for most objects moving through a fluid, the significant fluid force is drag
DESCRIPTION
LIFT. For most objects moving through a fluid, the significant fluid force is drag. However for some specially shaped objects the lift force is also important. LIFT. DRAG. TIME OUT. Newtonian Theory (1687) entire second book of Principia dedicated to fluid mechanics - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/1.jpg)
For most objects moving througha fluid, the significant fluid force is drag.
However for some specially shapedobjects the lift force is also important.
LIFT
LIFT
DRAG
![Page 2: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/2.jpg)
TIME OUT
![Page 3: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/3.jpg)
Newtonian Theory (1687) entire second book of Principia dedicated to fluid mechanics - assumed particles of fluid lose momentum
normal to plate but keep momentum parallel to plate.
p due to random motion of molecules
p p
Aside
![Page 4: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/4.jpg)
Force normal to plate, N = Time rate of change of the normal component of momentum =(mass flow) x change in normal component of velocity =
N = ( V A sin) x (V sin)N/A = p – p = (V sin)2
(p - p ) / (1/2 V 2) = Cp = 2 sin2
Cp = 1.86 as M Aside
![Page 5: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/5.jpg)
Forces on airplane atlevel speed and constantheight and speed.
Lift force is the component of R that is perpendicular tofree stream velocity, and drag is the component of R parallel to the free stream velocity. If planes height is not changing then: Lift = Weight
![Page 6: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/6.jpg)
Force generated if we brought fluid directly approaching area to rest
CL = FL/(1/2 V2Ap)
CD = FD /(1/2 V2A)
FL
FD
Note: FL is not parallel to N
![Page 7: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/7.jpg)
Chord of an airfoil is the straight line joining the leading and trailing edge. If mean line and chord do not coincide thenairfoil is cambered. Ap refers to planform area, maximum projectionof wing (independent of angle of attack, )
![Page 8: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/8.jpg)
Ap and c are independent of
CL =FL/(1/2 V
2Ap)
CD =FD/(1/2 V
2A)
Ap = planform areamax. proj. of wing
A
![Page 9: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/9.jpg)
Given: Kite in standard air, mass = 0.2 kg;CL = 2sin(); CL/CD = 4. Find
U= 10 m/s 0.2kg (g)
Area = 1 m2
5o
= ?
![Page 10: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/10.jpg)
Kite in standard air, CL = 2sin(); CL/CD = 4
FL
UFD
T
mg
= ?
CL = FL/(1/2 V2Ap)
CD = FD /(1/2 V2A)
CL/CD = FL/FD = 4(Ap = A)
![Page 11: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/11.jpg)
9.144
Fy = FL – mg –Tsin() = 0Fx = FD –Tcos() = 0FL = CL A ½ U
2
CL = 2sin(5o) = 0.548FL = 33.7NFD = FL/4 = 8.43N
FL
UFD
T
mg
= ?
![Page 12: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/12.jpg)
tan () = Tsin()/Tcos() = (FL – mg)/FD
= tan-1{{(FL – mg)/FD} = 75.1o
FL
UFD
T
mg
= ?
![Page 13: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/13.jpg)
For all cases angle of attack is 4o and aspect ratio is 6.Lift to drag ratios of about 20 are common for modern transport planes.
flat plate bent plate airfoil
IMPORTANCE OF CAMBER
![Page 14: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/14.jpg)
Otto Lilienthal on a monoplaneglider in 1893
Otto Lilienthal on a biplaneglider in 1893
Otto Lilienthal (1848-96) is universally recognized as as the first flying human. His wings were curved (camber – height = 1/12th of chord). On August 9th, 1896 Lilienthal suffered a fatal spinal injury, falling10-15 meters from the sky.
![Page 15: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/15.jpg)
Lift = U
![Page 16: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/16.jpg)
2-D potential flowLper unit span = U
= C vds
“In order for lift to be generated there must be a net circulation around the profile.”
