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PHYS 2

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Does the length of this hose have any effect on the water delivery process?
The answer is yes, longer hoses generally deliver less water. Moving water doesn't slide freely through a stationary hose. In reality, it experiences frictional forces that oppose its motion relative to the hose.
But this friction is unusual because most of the water in the hose never actually touches the hose itself. If water deep inside the hose is going to experience any forces due to relative motion, then those forces are going to have to occur within the water. Water must exert frictional forces on itself!
Sure enough, water does experience internal frictional forces. They're called viscous forces:
forces that appear whenever one layer of a fluid tries to slide across another layer of that fluid. Viscous forces oppose relative motion and you can observe their effects easily when you pour honey out of a jar. Typical of most liquids, this decrease in viscosity with increasing temperature reflects the molecular origins of viscous forces: the molecules in a liquid stick to one another, forming weak chemical bonds that require energy to break. In a hot liquid, the molecules have more thermal energy, so they break these bonds more easily in order to move past one another.
Viscosity slows the flow of water through your hose...
Chemical forces between the hose and the outmost layer of water hold that layer of water stationary, and this motionless layer exerts viscous forces on the layer of moving water inside it. As this second layer slows, it exerts viscous forces on yet another layer. Layer by layer, viscous forces hold back the moving water until even water at the center of the hose feels viscosity's slowing effects (Fig. 6.1.1). Although water at the center of the hose moves faster than water in any other layer, it's still affected by the stationary hose.
However, unlike the forces of ordinary sliding friction—which don't depend on relative velocities...
viscous forces become larger as the relative velocities within a fluid increase. That's because as two layers of water slide past one another faster, their molecules collide harder and more frequently. Since it experiences stronger viscous forces, fast-moving water wastes more energy per meter and needs a larger pressure gradient to keep it moving steadily through a hose than does slow-moving water.
Because of viscous forces, the amount of water flowing steadily through a hose depends on four factors:
1.
It's inversely proportional to the water's viscosity.
2.
It's inversely proportional to the length of the hose. The longer the hose, the more opportunity viscous forces have to slow the water down.
3. It's proportional to the pressure difference between the hose's inlet and its outlet. This pressure difference determines the water's pressure gradient and thus how hard the water is pushed forward through the hose. 4. It's proportional to the fourth power of the diameter of the hose. Tripling the hose's diameter provides the water with nine times as much room and also allows water near the hose's center to move nine times faster.
Poiseuille's Law
The volume of fluid flowing through a cylindrical pipe each second is equal to (π/128) times the pressure difference (Δp) across that pipe times the pipe's diameter to the fourth power, divided by the pipe's length times the fluid's viscosity (η).
To deliver large amounts of water at high pressure or velocity:
Fire hoses must have large diameters. When filled with high-pressure water, these wide hoses become stiff and heavy, making them difficult to handle. Chemical additives that decrease water's viscosity allow firefighters to use narrower, lighter, and more flexible hoses.
Just how much total energy the water retains depends on:
how fast it moves inside the hose. If you allow lots of water to leave the hose, water will move through it quickly and encounter large viscous forces. In the process, most of the water's total energy will be wasted as thermal energy and the water will pour gently out of the end of the hose.
But if you partially block the hose's opening with your thumb and reduce the flow, water will travel slowly through the hose and encounter smaller viscous forces. As a result, the water will retain most of its total energy and will still be at high pressure when it reaches your thumb. In contrast to a narrow hose, wide pipes can carry large amounts of water while letting that water travel slowly, experience weak viscous forces, and waste little of its total energy. In such energy efficient water delivery systems, friction is insignificant and Bernoulli's equation accurately predicts water's properties throughout its trip.
On reaching your garden, the hose bends toward the right and the flowing water bends with it...
Since the hose is motionless, the unbalanced pressures inside it must be caused by the water itself; the water is experiencing dynamic pressure variations.
Since the hose is motionless, the unbalanced pressures inside it must be caused by the water itself; the water is experiencing dynamic pressure variations...
can't vary and the only interchanges we'll see are between pressure potential energy and kinetic energy.
Water approaches the bend through a straight section of hose in which it travels at:
velocity and has a uniform pressure. Its velocity is constant because the straight hose directs all the streamlines forward and because the water moving along a given streamline can't change its speed.
Bends and Pressure Imbalances
When the path of a fluid in steady-state flow bends, the pressure on the outside of the bend is always higher than the pressure on the inside of the bend.
Common Misconceptions: Speed and Pressure in Fluids
Misconception: A fast-moving fluid always has a low pressure.
Resolution: The pressure of a specific portion of fluid depends on its circumstances and can take any value, high or low. However, if a fluid speeds up without descending as it flows along a streamline in steady-state flow, then its pressure will decrease. In that special context, the faster moving fluid has a lower pressure.
As you direct the stream of water at the plants in your garden, you notice two interesting phenomena: first, the stream pushes on any surface that slows it down and, second, it tends to break up into fragments as it flows around obstacles.
The pushing effect is another Bernoulli result: when the stream encounters a surface, it slows down and spreads out sideways. As the water slows close to the surface, its pressure there rises above atmospheric pressure and it is this elevated pressure that actually pushes the surface forward.
In trying to go around the obstacle, the stream of water loses its orderly structure and disintegrates into a swirling, hissing froth.
laminar flow:
smooth, silent flow that's characterized by simple streamlines. In laminar flow, adjacent regions of a fluid always remain adjacent. For example, if you place two drops of dye near one another in a smoothly flowing stream, they will remain close together indefinitely as they follow streamlines in the laminar flow. Laminar flow is the orderly result of viscous forces, which tend to bring adjacent portions of fluid to the same velocity. When viscosity dominates a fluid's motion, the flow is usually laminar.
Turbulent flow:
the disorderly consequence of inertia, which tends to propel each portion of fluid independently according to its own momentum. When inertia dominates a fluid's motion, the flow is usually turbulent.
The plants and faucet stopper are evidently initiating turbulence in what had been laminar flows; flows that were dominated by viscous forces are suddenly dominated by inertia instead. Whether a flow is laminar or turbulent depends on several characteris
1. The fluid's viscosity. Viscous forces tend to keep nearby regions of fluid moving together, so high viscosity favors laminar flow. 2.
