Separated ows : Wakes and Cavities

Transcription

Lecture Notes
Separated ows : Wakes and Cavities
Contents
1 Generalities
1.1
1.2
1.3
1.4
1.5
1.6
What is separation ? . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The separation mechanism from smooth walls . . . . . . . . . . . . . .
Local criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Low Reynolds number separation . . . . . . . . . . . . . . . . . . . . .
Intermediate Reynolds number separation . . . . . . . . . . . . . . . .
High Reynolds number : boundary layer separation and reattachment
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2 Separation in the theoretical frame of the Boundary Layer Theory
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3 Two dimensional Blu-body ows
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2.1 Boundary Layer Theory of Prandtl (BLT) . . . . . . . . . . . . . . . . . . . . .
2.2 Attached boundary layer : acceleration and deceleration of the external stream
2.2.1 External stream . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Boundary layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Flow over an inclined plate . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.4 Steady boundary layer over a body . . . . . . . . . . . . . . . . . . . . .
2.3 Separated boundary layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Limitation of the BLT approach : The Goldstein's singularity . . . . . .
2.3.2 The triple deck BLT theory . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Unsteady separation :growth of a boundary layer in initially irrotational ow .
2.4.1 Main features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 Threshold of the separation . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.3 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Mean properties . . . . . . . . . . . . .
3.1.1 Mean wake Topologies . . . . .
3.1.2 Stress around the body . . . .
3.1.3 Drag vs. Reynolds number . .
3.1.4 Drag Crisis . . . . . . . . . . .
3.2 Wake Dynamic . . . . . . . . . . . . .
3.2.1 Strouhal vs. Reynolds number
3.2.2 2D Dynamic . . . . . . . . . .
3.2.3 3D Dynamics . . . . . . . . . .
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4 Three dimensional Blu-body ows
4.1 2D bodies (Recall) . . . . . . . . . . . . . . . . . . .
4.1.1 Bifurcation scenario in the laminar regime . .
4.1.2 Periodic mode in the turbulent wake . . . . .
4.2 3D axisymmetric bodies . . . . . . . . . . . . . . . .
4.2.2 Steady modes in the turbulent wake . . . . .
4.2.3 Periodic modes in the turbulent wake . . . .
4.3 3D non-axisymmetric bodies . . . . . . . . . . . . . .
4.3.2 Steady modes in the turbulent wake . . . . .
4.3.3 Stochastic model for the long time dynamics .
4.4 Conclusion and perspectives for industrial ows . . .
5 Free streamline theory
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5.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1 Flow approximation: cavity model . . . . . . . . . . . . .
5.1.2 Complex potential ow for irrotationnal 2D ows . . . . .
5.1.3 Potential ow at separation . . . . . . . . . . . . . . . . .
5.2 Flow over a plate perpendicular to the main velocity (exercise) .
5.2.1 The Schwarz-Christoel theorem . . . . . . . . . . . . . .
5.2.2 Computation of the complex potential . . . . . . . . . . .
5.2.3 Shape of the free streamlines . . . . . . . . . . . . . . . .
5.2.4 Drag computation . . . . . . . . . . . . . . . . . . . . . .
5.3 General drag formula from Helmholtz's drag . . . . . . . . . . . .
5.4 Flow approximation with nite separated region : closure models
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6 Cavitation
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7 Conclusion
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6.1 Condition for cavitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
6.2 Pressure around the body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6.3 Gravity eect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2
Chapter 1
Generalities
Contents
1.1
1.2
1.3
1.4
1.5
1.6
What is separation ? . . . . . . . . . . . . . . . . . . . . . . . . . .
The separation mechanism from smooth walls . . . . . . . . . . .
Local criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Low Reynolds number separation . . . . . . . . . . . . . . . . . . .
Intermediate Reynolds number separation . . . . . . . . . . . . .
High Reynolds number : boundary layer separation and reattachment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1.1 What is separation ?
The phenomenon of separation is illustrated by the ow past a square block at low (gure 1.1)
and large (gure 1.2) Reynolds numbers for the two dimensional case (2D) and by the ow
past an ellipsoid for the 3D case. As can be seen, the ow do not follow the solid boundaries,
but separates from the solid wall and turns into the interior of the uid. It may reattaches to
the wall further downstream as in gure 1.1.
Separation is not a feature of large Reynolds number ows only and can occur under
widely varying conditions. In the following, we present a general description that applies
whatever the Reynolds number is (see the approach of D.J. Tritton, in his book, "Physical
Fluid Dynamics").
1.2 The separation mechanism from smooth walls
We conne attention on steady 2D ows separation as depicted in gure 1.4, where the ow
detaches from a smooth wall (experimental cases of gures 1.1 and 1.2). We call S the point
at which the ow separates. In 2D steady ows the vorticity ω is a scalar and its dynamic
reduces to the balance between advection and viscous diusion:
~
(~u.∇)ω
= ν∆ω
3
(1.1)
Figure 1.1: Flow past a 2D square block at Re = 0.02.
Figure 1.2: Flow past a 2D square block at Re = 3500.
Figure 1.3: Flow past a prolate spheroid at incidence.
Figure 1.4: Schematic view of the streamlines in the vicinity of a separation point.
4
Figure 1.5: Schematic view of the streamlines in front of the square block visualized in gures 1.1 and 1.2 showing the region of reversed vorticity in the recirculation region.
∂v
We start by writing the vorticity ω = ∂x
− ∂u
∂y and the normal vorticity gradient at the wall,
∂ω say ω|y=0 and ∂y y=0 respectively. Because at the wall the no slip condition requires u and
v to be zero constant, their gradient along the wall are also zero. The wall vorticity is then
equal to minus the normal velocity gradient ∂u
∂y y=0 :
ω|y=0 = −
∂u .
∂y y=0
(1.2)
Making use of the boundary conditions again together with the continuity equation, we can
show that the vorticity gradient at the wall reads:
∂ω ∂ 2 u =
−
.
∂y y=0
∂y 2 y=0
(1.3)
From equation 1.2 and the schematic view of gure 1.4, we can tell that vorticity changes sign
at S , and separation necessarily implies the existence of a region in which the vorticity has
opposite sign (referred as reversed vorticity later, see gure 1.5) from that associated with the
ow as a whole. The key to understanding when separation may occur is then transferred to
the fundamental question : How reversed vorticity is introduced into the ow ?
Let's consider the x-component of the momentum equation:
u
∂u
∂u
1 ∂p
∂2u
∂2u
+v
=−
+ν 2 +ν 2
∂x
∂y
ρ ∂x
∂x
∂y
(1.4)
At the wall (y = 0), this equation exactly becomes:
1 ∂p ∂ω +ν
=0
y=0
ρ ∂x
∂y y=0
(1.5)
We see that the pressure gradient is directly related to the vorticity gradient, which is the
starting point for introduction of reversed vorticity into the ow. Once the vorticity gradient
created, viscous diusion will transport the vorticity into the ow. For vorticity initially
negative in the oncoming ow (as in gure 1.4), to reverse thevorticity we need to add positive
vorticity at the wall (i.e. a negative vorticity gradient: ∂ω
∂y y=0 < 0). This added vorticity
will diuse by viscosity down this gradient and positive vorticity will then be introduced into
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Figure 1.6: Action of a pressure gradient at a wall. (b) ow without pressure gradient. (a)
favorable pressure gradient, in this case vorticity of same sign of that of the main ow is
introduced which magnies vorticity. (c) adverse pressure gradient but not enough to cause
separation. (d) adverse pressure gradient with separation (back ow)
the ow. From equation 1.5, negative vorticity gradient implies a positive pressure gradient
at the wall. This can be generalized by saying that to reverse the vorticity at a wall the
ow needs to develop an adverse pressure gradient at the wall (a gradient oriented as
the main velocity). However, the introduction of reversed vorticity into the main ow does
not necessarily mean that a region of opposite vorticity and hence a separation will appear.
Adverse pressure gradient at a wall is a necessarily but not sucient condition for separation.
Figure 1.6 summarizes the dierent options for a wall ow with pressure gradient.
1.3 Local criteria
Footprints on the body surface indicate whether the ow locally separates. These footprints
are dened from the wall shear stress :
∂u →
∂w →
−
−
−
τ→
e +µ
ez ,
W = µ
W x
∂y
∂y W
(1.6)
where µ is the dynamic viscosity and (u, v, w) the components of the velocity elds in the
−
−
−
−
(→
ex , →
ey , →
ez ) frame. Here, the normal to the wall is →
ey . For 2D ows, the skin friction is a scalar τW = µ ∂u
∂y W which displays the following special
behavior at separation S :
τW (S) = −µω(S) = 0
∂τW (S)
∂ω(S)
= −µ
< 0.
∂x
∂x
(1.7)
Eq 1.7 is known as the Prandtl criteria for separation. Vorticity changes sign across the point
S , from which a unique separation streamline initiated from S divides the ow.