Pg 448 ~ our book
ASIDE
![Page 17: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/17.jpg)
INVISCID INCOMPRESSIBLE FLOW AROUND AN 2-D CYLINDER
ASIDE
![Page 18: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/18.jpg)
p + (/2)U2 = p + (/2)(2U sin)2
(2U sin) comes from 2-D potential flowfor a source, sink and free stream flows
FREE STREAM ON SURFACE
ASIDE
![Page 19: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/19.jpg)
Governing equation for incompressible, irrotational flow:
•V = 0; x V = 0; V = ; 2 = 0
Laplace’s Equation, 2 = 0, is a second order, linear partial differential equation.
Solutions can be added!
Turns out that a 2-D source, 2-D sink and free stream makes a flowlike that over a 2-D ellipse.
ASIDE
![Page 20: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/20.jpg)
m/(Ua) =1
Potential flow solutions: Source + Sink + Uniform flow
ASIDE
![Page 21: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/21.jpg)
Doublet: 2(a)m =
constant as a shrinks to zero
ASIDE
![Page 22: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/22.jpg)
add circulation: u = ro
p + (/2)U2 = p + (/2)(2U sin + ro)2
= C(v+u)ds = 2rou
ASIDE
Lift is integral of pressure around cylinder
![Page 23: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/23.jpg)
= C vds = ro2ro; Lper unit span = U
ASIDE
![Page 24: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/24.jpg)
Bernoulli’s Equation
![Page 25: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/25.jpg)
We have just shown thatif there is a net circulation,
then there is lift.
SO WHAT IS WRONG WITH THIS PICTURETAKEN FROM A POPULAR TEXTBOOK?
![Page 26: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/26.jpg)
Both figures claim lift, which figure’s streamlines are consistent with lift?
![Page 27: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/27.jpg)
U = 4 m/sR = 7.7 cmRe = 4 x 104
= 0
U = 4 m/sR = 7.7 cmRe = 4 x 104
= 4U/R
![Page 28: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/28.jpg)
Both develop lift, see streamlines pinched on top (faster speeds, lower pressure)and wider on bottom (lower speeds andhigher pressure)
![Page 29: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/29.jpg)
Kelvin’s theorem showed that the circulation around any closed curve in an inviscid, isentropic fluid is zero. Consequently there must be circulation around the airfoil in which the magnitude is the same as and whose rotation is opposite to that of the starting vortex.
A CONSEQUENCE OF CIRCULATION AROUND WING IS STARTING VORTEX
![Page 30: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/30.jpg)
U = 30 cm/sChord = 180 mmRe = 5 x 105
Floating tracer method
Starting vortex
“Trailing vortices can be very strongand persistent, possibly being a hazard to other aircraft for 5 to 10 miles behinda large plane – air speeds of greater than 200 miles have been measured.”
![Page 31: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/31.jpg)
Bernoulli’s Equation
![Page 32: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/32.jpg)
HINT: SAME THING THAT IS WRONG WITH THIS PICTURE?
“The phenomenon of aerodynamic list is commonly explained by the velocity increase causing pressure to decrease (Bernoulli effect) overthe top surface of the airfoil..” ~ YOUR BOOK PG 448
![Page 33: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/33.jpg)
Norman Smith: Physics Teacher, Nov. 1972, pg 451-455.
“In spite of popular support, Bernoulli’s Theorem is not responsible for the lift on an airplane wing.”
“The phenomenon of aerodynamic list is commonly explained by the velocity increase causing pressure to decrease (Bernoulli effect) overthe top surface of the airfoil..” ~ YOUR BOOK PG 448
![Page 34: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/34.jpg)
Lift is a result of Newton’s 3rd law. Lift must accompany a deflection of air downward.
![Page 35: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/35.jpg)
BERNOULLI EQUATION GOOD FOR STREAM TUBES WHEREENERGY IS NOT BEING ADDED OR SUBTRACTED
Yet one can argue that B.E. is valid
for outer stream tubesso book not wrong.