The fluid's speed past a stationary obstacle. The faster the fluid is moving, the more quickly two nearby regions of fluid can become separated and the harder it is for viscous forces to keep them together.

3.
The size of the obstacle the fluid encounters. The larger the obstacle, the more likely that it will cause turbulence because viscous forces will be unable to keep the fluid ordered over such a long distance.

4.
The fluid's density. The denser the fluid, the less it responds to viscous forces and the more likely it is to become turbulent.
Reynolds number:
A dimensionless number that characterizes fluid flow through a system. At low Reynolds numbers a fluid's viscosity dominates the flow, while at high Reynolds numbers a fluid's inertia dominates.
One of the most common features of turbulent flows is the vortex:
a whirling region of fluid that moves in a circle around a central cavity. A vortex resembles a miniature tornado, with its cavity created by inertia as the fluid spins. Vortices are easily visible behind a canoe paddle or in a mixing bowl. Once an object moves fast enough through a fluid to create turbulence, these vortices begin to form. Each vortex builds up behind the object but is soon whisked away to form a wake of shed vortices.
While laminar flow is fully predictable, turbulent flow exhibits chaotic behavior or chaos: 1)
This pressure surge is what accelerates the water backward to slow it down and also what leads to the loud “thump” sound you hear as the water stops.
you can no longer predict exactly where any particular drop of water will go.
2) water hammer, the surging pressure in front of stopping water jerks the nozzle, swells the hose, and may even rattle the pipes in your home.
Faucets and Water Flow
􀂄􀂄 In going through a faucet, water must
􀂄􀂄 flow through a narrow passage
􀂄􀂄 and pass close to the faucet’s stationary surfaces
􀂄􀂄 Total energy limits flow speed through passage
􀂄􀂄 The water turns its total energy into kinetic energy,
􀂄􀂄 but its peak speed is limited by its initial pressure
􀂄􀂄 Motion near the surfaces slows the water
􀂄􀂄 Because water at the walls is stationary,
􀂄􀂄 viscous forces within the water slow all of it
Viscous Forces and Viscosity
􀂄􀂄 Viscous forces
􀂄􀂄 oppose relative motion within a fluid
􀂄􀂄 and are similar to sliding friction: they waste energy
􀂄􀂄 Fluids are characterized by their viscosities
􀂄􀂄 the measure of the strength of the viscous forces
􀂄􀂄 and caused by chemical interactions with the fluids
Hoses and Water Flow (part 1)
􀂄􀂄 The rate at which water flows through a hose,
􀂄􀂄 increases as end-to-end pressure difference increases,
􀂄􀂄 decreases as water’s viscosity increases,
􀂄􀂄 decreases as hose becomes longer,
􀂄􀂄 and increases dramatically as hose becomes wider
􀂄􀂄 Increasing the hose width
􀂄􀂄 enlarges cross-sectional area through which to flow
􀂄􀂄 and lets water get farther from the walls of the hose
Hoses and Water Flow (part 2)
􀂄􀂄 Water flow through a hose is proportional to
􀂄􀂄 pressure difference
􀂄􀂄 1/viscosity
􀂄􀂄 1/hose length
􀂄􀂄 (pipe diameter)4
􀂄􀂄 Poiseuille’s law
Wasting Energy in a Hose
􀂄 Viscous effects
􀂄􀂄 waste water’s total energy as thermal energy
􀂄􀂄 and become stronger with increased flow speed
􀂄􀂄 Faster flow leads to more viscous energy loss
􀂄􀂄 Faster flow causes quicker loss of pressure
Making Water Accelerate
􀂄 Even in steady-state, water can accelerate
􀂄􀂄 but forward acceleration would leave gaps
􀂄􀂄 and backward acceleration would cause jams,
􀂄􀂄 so the acceleration must involve turning
􀂄􀂄 Acceleration toward the side (turning)
􀂄􀂄 requires obstacles,
􀂄􀂄 and involves pressure imbalances
􀂄􀂄 and changes in speed
Bending the Flow in a Hose
􀂄􀂄 Since water accelerates toward lower pressure,
􀂄􀂄 water flow needs a pressure imbalance to bend
􀂄􀂄 The flow naturally develops a pressure gradient
􀂄􀂄 higher pressure & lower speed
on the outside of the bend
􀂄􀂄 lower pressure & higher speed
on the inside of the bend
􀂄􀂄 and water accelerates from
high pressure to lower pressure
Speeding the Flow in a Nozzle
􀂄 Since water must speed up through a nozzle,
􀂄􀂄 it needs a pressure imbalance to push it forward
􀂄􀂄 The flow naturally develops a pressure gradient
􀂄􀂄 lower pressure & higher speed
as the neck narrows
Water Flow Isn’t Always Smooth
􀂄􀂄 We’ve been examining laminar flow
􀂄􀂄 in which viscosity dominates the flow’s behavior
􀂄􀂄 and nearby regions of water remain nearby
􀂄􀂄 Now we’ll also consider turbulent flow
􀂄􀂄 in which inertia dominates the flow’s behavior
􀂄􀂄 and nearby regions of water become separated
Water and Momentum
􀂄􀂄 Water carries momentum
􀂄􀂄 Water transfers its momentum via impulses:
impulse = pressure· surface area· time
􀂄􀂄 Large momentum transfers requires
􀂄􀂄 large pressures,
􀂄􀂄 large surface areas,
􀂄􀂄 and/or long times.
􀂄􀂄 Moving water can be surprisingly hard to stop
Summary about Garden Watering
􀂄􀂄 Total energy limits speed, height, and pressure
􀂄􀂄 Bending water flows develop pressure gradients
􀂄􀂄 Nozzles exchange pressure for speed
􀂄􀂄 Viscosity wastes flowing water’s total energy
􀂄􀂄 Turbulence wastes flowing water’s total energy
􀂄􀂄 Wasted total energy because thermal energy
􀂄􀂄 Moving water has momentum, too
In air, a moving ball experiences aerodynamic forces:
forces exerted on it by the air because of their relative motion. These consist of drag forces (aka “air resistance”) that push the ball downwind and lift forces that push the ball to one side or the other.