The situation is more complicated for 3D ows. Figure 1.3 shows a numerical simulation
of the streamlines in a laminar ow over a spheroid. One clearly sees a detached layer (the
darkest) rolling up into a concentrated vortex. The detached layer denes a separation streamsurface that intersects the body in a separation line. The separation line plays the same role
as the separation point S in the 2D case. Unlike Eq.(1.7), the numerical simulation shows that
→
−
−
τ→
W 6= 0 in the major portion of the separated line. To understand how separation happens,
6
Figure 1.7: 3D separation scenario with skin friction lines converging.
we will consider the skin-friction lines. The separation line dened as the intersection of the
body with the separated stream-surface is a special skin friction line. If two skin friction lines
converge as depicted in gure 1.7, then the incompressibility (that dictates that the section
of a streamline tube is constant along the tube) obliges the uid to be ejected from the body
surface. For h small the velocity proles are linear, u(h) = τµW h. Taking ū for the mean
velocity averaged over the surface hl, one approximately have that ū = u(h)/2 if l small
enough. Since the mass ux through the surface hl, ρūhl is constant, one obtain
h = C(
ν 1
)2 .
lτW
(1.8)
Therefore, h will grow unboundely not only as τW → 0 as in 2D separation but also as l → 0.
Namely, a convergence of skin friction lines creates a uid ejection and then separation from
the body. This is known as the Lighthill criteria.
1.4 Low Reynolds number separation
When inertia is negligible, the ow over a symmetric body has the upstream-downstream
symmetry. This is also the case for the separation as we can see in gure 1.1 where the
recirculation region in front of the block is matched by a similar one behind it. If separation
is present, it is because this low Re ow develops an adverse pressure gradient at the walls.
With no inertia, the pressure gradient is only balanced by viscous forces. We will not discuss
any further low Reynolds number separations, we just mentioned it to emphasize the fact that
separation is not only an inertial eect as it is commonly thought. This biased idea is due to
a screening eect of the large amount of work that has been devoted to the boundary layer
detachment that covers so many applications in industrial ows of so great importance.
1.5 Intermediate Reynolds number separation
The limit at zero Reynolds number is called the Stokes ow, with strictly no inertia eect.
Particularly, the ow have the upstream-downstream symmetry because the vorticity diuses
by viscosity equivalently in all directions. For bodies plunged in a uniform ow, the pressure
gradient at the wall is balanced by viscous forces. Since this viscous forces are oriented as the
7
Figure 1.8: Steady separation around a circular cylinder for intermediate Reynolds numbers
main velocity, the pressure gradient is generally always negative or favorable. This applies for
spheres or cylinders and there cannot be any separations at low Reynolds numbers here. We
will take the example of the circular cylinder for which the ow is fully attached for Re < 4.
When the Reynolds number is increased, advection will break the symmetry by transporting
vorticity downstream. Consequently, vorticity created at the cylinder's front will ll the region
at the rear of the cylinder. As Re increases, the vorticity at the rear increases so much that
it eventually produces a back ow for Re > 4 and two counter rotating recirculation zones
appear (see gure 1.8). For Re < 47, the ow remains steady but the characteristics of the
separated region evolves as Re increases. The separation points repaired as the separation
angle θS moves upstream from 180◦ to 120◦ . This variation is given by the empirical law
:θS = 101.5 + 155Re−1/2 . The length of the recirculation region L increases linearly as Re (see
gures 1.9).
1.6 High Reynolds number : boundary layer separation and
reattachment
At suciently large Reynolds numbers, the ow can be divided into two parts, an inviscid
irrotationnal part and a boundary layer in which the vorticity is conned. We will see that the
pressure gradient is imposed by the inviscid irrotationnal part. Both laminar and turbulent
boundary layers can separate. Once the boundary layer is detached from the wall, vorticity
is introduced in the bulk in a sheet shape of nite thickness (see gure 1.10). This sheet
8
Figure 1.9: Length of the recirculation zone and position of the separation points vs. Re
9
Figure 1.10: Separation sketch with vorticity sheet (the grayscale corresponds to the vorticity
sign). (a) laminar separation. (b) turbulent separation
Figure 1.11: Mixing layer with main velocity U at the top and zero velocity at the bottom
is equivalent to a inexional velocity prole similar to a mixing layer having the main ow
velocity on the top side and approximately a zero velocity underneath (see gure 1.11). So the
result of the well known mixing layer is important for this purpose and we recall few important
results for laminar and turbulent mixing layers.
It is useful to remember that depending on the nature of the mixing layer (laminar or
turbulent) the mean ow properties will be signicantly dierent. For the laminar case, the
growth of the mixing layer thickness ∆ results from the combined eect of viscous diusion
and advection. It evolves with distance as:
1
∆lam (x) ∝ (x − x0 ) 2 .
(1.9)
From dimensional analysis, it is easy to show that ∆lam depends on the Reynolds number
and tends to zero as Re tends to innity. This is the vortex sheet (or vorticity sheet) that is
innitely thin.
For the turbulent case, ∆ results from turbulent diusion and advection :
∆turb (x) ∝ (x − x0 ).
(1.10)
Hence, on the contrary to the laminar ow, it does not depend on the Reynolds number.
10
For this mixing layer, an entrainment occurs on the side with the zero velocity only. Some
uid from the region at rest is then entrained by the mixing layer as depicted in gure 1.11.
The entrainment velocity Ve for the laminar case decreases with distance as :
1
Velam (x) ∝ (x − x0 )− 2 ,
(1.11)
and also tends to zero when Re tends to innity. However, for the turbulent mixing layer the
entraiment remains constant with distance:
Veturb (x) ∝ (x − x0 )0 ,
(1.12)
and does not depends on Re neither.
The ow behind a separation point will depend crucially on the nature of the detached
boundary layer. For instance, the presence of an entrained velocity is often the cause for a
premature reattachment (unless the adverse pressure gradient continues to prevent it). This
eect, known as the Coanda eect, is explained by the presence of the wall that avoids the
mixing layer to draw uid into itself. Instead, the mixing layer is drawn to the wall.
It is important to discuss the stability properties of mixing layer: under which condition
the layer is whether laminar or turbulent? Theoretically, the laminar mixing layer is always
potentially unstable (presence of an inexional point in the velocity prole); it means that
suciently far downstream unsteadiness will develop. This is known as the Kelvin-Helmholtz
instability (see gure 1.12). The turbulent transition will then irremediably occurs whatever
the Reynolds number is. For a mixing layer originated from a separation at the wall, the
relevant question is how far from the wall this transition will take place ? At low Reynolds
number, the layer can reattach before that transition occurs. A separated boundary layer can
stay steady and laminar from the separation point to the reattachment point (see step ow or
blu body ows as in gure 1.8). As Re increases unsteadiness will develop closer and closer
to the separation point and hence to the wall.
We should make a comment about dierences between laminar and turbulent mixing layer.
Turbulence can be viewed as an ultimate state of unsteadiness where the layer breaks into
several eddies (or vorticity patches) with more or less disordered motion (see gure 1.12), this
type of ow is described by equations 1.12 and 1.10. Before this ultimate state we can have
"organized" unsteady motion corresponding to an oscillation of the vorticity sheet without
formation of distinct vorticity patches. In this case the ow is considered as laminar and
described by equations 1.11 and 1.9.
11
Figure 1.12: Mixing layer originated from the separation of a circular cylinder at Re = 10000.
After a short distance from the separation point, some well dened Kelvin-Helmholtz vortices
appear that will increases considerably the entrainment velocity.
12
Chapter 2
Separation in the theoretical frame of
the Boundary Layer Theory
Contents
2.1 Boundary Layer Theory of Prandtl (BLT) . . . . . . . . . . . . . . 13
2.2 Attached boundary layer : acceleration and deceleration of the
external stream . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.1
2.2.2
2.2.3
2.2.4
External stream . . . . . . . . . .
Boundary layer . . . . . . . . . . .
Flow over an inclined plate . . . .
Steady boundary layer over a body
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2.3.1 Limitation of the BLT approach : The Goldstein's singularity . . . .
2.3.2 The triple deck BLT theory . . . . . . . . . . . . . . . . . . . . . . .
18
18
2.4.1 Main features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 Threshold of the separation . . . . . . . . . . . . . . . . . . . . . . .
2.4.3 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2.3 Separated boundary layer . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4 Unsteady separation :growth of a boundary layer in initially irrotational ow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1 Boundary Layer Theory of Prandtl (BLT)
In the following, we recall the main results of the theory that will be useful for separation
considerations.
The boundary layer theory suggests very important hypothesis that under rather broad
conditions, viscosity eect (stresses, forces due to viscosity, vorticity diusion...) are signicant
and comparable in magnitude with convection and any other manifestation of intertia forces, in
layers adjoining solid boundaries and in certain other layers, the thicknesses of which approach
zero as Re goes to innity, and are small outside the layers. The boundary layer is the layer
13
in which the uid makes a transition from the required value zero at the solid boundary to a
nite value in the bulk appropriate to an inviscid uid. The fact that the boundary layer is
thin, compared with linear dimensions of the boundary, makes possible certain approximations
in the Navier-Stokes equations. We take the boundary to be a plane wall at y = 0 and the
ow to be two-dimensional. Now if U0 is representative of the magnitude of the ow eld ~u as
a whole, and L a length in the x−direction over which ~u changes appreciably, the Reynolds
number of the ow as a whole is :
U0 L
Re =
(2.1)
ν
U0 and L are characteristics of the inviscid motion. Comparable magnitude of viscosity eect
with convection in the boundary layer of thickness δ0 implies that :
(2.2)
1
δ0 ∼ LRe− 2
to :
From the equation of motion in the y− direction, the boundary layer approximation yields
∂p
= 0.