![Page 36: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/36.jpg)
BERNOULLI’S EQUATIONX-MOMENTUM EQUATION:INVISCID and NO BODY FORCES:
(Du[t,x,y,z]/dt) = - p/x (u/t) + u(u/x) + v(u/y) + w(u/z) = - p/x
STEADY: u(u/x) + v(u/y) + w(u/z) = - p/x
dx[u(u/x) + v(u/y) + w(u/z) = - p/x]
aside
![Page 37: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/37.jpg)
BERNOULLI’S EQUATION
CONSIDER FLOW ALONG A STREAMLINE:
ds x V = 0
udz-wdx = 0; vdx-udy = 0
u(u/x)dx + v(u/y)dx + w(u/z)dx = - p/xdx
u(u/x)dx + u(u/y)dy + u(u/z)dz = - p/x dx
aside
![Page 38: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/38.jpg)
BERNOULLI’S EQUATION
u(u/x)dx + u(u/y)dy + u(u/z)dz = - p/x dx
u{(u/x)dx + (u/y)dy + (u/z)dz} = - (1/)p/x dx
du
udu - (1/) p/x dx ½ d(u2) = - (1/) p/x dx
aside
![Page 39: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/39.jpg)
BERNOULLI’S EQUATION
X-MOMENTUM EQUATION: ½ d(u2) = - (1/) p/x dxY-MOMENTUM EQUATION: ½ d(v2) = - (1/) p/y dyX-MOMENTUM EQUATION: ½ d(w2) = - (1/) p/z dz
½ d(V2) = - (1/) dpdp = - V dV Euler equation
u2 + v2 + w2 = V2
p/x dx + p/y dy + p/z dz = dp
aside
![Page 40: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/40.jpg)
BERNOULLI’S EQUATION
½ d(V2) = - (1/) dp {½ d(V2) = - dp}
INCOMPRESSIBLE:
½ (V22) - ½ (V1
2) = - (p2 – p1)p2 + ½ (V2
2) = p1 + ½ (V12)
= constant along streamline
If irrotational each streamline has same constant.
aside
![Page 41: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/41.jpg)
BERNOULLI’S EQUATION
p2 + ½ (V22) = p1 + ½ (V1
2)
Momentum equation and steady, no body forces, inviscid, incompressible along a streamline.
Kinetic Energy / unit volume
If multiply by volume have balance between work done by pressure forces and change in kinetic energy.
(interesting that for an incompressible, inviscid flow energy equation is redundant for the momentum equation)
aside
![Page 42: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/42.jpg)
LIFT ‘Measurements’
Lift = U= unsymmetrical flow field
p2 + ½ (V22) = p1 + ½ (V1
2) = constant along streamline
CL = FL/(½ U2Ap)= f(shape, Re, Ma, Fr, /l)
Except at very low Rew does not contribute directly to lift
![Page 43: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/43.jpg)
Calculated (dots) and measured (circles) pressure coefficients for airfoil at = 7o.
= (p-p)/(1/2 U2)
Juan Lopez, which curve is for the upper surface?
![Page 44: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/44.jpg)
Calculated (dots) and measured (circles) pressure coefficients for airfoil at = 7o.
= (p-p)/(1/2 U2)
_________ pressure gradient
__________ pressure gradient
![Page 45: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/45.jpg)
Calculated (dots) and measured (circles) pressure coefficients for airfoil at = 7o.
= (p-p)/(1/2 U2) unfavorable pressure gradient
favorable pressure gradient
First person in 3rd row, which surface iis flow likely to separate?
![Page 46: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/46.jpg)
Stall results from flow separation
over a major section of the upper surface
of airfoil
Rec = 9 x 106
CL = FL/( ½ V2Ap)
![Page 47: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/47.jpg)
Because of the asymmetry of a
cambered airfoil, the pressure
distribution on the upper and lower surfaces are different.Must have camber to
get lift at zero angle of attack.
Rec = 9 x 106
CL = FL/( ½ V2Ap)
![Page 48: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/48.jpg)
Rec = 9 x 106
CL = FL/( ½ V2Ap)
Typical lift force is of the order unity. (dynamic pressure x planform area)Consequently, FL ~ ½ V2Ap
Wing loading = FL/Ap
= 1.5 lb/ft2 1903 Wright Flyer= 150 lb/ft2 Boeing 747= 1 lb/ft2 bumble bee
![Page 49: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/49.jpg)
As angle of attack increasesstagnation point moves
downstream along bottom surface, causing an
unfavorable pressure gradient at the nose*.
