Outline
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Introduction
General Characteristics of External Flow
Boundary Layer Characteristics
Drag
Lift
Introduction
Introduction
• Objects are completely surrounded by the fluid and the flows are termed
external flows.
• Examples include the flow of air around airplane, automobiles, and falling
snowflakes, or the flow of water around submarines and fish.
• External flows involving air are often termed aerodynamics in response to
the important external flows produced when an object such as an airplane
flies through the atmosphere.
Approaches to External Flows
• Theoretical
– The information obtained from theoretical methods is limited due to the
complexities of the governing equations and geometry of objects.
• Experimental
– Much of the information about external flows comes from experiments
carried out on scale models of the actual objects.
• Numerical
– Computational techniques are also capable of predicting external flows
past objects.
General Characteristics
General Characteristics
• A body immersed in a moving fluid experiences a resultant
force due to the interacting between the body and the fluid
surrounding:
– The body is stationary and the fluid flows past the body with velocity
U.
– The fluid far from the body is stationary and the body moves through
the fluid with velocity U.
– In any case, we can fix the coordinate system in the body and treat the
situation as fluid flowing past a stationary body with constant upstream
velocity U
• For a given-shaped object, the characteristics of the flow
depend very strongly on various parameters such as size,
orientation, speed, and fluid properties.
General Characteristics
• The structure of an external flow and the ease with which the
flow can be described and analyzed often depend on the nature
of the body in the flow.
• Three general categories of bodies include
– (a) two-dimensional objects,
– (b) axisymmetric bodies, and
– (c) three-dimensional bodies.
• Another classification of body shape can be made depending
on whether the body is streamlined or blunt.
General Characteristics
Two-dimensional, axisymmetric,
and three- dimensional flows.
Lift and Drag
• The interaction between the
body and the fluid:
– Stresses-wall shear stresses, w
,due to viscous effects.
– Normal stresses, due to the
pressure p.
• Both w and p vary in
magnitude and direction
along the surface.
• The detailed distribution of
w and p is difficult to obtain.
• However, only the integrated
or resultant effects of these
distributions are needed.
Lift and Drag
• The resultant force in the direction of the upstream velocity is
termed the Drag, and the resultant force normal to the
upstream velocity is termed the Lift.
• The x and y components of the fluid force on the small area
element dA are:
• the net x and y components of the force
Lift and Drag
• Without detailed information concerning the shear
stress and pressure distributions on a body, the
drag and the lift are difficult to obtain by
integration.
• A widely used alternative is to define
dimensionless lift and drag coefficients and
determine their approximate values by means of
either a simplified analysis, some numerical
technique, or an appropriate experiment.
Lift and Drag
It is much easier to force a streamlined
body than a blunt body through a fluid.
Example
• Air at standard conditions flows past a flat plate as is indicated
in Figure E9.1. In case (a) the plate is parallel to the upstream
flow, and in case (b) it is perpendicular to the upstream flow.
If the pressure and shear stress distributions on the surface are
as indicated (obtained either by experiment or theory),
determine the lift and drag on the plate.
Example
• With the plate parallel to the upstream flow we have θ=90゜
so that the lift and drag are given by
Example
• With the plate perpendicular to the upstream flow we have
θ=0゜so that the lift and drag are given by
Characteristics of Flow Past an Object
• For a given-shaped object, the characteristics of the flow
depend very strongly on various parameters such as size,
orientation, speed, and fluid properties.
• According to dimensional analysis arguments, the character of
flow should depend on the various dimensionless parameters.
• For typical external flows the most important of these
parameters are the Reynolds number, the Mach number, and
for the flow with a free surface, the Froude number.
Characteristics of Flow Past an Object
• For the present, we consider how the external flow and its
associated lift and drag vary as a function of Reynolds number
• For most external flows, the characteristic length of objects are
on the order of 0.10m~10m. Typical upstream velocities are
on the order of 0.01m/s~100m/s. The resulting Reynolds
number range is approximately 10~109.
• Re > 100. The flows are dominated by inertial effects.
• Re < 1. The flows are dominated by viscous effects.
Re Effect
• The general differences between small and large Reynolds
number flow past streamlined and blunt objects can be
illustrated by considering flows past two objects—
– A flat plate parallel to the upstream velocity
– A circular cylinder.
