be produced either by the solid fixed to the ground or by the fluid in ambient condition.
If we look at the water element in the diffuser, it is laterally stretched and longitudinally contracted at a rate. Over an infinitesimally small control volume fixed in space, the fluid is being decelerated in the streamwise direction, and in reaction to this deceleration, the fluid downstream is being forced forward (i.e. a principle of drag force). As discussed in the previous cases, the magnitude of this inertial force depends on the deceleration of the fluid as well as the mass of water being decelerated in the diffuser. Hence, the pressure is increased in the downstream direction due to an accumulation of the decelerating fluid in that direction (Figure 1.10b).
Figure 1.11 Diffuser-augmented wind turbines (DAWT)
1.5.2.3 Venturi Effect
If the diffuser exit is open to the ambient fluid, the inertial force of the decelerating fluid causes the upstream pressure at the throat to be lower than the ambient pressure. This occurrence of negative pressure at the throat is called Venturi effect. In fact, the Venturi effect is used in many applications to readily suck the fluid in the area close to the neck.
One example is a Venturi tunnel under a car with a carefully shaped under-tray for additional downforce. The wind tamer (or skywolf wind turbine), a commercial product of “diffuser-augmented wind turbines” (DAWT) that install diffusers at the turbine blades, utilizes the Venturi effect to lower the pressure at the blades to increase the efficiency of the wind turbines (Figure 1.11).
Example 1.5 Squid propulsion
A squid propels by transforming itself into an instant diffuser and nozzle. In the refilling phase, external water enters the inflated tube, which we call the mantle, through small openings around the squid's head. The mantle forms a nozzle-diffuser-like flow passage with flaps that build a negative pressure at the throat (Figure 1.12a). In the ejection phase, the squid forcefully expels water through a small orifice by forming a funnel. It clamps the openings shut by contracting the mantle, as sketched in Figure 1.12b. The squid is propelled by the conservation of momentum.
Figure 1.12 Water jet propulsion of squid by shaping an instant diffuser (a) and nozzle (b) with its body: mantle, funnel, and flaps
1.5.3 Shear Resistance Force
We have shown so far that pressu