*
*
![Page 50: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/50.jpg)
stagnation point
Stream tube narrows then widens
unfavorable
favorable
favorable to unfavorablecauses lam. to turb. trans.
![Page 51: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/51.jpg)
= 2o
*
![Page 52: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/52.jpg)
= 10o
*
![Page 53: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/53.jpg)
*
= 15o -
= 15o +
separation at leading edge
Check angle =15*
![Page 54: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/54.jpg)
![Page 55: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/55.jpg)
Transition from laminar to turbulent flow on
upper surface.
Laminar flow sections designed to fly at low
angles of attack,but has less
maximum lift.Re = 9 x 106
Turbulent Lam.Turbulent
CD = FD/( ½ V2Ap)
CL = FL/( ½ V2Ap)
![Page 56: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/56.jpg)
Lift-Drag Polars are often used to present airfoil data.
Plot is for one particular Rec number
X
X is where you find the most efficient angle of attackHigh performance airfoils can generate lift that is 100 times Their drag – so can flide 100m for every m they drop in altitude.
![Page 57: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/57.jpg)
CL related to load
CD related (mustinclude fuselage drag,
etc.) to drag thatplane must overcome
to achieve lift.
Higher the CL/CD the better!
LIFT – DRAG POLAR
Graph for one Re #different angles
of attack
Different angle of attack
Note that x and y axisHave different scales
![Page 58: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/58.jpg)
Wing tip vortices
crop duster
![Page 59: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/59.jpg)
All real airfoils of finite span, wings, have more drag and less lift than what 2-D airfoil section would indicate.
Trailing vortices reduce lift because pressure difference is reduced.
![Page 60: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/60.jpg)
LIFT = U
U
![Page 61: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/61.jpg)
Two primary leading edgevortices made visible by
air bubbles in water.(Van Dyke Album of Fluid Motion)
Schematic of subsonic flow over the top of a delta wing at an angle
of attack.
![Page 62: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/62.jpg)
The tendency for flow to leak around the wing tipsgenerally cause streamlines over the top surface ofthe wing to veer to the wing root and streamlinesover the bottom surface veer to the wing tips.
Endplates (winglets) at end of wing reduces tip vortex
![Page 63: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/63.jpg)
Winglet is a vertical or angled extension of the wing tips for reducing lift-induced drag.
![Page 64: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/64.jpg)
CD = CL CL2/(ar)
ar = b2/Ap
Winglets work by increasing the effective ar of the wing without increasing the span.
The vortex which rotates aroundfrom below the wing strikes thewinglet, generating a small lift force.
![Page 65: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/65.jpg)
The tendency for flow to leak around the wing tipsalso produces wing tip vortices downstream of the wing which induce a small downward component of air velocity in the neighborhood of the wing itself.
Not all same strength
![Page 66: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/66.jpg)
Lift force is perpendicular to local relative wind.eff is angle that wing sees, angle between chord line and relative wind.
Local lift vector is aligned perpendicular
to the local relative wind.
“This causes the lift force to lean backwards a little, resulting in some of the lift appearing as drag.” Fox et al.
V=V
![Page 67: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/67.jpg)
Di = L sin(i) ~ L i (or L )i ~ CL/( ar) [theory/exp]
CD,I ~ CL i ~ CL2/( ar)
* From Fundamentals of Aerodynamics by Anderson
*
![Page 68: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/68.jpg)
• Induced drag was derived from inviscid, incompressible flow theory –
• Induced drag only for finite wing
• No skin friction or separation
• D’Alembert’s paradox does not occur for finite wing!
![Page 69: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/69.jpg)
![Page 70: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/70.jpg)
![Page 71: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/71.jpg)
Loss of lift and increase in drag caused by finite-span effects are concentrated near the tip of the wing; hence short stubby wingswill experience these effects more severely than a very long wing.