Re Effect
• With Re = 0.1, the viscous effects are relatively strong and the
plate affects the uniform upstream flow far ahead, above,
below, and behind the plate. In low Reynolds number flows
the viscous effects are felt far from the object in all directions.
Re Effect
• With Re = 10, the region in which viscous effects are important become
smaller an all directions except downstream.
• One does not need to travel very far ahead, above, or below the plate to
reach areas in which the viscous effects of the plate are not felt.
• The streamlines are displaced from their original uniform upstream
conditions, but the displacement is not as great as for the Re=0.1 situation.
Re Effect
• With Re = 107 , the flow is dominated by inertial effects and the viscous
effects are negligible everywhere except in a region very close to the plate
and in the relatively thin wake region behind the plate.
• Since the fluid must stick to the solid surface, there is a thin boundary
layer region of thickness  <<  next to the plate in which the fluid velocity
changes from U to zero on the plate.
• The thickness of boundary layer increases in the direction of flow, starting
from zero at the forward or leading edge of the plate.
Re Effect
• The flow within the boundary layer may be laminar or turbulent depending
on various parameters involved.
• The streamline of the flow outside of the boundary layer are nearly parallel
to the plate.
• The existence of the plate has very litter effect on the streamline outside of
the boundary layer – either ahead, above, and below the plate.
Flow Past a Circular Cylinder
• When Re = 0.1, the viscous effects are important several diameters in any
direction from the cylinder.
• A characteristic of this flow in this case is that the streamlines are
essentially symmetric about the center of the cylinder-the streamline
pattern is the same in front of the cylinder as it is behind the cylinder.
Flow Past a Circular Cylinder
• As Reynolds number is increased (Re =50), the region ahead of the
cylinder in which viscous effect are important becomes smaller, with the
viscous region extending only a short distance ahead of the cylinder.
• The flow separates from the body at the separation point.
• With the increase in Reynolds number, the fluid inertia becomes more
important and at the some on the body, denoted the separation location, the
fluid’s inertia is such that it cannot follow the curved path around to the
rear of the body.
Flow Past a Circular Cylinder
• With larger Reynolds numbers (Re=10^5), the area affected by the viscous
forces is forced farther downstream until it involve only a thin (δ<<D)
boundary layer on the front portion of the cylinder and an irregular,
unsteady wake region that extends far downstream of the cylinder.
• The velocity gradients within the boundary layer and wake regions are
much larger than those in the remainder of the flow field.
Boundary Layer Characteristics
Boundary Layer Characteristics
• The flow past an object can be treated as a combination of
viscous flow in the boundary layer and inviscid flow
elsewhere.
• Inside the boundary layer the friction is significant and across
the width of which the velocity increases rapidly from zero (at
the surface) to the value inviscid flow theory predicts.
• Outside the boundary layer the velocity gradients normal to the
flow are relatively small, and the fluids acts as if it were
inviscid, even though the viscosity is not zero.
Flat Plate Boundary Layer
• The boundary layer is laminar for a short distance downstream from the
leading edge; transition occurs over a region of the plate rather than at a
single line across the plate.
• The transition region extends downstream to the location where the
boundary layer flow becomes completely turbulent.
• For a finite length plate, it is clear that the plate length, l, can be used as
the characteristic length, with the Reynolds number as Re=U l /ν.
• For the infinitely long flat plate extending from x=0 to x→∞, it is not
obvious how to define the Reynolds number because there is no
characteristic length.
Flat Plate Boundary Layer
• For the infinitely long flat plate we use x, the coordinate
distance along the plate from the leading edge, as the
characteristic length and define the Reynolds number as
Rex=Ux/ν.
• For any fluid or upstream velocity the Reynolds number will
be sufficiently large for boundary layer type flow if the plate
is long enough.
Boundary Layer Thickness
• Standard Boundary layer thickness
• Boundary layer displacement thickness
• Boundary layer momentum thickness
Standard Boundary Layer Thickness
• The standard boundary layer thickness is the distance from the
plate at which the fluid velocity is within some arbitrary value
of the upstream velocity.
Boundary Layer Displacement Thickness
• The boundary layer retards the fluid, so that the mass flux and momentum
flux are both less than they would be in the absence of the boundary
layer.
• The displacement distance is the distance the plate would be moved so
that the loss of mass flux (due to reduction in uniform flow area) is
equivalent to the loss the boundary layer causes.
b is the plate width
Boundary Layer Displacement Thickness
The displacement thickness represents the amount that the thickness of the body
must be increased so that the fictitious uniform inviscid flow has the same mass
flowrate properties as the actual viscous flow.
