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57:020 Mechanics of Fluids and Transport Processes Chapter 11 Professor Fred Stern Typed by Stephanie Schrader Fall 2005 1
Chapter 11: Drag and Lift 11.1 Basic Considerations Recall separation of drag components into form and skin-friction
( )
⎭⎬⎫
⎩⎨⎧
∫ ⋅τ+∫ ⋅−ρ
= ∞S
wS2
D dAitdAinppAV
21
1C
CDp Cf
( )⎭⎬⎫
⎩⎨⎧ ⋅∫ −
ρ= ∞ dAjnpp
AV21
1CS2
L
ct << 1 Cf > > CDp streamlined body
ct ∼ 1 CDp > > Cf bluff body
57:020 Mechanics of Fluids and Transport Processes Chapter 11 Professor Fred Stern Typed by Stephanie Schrader Fall 2005 2
Streamlining: One way to reduce the drag Make a body streamlined: reduce the flow separation reduce the pressure drag increase the surface area increase the friction drag
Trade-off relationship between pressure drag and friction drag
Trade-off relationship between pressure drag and friction drag
Benefit of streamlining: reducing vibration and noise
57:020 Mechanics of Fluids and Transport Processes Chapter 11 Professor Fred Stern Typed by Stephanie Schrader Fall 2005 3
11.2 Drag of 2-D Bodies First consider a flat plate both parallel and normal to the flow
( ) 0inppAV
21
1CS2
Dp =∫ ⋅−ρ
= ∞
∫ ⋅τρ
=S
w2
f dAitAV
21
1C
= 2/1LRe33.1 laminar flow
= 5/1LRe
074. turbulent flow
where Cp based on experimental data
vortex wake typical of bluff body flow
flow pattern
57:020 Mechanics of Fluids and Transport Processes Chapter 11 Professor Fred Stern Typed by Stephanie Schrader Fall 2005 4
57:020 Mechanics of Fluids and Transport Processes Chapter 11 Professor Fred Stern Typed by Stephanie Schrader Fall 2005 5
( )∫ ⋅−ρ
= ∞S2
Dp dAinppAV
21
1C
= ∫S
pdACA1
= 2 using numerical integration of experimental data Cf = 0 For bluff body flow experimental data used for cD. In general, Drag = f(V, L, ρ, µ, c, t, ε, T, etc.) from dimensional analysis c/L
⎟⎠⎞
⎜⎝⎛ ε
=ρ
= .etc,T,L
,Lt,ArRe,f
AV21
DragC2
D
scale factor
57:020 Mechanics of Fluids and Transport Processes Chapter 11 Professor Fred Stern Typed by Stephanie Schrader Fall 2005 6
Potential Flow Solution: θ⎟⎟⎠
⎞⎜⎜⎝
⎛−−=ψ ∞ sin
rarU
2
22 U21pV
21p ∞∞ ρ+=ρ+
2
22r
2p U
uu1U
21
ppC∞
θ
∞
∞ +−=
ρ
−=
( ) θ−== 2p sin41arC surface pressure
ru
r1ur
∂ψ∂
−=
θ∂ψ∂
=
θ
57:020 Mechanics of Fluids and Transport Processes Chapter 11 Professor Fred Stern Typed by Stephanie Schrader Fall 2005 7
Flow Separation Flow separation:
The fluid stream detaches itself from the surface of the body at sufficiently high velocities. Only appeared in viscous flow!! Flow separation forms the region called ‘separated region’
Inside the separation region: low-pressure, existence of recirculating/backflows viscous and rotational effects are the most significant!
Important physics related to flow separation:
’Stall’ for airplane (Recall the movie you saw at CFD-PreLab2!) Vortex shedding
(Recall your work at CFD-Lab2, AOA=16°! What did you see in your velocity-vector plot at the trailing edge of the air foil?)
57:020 Mechanics of Fluids and Transport Processes Chapter 11 Professor Fred Stern Typed by Stephanie Schrader Fall 2005 8
57:020 Mechanics of Fluids and Transport Processes Chapter 11 Professor Fred Stern Typed by Stephanie Schrader Fall 2005 9
57:020 Mechanics of Fluids and Transport Processes Chapter 11 Professor Fred Stern Typed by Stephanie Schrader Fall 2005 10
57:020 Mechanics of Fluids and Transport Processes Chapter 11 Professor Fred Stern Typed by Stephanie Schrader Fall 2005 11
57:020 Mechanics of Fluids and Transport Processes Chapter 11 Professor Fred Stern Typed by Stephanie Schrader Fall 2005 12
57:020 Mechanics of Fluids and Transport Processes Chapter 11 Professor Fred Stern Typed by Stephanie Schrader Fall 2005 13
57:020 Mechanics of Fluids and Transport Processes Chapter 11 Professor Fred Stern Typed by Stephanie Schrader Fall 2005 14
57:020 Mechanics of Fluids and Transport Processes Chapter 11 Professor Fred Stern Typed by Stephanie Schrader Fall 2005 15
57:020 Mechanics of Fluids and Transport Processes Chapter 11 Professor Fred Stern Typed by Stephanie Schrader Fall 2005 16
57:020 Mechanics of Fluids and Transport Processes Chapter 11 Professor Fred Stern Typed by Stephanie Schrader Fall 2005 17
Magnus effect: Lift generation by spinning Breaking the symmetry causes the lift!
Effect of the rate of rotation on the lift and drag coefficients of a smooth sphere:
57:020 Mechanics of Fluids and Transport Processes Chapter 11 Professor Fred Stern Typed by Stephanie Schrader Fall 2005 18
Lift acting on the airfoil Lift force: the component of the net force (viscous+pressure) that is perpendicular to the flow direction
Variation of the lift-to-drag ratio with angle of attack:
The minimum flight velocity:
Total weight W of the aircraft be equal to the lift
ACWVAVCFW
LLL
max,min
2minmax,
221
ρρ =→==
57:020 Mechanics of Fluids and Transport Processes Chapter 11 Professor Fred Stern Typed by Stephanie Schrader Fall 2005 19
< 0.3 flow is incompressible, i.e., ρ ∼ constant
11.3 Effect of Compressibility on Drag: CD = CD(Re, Ma)
aUMa ∞=
speed of sound = rate at which infinitesimal disturbances are propagated from their source into undisturbed medium
Ma < 1 subsonic Ma ∼ 1 transonic (=1 sonic flow) Ma > 1 supersonic Ma >> 1 hypersonic CD increases for Ma ∼ 1 due to shock waves and wave drag
Macritical(sphere) ∼ .6 Macritical(slender bodies) ∼ 1 For U > a: upstream flow is not warned of approaching
disturbance which results in the formation of shock waves across which flow properties and streamlines change discontinuously
57:020 Mechanics of Fluids and Transport Processes Chapter 11 Professor Fred Stern Typed by Stephanie Schrader Fall 2005 20
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