“New” glider by Wright brothers which was astoundingly successfulhad an increase in wingspan to chord ratio from 3 to 6.
![Page 72: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/72.jpg)
Expect induced drag effects to scale with wing aspect ratio = b2/Ap
ar = b/cc
ar = b2/Ap
![Page 73: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/73.jpg)
L/D ~ 400 for ar = L/D ~ 40 for sailplane with ar = 40L/D ~ 20 for typical light plane
with ar = 12
Soaring birds(but large wings have more inertia,
harder to turn quickly)
Maneuvering birds
![Page 74: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/74.jpg)
Tuna Butterfly Fish
Pike
Bass
![Page 75: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/75.jpg)
Di = L sin(i) ~ L i (or L ) ~ CL/( ar)
( = 0; ar = )Di = D = L = LCL/( ar)
CD = CL2/( ar)
CD = CD, + CD,i = CD, + CL2/( ar)
FINITEWING
![Page 76: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/76.jpg)
CD = CD, + CD,i = CD, + CL2/( ar)
CL = FL/( ½ U2Ap) and W = or ~ L
FOR AIRCRAFTCD = CD,0 + CD,i = CD,0 + CL
2/( ar)CL = FL/( ½ U2Ap) and W = or ~ L
FOR WING
FOR INFINITE WINGCD, = FD/( ½ U2Ap)CL, = FL /( ½ U2Ap)
FROM DATA FIGURES
![Page 77: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/77.jpg)
To get same lift (same CL) as infinite armust increase ~ CL/( ar); [linear]
To get same drag (same CD) as infinite ar must increase CD ~ CL = CL
2/(ar); [quadratic]
![Page 78: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/78.jpg)
EXAMPLE
PROBLEMS
![Page 79: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/79.jpg)
A light plane has 10 m effective wingspan and 1.8mchord (regardless or airfoil chosen). It was originally designed to use a conventional (NACA 23015) airfoil section. With this airfoil, its cruising speedon a standard day near sea level is 225 km/hr. Aconversion to a laminar flow (NACA 662-215) section airfoil is proposed. Determine cruising speed that could be achieved for the same power.
![Page 80: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/80.jpg)
A light plane has 10 m effective wingspan and 1.8m chord (regardless or airfoil chosen). It was originally designed to use aconventional (NACA 23015) airfoil section. With this airfoil, itscruising speed on a standard day near sea level is 225 km/hr. Aconversion to a laminar flow (NACA 662-215) section airfoil isproposed. Determine cruising speed that could be achieved for thesame power.
{FDV}23015 = P p = {FDV}66-215
FD = CD ½ V2A
assume efficiency same
{CD ½ V3A}23015 = {CD ½ V3A}66-215
![Page 81: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/81.jpg)
CD = CD, + CD,i = CD, + CL2/(ar)
CD , CL for airfoil
for plane need CD,0
Assume airfoils should operate near design liftcoefficients.
(~0.3/47.6)(~0.2/59.5)
![Page 82: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/82.jpg)
23015
662-215
CD = CD, + CD,i = CD, + CL2/(ar)
{~28% increase}
{CD ½ V3A}23015 = {CD ½ V3A}66-215
V66-215 = VD23015 (CD23015/CD66-215)
![Page 83: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/83.jpg)
EXAMPLE
PROBLEMS
![Page 84: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/84.jpg)
Aircraft with gross mass, m=4500 kg, flown in a circular path of 1 km radius at 250 kph. The plane has a NACA 23015With ar = 7 and lift area = 22 m2.
Find: Power to maintain level flight. P = FDV
Fig from 9.151
R = 1km
![Page 85: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/85.jpg)
P = FDV
FD = CD (1/2V2Ap)
CD = CD, + CD,i = CD, + CL2/(ar)
Determine from force balance.Once know CL, can find CD, from Fig. 9-19
CL = FL / (1/2V2Ap)
FL = mg / cos()
![Page 86: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/86.jpg)
FL cos ()
mV2/R
R = 1 km
FL sin ()
![Page 87: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/87.jpg)
Fy = FLcos() – mg = 0Fr = -FLsin() = mar = -mV2/RFLsin() / FLcos() = (mV2/R) / mgtan () = V2/(Rg); = 26.2o
FL = mg / cos() = 49.2 kN CL = FL / (1/2V2Ap) = 0.754
Don’t know if flying at design CL, (and corresponding CD)but know weight and speed so can figure out CL, then find CD from graph.