This concept can be used to simulate the presence that the boundary layer
has on the flow outside of the boundary layer by adding the displacement
thickness to the actual wall and treating the flow over the thickened body as an
inviscid flow.
Boundary Layer Momentum Thickness
• The momentum thickness is the distance the plate would be
moved so that the loss of momentum flux is equivalent to the
loss the boundary layer actually causes.
Example
Air flowing into a 2-ft-square duct with a uniform velocity of 10
ft/s forms a boundary layer on the walls as shown in Fig. The
fluid within the core region (outside the boundary layers) flows
as if it were inviscid. From advanced calculations it is determined
that for this flow the boundary layer displacement thickness is
given by
where
and x are in feet. Determine
the velocity of the air within the duct but outside of the boundary
layer
Example
• the volume flow rate across any section of the duct is equal to
that at the entrance (i.e., Q1=Q2). That is,
• According to the definition of the displacement thickness, the
flowrate across section (2) is the same as that for a uniform
flow with velocity U through a duct whose walls have been
moved inward by δ*
• combining Eqs.
Example
Calculate the displacement and momentum thickness of the
boundary layer if the Velocity profile in boundary layer is
Here 𝛿 represents the average thickness of the boundary layer.
Solution
Example
• Calculate the displacement thickness and momentum thickness
of the boundary layer if the velocity distribution is:
Solution
Boundary Layer Equations
• The equations for the boundary layer thickness, displacement thickness and
momentum thickness can only be evaluated if we have a solution for the
velocity of the fluid as a function of y.
• Requires a solution of the Navier-Stokes equations ...
• For steady, incompressible, two-dimensional laminar flows with negligible
gravitational effects, the NS Eqns can be simplified for boundary layer flow
analysis:
Parabolic
Prandtl/Blasius Boundary Layer Solution
• In theory, the details of viscous, incompressible flow past any object can be
obtained by solving the governing Navier–Stokes equations.
• Although the mathematical problem is well-posed, no one has obtained an
analytical solution to these equations for flow past any shaped body.
• By using boundary layer concepts introduced in the previous sections,
Prandtl was able to impose certain approximations (valid for large
Reynolds number flows), and thereby to simplify the governing equations.
• In 1908, H. Blasius (1883 –1970), one of Prandtl’s students, was able to
solve these simplified equations for the boundary layer flow past a flat
plate parallel to the flow.
Prandtl/Blasius Boundary Layer Solution
Momentum
Integral BL Eq.
for a Flat Plate
Local friction coefficient,
laminar flat plate
Transition from Laminar to Turbulent Flow
• The parameter that governs the transition to turbulent flow is the Reynolds
numbers – in this case, the Reynolds number based on the distance from
the leading edge of the plate, Rex=Ux/.
• The value of the Reynolds number at the transition location is a rather
complex function of various parameters involved, including the roughness
of the surface, the curvature of the surface, and some measure of the
disturbances in the flow outside the boundary layer.
• On a flat plate with a sharp leading edge in a typical air-stream, the
transition takes place at a distance x from the leading edge given by
Rexcr=2×105 to 3×106. Rexcr= 5×105 is used.
Transition from Laminar to Turbulent Flow
• The actual transition from laminar to turbulent boundary layer flow may
occur over a region of the plate, not a specific single location.
• Typical, the transition begins at random location on the plate in the vicinity
of Rex= Rexcr
• The complex process of transition from laminar to turbulent flow involves
the instability of the flow filed.
Transition from Laminar to Turbulent Flow
• Small disturbances imposed on the boundary layer flow will
either grow or decay, depending on where the disturbance is
introduced into the flow.
• If the disturbances occur at a location with Rex<Rexcr , they
will die out, and the boundary layer will return to laminar
flow at that location.
• If the disturbances occur at a location with Rex>Rexcr , they
will grow and transform the boundary layer flow downstream
of this location into turbulence.
• The boundary layer on a flat plate will become turbulent if the
plate is long enough.
Transition from Laminar to Turbulent Flow
Turbulent spots and the transition from laminar to turbulent boundary
layer flow on a flat plate. Flow from left to right.