The airflow around a slowly moving ball is:
laminar. Air slows down in front of and behind the ball and its pressure increases. Air speeds up at the sides of the ball, and its pressure decreases. However, the pressure forces on the ball balance one another perfectly, and it experiences no pressure drag. Only viscous drag is present to affect the ball.
The slow-moving air separates neatly around the front of the ball and comes back together behind it. It produces a:
wake, an air trail behind the ball, that's smooth and free of turbulence. But the air's speed and pressure aren't uniform all the way around the ball. Whenever air bends away from the ball, so that the ball is on the outside of a bend, the pressure near the ball must be higher than atmospheric. And whenever the air bends toward the ball, so that the ball is on the inside of a bend, the pressure near the ball must be lower than atmospheric.
With that introduction, let's examine the slow-moving airflow around the ball. Air heading toward the ball's front bends away from it, so the pressure near the front of the ball must be higher than atmospheric. This rise in air pressure is accompanied by
airspeed—the air's speed relative to the ball.
Air rounding the ball's sides bends toward it, so the pressure near the sides of the ball must be:
atmospheric. This drop in air pressure is accompanied by an increase in airspeed.
particularly remarkable that low-pressure air at the sides of the ball is able to flow around to the back of the ball, where the pressure is higher. This air is experiencing:
a pressure imbalance that pushes it backward, opposite its direction of travel. But a pressure imbalance causes acceleration, not velocity, and the low-pressure air flowing past the sides of the ball has enough energy and forward momentum to carry it all the way to the back of the ball.
The airflow around the ball is symmetric, and the forces that air pressure exerts on the ball are also symmetric. These pressure forces:
cancel one another perfectly so that the ball experiences no overall force due to pressure. Most importantly, the high pressure in front of the ball is balanced by the high pressure behind it. As a result of this symmetric arrangement, the only aerodynamic force acting on the ball is viscous drag—the downstream frictional force caused by layers of viscous air sliding across the ball's surface.
Balls don't always experience laminar airflow. Turbulence is common, particularly in sports, and brings with it a new type of drag force. When the air flowing around a ball is turbulent, the air pressure distribution is no longer symmetric and the ball e
pressure drag—the downstream force exerted by unbalanced pressures in the moving air. These unbalanced pressures exert an overall force on the ball that slows its motion through the air.
A ball can experience turbulent airflow and pressure drag when the Reynolds number exceeds about:
2000. The Reynolds number, introduced in the previous section, combines the ball's size and speed with the air's density and viscosity to give an indication of whether the airflow is dominated by viscosity or inertia. At low Reynolds numbers, the air's viscosity dominates over its inertia and the airflow is laminar. But at high Reynolds numbers, air's inertia dominates over its viscosity and the airflow tends to become turbulent. This turbulence, however, won't start until something triggers it and viscosity provides that trigger.
To understand viscosity's role, we must look at the air near the ball's surface. Even in a strong wind, viscous forces slow down a thin:
boundary layer of air near the ball's surface. Discovered by Ludwig Prandtl with help from Gustave Eiffel, this boundary layer moves more slowly and has less total energy than the freely flowing air farther from the surface.
As air flows toward the back of the ball, it travels through an:
adverse pressure gradient—a region of rising pressure that pushes backward on the air and causes it to decelerate. While the freely flowing airstream outside the boundary layer has enough energy and forward momentum to continue onward and reach the back of the ball on its own, air in the boundary layer does not. It needs a forward push.
At low Reynolds numbers, the entire airstream:
helps to push that boundary layer all the way to the back of the ball and the airflow remains laminar. But at high Reynolds numbers, viscous forces between the freely flowing airstream and the boundary layer are too weak to keep the boundary layer moving forward into the rising pressure behind the ball.
Without adequate help, the boundary layer eventually:
stalls—it comes to a stop and thereby spoils steady-state flow. More horrible still, this stalled boundary layer air is pushed backward by the adverse pressure gradient and it returns all the way to the sides of the ball. As it does, it cuts like a wedge between the ball and the freely flowing airstream. The result is an aerodynamic catastrophe: the airstream separates from the ball, leaving a huge turbulent wake or air pocket behind the ball.
Because of this turbulent wake, air no longer bends smoothly away from the back of the ball and there is no rise in pressure there. Instead:
the pressure behind the ball is roughly atmospheric. The absence of a high-pressure region behind the ball spoils the symmetry of pressure forces on the ball and those forces no longer cancel. The ball experiences an overall pressure force downwind—the force of pressure drag. In effect, the ball is transferring forward momentum to the air in its turbulent wake and dragging that wake along with it. Pressure drag slows the flight of almost any ball moving faster than a snail's pace. The pressure drag force is roughly proportional to the cross-sectional area of the turbulent air pocket and to the square of the ball's speed through the air.
At very high Reynolds numbers the boundary layer itself becomes:
turbulent. It loses its laminar streamlines and begins to mix rapidly within itself and with the freely flowing airstream nearby. This mixing brings additional energy and forward momentum into the boundary layer and makes it both harder to stop and more resistant to reversed flow. Although this turbulent boundary layer still stalls before reaching the back of the ball, the stalled air flows upstream only a short distance. While the freely flowing airstream still separates from the ball, that separation occurs far back on the ball and the resulting turbulent wake is relatively small.
As a result of this smaller air pocket, the pressure drag is:
reduced from what it would be without the turbulent boundary layer. The effect of replacing the laminar boundary layer with a turbulent one is enormous; it's the difference between a golf drive of 70 yards and one of 250 yards!
Delaying the airflow separation behind the back of the ball is so important to distance and speed that the balls of many sports are designed to encourage a:
turbulent boundary layer. Rather than waiting for the Reynolds number to exceed 100,000, the point near which the boundary layer spontaneously becomes turbulent, these balls “trip” the boundary layer deliberately (Fig. 6.2.9). They introduce some impediment to laminar flow, such as hair or surface irregularities, which causes the air near the ball's surface to tumble about and become turbulent. The drop in pressure drag more than makes up for the small increase in viscous drag. That's why a tennis ball has fuzz and a golf ball has dimples.
