Angular Slip Velocity and Lift
D.D. Joseph, D. Ocando, P.Y. Huang
For the flow around a circular cylinder, the Kutta-Joukowski law states that, there
will be no lift force on the cylinder if there is no circulation around it, and if there is
a circulation, the value of the lift force will be given by the product of the
magnitude of this circulation with the free stream velocity and the density of the
fluid:
FL  U
(1)
The lift force is either positive or negative, depending on the sign of the
circulation.
When a circular cylinder is released in a 2-D channel Poiseuille flow, it will have
three degrees of freedom: translational motion on a plane and angular motion.
Therefore, the free stream velocity on the K-J law will be related to the axial slip
velocity (Fig.2), defined as the fluid axial velocity minus the cylinder axial velocity:
U  u s  u f  uc
(2)
On the other hand, the circulation on the K-J law will be related to the angular slip
velocity (Fig.2), defined as half the fluid shear rate minus the particle angular
velocity:
  s 

2
 c
(3)
In dynamic simulations when the cylinder is above the vertical equilibrium
position (Fig.3), the value of the angular slip velocity is less than the angular slip
velocity at equilibrium (Fig.4). On the other hand, when the cylinder is below the
vertical equilibrium position (Fig.3), the value of the angular slip velocity is higher
updated 07/12/16
1
219474817.doc
than the angular slip velocity at equilibrium (Fig.4). In the other hand, there is no
clear relation between the evolution of the axial slip velocity and the evolution of
the particle’s vertical position (Fig.5). Therefore, the relevant variable in the
analysis of the evolution of the particle’s vertical position, appears to be the
difference between the angular slip velocity and the angular slip velocity at
equilibrium:

   s   es

(4)
This assumption was verified for the case of a particle with higher density than
the fluid (Fig.6 and Fig.7). In this case, when the particle starting close to the
centerline crosses the equilibrium position, the angular slip velocity also crosses
its equilibrium value.
The evolution of the forces (vertical and horizontal) and torque for the benchmark
case are shown in Fig.8 through Fig.10. These transient results show no clear
relation with the slip parameters. Therefore, the slip variables can be used to
explain (or predict) the steady state lift force or, the evolution on the particle’s
position, but cannot be used to predict the evolution of the forces and torque
acting on the particle.
Shear Slip Model
The forces acting on the long particle are the force due to pressure acting on the
sides perpendicular to the flow, and the force due to shear stress acting on the
sides parallel to the flow (Fig.1). The former force is always positive, while the
latter may be positive or negative depending if the fluid is faster than the particle
or viceversa:
 A  B
 A   B l   p1  p2 d  0
 p  p2 
 pd  0
p 1
l
(5)
(6)
where the shear stresses are defined by:
 A  
du A
(h A )
dy 
 B  
du B
( hB )
dy
The velocity profiles above and below the long particle are given by:
2
(7)
y’
p1 >p2
uA(y’)
p1
p2
A
UA
B
UB
y
uB(y)
x
Fig 1. Sketch of flow field under consideration and variables involved in the slip models
u A  y  
U y
p
y hA  y   A
2
hA
(8)
uB y 
U y
p
yhB  y   B
2
hB
(9)
where different velocities ( U A , U B ) were assumed for the top and bottom walls to
take into account the angular speed of the circular particle. The relation between
them is given by:
U A UB 
 (hB  d 2)
2
d
(10)
where   y  is the shear rate for the undisturbed flow (without the particle), given
by:
 ( y ) 
du
p
h  2 y 

dy 2
(11)
The shear rate on the particle’s sides parallel to the flow may be evaluated from
Eqns. (8), (9) and (10):
3

du A
hA    p hA  U B   (hB  d 2) d
dy 
2
hA
2
hA
(12)
du B
hB    p hB  U B
dy
2
hB
(13)
Substituting, recursively, Eqns. (12) and (13) in Eqn. (7), and then the resultant
equation in Eqn. (6), gives the formula to obtain U B :
p
UB 

2d  hA  hB hA hB   (hB  d
2h A  hB 
2) hB d
(14)
when the top and bottom walls have the same velocity ( U A  U B ), the formula
reduces to:
p
UA UB 