Transition from Laminar to Turbulent Flow
• Transition from laminar to
turbulent flow involves a
noticeable change in the shape
of the boundary layer velocity
profiles.
• The turbulent profiles are flatter,
have a large velocity gradient at
the wall, and produce a larger
boundary layer thickness than do
the laminar profiles.
Example
A fluid flows steadily past a flat plate with a velocity of U = 10 ft/s. At
approximately what location will the boundary layer become turbulent,
and how thick is the boundary layer at that point if the fluid is (a) water
at 60 °F (b) standard air, or (c) glycerin at 68 °F?
For any fluid, the laminar boundary layer thickness is found from
The boundary layer remains laminar up to
Assume
Example
Turbulent Boundary Layer Flow
• The structure of turbulent boundary layer
flow is very complex, random, and
irregular.
• The velocity at any location in the
turbulent flow is unsteady in a random
fashion.
• The flow can be thought of as a jumbled
mix of intertwined, finite-sized eddies (or
swirls) – considerably more than is
associated with the mixing found in
laminar flow where it is confined to the
molecular scale.
Turbulent Boundary Layer Flow
• Although there is considerable random motion of fluid
particles perpendicular to the plate, there is very little net
transfer of mass across the boundary layer – the largest flow
rate by far is parallel to the plate.
• Momentum? There is, however, a considerable net transfer of
x component of momentum perpendicular to the plate because
of the random motion of the particle.
• Fluid particles moving toward the plate (in the –y direction)
have some of their excess momentum (come from higher
velocity area) removed by the plate.
Turbulent Boundary Layer Flow
• Conversely, particles moving away from plate gain momentum from
the fluid. The net result is that the plate acts as a momentum sink,
continually extracting momentum from the fluid.
• For laminar flows, such cross-stream transfer of these properties
takes place solely on the molecular scale.
• For turbulent flow the randomness is associated with fluid particle
mixing.
• Consequently, the shear force for turbulent boundary layer flow is
considerably greater than it is for laminar boundary layer flow.
• Since there is no precision expression for the shear stress in the
turbulent flow, there are no “exact” solutions for turbulent boundary
layer flows.
Turbulent Boundary Layer Flow
Effect of Surface Roughness on Drag
• In general, the drag coefficient for
a flat plate of length l is a function
of the Reynolds number Rel, and
the relative roughness, ε/l
Friction drag coefficient for a flat
plate parallel to the upstream flow
Example
The water ski shown in Figure moves through 70℉ water with a
velocity U. Estimate the drag caused by the shear stress on the
bottom of the ski for 0 < U < 30 ft/s.
Example
Effects of Pressure Gradient
• In general, when a fluid flows past an object other than a
flat plate, the pressure field is not uniform.
• For bodies of nonzero thickness, the upstream velocity
(that far ahead of the plate) and the free-stream velocity
(that at the edge of the boundary layer) are not equal.
• Due to this the, the component of the pressure gradient in
the streamwise direction is not zero, within the BL.
d’Alembert’s paradox can be described in terms
of the effect of the pressure gradient on
boundary layer flow.
Effects of Pressure Gradient
Because of the viscous effects involved, the
particle in the boundary layer experiences a
loss of energy as it flows along.
This loss means that the particle does not
have enough energy to coast all of the way
up the pressure hill (from C to F) and to
reach point F at the rear of the cylinder.
Thus, the fluid flows against the increasing
pressure as far as it can, at which point the
boundary layer separates from the surface.
Because of the boundary layer separation, the
average pressure on the rear half of the cylinder
is considerably less than that on the front half.
Thus, a large pressure drag is developed, even
though (because of small viscosity) the viscous
shear drag may be quite small.
Effects of Pressure Gradient
Compared with a laminar boundary layer, a turbulent boundary layer flow has more
kinetic energy and momentum associated with it because:
The velocity profile is fuller, more nearly like the ideal uniform profile, The
turbulent boundary layer can flow farther before it separates than can the laminar
boundary layer.
Effects of Pressure Gradient
• 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.
• When the outer flow decelerates, U(x) decreases, P(x) increases, and
we have an unfavorable or adverse pressure gradient.
• 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 from the wall.
Effects of Pressure Gradient
• If the adverse pressure gradient is strong enough
(dP/dx = −U dU/dx is large), the boundary layer is likely to
separate off the wall.
• The boundary layer equations are not valid downstream of a
separation point because of reverse flow in the separation
bubble.