∂y
(2.3)
The pressure is thus approximately uniform across the boundary layer. This is the key of the
theory for which the pressure inside the boundary layer is given by the inviscid ow outside
the boundary layer. So, the boundary conditions are rst that :
u = v = 0 at y = 0,
(2.4)
and second that the boundary layer must join smoothly with the region of the ow outside it.
If U is the x−component of the velocity just outside the boundary then, this second condition
can be expressed as :
u(x, y, t) → U (x, t) as y/δ0 → ∞
(2.5)
Like the pressure p, U must be considered as given, and both are related by the approximate
equation:
∂U
∂U
1 ∂p
+U
=−
(2.6)
∂t
∂x
ρ ∂x
describing inviscid ow in the x−direction just outside the boundary layer.
We dene dimensionless quantities adapted to the boundary layer problem :
x0 =
1 y
1 v
tU0
u
p − p0
x 0
, y = Re 2 , t0 =
, u0 =
, v 0 = Re 2 , p0 =
L
L
L
U0
U0
ρU02
(2.7)
An approximate form of the boundary layer equation that becomes exact in the limit
Re → ∞, is:
0
0
∂u0
0 ∂u
0 ∂u
+
u
+
v
∂t0
∂x0
∂y 0
=
0
=
∂p0
∂ 2 u0
+
∂x0
∂y 02
0
∂p
∂y 0
−
∂u0 ∂v 0
+
= 0
∂x0 ∂y 0
U
u0 →
as y 0 → ∞
U0
at y 0 = 0 , u0 = v 0 = 0.
14
(2.8)
The pressure gradient is given by equation 2.6 and the velocity U (x) is the inviscid solution
of the problem given by the potential ow theory. The pressure gradient is then prescribed in
the theory which means that it is unable to react to the dynamics inside the boundary layer.
This lack of coupling is a loss compared to the full Navier-Stokes equation which will explain
the inconsistency of the Prandtl system 2.8 in most cases of boundary layer separation.
2.2 Attached boundary layer : acceleration and deceleration of
the external stream
2.2.1 External stream
In the inviscid ow, the pressure gradient is governed by the ow acceleration (Euler equation).
For the steady case, the energy budget with respect to distance s along a streamline is given
by the Bernoulli's Theorem :
1
p(s) + ρq(s)2 = constant,
2
(2.9)
where q is the velocity modulus. Thus the rate of change of the speed q along the streamline
is given by:
∂q
1 ∂p
=−
(2.10)
∂s
ρq ∂s
At a given pressure gradient, the rate of change of the speed q is much larger for lower velocities
than for the larger ones. Starting from a velocity prole without pressure gradient such as in
gure 1.6(b), we see that if we add a negative pressure gradient (an acceleration), the velocity
will increase more closer to the wall than the velocity far from it. Hence the velocity prole will
be deformed such that the boundary layer thickness will decrease (gure 1.6(a)). A positive
or adverse pressure gradient will have the opposite tendency: a thickening of the boundary
layer (gure 1.6(c)).
2.2.2 Boundary layer
We are going to study the viscous eect related to the acceleration and deceleration of the
external stream. We present a completely equivalent mechanism than that presented in 1.2.
Let's consider the velocity prole without pressure gradient such as in gure 1.6(b). In this
case, the velocity prole at the wall is linear, which means that there is no volume viscous
forces at the wall (the volume viscous force is given by the curvature of the velocity prole).
With a negative pressure gradient, the steady ow implies that the pressure gradient has been
balanced by viscous forces. To produce the necessary viscous forces, some negative curvature
in the velocity prole has to be introduce close to the wall. In consequence the boundary layer
will have larger velocity and be thinner ((gure 1.6(a)). On the contrary, with an adverse
pressure gradient, the introduced curvature that is needed to balance this pressure gradient
has to be positive. Since the curvature is negative in the bulk, an inexional point appears in
the vicinity of the wall (see gure 1.6(c)). The condition for steadiness is that an increase of
the adverse pressure gradient has to be followed by an increase of the positive curvature. At
some point, this curvature increase will force the velocity to be negative (see gure 1.6(d)).
This situation corresponds to the presence of the region of reversed vorticity as mentioned in
1.2.
15
2.2.3 Flow over an inclined plate
This problem gives rise to an important family of self-similar velocity proles (Falkner Skan)
for the boundary layer with pressure gradient, including the Blasius velocity prole similar as
those displayed in gure 1.6. It is the solution of the boundary layer equation in the case of
the external stream velocity given by:
U = cxm
(2.11)
Physically, it corresponds to the ow past an inclined plate with angle of attack α = βπ/2 =
2m
m+1 π/2 as depicted in gure 2.1. Zero incidence corresponds to the Blasius ow (no pressure
gradient), positive incidence to an accelerated ow and negative incidence to a decelerated ow
(Figure 2.1). For any values of positive m, the self-similar solution is unique and corresponds
to accelerated boundary layers. For negative values of m, the solution is not unique anymore.
The physical solution is chosen to smoothly join the solution obtained for m > 0. It is found
that the ow over the plate changes direction for m < −0.0904. At m = −0.0904 the friction
is zero for all x at the wall (no vorticity at the wall)). This correspond to the limit case that
can be "physically" supplied by the BLT of Prandtl in this problem. Actually for any values
of m < −0.0904 (ie β = −0.199 or α = −17.9o ), the mathematical solution of system 2.8 with
back ow is not admissible as discussed later.
It is easy to estimate the eect of the external stream acceleration on the boundary layer
thickness by the calculation of the boundary layer displacement dened as :
Z
∞
(1 −
δ1 =
0
1
u
)dy ∝ x 2 (1−m)
U
(2.12)
For m = 0 (i.e. α = 0), the boundary layer thickness naturally grows as x increases because
of the viscous diusion. One needs to accelerate the external stream until m reaches 1 to stop
this growth. For m = 1 (i.e. α = 90o ), which corresponds the stagnation point on the plate,
the displacement is constant for all x but not the frictionnal stress at the wall that is given by
:
1
∂u τ0 = µ y=0 ∝ x 2 (3m−1)
(2.13)
∂y
The wall shear stress (or the vorticity at the wall) is uniform for m = 31 . In this case, the
eect of acceleration is exactly counter-balanced by viscous diusion.
2.2.4 Steady boundary layer over a body
When a body moves with constant velocity in a uid at rest, the boundary layer will vary
with position on its surface. For 2D ows, the boundary layer begins at the point where
the dividing streamline coming for far upstream intersects the body surface. For a rounded
forehead body (locally plane), the initial development is similar to the "stagnation point ow"
corresponding to the case m = 1 in gure 2.1. In the case of a sharp forward-facing edge,
the initial velcocity prole will be one member of the family m ≥ 0 in gure 2.1. Further
downstream the origin of the boundary layer, the velocity of the external stream varies in a
dierent manner, determined by the body shape as a whole. There will be both acceleration
and deceleration with consequent variation of the boundary layer thickness and velocity within
16
Figure 2.1: The Falkner-Skan problem.
17
it. This framework is very successful as long as local adverse pressure gradients are more than
slight (to not have separation).
If the ow separates, this framework is still valid only if the boundary layer before the
separation is considered (not in the reversed ow region).
2.3 Separated boundary layer
2.3.1 Limitation of the BLT approach : The Goldstein's singularity
The boundary layer theory is unfortunately unable to describe a separated ow. There are few
reasons. For a main stream in the x direction (as in gure 1.2), the approximated system 2.8
can only give the ow in the boundary layer at a position x = 0 by integration of its past
history for x < 0. Hence, when a back ow occurs, history from x > 0 is also required, that is
not determined by the condition x < 0, contrary to the assumed basis of the integration. It
is then important to remember that if the boundary layer theory gives as a solution a backow, it is obviously erroneous and would never describe a physical situation (see example
of the previous ow over a plate). Secondly, even before that the boundary layer separates,
the Prandtl system often presents a nite time singularity known as a Goldstein's singularity.
Even before that separation occurs (say at xS ), the vertical velocity component blows as :
1
v 0 ∝ u0S (y 0 )(x0S − x0 )− 2
(2.14)
Of course such solution at the separation point does not exist. It is due to the fact that
the Prandtl system, which is an approximation of the Navier Stokes equation, is missing
important physical eects. The missing part is due to the normal velocity v around the
separation point. A ow leaving a wall must have a strong normal velocity component, with
an order of magnitude larger than the expected one from the BLT theory (2.7):
1
v(x, y) >> U0 Re− 2
(2.15)
When this happens, the Prandtl system (2.8) has neither normal viscous diusion nor normal
pressure gradient to balance it, then the normal velocity grows unboundedly which causes the
singularity (2.14). Furthermore, Goldstein showed that for a given adverse pressure gradient,
as x → xS the boundary layer has a two 1layer structure
shown in gure 2.2. The sublayer
1
1
−2
4
adjacent to the wall decays as δsub ∼ LRe (xS − x) and the outer main layer is δ ∼ LRe− 2 .
The singularity develops in the viscous sublayer and it is generally impossible to continue the
solution through the separation point.
2.3.2 The triple deck BLT theory
This new generation of the boundary layer theory (due to Sychev) removes the Goldstein
singularity and gives physical insights on the structure of a boundary layer at the separation.