2d  hA  hB hA hB
2h A  hB 
(15)
Both models, same or different velocity on top and bottom walls, were plotted
against the DNS results, and also the undisturbed velocity profile was included to
note the effect of the particle on the velocity profile (Fig. 15 through Fig. 22). In
general, the shear slip model gives a better approximation to the DNS results
than the model with same top and bottom wall velocity. That is especially true
when the particle is close to the wall (Fig.19 and Fig.20). The reason for that is
that close to the wall the fluid shear rate is higher and, therefore the particle’s
angular velocity is also higher. The same velocity model does not include this
angular motion and there, it is the reason for its difference with the DNS results.
This result was validated using a smaller cylinder (Fig.21 and Fig.22). The only
difference with the larger-particle case was that the DNS result and the
undisturbed velocity profile are closer than before. This was expected, because
when the particle is smaller the disturbance in the fluid velocity profile must be
also smaller.
4
List of symbols
d : distance between particle’s top and bottom walls.
FL : lift force.
Fx : horizontal force acting on the particle.
Fy : vertical force acting on the particle.
hA : distance between channel’s top wall and particle’s top wall.
hB : distance between channel’s top wall and particle’s top wall.
p : absolute value of the pressure gradient.
t : time.
u A : fluid velocity above particle’s top wall.
u B : fluid velocity below particle’s bottom wall.
u c : cylinder’s axial velocity.
u f : cylinder’s axial velocity.
u s : axial slip velocity.
U : free stream velocity.
U A : particle’s top wall velocity.
U B : particle’s bottom wall velocity.
y : vertical coordinate pointing upward.
y  : vertical coordinate pointing downward.
y p : particle’s vertical coordinate.
Greek letters
 : dynamic viscosity.
 : shear rate.
 s : angular slip velocity.
 : circulation around cylinder.
 c : cylinder’s angular velocity.
 s : angular slip velocity.
 es : angular slip velocity at equilibrium.
 A : shear stress on particle’s top wall.
 B : shear stress on particle’s bottom wall.
5
100
ANGULAR Slip Velocity
HORIZONTAL Slip Velocity
Lift Force
90
80
Relative value [%]
70
60
50
40
30
20
10
0
-10
6
7
8
CENTERLINE
Fig. 2. FIXED PARTICLE:
9
10
11
12
WALL
y [cm]
R  20 ,  S  F  1 . Steady state relative values for the lift force
and the slip velocities. In the region close to the wall, the lift force and the horizontal slip
velocity have a similar non-linear behavior. In the region close to the centerline, the lift
force appears to be related with the angular slip velocity. Therefore, the lift force may be
expressed as a function of the slip velocities product L  Lu s ,  s  .
6
5
y p [cm]
4
Starting at the CENTERLINE
3
Starting at the WALL
2
1
0
0
20
40
60
80
100
120
t [s]
Fig. 3. BENCHMARK CASE:
R  10 ,  S  F  1 . Evolution of the particle’s vertical position.
The particle moves from the starting position to the final position without crossing the
equilibrium position
6
10
Starting at the CENTERLINE
Starting at the WALL
-1
s [s ]
1
0.1
0.01
0
20
40
60
80
100
120
t [s]
Fig. 4. BENCHMARK CASE:
R  10 ,  S  F  1 . Evolution of the particle’s ANGULAR slip
velocity. The angular slip velocity function evolves without crossing the equilibrium value.
When the angular slip velocity is BELOW the equilibrium value, the particle moves
DOWNWARD. When the angular slip velocity is ABOVE the equilibrium value, the
particle moves UPWARD.
30
25
u s [cm/s]
20
Starting at the CENTERLINE
Starting at the WALL
15
10
5
0
-5
0
20
40
60
80
100
120
t [s]
Fig. 5. BENCHMARK CASE:
R  10 ,  S  F  1 . Evolution of the particle’s AXIAL slip velocity.