So how much does drag affect balls in various sports? For those that involve rapid movements through air or water, the answer is:
quite a bit. Drag forces increase dramatically with speed; as soon as a turbulent wake and pressure drag appear, the drag force increases as the square of a ball's speed. As a result, baseball pitches slow significantly during their flights to home plate, and the faster they're thrown, the more speed they lose. A 90-mph fastball loses about 8 mph en route, while a 70-mph curveball loses only about 6 mph.
A batted ball fares slightly better because it travels fast enough for the boundary layer around it to become turbulent, an effect that appears at around 160 km/h (100 mph). While the resulting reduction in drag explains why it's possible to hit a home r
the presence of air drag still shortens the distance the ball travels by as much as 50%. Without air drag, a routine fly ball would become an out-of-the-park home run. To compensate for air drag, the angle at which the ball should be hit for maximum distance isn't the theoretical 45° above horizontal discussed in Section 1.2. Because of the ball's tendency to lose downfield velocity, it should be hit at a little lower angle, about 35° above horizontal.
Since the ball loses much of its horizontal component of velocity during its trip to the outfield, a long fly ball tends to drop almost straight down as you catch it. Gravity causes it to move downward, but drag almost stops its horizontal motion away fr
terminal velocity, the downward velocity at which the upward drag force exactly balances its downward weight and it stops accelerating. Even if you drop a baseball from an airplane, its velocity will not exceed this value.
The drag forces on a ball push it downstream, parallel to the onrushing air. But in some cases, the ball may also experience lift forces:
forces that are exerted perpendicular to the airflow. To experience drag, the ball only has to slow the airflow down; to experience lift, the ball must deflect the airflow to one side or the other. Although its name implies an upward force, lift can also push the ball toward the side or even downward.
Curveballs and knuckleballs both use lift forces. In each of these famous baseball pitches, the ball deflects the airstream toward one side and the ball accelerates toward the other. Again we have:
action and reaction—the air and the ball push off one another. Getting the air to push the ball sideways is no small trick. Explaining it isn't easy either, but here we go. A curveball is thrown by making the ball spin rapidly about an axis perpendicular to its direction of motion. The choice of this axis determines which way the ball curves. In Fig. 6.2.11, the ball is spinning clockwise, as viewed from above. With this choice of rotation axis, the ball curves to the pitcher's right because the ball experiences two lift forces to the right. One is the Magnus force. The other is a force we will call the wake deflection force.
The Magnus force occurs because:
the spinning ball carries some of the viscous air around with it. The steady-state flow pattern that forms around this ball is asymmetric: the airstream that moves with the turning surface is much longer than the airstream that moves opposite that surface. Since the longer airstream bends mostly toward the baseball, the average pressure on that side of the ball must be below atmospheric. The shorter airstream bends mostly away from the ball, so the average pressure on that side of the ball must be above atmospheric. Because the pressure forces on the ball's sides don't balance one another, the ball experiences the Magnus force toward the low-pressure side—the side turning toward the pitcher—and deflects in that direction. The airflow deflects in the opposite direction.
In laminar flow, the Magnus force is the only lift force acting on a spinning object. But a pitched baseball has a turbulent wake behind it and is also acted on by the:
wake deflection force. This force appears when the ball's rapid rotation deforms the wide, symmetric wake (Fig. 6.2.5) that develops behind it at high Reynolds numbers. When the ball isn't spinning, the freely flowing airstream separates from the ball approximately at its side and that separation is symmetric all the way around the ball's middle.
But when the ball is spinning, the moving surface pushes on the airstream with...
viscous forces. As a result, airstream separation is delayed on one side of the ball and hastened on the other. The overall wake of air behind the ball is thus deflected to one side and the ball experiences the wake deflection force toward the opposite side—the side turning toward the pitcher. The wake deflection force and the Magnus force both push the ball in the same direction.
Of these two forces, the wake deflection force is probably the more important for a curveball. Although:
A skillful pitcher can make a baseball curve about 0.3 m (12 inches) during its flight from the mound to home plate—the more spin, the more curve. The pitcher can also choose the direction of the curve by selecting the axis of the ball's rotation. The ball will always curve toward the side of the ball that is turning toward the pitcher.
When the pitcher throws a ball with backspin, so that the top of the ball turns toward the pitcher, the ball experiences:
an upward lift force. In baseball this force isn't strong enough to overcome gravity, but it does make the pitch hang in the air unusually long. And in golf, where the club can give the ball enormous backspin, the ball really does lift itself upward so that it flies down the fairway like a glider.
However, there are some cases when a ball's behavior stems from its lack of spin.
In baseball, for example, a knuckleball is thrown by giving the ball almost no rotation. Its seams are then very important. As air passes over a seam, the flow is disturbed so that the ball experiences a sideways aerodynamic force: a lift force. The ball flutters about in a remarkably erratic manner. Releasing the ball without making it spin is difficult and requires great skill. Pitchers who are unable to throw a knuckleball legally sometimes resort to lubricating their fingers so that the ball slips out of their hands without spinning. Like its legal relative, the so-called spitball dithers about and is hard to hit. The same is true for a scuffed ball.
Observations about Balls and Air
􀂄􀂄 Air resistance slows a ball down
􀂄􀂄 The faster a ball moves, the quicker it slows
􀂄􀂄 Some balls have deliberately roughened surfaces
􀂄􀂄 Spinning balls curve in flight
Aerodynamic Forces: Drag
􀂄􀂄 Air resistance is technically called “drag”
􀂄􀂄 When a ball moves through air, drag forces arise
􀂄􀂄 The air pushes the ball downstream
􀂄􀂄 and the ball pushes the air upstream
􀂄􀂄 Drag forces transfer momentum
􀂄􀂄 air transfers downstream momentum to ball
􀂄􀂄 ball transfers upstream momentum to air
Aerodynamic Forces: Lift
􀂄􀂄 When a ball deflects passing air, lift forces arise
􀂄􀂄 The air pushes the ball to one side
􀂄􀂄 and the ball pushes the air to the other side
􀂄􀂄 Lift forces transfer momentum
􀂄􀂄 air transfers sideways momentum to ball
􀂄􀂄 ball transfers sideways momentum to air
􀂄􀂄 Lift forces don’t always point upward!