To remove the singularity, one have to give up (2.7) around the separation point in order to
obtain a new approximation of the Navier Stokes equations that allows a two-ways coupling
between the external ow and the wall ow. In this context the pressure gradient cannot be
prescribed in advance as for the Prandtl system, but has to be solved from the interaction.
The technic is to introduce a new length scale corresponding to a region of width ` in the
18
Figure 2.2: Two layers structures of the boundary layer before a separation in the frame of
the Prandtl theory (2.8)
Figure 2.3: Triple-deck structure of a boundary layer at separation.
19
streamwise direction and centered at xS such that `0 = `/L << 1. We are now focussing on
this region by presenting the main steps of the triple deck theory using simplied arguments
(see the book of Wu, Ma & Zhou).
The interaction of the strong ejection (2.15) in the separating ow with the external pressure gradient causes a viscous response in the sublayer δsub adjacent to the wall. Let this occur
0 ∼ in the boundary layer
in a lower deck of normal thickness δld ∼ δ in the global scale or δld
scale with << 1 to be determined. Since adjacent to the wall, the streamwise velocity prole
can be represented by a uniform shear ow u0 (y 0 ) ∼ . Then by the momentum equation (2.8),
the balance between the inertial force, the interactive pressure increment ∆p and the viscous
force requires that :
2
∆p
∼ 0 ∼ 2
(2.16)
0
`
`
which gives ∆p = O(2 ) and `0 = O(3 ). To determine , we note that the appearance of
the lower deck raises the rest of the boundary layer (the main deck ) up by an additional
displacement of δ in global scale. The slope of this displacement is of O( δ` ) in global scale.
On the other hand, across the main deck the pressure remains unchanged, so that ∆p = O(2 )
propagates all the way to the outer edge of the boundary layer and alters the external potential
ow in a zone called the upper deck. The ow is therein inviscid and irrotationnal without any
preferred direction, so the upper deck thickness should be of the same order as its streamwise
length `. In the upper deck, the displacement slope must 1be balanced by the
interactive
3
pressure increment. This yields δ` ∼ 2 , and hence ∼ O(Re− 8 ) and `0 ∼ O(Re− 8 ). Therefore
a triple-deck structure as summarized in gure 2.3 is established, which should replace and
complete the two layer structure in gure 2.2.
2.4 Unsteady separation :growth of a boundary layer in initially
irrotational ow
2.4.1 Main features
See gures 2.6, 2.4 for the wake dynamics and 2.5 for drag evolution. The separation occurs
after a laps of time from the impulse motion.
2.4.2 Threshold of the separation
We will consider the temporal development of boundary layer ow from an initial state of zero
vorticity everywhere, and with a given distribution of the external stream velocity U . For this
purpose, let's consider a body immersed in an innite uid at rest that is made to move in
translation at t = 0. The motion in the boundary layer is governed by equations 2.8 with
equation 2.6 to give the pressure gradient and the new requirement
u(x, y, t) = U (x, t) at t = 0.
(2.17)
In the simple case of a constant motion of the body, U is independent of the time t. The
dimensional form of the x−component of the momentum equation of the boundary layer is:
∂u
∂u
∂u
dU
∂2u
+u
+v
=U
+ν 2
∂t
∂x
∂y
dx
∂y
20
(2.18)
Figure 2.4: Formation of the separated region behind a cylinder after acceleration from rest.
21
Figure 2.5: Drag evolution.
22
Figure 2.6: secondary separations of an impulsively started cylinder at high Re
23
Figure 2.7: Streamlines and vorticity in a numerical simulation at Re = 100 for dierent times
in units of Ua0 after impulsively starting the cylinder from rest to constant motion at velocity
U0 .
We would like to know what happens over the rear portion of the body where the pressure
gradient is adverse ( dU
dx < 0).
Initially, the boundary layer is very thin so that the main contribution to ∂u
∂t in equation 2.18 comes from viscous term. The rst approximation for u, that we denote by u1
satises:
∂u1
∂ 2 u1
=ν
(2.19)
2
∂t
∂y
of which the solution is :
2
u1 (x, y, t) = U (x) √
π
Z
η
e
−η 2
dη = U (x)erf(η)
0
(2.20)
where η = 12 y/(νt) 2 . We use this rst approximation to estimate the convection term in
equation 2.18. The second approximation is u = u1 + u2 , and u2 satises:
1
∂ 2 u2
dU
∂u1
∂u1
∂u2
−ν
=U
− u1
− v1
2
∂t
∂y
dx
∂x
∂y
(2.21)
with the boundary conditions :
u2 (x, y, 0) = 0, u2 (x, 0, t) = 0, u2 (x, y, t) → 0 as y → ∞
(2.22)
The solution for u2 is of the form tU dU
dx f (η). The second approximation is then :
u = U (x)erf(η) + tU
24
dU
f (η)
dx
(2.23)
The two functions f (η) and erf(η) are everywhere non-negative and the ratio f (η) to erf(η)
is maximum at η = 0. Consequently the necessary condition to have a back-ow is dU
dx < 0.
If the condition is satised, and that the gradient is large enough, a back-ow will occur rst
at η = 0. The interval of time before back-ows occur at any position x is the value of t
which makes ∂u
∂y y=0 zero. With equation 2.23 and the knowledge of the function f (η), the
time interval is :
0.70
tS (x) = −
(2.24)
dU/dx
The exact time and place on the body surface at which back-ow begins depend on the function
dU (x)/dx, which is determined completely by the shape of the body.
2.4.3 Exercise
• Compute tS for a circular cylinder in an innite uid at rest that is made to move in
translation with a constant velocity U0 at t = 0. For the inviscid motion, the general
2
result of the velocity potential of a cylinder of radius a is φ(~r) = −U0 cos θ(r + ar ).
2.5 Conclusion
Boundary layers separates for strong enough adverse pressure gradient. While the boundary
layer theory of Prandtl is hopeless to describe a separated boundary layer, the triple deck
theory gives physical insight on the ow near separation. Whatever the theory, the potential
ow that is used as an external ow is crucial. The theories are useful only if the external
ow is known. It means that for a separated ow like in gure 2.7, the potential ow with
separation needs to be computed, as it will be done by the free streamline theory.
25
Chapter 3
Two dimensional Blu-body ows
Contents
3.1 Mean properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.1.1
3.1.2
3.1.3
3.1.4
Mean wake Topologies . . .
Stress around the body . .
Drag vs. Reynolds number
Drag Crisis . . . . . . . . .
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26
28
28
30
3.2.1 Strouhal vs. Reynolds number . . . . . . . . . . . . . . . . . . . . .
3.2.2 2D Dynamic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.3 3D Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
36
41
3.2 Wake Dynamic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.1 Mean properties
3.1.1 Mean wake Topologies
We will consider 2D blu bodies (gure 3.1) in a uniform stream of velocity U0 and pressure
p0 at ∞. They are characterized by the size d of the projected area of the body on a plane
perpendicular to the main velocity. The Reynolds number is usually dened as :
Re =
U0 d
ν
(3.1)
For Reynolds number larger than 100, the unsteady ow around blu bodies can be
sketched as in gure 3.2. The main characteristics of the recirculation bubble is the base
pressure pb (the pressure at the rear of the body), its length L (dened by the mean stagnation point at the closure), its width d0 (dened as the maximum width of the mean recirculation
region). The main characteristic of the unsteady wake is the frequency f at which the velocity
oscillates.
26
Figure 3.1: Flow past a circular cylinder and a plate at Re∼ 104 .
Figure 3.2: Sketch of the ow around a blu-body. The wake bubble or cavity is averaged in
time.
27
3.1.2 Stress around the body
The local stress in a Newtonian uid is of the two kinds, the viscous stress and the pressure
stress :
σ = 2µe − pδ
(3.2)
Both will contribute to the stress around the body in a dierent manner depending on the
Reynolds number and of course whether the ow is separated or not. The local √
viscous stress
is completely governed by the boundary layer on the body and varies as ρU02 / Re. On the
other hand, the local pressure is determined by the inviscid motion around the body and does
not depend on the Reynolds number as we saw in the preceding chapter. The local pressure
is given by Bernoulli theorem and evolves as ρU02 . Hence, the viscous contribution to the
aerodynamic force tends to become negligible as Re increases (see gure 3.3).
The mean pressure coecient distribution
p(s) − p0
1
2
2 ρU0
Cp (s) =
(3.3)
on the body surface is given in gure 3.3 for circular cylinders. In the separated region,
repaired by the zero wall shear stress, the pressure remains fairly well constant. This constant
value is called the base pressure coecient or base suction Cpb , it is responsible for the large
resistance that the uid exerts on the body (drag). The base pressure is not the lowest in the
wake which is inside the recirculation bubble.
For the circular cylinder (smooth body), the separation is preceded by an adverse pressure
gradient together with a reduction of the skin friction as expected by the BLT (see preceding
chapter). However, in case of separation at salient edges (see gure 3.4), there is no adverse
pressure gradient before separation at a salient edge. There is then no evidence in the upstream boundary layer for an imminent separation. It is important then to distinguish sharp
separation from smooth separation.