There is no clear relation between the particle’s vertical position and the axial slip
velocity.
7
6
5
4
y P [cm]
Starting at the CENTERLINE
Starting at the WALL
3
2
1
0
0
20
40
60
80
100
120
t [s]
Fig. 6. HEAVY PARTICLE:
R  10 ,  S  F  1.01 . Evolution of the particle’s vertical position.
The particle starting at the centerline crosses the equilibrium position and then moves
upward.
5
4
Starting at the CENTERLINE
-1
s s ]
3
Starting at the WALL
2
1
0
0
20
40
60
80
100
120
t [s]
Fig. 7. HEAVY PARTICLE:
R  10 ,  S  F  1.01 . Evolution of the particle’s angular slip
velocity. For the particle starting at the centerline, the angular slip velocity function
crosses the equilibrium value. When the angular slip velocity is BELOW the
equilibrium value, the particle moves DOWNWARD. When the angular slip velocity
is ABOVE the equilibrium value, the particle moves UPWARD.
8
0.002
Vertical force [dyn]
Torque [dyn.cm]
F y [dyn],
T [dyn.cm]
0.001
0
-0.001
-0.002
-0.003
-0.004
0
20
40
60
80
100
120
t [s]
Fig. 8. BENCHMARK CASE:
R  10 ,  S  F  1 . Evolution of the vertical force and torque
acting on a particle starting close to the CENTERLINE. There is no clear relation between
the slip parameters and the vertical force.
0.03
Vertical force [dyn]
Torque [dyn.cm]
F y [dyn],
T [dyn.cm]
0.02
0.01
0
-0.01
-0.02
-0.03
10
30
50
70
90
110
t [s]
Fig. 9. BENCHMARK CASE:
R  10 ,  S  F  1 . Evolution of the vertical force and torque
acting on a particle starting close to the WALL. There is no clear relation between the slip
parameters and the vertical force.
9
1.4
1.2
1
Starting at the CENTERLINE
F x [dyn]
0.8
Starting at the WALL
0.6
0.4
0.2
0
-0.2
0
20
40
60
80
100
120
t [s]
Fig. 10. BENCHMARK CASE:
R  10 ,  S  F  1 . Evolution of the horizontal force acting the
particle. The horizontal force acting on a particle starting close to the WALL is always
positive, or in other words the particle is dragged from the wall to the equilibrium position.
The force for the particle starting from the CENTERLINE is first positive and then
negative, because the particle is moving from a higher fluid velocity region to a lowerfluid-velocity region.
10
0.05
0
F y [dyn]
-0.05
-0.1
-0.15
FIXED position ABOVE equilibrium
-0.2
FIXED position BELOW equilibrium
-0.25
-0.3
10
20
30
40
50
60
70
80
90
100 110 120
t [s]
Fig. 11. FIXED PARTICLE AROUND EQUILIBRIUM:
R  10 , y p  4.16 [cm], y p  4.19 [cm].
Evolution of the vertical force on the particle. The vertical force pulls the particle towards
the equilibrium position. This result shows that this position is a STABLE equilibrium
position.
1
FIXED position ABOVE equilibrium
-1
s [s ]
FIXED position BELOW equilibrium
0.1
0.01
10
20
30
40
50
60
70
80
90
100
110
120
t [s]
Fig. 12. FIXED PARTICLE AROUND EQUILIBRIUM:
R  10 , y p  4.16 [cm], y p  4.19 [cm].
Evolution of the particle’s angular slip velocity. At STEADY STATE: When the angular
slip velocity is BELOW the equilibrium value, the particle moves DOWNWARD
(negative force). When the angular slip velocity is ABOVE the equilibrium value,
the particle moves UPWARD (positive force).
11
0.06
0.04
F y [dyn]
0.02
FIXED position ABOVE centerline
0
FIXED position BELOW centerline
-0.02
-0.04
-0.06
0
20
40
60
80
100
120
t [s]
Fig. 13. FIXED PARTICLE AROUND CENTERLINE:
R  10 , y p  5.95 [cm], y p  6.05 [cm].
Evolution of the vertical force on the particle. The vertical force pushes the particle away
from the centerline. This result shows that the centerline is an UNSTABLE equilibrium
position. If the particle starts at the centerline, it will stay there, but the slightest
perturbation will force the particle away.
0.06
FIXED position ABOVE centerline
0.04
FIXED position BELOW centerline
-1
s [s ]
0.02
0
-0.02
-0.04
-0.06
0
20
40
60
80