CD = CD, + CL2/(ar)
![Page 88: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/88.jpg)
CD ~ 0.007 for CL = 0.754 from Fig 9.19
CL = 0.754
CD = 0.007
![Page 89: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/89.jpg)
CD = CD, + CD,i = CD, + CL2/(ar)
CD ~ 0.007 for CL = 0.754 from Fig 9.19
CD = 0.007 + (0.754)2/(7) = 0.0329
FD = FLCD/CL = 49.2kN x 0.0329 / 0.754 = 2.15kN
Power = FD V = 2.15kN x 250[km/hr] [1000(m/km)/3600(s/hr)]= 149 kW
![Page 90: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/90.jpg)
W = FL = CL1/2 V2A; Vmin occurs for CLmax;
Vmin = [2W/ CLmaxA]1/2
FLAPS
![Page 91: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/91.jpg)
W = FL = CL1/2 V2A; Vmin occurs for CLmax; Vmin = [2W/ CLmaxA]1/2
increase A to reduce Vmin: Vmin Vstall
![Page 92: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/92.jpg)
= 0 = 0
= 0 = 15
TRAILING EDGE FLAPS-VARIES CAMBER
![Page 93: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/93.jpg)
LEADING EDGE SLATS-POSTPONES STALL = 10o
= 30o+
= 25o
= 30o-
![Page 94: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/94.jpg)
LEADING EDGE SLATS-POSTPONES STALL
Stall at 15o+
without leading edge slats
Not stalling yetwith leading edge slats
![Page 95: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/95.jpg)
25% of c
40% of c
![Page 96: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/96.jpg)
Maximum Lift: = 20o CL ~ 4 – 4.5
![Page 97: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/97.jpg)
EXAMPLE
PROBLEMS
![Page 98: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/98.jpg)
9.143
Airplane with effective lift area of 25 m2 is fitted with airfoils of NACA 23012 Section – conf. 2 (Fig. 9.23). Neglecting added lift due to ground effects determine the maximum mass of airplane if takeoff speed is 150 km/hr?
CL = FL/(1/2 V2Ap)
WFig. 9.2325 m21.23 kg/m3
![Page 99: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/99.jpg)
Assume CL at lift off is CL max.
CL = 2.67; CL (1/2 V2Ap) = Wm = CL (1/2 V2Ap)/g = 7260 kg
NACA 23012
![Page 100: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/100.jpg)
INTERESTING
FIGURE
![Page 101: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/101.jpg)
1st jetliner
(1903, 30mph)
x
![Page 102: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/102.jpg)
EXAMPLE
PROBLEMS
![Page 103: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/103.jpg)
Lift Problem Examples – Relevant Equations
CL = FL/( ½ V2 Ap)CD = FD/( ½ V2 Ap)
Ap = max projection of wing
CL and CD values from wind tunnels are for 2-D airfoils
CD = CD, + CD,I = CD, + CL2/(ar)
CD = CD,0 + CD,I = CD,0 + CL2/(ar)
ar = b2/Ap
If steady flight: T = D and W = L = CL ½ V2Ap
CD for finite wing
![Page 104: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/104.jpg)
Ex. 9.8: Given: W=150,000lbf, A=1600ft2, ar=6.5, CD,0=0.0182, =.00238 slug/ft2, Vstall=175mph, M0.6, c=759mph; steady level flight
Find optimum cruise speed.
(1) FD = CD ( ½ V2 Ap)(2) CD = CD,0 + CL
2/(ar)(3) CL = W/( ½ V2 Ap)
Optimum cruise speed = speed when FD/V vs V is minimum.
