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57:020 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 2006 1
Chapter 9 Flow over Immersed Bodies 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 9 Professor Fred Stern Fall 2006 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 9 Professor Fred Stern Fall 2006 3
Qualitative Description of the Boundary Layer Recall our previous description of the flow-field regions for high Re flow about slender bodies
57:020 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 2006 4
τw = shear stress τw ∝ rate of strain (velocity gradient)
=0yy
u
=∂∂
µ
large near the surface where
fluid undergoes large changes to satisfy the no-slip condition
Boundary layer theory is a simplified form of the complete NS equations and provides τw as well as a means of estimating Cform. Formally, boundary-layer theory represents the asymptotic form of the Navier-Stokes equations for high Re flow about slender bodies. As mentioned before, the NS equations are 2nd order nonlinear PDE and their solutions represent a formidable challenge. Thus, simplified forms have proven to be very useful. Near the turn of the century (1904), Prandtl put forth boundary-layer theory, which resolved D’Alembert’s paradox. As mentioned previously, boundary-layer theory represents the asymptotic form of the NS equations for high Re flow about slender bodies. The latter requirement is necessary since the theory is restricted to unseparated flow. In fact, the boundary-layer equations are singular at
57:020 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 2006 5
separation, and thus, provide no information at or beyond separation. However, the requirements of the theory are met in many practical situations and the theory has many times over proven to be invaluable to modern engineering. The assumptions of the theory are as follows: Variable order of magnitude u U O(1) v δ<<L O(ε) ε = δ/L
x∂∂ L O(1)
y∂∂ 1/δ O(ε-1)
ν δ2 ε2 The theory assumes that viscous effects are confined to a thin layer close to the surface within which there is a dominant flow direction (x) such that u ∼ U and v << u. However, gradients across δ are very large in order to satisfy the no slip condition. Next, we apply the above order of magnitude estimates to the NS equations.
57:020 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 2006 6
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
+∂∂
µ+∂∂
−=∂∂
+∂∂
2
2
2
2
yu
xu
xp
yuv
xuu
1 1 ε ε-1 ε2 1 ε-2
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
+∂∂
µ+∂∂
−=∂∂
+∂∂
2
2
2
2
xv
xv
yp
yvv
xvu
1 ε ε 1 ε2 1 ε-1
0yv
xu
=∂∂
+∂∂
1 1
Retaining terms of O(1) only results in the celebrated boundary-layer equations
2
2
yu
xp
yuv
xuu
∂∂
µ+∂∂
−=∂∂
+∂∂
0yp=
∂∂
0yv
xu
=∂∂
+∂∂
Some important aspects of the boundary-layer equations:
1) the y-momentum equation reduces to
0yp=
∂∂
elliptic
parabolic
57:020 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 2006 7
i.e., p = pe = constant across the boundary layer
from the Bernoulli equation:
=ρ+ 2ee U
21p constant
i.e., x
UUxp e
ee
∂∂
ρ−=∂∂
Thus, the boundary-layer equations are solved subject to a specified inviscid pressure distribution
2) continuity equation is unaffected 3) Although NS equations are fully elliptic, the
boundary-layer equations are parabolic and can be solved using marching techniques
4) Boundary conditions
u = v = 0 y = 0 u = Ue y = δ
+ appropriate initial conditions @ xi There are quite a few analytic solutions to the boundary-layer equations. Also numerical techniques are available for arbitrary geometries, including both two- and three-dimensional flows. Here, as an example, we consider the
edge value, i.e., inviscid flow value!