100
120
t [s]
Fig. 14. FIXED PARTICLE AROUND CENTERLINE:
R  10 , y p  5.95 [cm], y p  6.05 [cm].
Evolution of the particle’s angular slip velocity. At STEADY STATE: When the angular
slip velocity is BELOW the equilibrium value, the particle moves UPWARD. When the
angular slip velocity is ABOVE the equilibrium value the, particle moves DOWNWARD.
This behavior is the OPPOSITE of the previous cases, because the previous cases
were STABLE equilibrium positions, and therefore THE FORCE FIELD around them
is the OPPOSITE.
12
60
50
u [cm/s]
40
30
Undisturbed
DNS
Slip model
20
10
0
TOP
0 WALL
2
4
6
8
10
BOTTOM
WALL 12
y [cm]
Fig. 15. FIXED PARTICLE AT THE CENTERLINE: y p  6.0 [cm],
R  20 . Axial velocity
profiles at steady state. The slip model based on the long particle assumption, gives
similar results as the undisturbed velocity profile. When the particle is at the centerline
the perturbation on the flow field is minimal.
60
50
u [cm/s]
40
30
Undisturbed
DNS
Shear slip model
20
10
0
TOP
0 WALL
2
4
6
8
10
BOTTOM
WALL 12
y [cm]
Fig. 16. FIXED PARTICLE AT THE CENTERLINE: y p  6.0 [cm],
R  20 . Axial velocity
profiles at steady state. The shear slip model gives the same results as the slip model
(Fig. 9). The shear slip model includes the effect of the fluid shear rate at the particle
position. The undisturbed velocity profile is used to get the shear rate. At the centerline
the shear rate value is zero, therefore both slip models give identical results.
13
60
50
u [cm/s]
40
30
Undisturbed
DNS
Slip model
20
10
0
TOP
0 WALL
2
4
6
8
10
BOTTOM
WALL 12
y [cm]
Fig. 17. FIXED PARTICLE AT y p  3.0 [cm]:
R  20 . Axial velocity profiles at steady state. The
slip model based on the long particle assumption gives lower velocity values than the
DNS results. The long particle imposes a greater restriction on the flow than the circular
particle. On the other hand, the circular particle rotates, while the long particle does not.
60
50
u [cm/s]
40
30
Undisturbed
DNS
Shear slip model
20
10
0
0
TOP
WALL
2
4
6
8
10
BOTTOM
WALL 12
y [cm]
Fig. 18. FIXED PARTICLE AT y p  3.0 [cm]:
R  20 . Axial velocity profiles at steady state. The
shear slip model gives a better approximation to the DNS result. The particle’s angular
rotation is approximated as half of the shear rate at the particle position on the
undisturbed fluid velocity profile.
14
60
50
u [cm/s]
40
30
Undisturbed
DNS
Slip model
20
10
0
0
TOP
WALL
2
4
6
8
10
BOTTOM
WALL 12
y [cm]
Fig. 19. FIXED PARTICLE AT y p  1.5 [cm]:
R  20 . Axial velocity profiles at steady state. As
the particle is closer to the wall, the difference between the slip model results and the
DNS values is larger. As the particle is closer to the wall, the particle’s rotation speed is
higher (the undisturbed fluid shear rate is maximum at the wall). Again, this rotation it is
60
50
u [cm/s]
40
Undisturbed
DNS
Shear slip model
30
20
10
0
0
TOP
WALL
2
4
6
8
10
BOTTOM
WALL 12
y [cm]
not included in the slip model based on the long particle.
Fig. 20. FIXED PARTICLE AT y p  1.5 [cm]:
R  20 . Axial velocity profiles at steady state. The
shear slip model gives an excellent approximation to the DNS result.
15
60
50
u [cm/s]
40
Undisturbed
DNS
Slip model
30
20
10
0
0
TOP
WALL
2
4
6
8
10
BOTTOM
WALL 12
y [cm]
Fig. 21. SMALL PARTICLE AT y p  1.5 [cm]:
d  0.5 [cm], R  20 . Axial velocity profiles at
steady state. As the particle is smaller, the difference between disturbed and undisturbed
velocity profiles is smaller.
60
50
u [cm/s]
40
30
Undisturbed
DNS
Shear slip model
20
10
0
0
TOP
WALL
2
4
6
8
10
BOTTOM
WALL 12
y [cm]
Fig. 22. SMALL PARTICLE AT y p  1.5 [cm]:
d  0.5 [cm], R  20 . Axial velocity profiles at
steady state. The shear slip model gives a better approximation to the DNS result.
16