Use eq 3 to plug CL into eq 2, then plug CD from eq 2 into eq 1 Plot FD/V as a function of V between 175-455 mph (stall – 0.6 x c)and find peak.
![Page 105: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/105.jpg)
optimum cruise speed
0
10
20
30
40
50
60
70
0 100 200 300 400 500 600
velocity (mph)
dra
g/v
elo
city
0
5000
10000
15000
20000
25000
0 100 200 300 400 500 600
level flight speed (mph)
Th
rust
= D
rag
(lb
f)
~ 325mph for optimum cruising
![Page 106: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/106.jpg)
Ex. 9.8
GIVEN: W = 150,000 lbs; A = 1600 ft2; ar = 6.5; CD,0 = 0.0182; Vstall = 175 mph
FIND: (a) Drag from 175 mph to M = 0.6 (b) optimum cruise speed at sea level (c) Vstall and optimum cruise speed at
30,000 ft altitude
![Page 107: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/107.jpg)
(a) Drag from 175 mph to M = 0.6
FDRAG = CD A (1/2) V2
CD = CD,0 + CL2/(ar)
CL = W/(1/2 V2)
150,000 lbf
0.00238 slug/ft3
175,…., 455 mph (M=0.6)
6.50.0182
1600 ft2 0.00238 slug/ft3
175,…., 455 mph (M=0.6)
![Page 108: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/108.jpg)
Aircraft Characteristics
0
5
10
15
20
0 100 200 300 400 500
Speed V (mph)
Dra
g FD
(10
00 lb
f)
0
5
10
15
20
25
30
Po
we
r P
(10
00 h
p)
Drag force (1000 lbf)
Optimum line
P ower (1000 hp)
Optimum cruise speed at sea level, minimize FD/V
V (mph) 175 200 225 250 275 300 325 350 375 400 425 450 455CL 1.195874 0.915591 0.72343 0.585978 0.484279 0.406929 0.346733 0.298968 0.260435 0.228898 0.202761 0.180857 0.176904CD 0.088234 0.059253 0.043829 0.035015 0.029685 0.026309 0.024087 0.022577 0.021522 0.020766 0.020213 0.019802 0.019733FD (1000 lbf)11.06728 9.707257 9.087726 8.963245 9.19457 9.697927 10.42047 11.3275 12.39552 13.60812 14.95355 16.42327 16.73153P (1000 hp) 5.164729 5.177204 5.452635 5.975497 6.742685 7.758342 9.03107 10.57234 12.39552 14.51533 16.94736 19.70792 20.30093
![Page 109: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/109.jpg)
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0 50 100 150 200 250 300 350 400 450 500
velocity (mph)
Drag
/ Ve
loci
ty 323 mph
(b) optimum cruise speed at sea level
![Page 110: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/110.jpg)
(c) optimum cruise and stall speed at 30,000 ft
FLIFT = W = CL A (1/2) SL VSL2
FLIFT = W = CL A (1/2) 30,000 V30,000
2
V30,000/VSL = [SL/ 30,000]1/2 = 1.63
V30,000 stall = 1.63 VSL stall; V30,000 op. cr. = 1.63 VSL op. cr.
![Page 111: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/111.jpg)
THE
END
![Page 112: For most objects moving through a fluid, the significant fluid force is drag](https://reader036.vdocuments.us/reader036/viewer/2022062314/56813c0c550346895da5785f/html5/thumbnails/112.jpg)
Lift force acting on an airfoil section can be evaluated using circulation theory (Kutta-1902;Joukowski-1906)
For an ideal fluid with no viscosityand a thin uncambered airfoil ofchord length c : Lper unit span = U
=circulation (Eq. 5-17; V•ds) = Uc[sin()]* Uc for small = density of fluidU = velocity of uniform flowL = U2c CL = U2c/(½ U2c) = 2
If no camber thenL = 0 at = 0
In ideal fluid slope = 2, viscosity reduces slope
separation
separation
*Proving this is beyond our scope but can be found in Anderson’s book: Fundamentals of Aerodynamics, pg 272
ASIDE