3.1.3 Drag vs. Reynolds number
Let's consider the total force exerted on the body. It can be computed from the sum of the
local stress acting on the body surface Σ:
F~ =
ZZ
ZZ
Σ
ZZ
2µe~n.ds −
σ~n.ds =
Σ
p~n.ds
Σ
(3.4)
where we use the usual decomposition of the local total stress into viscous and pressure con~ 0 /U0 ,
tribution. The drag D which is the component, parallel to the main velocity D = F~ .U
is associated to kinetic energy variations of the ow. The pressure term in the drag is called
the form drag. If the ow remains attached to the body, the form drag would be zero and
only the viscous term would contribute to the total force. In this case, the drag would be zero
at ν → 0. This is not observed experimentally (D'Alembert's paradox), because separation
provokes a zone of suction behind the body making the pressure distribution asymmetric (see
gure 3.3) giving rise to a nite form drag. The drag coecient is dened as:
1
CD = D/( ρU02 dl0 ),
2
28
(3.5)
Figure 3.3: Pressure coecient and skin friction distribution around a circular cylinder for
dierent Reynolds number.
29
Figure 3.4: Pressure distribution around a plate.
l0 beeing the spanwise length of the cylinder. If we neglect the viscous contribution to the
drag, which is reasonable at large Reynolds number, we have from equation 3.4 that
I
CD '
Cp~n.~ex d`
(3.6)
for a 2D body in a stream U0~ex .
We show in gure 3.5 the drag coecient and in gure 3.6 the base pressure coecient vs.
Re for a circular cylinder over a very wide range of Reynolds numbers. The main characteristic
of the ow are summarized in gure 3.7
3.1.4 Drag Crisis
The drastic drag decrease around Re = 5106 is called the drag crisis. The eect is associated
to a strong eect of the position of the separation point, that abruptly moves downstream (see
gure 3.8). It is the consequence of the turbulent transition of the laminar separation. As
said in the chapter before, laminar free shear layer after separation are unstable (similarly to
mixing layers) and will transit to turbulence at a distance xt downstream from the separation
point. If we look in gure 3.5, the drag coecient increases continuously from Re = 2000 to
Re = 200000 before the abrupt decrease. Same observation can be made in gure 3.6 where
−Cpb increases continuously. Both observations are related to the upstream progression of
the transition point xt in the free shear layer as Re increases. Once the transition is close to
the wall, some turbulent reattachment takes place because of the Coanda eect (entrainment
velocity). This is the critical régime. The reattachment reduces considerably the size of
the region of low pressure at the rear , hence the drag will be very low (compare pressure
distributions before and after the drag crisis in gure 3.3). For larger Re, the boundary layer on
the cylinder becomes turbulent, and the free shear layer separates in a turbulent régime : there
is no reattachment anymore, but the separation is now shifted downstream at an angle close
to 120o (as depicted in gure 3.10 and visualized in gure 3.9). A turbulent boundary layer
has a better resistance to an adverse pressure gradient and then delays separation compared
to that of a laminar boundary layer. The reason is that turbulent boundary layer presents
a much larger wall friction (see gure 3.11). Hence for a given adverse pressure gradient, a
larger distance is needed to reduce the friction to zero, and then for the ow to separate.
30
Figure 3.5: Drag coecient for a circular cylinder
31
Figure 3.6: Base pressure coecient for a circular cylinder
32
Figure 3.7: Flow past a circular cylinder vs. Re
33
Figure 3.8: Separation points position on a circular cylinder around the drag crisis.
Figure 3.9: Laminar separation vs. fully turbulent separation.
34
Figure 3.10: Mean separation modication before and after the drag crisis.
Figure 3.11: Friction law vs. Re for a at plate at zero incidence.
35
Figure 3.12: Strouhal number evolution with Reynolds number for a circular cylinder.
3.2 Wake Dynamic
3.2.1 Strouhal vs. Reynolds number
For 2D blu-bodies, the wake dynamic is governed by a strong periodic oscillation. The
associated frequency f is constant whatever the location around the body, which is then
referred as a global frequency. This frequency appears at Re = 50 and is still present until
the drag crisis and even beyond. The underlying mechanism for the ow synchronization
phenomenon will be discussed in the next part. The Strouhal number dened as
S=f
d
U0
(3.7)
is measured experimentally for circular cylinder and plotted in Figure 3.12. This curves show
that the Strouhal number changes of variation law with Re each time the wake presents a new
instability. We will try to understand these variations in the following. There is a good tting
up to Re < 2.105 (before the drag crisis) if one assumes :
m
S = S∗ + √
Re
(3.8)
with dierent constant S ∗ and m in dierent regimes
3.2.2 2D Dynamic
3.2.2.1 Origin of the Kármán street
The 2D dynamic is the one that can be obtained with only a 2 dimensional ow. This can
be easily realized with numerical simulation or experimentally in soap lms. In this case, the
vorticity is a scalar and obeys :
∂ω
~
+ (~u.∇)ω
= ν∆ω
(3.9)
∂t
36
For physical purpose it is useful to approach this dynamic with the Vortex Method. This
method is based on the discretization of the initial vorticity eld ω0 (~r) into particles having a
given circulation Γp :
X
ω0 (~r) ≈
Γp δ(~r − ~rp )
(3.10)
p
In this case, it can be shown that solving equation 3.9 is equivalent to solve:
d~rp
dt
dΓp
dt
~p
= V
(3.11)
= 0
(3.12)
~p =
V
X −Γi (~rp − ~ri )
∧
~
e
(Biot & Savart)
z
2π|~rp − ~ri |2
(3.13)
i6=p
If the motion is inviscid and 2D, the viscous diusion term and the 3D stretching term
are both zero which means that the circulation carried by each particle is conserved during
the dynamic. The dynamic described by equation 3.11 can then by numerically computed
from an initial discrete vorticity distribution. For this, we need to compute at each particle
location, the total velocity resulting from the sum of the velocities induced by all of the other
particles deduced from the Biot & Savart law. Once thais velocity obtained, one just have to
displace the particle with this velocity. A new particles distribution is then obtained and the
process is iterated again. The basic dynamic is easily understandable from only two particles
having either equal or opposite circulation (two co- or counter rotating vortices). A linear
distribution of particle having equal circulation is called a vortex sheet. It is easy to see that
a vortex sheet will roll-up with this inviscid motion.
For blu-bodies with separation, we have two vortex sheets of opposite circulation per
length unit. Each particle will have a velocity induced by the contribution of all the other
particles of both sheets, making a non-linear coupling between the sheets. This dynamic leads
to a limit cycle with a well dened frequency (see gure 3.13). This model reproduces well the
Kármaán street dynamic which then results from the interaction of the two vorticity sheets
originated from the two separation points on the body. Whatever the body shape (symmetric
or not) the circulation of the sheet are equal and opposite. However, the detail of the vorticity
distribution in the sheet at separation depends on the history of the boundary layer over the
body that began at the forehead stagnation point (i.e. at the dividing streamline). This
"detail" will produce eect on the wake frequency selection and will be discussed in the next
section.
3.2.2.2 Pressure dynamic
The presence of the body with separation will create a low pressure due to the contraction
of the streamlines passing the body. There is another eect that creates low pressures, which
corresponds to the vortex formation. Each vortex creates a low pressure, which then might
contribute to the drag.
37
Figure 3.13: Simple vortex method simulation. At the top, passive particles (like a dye) are
~ 0 . Particles are simply displaced from left to right at
injected at A and B in a uniform ow U
constant velocity and form two emission lines. At the bottom each particle injected at A carries
a clockwise circulation and each particle injected at B carries an anti-clockwise circulation.
At each time step, we compute the resulting velocity (Biot and Savart) on each particle p,
−Γi (~
rp −~
ri )
~p = P
V
∧~
e
induced by all the other particles i. Once all the particle's velocity
z
2
i6=p 2π|~
rp −~
ri |
computed, we displace them by the quantity V~p ∆t. For this simulation there is no blu-body.
38
Figure 3.14: Pressure and vorticity in 2D dynamic. Blue : positive pressure coecient, red
: negative pressure coecient, white : zero pressure coecient. Iso-lines are the vorticity
contours
Figure 3.15: Denition of an idealized wake width d0 with velocity US at separation (see later
free streamline theory).
3.2.2.3 Physical models for global frequency selection
A fundamental question is what determines the Strouhal frequency? A rst answer provided
by A. Roshko (1954) is that the frequency must be determined by the duration time for the
two sheets to meet by self induced velocity. Let's call d0 the separating distance between the
sheets and Us the velocity at separation and all along the sheet (see sketch in gure 3.15).
0
0
This time is Uds , one should have a universal Strouhal number given by S ∗ = f Uds . For a given
body, the Strouhal is then written:
d US
S = S∗ 0
.
(3.14)
d U0
The ratio is called the bluness, one can see, as a rule of the thumb that the larger the
bluness of the body the lower the Strouhal number. This is actually observed in gure 3.16,
a square cylinder is more blu than the wedge cylinder which is more blu than the circular
cylinder. Formula 3.14 does not take into account the thickness of the free shears which plays
an important role in the frequency selection. Gerrard (1966) proposed a physical mechanism
based on two characteristic lengths of the wake : the formation length L (comparable to the
bubble recirculation length, and the thickness width of the vortex sheet at a distance L from
separation points. The basic mechanism for a period of vortex shedding is described in term of
vortex sheet interaction (see above). The process of vortex formation (the roll-up of a sheet)
is interrupted by the proximity of the other sheet having the opposite circulation. When both
d0
d
39
Figure 3.16: Strouhal measurements for dierent cylinder shape
sheets are close together, vorticity of opposite signs are cancelled and the roll-up stops. The
characteristic length L is the distance at which the sheets meet. In this view, the thicker the
sheets at L, the larger the time interval to cancel the vorticity and then the larger the period
for vortex shedding.