Types of Drag & Lift
􀂄􀂄 Surface friction causes viscous drag
􀂄􀂄 Turbulence causes pressure drag
􀂄􀂄 Deflected flow causes lift
􀂄􀂄 Deflected flow also leads to induced drag
Perfect Flow Around a Ball
􀂄 Air bends away from ball’s front
􀂄􀂄 At front: high pressure, slow flow
􀂄􀂄 Air bends toward ball’s sides
􀂄􀂄 At side: low pressure, fast flow
􀂄􀂄 Air bends away from ball’s back
􀂄􀂄 At back: high pressure, slow flow
􀂄􀂄 Pressures on opposite sides balance perfectly,
􀂄􀂄 so ball experiences only viscous drag.
The Onset of Turbulence
􀂄 Air flowing into the rising pressure behind ball
􀂄􀂄 accelerates backward (decelerates)
􀂄􀂄 and converts kinetic energy into pressure potential.
􀂄􀂄 Air flowing nearest the ball’s surface
􀂄􀂄 also experiences viscous drag
􀂄􀂄 and converts kinetic energy into thermal energy.
􀂄􀂄 If it runs out of total energy, it stops or “stalls”
􀂄􀂄 If air nearest the ball stalls, turbulence ensues
Imperfect Flow Around Slow Ball
􀂄􀂄 Air flowing near ball’s surface
􀂄􀂄 stalls beyond ball’s sides
􀂄􀂄 and peels main air flow off of ball.
􀂄􀂄 Big wake forms behind ball
􀂄􀂄 Since wake pressure is ambient,
􀂄􀂄 ball experiences unbalanced pressures.
􀂄􀂄 Ball experiences a large pressure drag force
Boundary Layer
􀂄􀂄 Flow near the surface forms a “boundary layer”
􀂄􀂄 At low Reynolds number (<100,000)
􀂄􀂄 the boundary layer is laminar,
􀂄􀂄 so closest layer is slowed relentlessly by viscous drag
􀂄􀂄 At high Reynolds number (>100,000)
􀂄􀂄 the boundary layer itself is turbulent,
􀂄􀂄 so tumbling continually renews closest layer’s energy
􀂄􀂄 boundary layer penetrates deeper into rising pressure
Imperfect Flow Around Fast Ball
􀂄􀂄 Air flowing near ball’s surface
􀂄􀂄 stalls beyond ball’s sides
􀂄􀂄 and peels main air flow off of ball.
􀂄􀂄 Boundary layer is turbulent
􀂄􀂄 and retains total energy farther,
􀂄􀂄 so it resists peeling better.
􀂄􀂄 Small wake forms behind ball
􀂄􀂄 Ball experiences a small pressure drag force
Tripping the Boundary Layer
􀂄􀂄 To reduce pressure drag, some balls have fuzz
􀂄􀂄 Fuzz “trips” the boundary layer
􀂄􀂄 and causes boundary layer to become turbulent.
􀂄􀂄 Since turbulent boundary layer resists peeling better,
􀂄􀂄 ball’s main airflow forms smaller turbulent wake.
􀂄􀂄 Examples: Tennis balls and Golf balls
Spinning Balls, Magnus Force
􀂄􀂄 Turning surface pushes/pulls on the air flow
􀂄􀂄 Air on one side makes long bend toward ball
􀂄􀂄 Air on other side makes shorter bend away from ball
􀂄􀂄 Pressures are unbalanced
􀂄􀂄 The overall air flow is deflected
􀂄􀂄 Ball pushes air to one side
􀂄􀂄 Air pushes ball to other side
􀂄􀂄 Ball feels Magnus lift force
Spinning Balls, Wake Force
􀂄􀂄 Turning surface alters point of flow separation
􀂄􀂄 Flow separation is delayed on one side
􀂄􀂄 and hastened on the other side,
􀂄􀂄 so wake is asymmetric
􀂄􀂄 The overall air flow is deflected
􀂄􀂄 Ball pushes air to one side
􀂄􀂄 Air pushes ball to other side
􀂄􀂄 Ball feels Wake lift force
Summary about Balls and Air
􀂄􀂄 The air pressures around these objects are not
uniform and result in drag and lift
􀂄􀂄 Balls experience mostly pressure drag
􀂄􀂄 Spinning balls experience Magnus and Wake
Deflection lift forces
an airplane is supported in flight by an upward lift force on its wings and that this lift force comes from deflecting the passing airflow downward. Each wing is an airfoil. . .
an aerodynamically engineered surface that's designed to obtain particular lift and drag forces from the air flowing past it.
(More specifically, each wing is shaped and oriented so that, during flight, the airstream flowing over the wing bends downward toward its top surface while the airstream under the wing bends downward away from its bottom surface. These bends are associated with pressure changes near the wing itself and are responsible for the upward lift force that suspends the airplane in the sky.)
yourself in an airplane that has just begun rolling down the runway. From your perspective, air is beginning to flow past each of the airplane's wings. When this moving air encounters the wing's leading edge, it separates into two airstreams:
one traveling over the wing and the other under it (Fig. 6.3.1). These airstreams continue onward until they leave the wing's trailing edge. Since the airplane's nose is still on the ground, the wing is essentially horizontal and the airflow around it is simple and symmetric.
Since the wing isn't deflecting the airflow yet, it's experiencing no lift, only drag. But while this drag:
pushes the airplane downwind and thus opposite its forward motion along the runway, it's unusually weak. The wing produces almost no turbulent wake and thus experiences almost no pressure drag. What little drag it does experience is mostly viscous drag, essentially surface friction with the passing air.
What makes the horizontal wing streamlined is:
the extremely gradual rise in air pressure after its widest point. While this gently rising pressure pushes the wing's boundary layer backward, opposite the direction of flow, the force it exerts is so weak that the layer doesn't stall. Driven onward by viscous forces from the freely flowing airstream, the wing's boundary layer manages to keep moving forward all the way to the wing's trailing edge and never triggers flow separation. The wing produces almost no turbulent wake and experiences almost no pressure drag.