Effects of Pressure Gradient
Effects of Pressure Gradient
• Turbulent boundary layers tend to be able to sustain an
adverse pressure gradient better than an equivalent laminar
boundary layer.
• The more efficient mixing which occurs in a turbulent
boundary layer transports kinetic energy from the edge of the
boundary layer to the low momentum flow at the solid surface,
often preventing the separation which would occur for a
laminar boundary layer under the same conditions.
Effects of Pressure Gradient
Drag
• Drag is the component of force on a body acting parallel to the
direction of relative motion.
Drag coefficient
• The drag on a complex body can be approximated as the sum of the drag on
its parts.
Friction Drag
• Friction drag, is that part of the drag that is due directly to the shear
stress (magnitude + direction), on the object.
• Because the viscosity of most common fluids is small, the
contribution of the shear force to the overall drag on a body is often
quite small. (Large Re)
• For highly streamlined bodies or for low Reynolds number flow,
however, most of the drag may be due to friction drag.
• Friction drag coefficient is a function of both Re and relative
surface roughness ε/l for turbulent flows.
Pressure Drag
• Pressure drag, is that part of the drag that is due directly to the
pressure, p, on an object.
• It is often referred to as form drag because of its strong
dependency on the shape or form of the object.
is the reference pressure
is the pressure coefficient
Drag
Shape Dependence
Circular Cylinder
Drag obtained by using either of these drag
coefficients would be the same
Drag
Shape Dependence
Two objects of considerably different size that have the same drag force:
(a) circular cylinder CD = 1.2; (b) streamlined strut CD = 0.12.
Drag
Reynolds Number Dependence
(1) very low Reynolds number flow
–
governed by a balance between viscous and pressure forces.
For very small Reynolds number flows, the drag coefficient varies inversely with
the Reynolds number
Drag
Reynolds Number Dependence
(1) very low Reynolds number flow
Drag
Reynolds Number Dependence
(2) moderate Reynolds number flow Laminar BL
•
dependence for laminar boundary layer on a flat
plate
• Moderate Reynolds number flows past blunt bodies generally
produce drag coefficients that are relatively constant.
Drag
Reynolds Number Dependence
(2) moderate Reynolds number flow Laminar BL
Drag
Reynolds Number Dependence
(3) very large Reynolds number flow Turbulent BL.
• For streamlined bodies, the drag coefficient increases when the boundary layer
becomes turbulent (increase in shear stress).
• The drag coefficient for a relatively blunt object, such as a cylinder or sphere,
actually decreases when the boundary layer becomes turbulent due to delayed
BL separation.
Drag
Reynolds Number Dependence
Drag
Compressibility Effects
• The drag coefficient is usually independent of Mach number
for Mach numbers up to approximately 0.5.
• For larger Mach number flows, the drag coefficient can be
strongly dependent on Ma, with only secondary Reynolds
number effects.
• For most objects, values of CD increase dramatically in the
vicinity of Ma = 1 (existence of shock wave)
Drag
Compressibility Effects
Drag
Surface Roughness
• Depending on the body shape, an increase in surface
roughness may increase or decrease drag.
• Undesirable for SL bodies
• For sphere:
Drag
Drag
The drag on a complex body can be
approximated as the sum of the drag
on its parts.
Lift
• Any object moving through a fluid will experience a net force
of the fluid on the object.
• For objects symmetrical perpendicular to the upstream flow,
this force will be in the direction of the free stream—a drag.
• If the object is not symmetrical (or if it does not produce a
symmetrical flow field, such as the flow around a rotating
sphere), there may also be a force normal to the free stream—a
lift.
• The most important parameter that affects the lift coefficient is the shape of the
object.
• Usually most lift comes from pressure forces, not viscous forces.
Lift
• A typical device designed to produce lift does so by generating
a pressure distribution that is different on the top and bottom
surfaces.
• For large Reynolds number flows these pressure distributions
are usually directly proportional to the dynamic pressure, with
viscous effects being of secondary importance.
Lift
• The angle of attack is the angle between the airfoil chord and
the free stream velocity vector.
• The chord length (c) of an airfoil is the straight line joining the
leading edge and the trailing edge.
• The aspect ratio is defined as the ratio of the square of the
length of the airfoil (b) to the planform area
• The wing loading, defined as the average lift per unit area of
the wing, increases with speed.