57:020 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 2006 8
simple, but extremely important case of the boundary layer development over a flat plate. Quantitative Relations for the Laminar Boundary Layer Laminar boundary-layer over a flat plate: Blasius solution (1908) student of Prandtl
0yv
xu
=∂∂
+∂∂
2
2
yu
yuv
xuu
∂∂
ν=∂∂
+∂∂
u = v = 0 @ y = 0 u = U∞ @ y = δ We now introduce a dimensionless transverse coordinate and a stream function, i.e.,
δ
∝ν
=η ∞ yx
Uy
( )ην=ψ ∞ fxU
( )η′=∂η∂
η∂ψ∂
=∂ψ∂
= ∞fUyy
u ∞=′ U/uf
Note: xp∂
∂ = 0
for a flat plate
57:020 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 2006 9
( )ffxU
21
xv −′η
ν=
∂ψ∂
−= ∞
substitution into the boundary-layer equations yields 0f2ff =′′′+′′ Blasius Equation 0ff =′= @ η = 0 1f =′ @ η = 1 The Blasius equation is a 3rd order ODE which can be solved by standard methods (Runge-Kutta). Also, series solutions are possible. Interestingly, although simple in appearance no analytic solution has yet been found. Finally, it should be recognized that the Blasius solution is a similarity solution, i.e., the non-dimensional velocity profile f′ vs. η is independent of x. That is, by suitably scaling all the velocity profiles have neatly collapsed onto a single curve. Now, lets consider the characteristics of the Blasius solution:
∞U
u vs. y
VU
Uv ∞
∞
vs. y
Rex5
=δ value of y where u/U∞ = .99
ν∞=
xUxRe
57:020 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 2006 10
∞ν
′′∞µ=τ
U/x2
)0(fUw
i.e., xRe
664.0U
2cx
2w
fθ
==ρτ
=∞
see below
∫ ==L
0fff )L(c2dxc
L1C
57:020 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 2006 11
=LRe
328.1
ν∞LU
Other:
∫ =⎟⎟⎠
⎞⎜⎜⎝
⎛−=δ
δ
∞0 x
*
Rex7208.1dy
Uu1 displacement thickness
measure of displacement of inviscid flow to due boundary layer
∫ =⎟⎟⎠
⎞⎜⎜⎝
⎛−=θ
δ
∞∞0 xRex664.0dy
Uu
Uu1 momentum thickness
measure of loss of momentum due to boundary layer
H = shape parameter = θδ*
=2.5916
Quantitative Relations for the Turbulent Boundary Layer 2-D Boundary-layer Form of RANS equations
0yv
xu
=∂∂
+∂∂
57:020 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 2006 12
For flat plate or δ for general case
( )vuyy
upxy
uvxuu 2
2e ′′
∂∂
−∂∂
ν+⎟⎠
⎞⎜⎝
⎛ρ∂
∂−=
∂∂
+∂∂
requires modeling Momentum Integral Analysis Background: History and Modern Approach: FD To obtain general momentum integral relation which is valid for both laminar and turbulent flow
( )dycontinuity)vu(equationmomentum0y∫ −+∞
=
( )dxdU
UH2
dxdc
21
U f2w θ
++θ
==ρτ
dxdUU
dxdp
ρ=−
flat plate equation 0dxdU
=
∫ ⎟⎠⎞
⎜⎝⎛ −=θ
δ
0dy
Uu1
Uu momentum thickness
θδ
=*
H shape parameter
dyUu1
0
* ∫ ⎟⎠⎞
⎜⎝⎛ −=δ
δ displacement thickness
57:020 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 2006 13
y = h + δ*= streamline starts in uniform flow
merges with δ at 3
Steady ρ = constant neglect g v << u = uo ⇒ p = constant i.e., -∇p = 0
Can also be derived by CV analysis as shown next for flat plate boundary layer. Momentum Equation Applied to the Boundary Layer CV = 1, 2, 3, 4
∫ τ==−x
0wdxbdragD pressure force = 0 for v << Uo
force on CV wall shear stress u ∼ Uo
( ) ( )∫ ⋅ρ+∑ ∫ ⋅ρ=−=31
x dAVudAVuDF
= ( ) ∫ρ+−ρ
3
22o dyubbhU
∫ρ−ρ=δ
0
22o dyubbhU)x(D
57:020 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 2006 14
next eliminate h using continuity
∫=
∫ρ=ρ
∫ ⋅ρ+∫ ⋅ρ=
δ
δ
0o
0o
31
udyhU
udybbhU
dAVdAV0
( ) ∫ρ−∫ρ=δδ
0
2
0o dyubudybUxD
= ( )∫ −ρδ
0o dyuUub
∫ ⎟⎟⎠
⎞⎜⎜⎝
⎛−=
ρ=
δ
0 oo2o
D dyUu1
Uu
L2
bLU21
DC
θ = momentum thickness
L2CDθ
=
L2
bLU21
dxb
AU21
DC2o
x
0w
2o
Dθ
=ρ
∫ τ=
ρ=
depends on u(y)
57:020 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 2006 15
( ) ( )∫ θ=ρ
τx
0 2o
w x2dxxU
21
dxd
U212
12o
w θ=
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
ρ
τ
dxd
2cf θ
= cf = local skin friction coefficient
momentum integral relation for
flat plate boundary layer
∫ ⎟⎟⎠
⎞⎜⎜⎝
⎛−=θ
δ
0 oody
uu1
uu
Approximate solution for a laminar boundary-layer Assume cubic polynomial for u(y)
32 DyCyByAUu
+++=∞
0yuu 2
2=
∂∂
= y = 0 A = 0 B = δ23
0yu;Uu =∂∂
= ∞ y = δ C = 0 D = 3
21δ−
57:020 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 2006 16
i.e., 3y
21y
23
Uu
⎟⎠⎞
⎜⎝⎛δ
+δ
=
dxdc
21
U f2w θ
==ρτ momentum integral equation for 0
dxdp
=
dxd139.