At first, the airflow around the tilted wings continues to travel horizontally on average, although it develops...
peculiar shape (Fig. 6.3.2a). The two airstreams, one over the tilted wing and one under it, each bend twice—once up and once down. As we saw while studying balls, when an airstream bends toward the wing, the pressure near the wing is below atmospheric and when an airstream bends away from the wing, the pressure near the wing is above atmospheric. Since each airstream bends equally toward and away from the wing, it experiences no overall deflection or average pressure change and provides the wing with no overall lift.
But the lower airstream is making a sharp bend around the wing's trailing edge, essentially an:
an upward kink. Air's inertia makes such a kink unstable and it soon blows away from the wing's trailing edge as a swirling horizontal vortex of air. After shedding that vortex, the wing establishes a new, stable flow pattern in which both airstreams pass smoothly away from the wing's trailing edge, a situation named the Kutta condition.
In this new pattern, the airstream flowing over the wing is longer than the airstream flowing under it and both bend:
downward. The upper airstream bends primarily toward the wing, so the air's pressure just above the wing is below atmospheric (a shift toward red) and its speed is increased (narrowly spaced streamlines). In contrast, the lower airstream bends primarily away from the wing, so the air's pressure just below the wing is above atmospheric (a shift toward violet) and its speed is decreased (widely spaced streamlines). The air pressure is now higher under the wing than over it, so this new flow pattern produces upward lift. The air now supports your plane and up you go.
Another way to think about this lift is as a deflection of the airflow.
Air approaches the wing horizontally but leaves heading somewhat downward. To cause this deflection, the wing must push the airflow downward. In reaction, the airflow pushes the wing upward and produces lift. In other words, the wing transfers downward momentum to the air and is left with upward momentum as a result. These two explanations for lift—the Bernoullian view that lift is caused by a pressure difference above and below the wing and the Newtonian view that lift is caused by a transfer of momentum to the air—are perfectly equivalent and equally valid.
However, the overall aerodynamic force on the wing isn't quite perpendicular to the onrushing air:
it tilts slightly downwind. The perpendicular component of this aerodynamic force is lift, but the downwind component is a new type of drag force—induced drag.
induced drag
is a consequence of energy conservation: in addition to transferring momentum to the passing air, the wing also transfers some energy to it. The air extracts that energy from the wing by pushing the wing downwind with induced drag and thereby doing negative work on it. Since induced drag is undesirable, the airplane minimizes it by using as much air mass as possible to obtain its lift. A larger mass of air carries away the airplane's unwanted downward momentum while moving downward less quickly and with less kinetic energy. Since larger wings obtain their lift from larger air masses, they experience less induced drag.
Unfortunately, larger wings also have more surface area and experience more viscous drag, so bigger isn't always better. And because wing shape and airspeed affect aerodynamic forces, too...
wings must be carefully matched to their airplanes. Small propeller airplanes that move slowly through the air need relatively large, highly curved wings to support them. Those wings are often asymmetric—more curved on top than on bottom to make maximum use of the limited, low-speed air they encounter each second. Commercial and military jets fly faster and encounter far more high-speed air each second, so they can get by with relatively small, moderately curved wings.
But even at constant airspeed, a wing's lift can be adjusted by varying its:
angle of attack—the angle at which it approaches the onrushing air. The larger the angle of attack, the more the two airstreams bend and the greater the wing's lift. Because the wings are rigidly attached to the plane, the pilot tips the nose of the plane upward to increase the lift and downward to reduce the lift. That's why raising the plane's nose during takeoff is what finally makes the plane leap up into the air.
Since lift depends so strongly on a wing's _____, some planes can be flown _____:
angle of attack; upside down. As long as the inverted wing is tilted properly, it obtains upward lift and supports the plane. But this feat is easiest when a plane's wing has the same curvature, top and bottom. That's why stunt fliers who regularly fly upside down often use sport aircraft that have symmetric or nearly symmetric wings.
There's a limit to how much lift the pilot can obtain by increasing the wing's angle of attack because tilting the wing gradually transforms it from streamlined to:
blunt, that is, to having a rapid rise in air pressure after its widest point. As we saw for balls, blunt objects generally experience airflow separation and pressure drag. Indeed, beyond a certain angle of attack, the airstream over the top of a wing separates from its surface and the wing stalls. This separation starts when air in the upper boundary layer is brought to standstill by the rapidly rising pressure beyond the wing's widest point. Once this boundary layer stalls, it shaves most of the airstream away from the wing's upper surface.
The separated airstream over the top of the stalled wing leaves a billowing storm of turbulence beneath it. This airstream separation is:
an aerodynamic catastrophe for the airplane. Because the average pressure above the wing increases, the wing loses much of its lift. And the appearance of a turbulent wake heralds the arrival of severe pressure drag. The plane slows dramatically and drops like a rock.
To avoid stalling, pilots keep the angle of attack within a safe range. But the possibility of stalling also limits the...
minimum speed at which the airplane will fly. As the airplane slows down, the pilot must increase its angle of attack to maintain adequate lift. Below a certain speed, the airplane can't obtain that lift without tilting its wings until they stall. It can no longer fly.
To avoid stalling, a plane must never fly slower than this minimum speed, particularly during landings and takeoffs. For a small, propeller-driven plane with highly curved wings, the minimum flight speed...
is so low that it's rarely an issue. For a commercial jet, however, the minimum airspeed is about 220 km/h (140 mph). Airplanes taking off or landing this fast would require very long runways on which to build up or get rid of speed. Instead, commercial jets have wings that can change shape during flight. Slats move forward and down from the leading edges of the wings, and flaps move back and down from the trailing edges.
With both slats and flaps extended, the wing becomes larger and more strongly...
curved, similar to the wings of a small propeller plane, and the minimum safe airspeed drops to a reasonable 150 km/h (95 mph). Vanes near the flaps also emerge during landings to direct high-energy air from beneath the wings onto the flaps. These jets of air keep the boundary layers moving downstream and help prevent stalling.