For laminar or turbulent free shears the thickness increases with downstream distance.
The larger L the smaller the wake frequency. For circular cylinder, the formation length L
is found to increase with Re for Re < 2.103 and to decrease in the range Re = 2.103 to
Re = 2.105 before the drag crisis (gure 3.17). For Re < 2.103 , the free shears are laminar
1
and their thicknesses increase downstream as : δ(x) ∼ d 1 ( xd ) 2 . Hence, increasing Re will also
Re 2
reduce δ all along the shears. Since the wake frequency increases in this range (gure 3.12),
Figure 3.17: Formation length L vs. Re
40
Figure 3.18: Sketch of the sheets dynamic and entrained velocities
the shear thickness at L (although L increases) decreases. This is because the shear thinning
due to the Re increase is more important than the eect of the L increase. For the range
Re = 2.103 to Re = 2.105 , the shears change nature (become turbulent) before that the sheets
meet which provokes a sudden growth of the thickness, δ(x) ∼ x. The transition location xt
moves upstream as Re increases, meaning that at a given distance downstream, the thickness
will grow with Re. The resulting behavior with the fact that L increases in this range is a
wake frequency decrease.
The behavior of L with Re is directly related to an equilibrium between the entrained
velocity by the shears and the sheets dynamic (gure 3.18). When the shear are laminar, the
1
1
entrained velcocity, VE (x) ∼ U0 Re− 2 ( xd )− 2 is compensated by the velocity produced by the
inviscid 2D sheet dynamics. What happens if we increase Re, with identical inviscid dynamic?
The entrained velocity by the shears will decrease but the entrained velocity produced by the
dynamic will remain the same. As a result the bubble recirculation region will grow in size
until both uxes will compensate each other, and L will increase. This scenario is observed
for Re < 2000. For larger Re, the shears become turbulent and the entrained velocity become
very large just after the transition which will increase the entrained velocity. This time, the
recirculation bubble must shrink in size to compensate both uxes, this is the scenario for
Re > 2000 and before the drag crisis. In this range the vortices are formed closer and closer
to the rear of the body, which will create a lower base pressure and then a larger drag (see
gures 3.6, 3.5 and 3.14).
3.2.3 3D Dynamics
For (2D) cylinder bodies in a 3D ow, although the forcing is 2D, 3D perturbations of the
ow will become unstable for suciently large Re (Re = 150 for circular cylinders). The term
in the vorticity equation responsible for the 3D growth is the stretching term that has been
neglected so far :
∂~
ω
~ ω = ν∆~
~ u
+ (~u.∇)~
ω + (~
ω .∇)~
∂t
(3.15)
Vorticity will be orientated in stretching direction produced between each primary 2D
vorticies. Two (A-mode and B-mode) 3D patterns are observed after the 3D transition in the
wake (see gure 3.19). The eects of these transition are well observed in either the drag or
the Strouhal but are not yet fully understood.x
41
Figure 3.19: Mode A and Mode B in the wake
42
Chapter 4
Three dimensional Blu-body ows
Contents
4.1 2D bodies (Recall) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.1.1 Bifurcation scenario in the laminar regime . . . . . . . . . . . . . . .
4.1.2 Periodic mode in the turbulent wake . . . . . . . . . . . . . . . . . .
43
43
4.2.2 Steady modes in the turbulent wake . . . . . . . . . . . . . . . . . .
4.2.3 Periodic modes in the turbulent wake . . . . . . . . . . . . . . . . .
43
43
43
4.3.2 Steady modes in the turbulent wake . . . . . . . . . . . . . . . . . .
4.3.3 Stochastic model for the long time dynamics . . . . . . . . . . . . .
43
43
43
4.2 3D axisymmetric bodies . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.3 3D non-axisymmetric bodies . . . . . . . . . . . . . . . . . . . . . . . 43
4.4 Conclusion and perspectives for industrial ows . . . . . . . . . . . 43
One fundamental dierence with cylinders ows, is the ability of 3D blu-bodies to produce
a bifurcation at low Reynolds number between the trivial steady solution presenting the same
symmetries as the body and a steady solution that breaks one symmetry of the body. The
corresponding mode persists even at large Reynolds number. Unsteady periodic modes such
as the Karman shedding is also present but aects less the uid force properties of the 3D
bodies than for a 2D bodies.
The lecture notes are not completed yet, but the slides of this lecture part will be provided.
43
4.1 2D bodies (Recall)
4.1.1 Bifurcation scenario in the laminar regime
4.1.2 Periodic mode in the turbulent wake
4.2 3D axisymmetric bodies
4.2.2 Steady modes in the turbulent wake
4.2.3 Periodic modes in the turbulent wake
4.3 3D non-axisymmetric bodies
4.3.2 Steady modes in the turbulent wake
4.3.3 Stochastic model for the long time dynamics
4.4 Conclusion and perspectives for industrial ows
44
Chapter 5
Free streamline theory
Contents
5.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.1.1 Flow approximation: cavity model . . . . . . . . . . . . . . . . . . .
5.1.2 Complex potential ow for irrotationnal 2D ows . . . . . . . . . . .
5.1.3 Potential ow at separation . . . . . . . . . . . . . . . . . . . . . . .
44
45
46
5.2.1
5.2.2
5.2.3
5.2.4
48
49
51
51
5.2 Flow over a plate perpendicular to the main velocity (exercise) . 48
The Schwarz-Christoel theorem . . .
Computation of the complex potential
Shape of the free streamlines . . . . .
Drag computation . . . . . . . . . . .
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5.3 General drag formula from Helmholtz's drag . . . . . . . . . . . . . 52
5.4 Flow approximation with nite separated region : closure models 52
At large Reynolds numbers, viscous eects are negligible, and one could expect the ow
around a body to be given by the Euler equations. Considering steady ows around a vertical
plate, the fully potential solution for which the drag is zero, is very far from reality as can
be shown by the comparison in gure 5.1. Experimentally one nd a drag for the plate of
CD ≈ 2. This apparent contradiction known as the D'Alembert's paradox is the consequence
of the presence of the separation. The goal of this chapter is to nd other possible steady
solutions of the Euler equations, that are not irrotationnal every where, and that could model
reality. We will see such solutions (especially in the case of the plate) that allows to compute
non-zero drag even for an inviscid ow.
5.1 Theory
5.1.1 Flow approximation: cavity model
The idea is to introduce separations by "hand" specifying new boundary conditions to the
potential ows. The recirculation bubble behind the body is a slow uid motion region (its
velocity is taken as UC = 0 everywhere) compared to that of the ow outside the bubble. Both
potential ows are then delimited by a discontinuity line (a shear) presenting both velocities;
45
Figure 5.1: Plate ow at large Reynolds numbers.(a) : potential solution of the steady Euler
ow equations. (b) : experiment at Re=2000
US just outside the bubble and UC = 0 inside. Thus, the discontinuity lines are regions of
strong rotational called "vortex sheets". The velocity of a uid particle placed exactly on the
vortex sheet is half the velocity jump across the sheet, say US /2. Finally the absolute value of
the circulation per unit of length of the sheet is equal to the velocity jump : | γ |= US . Across
the sheet, the pressure is continuous, so that the pressure all along the sheets is constant and
given by the cavity pressure. The potential ow outside the cavity is then delimited by two
"free" streamlines of constant pressure ("free" means constant pressure along the streamline).
We introduce the cavity parameter, a dimensionless parameter that compares the pressure
in the cavity to the pressure of the incoming ow :
σ=
p∞ − pC
1
2
2 ρU∞
(5.1)
The base pressure of a blu body (see last chapter) should correspond to the cavity pressure, so that σ = −Cpb . As shown in the last chapter, Cpb is always negative implying that
σ > 0.
Bernoulli's theorem holds inside and outside the cavity, however their constant (or pressure
head) are dierent because of the presence of the vortex sheet. Outside the cavity we have,
1 2
1
p∞ + ρU∞
= pC + ρUS2 .
2
2
(5.2)
Since the cavity pressure is constant, the outside velocity of the free streamline is also constant
all along the sheet : US . If US < U∞ , then σ < 0 which is not the physical case we are looking
for. We want σ > 0 which is obtained for US > U∞ .
5.1.2 Complex potential ow for irrotationnal 2D ows
Due to the incompressibility and the irrotationnality:
The complex potentiel denoted as
W = Φ + iΨ is an analytic fonction of z = x + iy in the region of the z − plane occupied by the
46
potential
s
U∞,, p∞
UC=0
pC=const
s
US
potential
Figure 5.2: Approximation of the separated ow over a plate at large Reynolds numbers
ow meaning that W has a unique derivation with z at all points of that region. Conversely,
any analytic function of z can be regarded as the complex potential of a ow eld.
The ow velocity in cartesian and polar coordinates is given by
dW
= u − iv = qe−iθ ,
dz
(5.3)
where q is the velocity modulus and θ denes the direction of the velocity vector in the
z − plane.