23U
U1
2δ
=⎥⎦⎤
⎢⎣⎡
δµ
ρ dy
Uu1
Uu
0∫ ⎟
⎠⎞
⎜⎝⎛ −=θ
δ
dydu
w µ=τ
Compare with Exact Blassius
i.e., xRex65.4
=δ xRe
x5 7% ↓
x
2
w ReV323. ρ
=τ x
2
ReU332. ρ 3%↓
x
f Re646.c =
xRe664.
L
f Re29.1C =
LRe33.1
δ=⎟⎟
⎠
⎞⎜⎜⎝
⎛δ
+δ
==
23Uy
23
23Uu
0y
2
y
57:020 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 2006 17
( )∫ τρ
=L
0w
2f dxx
bLU21
1C
span length total skin-friction drag coefficient Approximate solution Turbulent Boundary-Layer Ret ∼ 3 X 106 for a flat plate boundary layer Recrit ∼ 500,000
dxd
2cf θ
=
as was done for the approximate laminar flat plate boundary-layer analysis, solve by expressing cf = cf (δ) and θ = θ(δ) and integrate, i.e. assume log-law valid across entire turbulent boundary-layer
Byuln1uu *
* +νκ
=
at y = δ, u = U
Buln1uU *
* +νδ
κ=
2/1
f
2cRe ⎟⎠⎞
⎜⎝⎛
δ
neglect laminar sub layer and velocity defect region
57:020 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 2006 18
or 52cReln44.2
c2 2/1
f2/1
f+
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛δ
6/1
f Re02.c −δ≅ power-law fit
Next, evaluate
∫ ⎟⎠⎞
⎜⎝⎛ −=
θ δ
0dy
Uu1
Uu
dxd
dxd
can use log-law or more simply a power law fit
7/1yUu
⎟⎠⎞
⎜⎝⎛δ
=
( )δθ=δ=θ727
⇒ dxdU
727
dxdUU
21c 222
fwδ
ρ=θ
ρ=ρ=τ
dxd72.9Re 6/1 δ
=−δ
or 7/1xRe16.
x−=
δ
7/6x∝δ almost linear
7/1x
f Re027.c =
( )LC67
Re031.C f7/1
Lf ==
cf (δ)
Note: can not be used to obtain cf (δ) since τw → ∞
i.e., much faster growth rate than laminar boundary layer
57:020 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 2006 19
Alternate forms given in text depending on experimental information and power-law fit used, etc. (i.e., dependent on Re range.) Some additional relations given in texts for larger Re are as follows:
( )2.5810
.455 1700Relog Ref
LL
C = − Re > 107
( )732.Relog98.cL Lf −=δ
( ) 3.2
xf 65.Relog2c −−= Finally, a composite formula that takes into account both the initial laminar boundary-layer (with translation at ReCR = 500,000) and subsequent turbulent boundary layer
is L
5/1L
f Re1700
Re074.C −= 105 < Re < 107
Total shear-stress coefficient
Local shear-stress coefficient
57:020 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 2006 20
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 9 Professor Fred Stern Fall 2006 21
( )∫ ⋅−
ρ= ∞
S2Dp dAinpp
AV21
1C
= ∫S
pdACA1
= 2 using numerical integration of experimental data
57:020 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 2006 22
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 9 Professor Fred Stern Fall 2006 23
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
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’
ru
r1ur
∂ψ∂
−=
θ∂ψ∂
=
θ
57:020 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 2006 24
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?)
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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 9 Professor Fred Stern Fall 2006 35
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 9 Professor Fred Stern Fall 2006 36
< 0.3 flow is incompressible, i.e., ρ ∼ constant
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 9 Professor Fred Stern Fall 2006 37