Once a commercial jet lands, flat panels on the top surfaces of its wings are:
tilted upward and cause the airflow to separate from the tops of the wings. The resulting turbulence created by these spoilers reduces the lift of the wings and increases their drag, so that the plane doesn't accidentally start flying again. Even before landing, the spoilers are sometimes used to slow the plane and help it descend rapidly toward an airport.
In flight, a wing does more than just push the passing air downward; it also:
twists the air near its tip. Since the air pressure below the wing is greater than the air pressure above it, air tends to flow around the wing's tip from bottom to top. The plane soon leaves this air behind, but not before the air has acquired lots of angular momentum and kinetic energy. A swirling vortex thus emerges from each wingtip and trails behind the plane for several kilometers, like an invisible tornado.
wingtip vortex from a jumbo jet can flip a small aircraft that flies through it or give passengers in a much larger plane an unexpected thrill. For safety...
air traffic controllers are careful to keep planes from flying through one another's wakes and schedule them at least 90 s apart on runways. Some modern airplanes have vertical wingtip extensions that reduce these vortices, both to save energy and to diminish the hazard.
For a plane to obtain lift, it needs airspeed:
air must flow across its wings. And since drag forces push it downwind, a plane in level flight can't maintain its airspeed unless something pushes it upwind. That's why a plane has propellers or jet engines: to push the air backward so that the air pushes the plane forward, action and reaction.
A propeller is an assembly of rotating wings. Extending from its central hub are:
two or more blades that together form a sophisticated fan. These blades have airfoil cross sections and are designed to create forward lift forces when the propeller turns and the blades move through the air.
As a propeller blade slices through the air, the airstreams bending around that blade experience:
pressure variations. The forward airstream bends toward the blade's front surface, so the pressure in front of the blade drops below atmospheric. And the rearward airstream bends away from the blade's rear surface, so the pressure behind the blade rises above atmospheric. The resulting pressure difference exerts a forward lift or thrust force on the propeller.
The propeller blades have all the features, good and bad, of airplane wings. Their thrust increases with...
size, front-surface curvature, pitch (i.e., angle of attack), and airspeed; in other words, the larger the propeller, the faster it turns, and the more its blades are angled into the wind, the more thrust it produces. The blades themselves have a twisted shape to accommodate the variations in airspeed along their lengths, from hub to tip.
And like a wing, a propeller stalls when the airflow separates from the front surfaces of its blades; it suddenly becomes:
more of an air-mixer than a propeller. This stalled-wing behavior was the standard operating condition for air and marine propellers (see ) before the work of Wilbur Wright. One of the principal sources of noise in submarines is the turbulence created by their propellers. To reduce this turbulence, the propellers of modern nuclear submarines are designed to avoid water flow separation and stalling.
propeller also experiences induced drag. As the propeller's thrust pushes...
the plane through the air, induced drag extracts energy from the propeller. To keep the propeller turning steadily, an engine must do work on the propeller. Propellers are driven by high-performance reciprocating engines, like those found in automobiles, or the turbojet engines that we'll discuss later.
Propellers aren't perfect; they have three serious limitations. First, a propeller exerts a:
torque on the passing air, so that air exerts a torque on the propeller. This reaction torque can flip a small plane. To minimize torque problems, some planes use pairs of oppositely turning propellers and single-propeller planes usually locate their propellers in front, so that the spinning air can return angular momentum to them while passing over their wings.
A second problem with a propeller is that its thrust:
diminishes as the plane's forward speed increases. When the airplane is stationary, a propeller blade moves through motionless air. But when the airplane is traveling fast, the air approaches that same propeller blade from the front of the plane (Fig. 6.3.11b). To retain its thrust at higher airspeeds, the propeller blade must increase its pitch—it must swivel forward—to meet this onrushing air.
The third and most discouraging problem with propellers, especially in high-speed aircraft:
is drag. To keep up with the onrushing air at high airspeeds, the propellers must turn at phenomenal rates. The tips of the blades travel so fast that they exceed the speed of sound—the fastest speed at which a fluid such as air can convey forces from one place to another. When the blade tip exceeds this speed, the air near the tip doesn't accelerate until the tip actually hits it. Instead of flowing smoothly around the tip, the air forms a shock wave—a narrow region of high pressure and temperature caused by the supersonic impact—and the propeller stalls. That's why propellers aren't useful on high-speed aircraft.
Unlike propellers, jet engines work well at:
high speeds. While a propeller tries to operate directly in the high-speed air approaching the plane, a jet engine first slows this air down to a manageable speed. To achieve this change in speeds, the jet engine makes wonderful use of Bernoulli's equation.
During flight, air rushes into the engine's inlet duct or:
diffuser at about 800 km/h (500 mph), the speed of the plane. Once inside that diffuser, the air slows down and its pressure increases, but its total energy is unchanged. The air then passes through a series of fanlike compressor blades that push it deeper into the engine, doing work on it and increasing both its pressure and its total energy. By the time the air arrives at the combustion chamber, its pressure is many times atmospheric.
Now fuel is added to the air and the mixture is ignited. Since hot air is less dense than cold air:
the hot exhaust gas takes up more space than it did before combustion. Furthermore, combustion subdivides the fuel molecules into smaller pieces that therefore take up still more volume. This hot exhaust gas pours out of the combustion chamber, traveling faster than when it entered.
The pressure of the exhaust gas is still very high as it streams through a windmill-like turbine. The air does work on that turbine and thereby...
spins the compressor for the incoming air. After the turbine, the high-pressure gas finally accelerates through the engine's outlet nozzle and emerges into the open sky at atmospheric pressure and extraordinarily high speed.
Overall, the engine slows the air down, adds energy to it, and then lets it...
accelerate back to high speed. Because the engine has added energy to the air, the air leaves the engine traveling faster than when it arrived. The air's increased backward velocity means that the jet engine has pushed it backward and the air has reacted by exerting a forward thrust force on the jet engine. In other words, the airplane has obtained forward momentum by giving the departing air backward momentum.