For the free streamline theory, the key of the problem is the introduction of a new complex
variable:
Ω = ln(US
dz
US
) = ln
+ iθ.
dW
q
(5.4)
On the free streamline where q = US , Re(Ω) = 0 and Im(Ω) is constant on any straight
streamlines as it is the case along the front vertical solid boundary of the plate.
5.1.3 Potential ow at separation
Before the ow leaves the body, it looks like that we locally have a wedge ow (see chapter 2).
In gure 2.1, the solid wall is the dashed horizontal streamline and the free streamline would
leave the wall at a positive angle βπ/2 > 0 . This correspond to m > 0 in Eq(2.11) meaning
that when the ow leaves the body at S the velocity is zero there. This leads to an inconsistency, since the free streamline which starts at S has a non zero constant velocity US > U∞
in Eq(5.2). The way to avoid this problem is to consider that the separation has to be smooth
(no wedge) (see gure 5.3). This solution is in agreements with experimental observation at
very large Reynolds number. Locally, the only way to satisfy the no-slip condition on the wall
before S located at z = 0 and the constant velocity US on the cavity, is to have :
√
Ω ≈ −ik z
47
(5.5)
Figure 5.3: A, wedge ow at separation. B, smooth separation.
using the denition of Eq(5.4). Close
to the separation point, y ≈ 0. On the cavity, x > 0
√
implies that Ω is not real Ω = −i x = iθ, then q = US everywhere on the cavity. The shape
of the leaving streamline is given by integrating its slope dh
dx = θ . The streamline leaves the
wall as
2 3
h(x) = − kx 2
3
(5.6)
dθ
k
=− √
dx
2 x
(5.7)
with a curvature
which is singular at the separation point.
p
At the wall and before S , x < 0 implies that Ω is real Ω = k | x |, and the velocity along
the wall evolves as
q = US e−k
√
|x|
(5.8)
The pressure gradient can be computed from Eq(5.2) for x suciently small
ρU 2
∂p
= −k p S .
∂x
2 |x|
(5.9)
The potential ow can be adapted to dierent ow separation congurations depending
on the value of k, but not all of them are physically possible as discussed in the following.
. The case k > 0 is in agreements with experiments where the
separation point is xed at a sharp edge of the body (square, plate ...) as described
by sketch A in gure 5.4. The potential ow gives an innite favorable pressure
gradient just at separation as nearly experimentally observed in gure 3.4. The case
of sharp edge separation is often called ow break-away rather than ow separation
because in this case the boundary layer separation is not due to an adverse pressure
gradient.
Smooth body separation . The solutions with k > 0 are impossible since the leaving
streamline has an innite negative curvature at S (see Eq 5.7). The streamline
then falls irremediably in the interior of the body instead of being ejected in the
uid (sketch B in gure 5.4). For a separation from a smooth body : k ≤ 0. For
Salient edge separation
48
Figure 5.4: Sharp edge: A, k>0 (favorable pressure gradient). Smooth wall: B, k>0 (favorable
pressure gradient); C, k<0 (adverse pressure gradient); D, Villat condition.
k 6= 0, the immediate consequence is that the pressure gradient is adverse before
separation, which is observed experimentally in gure 3.3, but becomes innite at
S (sketch C in gure 5.4). The Villat condition for separation from a smooth body
avoid the innite pressure gradient by searching the leaving streamline having the
same curvature as the body and computing the corresponding k → 0 when x → 0
(sketch D in gure 5.4). The consequence is that the lowest pressure is found in
the cavity instead of before the separation. Physically, the Villat condition is the
admitted theoretical solution for the ow at separation for an innite Reynolds
number. For the circular cylinder the Villat condition gives a separation angle
of 55o C when the velocity of the free streamline is taken as U∞ (or equivalently
σ = −Cpb = 0. This is the theoretical solution shown in gure 3.3.
5.2 Flow over a plate perpendicular to the main velocity (exercise)
With bodies having at solid walls, there exists a general theory to solve the potential ow
with free streamlines. The theory is based on the Schwarz-Christoel theorem conformal
transformation that is always able to map the ow on a half plane. The problem is solved
when the ow can be also mapped on a half plane using an other technique. Once the two
changes of variable are known, the equivalence between the two half planes allows to nd the
potential W (z) in the z − plane. Of course, the positions of the separations points have to be
known (even if it is not a problem for salient bodies) but moreover, the pressure in the cavity
has to be prescribed. This is unfortunately not a complete theory for the ow around a body.
When the pressure in the cavity is prescribed to p∞ (σ = 0), the ow is called Helmholtz ow,
the case we rst consider.
5.2.1 The Schwarz-Christoel theorem
. Irrotational ow in a region bounded externally by a closed polygon, of which
one or more vertices may be at innity can always be represented on a half plane where the
external boundary is mapped on the real axis only.
It is useful for us to apply the SC theorem to a ow comprised in a semi-innite strip
in Ω − plane (see gure 5.5). As the result, the ow becomes comprised if a half plane of
SC Theorem
49
Figure 5.5: SC Conformal transformation of the semi strip to a half plane
Figure 5.6: Flow in z − plane
the λ − plane (see gure 5.5). The correspondence between the variables is given by the SC
theorem :
1
1
iπ
λ = (b + c) + (b − c) cosh [
(Ω − Ω0 )]
2
2
Ω1 − Ω0
(5.10)
with b and c real as shown in gure 5.5. They can be chosen arbitrarily.
5.2.2 Computation of the complex potential
The rst way to map the ow on a half plane is to use directly the SC theorem. In the
z − space, the ow surrounds the front of the plate and both the free streamlines (gure 5.6).
Using the denition of Ω, it is easy to represent the same ow in the Ω − plane as depicted in
gure 5.7. By choosing b = +1 and c = −1, the change of complex variable is
λ = −i sinh Ω
(5.11)
A second way to map the ow on a half plane is to make a simple conformal transformation
of the ow in the W − plane. It is easy to deduce the ow of gure 5.6 in W − plane (see
1
gure 5.8). Taking W 2 , the ow becomes mapped on a half plane. However both half planes,
1
λ − plane and W 2 − plane are not completely equivalent; they are deduced from each other by
50
Figure 5.7: Conformal transformations of the ow to a half plane, deducing the ow of gure 5.6
in Ω − plane and then making use of the SC theorem
Figure 5.8: Conformal transformation of the W − plane into a half plane
an inversion. Moreover, λ is a dimensionless variable while W has the dimension of velocity
× length. Thus the correspondence between both variables is
1
W 1
=(
)2
λ
kU∞
(5.12)
with k, real and positive to be determined later. Using Eq(5.11), Eq(5.12) becomes
(
To extract
dW
dz ,
kU∞ 1
i
dz
dW
) 2 = − {U∞
−
}
W
2
dW
U∞ dz
(5.13)
we need to solve a second degree equation whose solutions are
dW
= −i
U∞ dz
r
kU∞
∓
W
r
1−
kU∞
W
(5.14)
To nd the right branch, we study the relation for the streamline AO (see gure 5.6) where
W = Φ = ρΦ eiπ . On the real axis Eq(5.13) that reads
s
dW
=−
U∞ dz
kU∞
∓
ρΦ
51
s
1+
kU∞
ρΦ
(5.15)
should be 1 when Φ → −∞ (i.e. ρΦ → ∞). The right branch is then
dW
= −i
U∞ dz
r
kU∞
+
W
r
1−
kU∞
W
(5.16)
The integration gives
1
1
z − z0
W 1
W 1 W
π
W 1
W
)2 + (
)2 (
− 1) 2 + i − ln{(
)2 + (
− 1) 2 }
= 2i(
k
kU∞
kU∞
kU∞
2
kU∞
kU∞
(5.17)
At z =1 0, W = 0 gives z0 = 0. The constant k can be computed now. At B, λ = +1
W
(( kU
) 2 = 1) in λ − plane and z = id
2 in z − plane (d is the breath of the plate). The result is
∞
k=
2
4+π
(5.18)
5.2.3 Shape of the free streamlines
On the streamline, Ψ = 0 and then W = Φ. As said just before at B , W = kU∞ . Since the
velocity is U∞ along the free streamline, ∂Φ
∂s = U∞ integrates into
Φ
W
k+s
=
=
kU∞
kU∞
k
(5.19)
that injected in Eq(5.17) gives a parametric form for the free streamline shape
1 1
s 1
s 1
x(s) = (k + s) 2 s 2 − ln{(1 + ) 2 + ( ) 2 }
k
k
1
1
π
y(s) = 2k 2 (k + s) 2 + k
2
(5.20)
(5.21)
For s → +∞, y 2 → 4kx. The free streamline tends asymptotically to a parabola. The cavity
has then an innite length.
5.2.4 Drag computation
The drag D = − body p(s).~nds can be deduced from the complex potential (for details see
Batchelor's book). The drag coecient for the Helmholtz ow on the vertical plate is
H
CD0 =
D
1 2
2 U∞ d
=
2π
= 0.88.
4+π
(5.22)
This is very far from the experimental value CD ≈ 2. For a a circular cylinder, with Villat
condition CD0 = 0.5 which is again much lower than the experimental value CD ≈ 1. The
explanation for the discrepancy is that for Helmholtz ows, the cavity number is σ = 0
(pressure cavity equal to p∞ ) and the cavity length innite. In experiment the cavity number
is larger (pressure cavity is lower than p∞ ), σ = −Cpb . The cavity is not steady and has
not an innite length. As we saw in the last chapter, the cavity is closed after few diameters
downstream the body due to the vortex sheet interactions. The closure is responsible for a
cavity number larger than zero.