Lift
• In general, the lift coefficient increases and the drag coefficient
decreases with an increase in aspect ratio.
• Long wings are more efficient because their wing tip losses are
relatively more minor than for short wings.
Lift
• Although viscous effects and the wall shear stress contribute
little to the direct generation of lift, they play an extremely
important role in the design and use of lifting devices.
• This is because of the viscosity-induced boundary layer
separation that can occur on nonstreamlined bodies such as
airfoils that have too large an angle of attack
Lift
• If alpha is too large, the boundary layer on the upper surface
separates, the flow over the wing develops a wide, turbulent
wake region, the lift decreases, and the drag increases. This
condition, as indicated by the figures in the margin, is termed
stall.
• In many lift-generating devices
the important quantity is the
ratio of the lift to drag developed
• High-performance airfoils
generate lift that is perhaps 100
or more times greater than their
drag.
Lift
• Changing the angle of attack represents a change in the shape of the object.
• Other shape changes can be used to alter the lift and drag when desirable.
• In modern airplanes it is common to utilize leading edge and trailing edge
flaps.
Circulation
• Since viscous effects are of minor importance in the generation
of lift, it should be possible to calculate the lift force on an
airfoil by integrating the pressure distribution obtained from
the equations governing inviscid flow past the airfoil.
(Potential Flow Theory).
Circulation
• Unrealistic flow situation can be corrected by adding an
appropriate clockwise swirling flow around the airfoil.
• The results are twofold: (1)The unrealistic behavior near the
trailing edge is eliminated (i.e., the flow pattern of Fig. b is
changed to that of Fig. c), and (2) the average velocity on the
upper surface of the airfoil is increased while that on the lower
surface is decreased.
• The addition of the clockwise swirl is termed the addition of
circulation.
Circulation
•
•
•
•
Near the tips of the wing this
pressure difference will cause
some of the fluid to attempt to
migrate from the lower to the
upper surface.
At the same time, this fluid is
swept downstream, forming a
trailing vortex (swirl) from each
wing tip
The trailing vortices from the
right and left wing tips are
connected by the bound vortex
along the length of the wing.
It is this vortex that generates the
circulation that produces the lift.
The combined vortex system (the
bound vortex and the trailing
vortices) is termed a horseshoe
vortex.
Circulation
Circulation
Induced Drag
• Tip vortices that interact with the free stream impose forces on
the wing tips in all directions, including the flow direction.
• The component of the force in the flow direction adds to drag
and is called induced drag.
• The total drag of a wing is then the sum of the induced drag
(3-D effects) and the drag of the airfoil section (2-D effects).
Lift Generated by Spinning
Magnus effect: The phenomenon of producing lift by the rotation of a solid body.
When the ball is not spinning, the lift is zero because of top–bottom symmetry.
But when the cylinder is rotated about its axis, the cylinder drags some fluid
around because of the no-slip condition and the flow field reflects the
superposition of the spinning and nonspinning flows.
Generation of lift on a rotating circular cylinder for the case of “idealized”
potential flow (the actual flow involves flow separation in the wake region).
Lift Generated by Spinning
Lift Generated by Spinning
•
Note that the lift
coefficient strongly
depends on the rate of
rotation, especially at
low angular velocities.
•
The effect of the rate of rotation
on the drag coefficient is small.
Roughness also affects the drag
and lift coefficients.
•
In a certain range of Reynolds
number, roughness produces
the desirable effect of
increasing the lift coefficient
while decreasing the drag
coefficient.
•
Therefore, golf balls with the
right amount of roughness
travel higher and farther than
smooth balls for the same hit.
Example
A well-hit golf ball (diameter D = 1.69 in., weight W = 0.0992 lb) can travel at U = 200
ft/s as it leaves the tee. A well-hit table tennis ball (Diameter D = 1.50 in., weight W =
0.00551 lb) can travel at U = 60 ft/s as it leaves the paddle. Determine the drag on a
standard golf ball, a smooth golf ball, and a table tennis ball for the conditions given.
Also determine the deceleration of each ball for these conditions.
For either ball, the drag can be obtained from
For the golf ball in standard air
Example
while for the table tennis ball
Example
Example
Example
Example
Example
Example
Example
Practice Problems
• 9.92, 9.95, 9.97, 9.87, 9.81, 9.77, 9.72, 9.69, 9.22, 9.19