The turbojet is less energy efficient than it could be. Since it gives backward momentum to a relatively small mass of air...
that air ends up traveling overly fast and with excessive kinetic energy. To make the engine more efficient, it should give backward momentum to a larger mass of air.
The turbofan engine solves this problem by using a turbojet engine to spin a huge fan. Since this fan is located in the engine's inlet duct...
the air's speed decreases and its pressure increases before it enters the fan. The fan then does work on the air and further increases its pressure. While about 5% of this air then enters the turbojet engine, the vast majority of it accelerates out the back of the fan duct and emerges into the open sky at atmospheric pressure and increased speed. Overall, the fan has pushed the air backward and the air has pushed the fan forward, producing forward thrust.
Like a turbojet, the turbofan slows air down, adds energy to it...
then lets it accelerate back to high speed. But because the turbofan engine moves more air than a turbojet engine, it gives that air less energy and uses less fuel. The huge fanlike engines on many jumbo jets are turbofans.
Observations about Airplanes
􀂄􀂄 Airplanes use the air to support themselves
􀂄􀂄 Airplanes need airspeed to stay aloft
􀂄􀂄 Airplanes seem to follow their nose, up or down
􀂄􀂄 Airplanes can rise only so quickly
􀂄􀂄 Airplane wings often change shape in flight
􀂄􀂄 Airplanes have various propulsion systems
Using a Wing to Obtain Lift (part 1)
􀂄􀂄 As air flows under a wing,
􀂄􀂄 air bends away from the wing
􀂄􀂄 air’s pressure rises, speed drops
􀂄􀂄 As air flows over the wing,
􀂄􀂄 air bends toward the wing
􀂄􀂄 air’s pressure drops, speed rises
􀂄􀂄 The wing experiences a pressure imbalance
􀂄􀂄 There is an upward pressure force on the wing
Using a Wing to Obtain Lift (part 2)
􀂄􀂄 The wing experiences
􀂄􀂄 a strong upward lift force
􀂄􀂄 a small downstream drag force
􀂄􀂄 Wing pushes air down, air pushes wing up!
􀂄􀂄 Downward momentum is transferred
􀂄􀂄 from the earth to the airplane by gravity,
􀂄􀂄 from the airplane to the air by lift forces, and
􀂄􀂄 from the air to the earth by pressure on the ground
At Take-Off
􀂄􀂄 As a wing starts moving in air:
􀂄􀂄 the airflow is symmetric
􀂄􀂄 and the wing experiences no lift
􀂄􀂄 However, this airflow is
􀂄􀂄 unstable at trailing edge kink
􀂄􀂄 and the wing sheds a vortex
􀂄􀂄 After the vortex leaves, the wing has lift
Angle of Attack
􀂄􀂄 A wing’s lift depends on...
􀂄􀂄 Tilting an airplane’s wings...
􀂄􀂄 changes the net force on the airplane
􀂄􀂄 and can make the airplane accelerate up or down
􀂄􀂄 but usually requires tilting the airplane’s fuselage
􀂄􀂄 Plane’s tilt controls lift, not direction of travel
Limits to Lift: Stalling
􀂄􀂄 At too great an angle of attack...
􀂄􀂄 the upper boundary layer stalls,
􀂄􀂄 the airstream detaches from wing,
􀂄􀂄 the lift nearly vanishes,
􀂄􀂄 and pressure drag appears
􀂄􀂄 Plane plummets abruptly
Wing Shape
􀂄􀂄 Asymmetric airfoils produce large lifts...
􀂄􀂄 They are well suited to low-speed flight
􀂄􀂄 Symmetric airfoils produce small lifts
􀂄􀂄 They are well suited to high-speed flight
􀂄􀂄 and allow plane to fly inverted easily
􀂄􀂄 High-speed planes often
change wing shape in flight
Turning and Orientation
􀂄􀂄 Airplanes also use lift to accelerate to the side
􀂄􀂄 Three orientation controls:
􀂄􀂄 Angle of attack controlled by elevators
􀂄􀂄 Left-right tilt controlled by ailerons
􀂄􀂄 Left-right rotation controlled by rudder
􀂄􀂄 Steering involves ailerons and rudder
􀂄􀂄 Elevation involves elevators and engine
Propellers
􀂄􀂄 Propellers are spinning wings
􀂄􀂄 They deflect air backward,
􀂄􀂄 do work on air (add energy),
􀂄􀂄 and pump air toward rear of plane
􀂄􀂄 Action-Reaction
􀂄􀂄 They push the air backward,
􀂄􀂄 so air pushes them forward
Jet Engines (Part 1)
􀂄􀂄 Jet engines pump air toward rear of plane...
􀂄􀂄 Engine consists of an oval “ball” with a complicated
duct or passageway through it
􀂄􀂄 Air passing through the duct exchanges first speed
for pressure and then pressure for speed
􀂄􀂄 Engine adds
energy to air
inside the duct
Jet Engines (Part 2)
􀂄􀂄 Air entering diffuser slows and its pressure rises
􀂄􀂄 Compressor does work on air
􀂄􀂄 Fuel is added to air and that mixture is burned
􀂄􀂄 Expanding exhaust gas does work on turbine
􀂄􀂄 As exhaust leaves
nozzle it speeds up
and its pressure drops
Jet Engines (Part 3)
􀂄􀂄 Turbojet obtains forward momentum by...
􀂄􀂄 moving relatively little air
􀂄􀂄 and giving that air too much energy
􀂄􀂄 Turbofan obtains forward momentum by
􀂄􀂄 moving much more air
􀂄􀂄 giving that air less energy
Summary about Airplanes
􀂄 Airplanes use lift to support themselves
􀂄􀂄 Propulsion overcomes induced drag
􀂄􀂄 Speed and angle of attack affect altitude
􀂄􀂄 Extreme angle of attack causes stalling
􀂄􀂄 Propellers do work on passing airstream
􀂄􀂄 Jet engines do work on slowed airstream
unlike the forces of ordinary sliding friction—which don't depend on...
relative velocities—viscous forces become larger as the relative velocities within a fluid increase. That's because as two layers of water slide past one another faster, their molecules collide harder and more frequently.
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