52
Figure 5.9: The pressure distribution obtained with the free streamline theory with σ = 0 is
stretched to have the desired base pressure coecient
5.3 General drag formula from Helmholtz's drag
From the drag computed with σ = −Cpb = 0, it is possible to derive a general formula for
the drag for a body with σ = −Cpb > 0. The idea is to match the real base pressure on the
pressure distribution found for the corresponding Helmholtz ow (see gure 5.9). If Cp0 (s) is
the pressure distribution of the Helmholtz ow around the body (the plate for example), we
match its zero base pressure coecient to Cpb by the operation:
Cp (s) = (Cp0 (s) − 1)(−Cpb + 1) + 1
(5.23)
CD = CD0 (1 − Cpb )
(5.24)
CL = CL0 (1 − Cpb )
(5.25)
which gives for the drag
and equivalently for the lift
This operation is possible only if the position of the separation points are the same in both
the real and the Helmholtz ow. The formula ts rather well the data, but it requires the base
pressure measurement. To complete the theory one need a theory for the base pressure ! We
next present some steady model giving a base pressure.
5.4 Flow approximation with nite separated region : closure
models
In order to take into account the bubble closure, some steady models with nite cavity of
positive cavity number (pC < p∞ ) have been elaborated. A closure cannot be simply realized
by a stagnation point (zero velocity) at the rear of the cavity because the velocity modulus
everywhere on the cavity is a non zero constant. The Riabouchinsky model closes the cavity
on another body which is a mirror of the main body. Whatever the model, it is found that
the smaller the cavity, L/d → 0 the larger the cavity number σ → ∞ (the larger the drag).
They are generally well tted by power laws, for the Riabouchinsky closure on a at plate (see
gure 5.10) :
2
L/d ≈ 2.78(−Cpb )− 3
53
(5.26)
Figure 5.10: Cavity closure model proposed by Riabouchinsky for to the at plate
The Helmholtz ow is recovered as σ → 0 for L/d → ∞. These models are not complete
again, since the base pressure can be computed only if the cavity length is known... However,
the relationship between the base pressure and the cavity length indicates that the drag of
blu-bodies is connected to the separated region shape. Even if real ows are unsteady, a
premature closure will impose strong curvatures of the ow around the separation while for
late closure, the curvature will be weaker. For Euler ows, strong curvature (centripetal
acceleration) is responsible for large pressure gradient and hence very low pressure in the
cavity.
54
Chapter 6
Cavitation
6.1 Condition for cavitation
We shall recall that pressure is by denition the isotropic part of the total strain rate tensor
that could be positive (compression) or negative (traction). The natural cavitation is the
phase transition at constant temperature of a liquid into a gas due to a pressure change. The
necessary but not sucient condition for cavitation is that the pressure should be below the
vapor pressure pV (T ) of the uid. Generally, cavitation in water is observed for negative
pressure, typically minus few bars at about 15o C . To model the cavitation inception, the
assumption of pure liquid have to be forgotten. Actually for pure water, cavitation would
imply the rupture of the Van der Waals molecular interactions that corresponds at 15o C to
a pressure of −300bars! The reason for the large discrepancy is the presence of germs in the
liquid. The Blake model assumes that germs are bubbles of radius R for which the pressure
dierence across the interface is balanced by the surface tension (γ ) eect. However, it can be
shown that below a critical pressure given by
pc (R) = pV (T ) −
4γ
,
3R
(6.1)
the surface tension is not able to counter balance the pressure dierence across the bubble
interface, as a consequence the germ grows unboundedly which provokes the cavitation. The
cavitation inception thus depends on the germs' size, the smaller the germ, the larger the
cavitation delay from pV (T ).
The necessary condition for cavitation p ≤ pV (T ) becomes in term of pressure coecient :
σV ≤ −Cp ,
(6.2)
V (T )
where σV = p∞1−p
is called the cavitation number also known as Thomas number. Hence,
2
ρU∞
2
the knowledge of the pressure eld (such as the minimum pressure coecient, Cpmin ) around
a body without cavitation, allows to predict that cavitation will never occur if σV >> −Cpmin
but that the ow will eventually cavitate for σV ≈ −Cpmin . Figure 6.1 displays the dierent
cavitation regimes that are observed around a circular cylinder initially in the subcritical
regime (Re < 200000).
55
Figure 6.1: Classication of the cavitation regimes in term of the cavitation number σV = KV .
The cavity parameter KC = −Cpb .
6.2 Pressure around the body
The modication of the pressure distribution around a circular cylinder at a supercritical
regime (Re > 300000) is shown in gure 6.2. The pressure becomes rigourously constant in
the separation zone for the smallest σV , the adverse pressure gradient disappears so that the
lowest pressure is found in the cavity. Actually, the base will be occupied by dry vapor only
when the cavitation number will be much lower than minus the base pressure coecient that is
measured with no cavitation. As soon as the base is occupied by vapor only, the base pressure
becomes equal to the vapor pressure. This is a strong simplication of the problem for the drag
of separated ows since the base pressure is now prescribed to a value that does not depend
on the ow dynamics. Consequently, the cavity parameter σC = σV . In gure 6.3, the semiempirical formula for the drag in Eq. 5.25 deduced from the Helmholtz drag is satisfactorily
recovered by setting −Cpb = σV . Another strong simplication with cavitation, is that for
supercavitating case, the ow is the same whatever the Reynolds number is: the distinction
between the supercritical and subcritical regimes does not hold anymore (see gure 6.4).
The unique solution of a supercavitating cylinder should have a separation at 55o when
the cavity parameter σv = 0 (Free streamline theory solution with Villat condition). In this
case the drag coecient should tend to 0.5 as expected by this theoretical solution as σv → 0.
This is actually what is observed in gure 6.5.
56
Figure 6.2: Mean pressure distribution in terms of K = σV , Re > 300000
57
Figure 6.3: Drag vs. the cavitation number σV = K for supercavitating blubodies.
58
Figure 6.4: Top: base pressure coecient (Cpb ) vs. cavitation number for initially super
critical ow (Kis , Kib ) and initially subcritical ow (Kiw ). Bottom: vapor visualization before
and after the jump in Cpb of the initially supercritical ows.
59
Figure 6.5: Cd vs. cavitation number.
6.3 Gravity eect
In section 5.4, we considered closure cavity models because the cavity cannot close freely in
the context of steady potential ow theory with positive cavity numbers. Actually, free steady
cavities can exist if gravity eect are present. This is observed in gure 6.6. The gas cavity
is formed behind a disk, here the cavitation is articial because the gas (air) is supplied at
the base of the disk while in natural cavitation the gas is supplied by the phase transition.
The cavity is terminated by two hollow tubes through which the injected air is evacuated. To
understand the shape of the cavity, one have to consider the free streamline that lives the disk
from the top and the other from the bottom. At the top separation, the velocity at separation
is smaller than the velocity at the bottom, this velocity dierence is at rst order :
Ub − Ut ≈
1 gd
2 U∞
(6.3)
This result can be found simply by writing the Bernoulli relationship on the dividing streamline. The velocity dierence will create a circulation around the cavity. Taking L as the length
of the cavity, the circulation will be
C ≈ L(Ub − Ut ) =
L gd
2 U∞
(6.4)
The circulation creates a vertical negative lift force Fy ≈ −ρC U∞ d that is counter balanced
by the buoyancy force FA ≈ ρgLd2 . The circulation around the cavity together with the side
eects of the cavity will create two counter vortices exactly in the same manner as for the
60
Figure 6.6: Gravity eect on an axisymmetric wake
longitudinal tip vortices of lifting wings of nite span. The circulation being conserved due to
the Kelvin's theorem for inviscid ows, the circulation of the vortices is the same as for the
cavity. Hence the vortices have a low pressure proportional to −ρC 2 /a2 . For the steady cavity
the equilibrium imposes that the pressure inside the vortices is equal to the base pressure. The
condition for this equilibrium also states that the Froude number based on the cavity length
∞
= cte. This cavity governed by gravity is observable for Froude number
is constant, say √UgL
(based on the bodies' diameter) lower than :
U∞
Fr = √ ≤ 2.5
gd
61
(6.5)
Chapter 7
Conclusion
This lecture is aimed at giving the basic ideas of the high drag origin of blu bodies due to ow
separation. We restricted ourselves to bidimensional geometries. Most of the physics has been
established during the beginning of the 20th century. However, there is a renewed interest
since the last 10 years. This correspond to a demand from the industries of car builders. The
reason is the obligation of the reduction of emission of CO2 . This reduction can be amazingly
achieved if cars become aerodynamics. At the moment, the drag coecient of cars are about
0.3. The consequence is that at 100km/h, 60% of the power delivered by the engine feeds
the turbulent ow around the vehicle. On the other hand, an aerodynamic car, as displayed
in gure 7.1 would have a drag coecient of 0.15. Unfortunately, these cars are not really
designed to be sold. Industrials need free design with low drag. The probable answer stands
in ow control, active are passive.
62
Figure 7.1: Evolution of the drag coecient of cars.
63

Separated ows : Wakes and Cavities

Transcription

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