boundary layer flow
DESCRIPTION
Boundary Layer Flow. p < 0. t w. U. U. p > 0. Drag force. The surrounding fluid exerts pressure forces and viscous forces on an object . The components of the resultant force acting on the object immersed in the fluid are the drag force and the lift force. . p < 0. t w. U. U. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Boundary Layer Flow](https://reader033.vdocument.in/reader033/viewer/2022061612/56816261550346895dd2bddb/html5/thumbnails/1.jpg)
Boundary Layer Flow
![Page 2: Boundary Layer Flow](https://reader033.vdocument.in/reader033/viewer/2022061612/56816261550346895dd2bddb/html5/thumbnails/2.jpg)
Drag force• The surrounding fluid exerts pressure forces
and viscous forces on an object.
• The components of the resultant force acting on the object immersed in the fluid are the drag force and the lift force.
p < 0U
p > 0
Utw
![Page 3: Boundary Layer Flow](https://reader033.vdocument.in/reader033/viewer/2022061612/56816261550346895dd2bddb/html5/thumbnails/3.jpg)
Drag prediction• The drag force is due to the pressure and shear
forces acting on the surface of the object.
• The tangential shear stresses acting on the object produce friction drag (or viscous drag).
p < 0U
p > 0
Utw
![Page 4: Boundary Layer Flow](https://reader033.vdocument.in/reader033/viewer/2022061612/56816261550346895dd2bddb/html5/thumbnails/4.jpg)
Drag prediction• Friction drag is dominant in flow past a flat plate
and is given by the surface shear stress times the area:
• Pressure or form drag results from variations in the normal pressure around the object:
A
npressured dapF ,
wviscousd AF t.,
p < 0U
p > 0
Utw
![Page 5: Boundary Layer Flow](https://reader033.vdocument.in/reader033/viewer/2022061612/56816261550346895dd2bddb/html5/thumbnails/5.jpg)
Viscous boundary layer• An originally laminar
flow is affected by the presence of the walls.
• Flow over flat plate is visualized by introducing bubbles that follow the local fluid velocity.
• Most of the flow is unaffected by the presence of the plate.
![Page 6: Boundary Layer Flow](https://reader033.vdocument.in/reader033/viewer/2022061612/56816261550346895dd2bddb/html5/thumbnails/6.jpg)
Viscous boundary layer• However, in the region
closest to the wall, the velocity decreases to zero.
• The flow away from the walls can be treated as inviscid, and can sometimes be approximated as potential flow.
![Page 7: Boundary Layer Flow](https://reader033.vdocument.in/reader033/viewer/2022061612/56816261550346895dd2bddb/html5/thumbnails/7.jpg)
Viscous boundary layer
• The region near the wall where the viscous forces are of the same order as the inertial forces is termed the boundary layer.
![Page 8: Boundary Layer Flow](https://reader033.vdocument.in/reader033/viewer/2022061612/56816261550346895dd2bddb/html5/thumbnails/8.jpg)
Viscous boundary layer
• The distance over which the viscous forces have an effect is termed the boundary layer thickness.
• The thickness is a function of the ratio between the inertial forces and the viscous forces- i.e., the Reynolds number. As NRe increases, the thickness decreases.
![Page 9: Boundary Layer Flow](https://reader033.vdocument.in/reader033/viewer/2022061612/56816261550346895dd2bddb/html5/thumbnails/9.jpg)
Effect of viscosity• The layers closer to the wall start moving right
away due to the no-slip boundary condition. The layers farther away from the wall start moving later.
• The distance from the wall that is affected by the motion is also called the viscous diffusion length. This distance increases as time goes on.
![Page 10: Boundary Layer Flow](https://reader033.vdocument.in/reader033/viewer/2022061612/56816261550346895dd2bddb/html5/thumbnails/10.jpg)
Moving plate boundary layer• Consider an impulsively
started plate in a stagnant fluid.
• When the wall in contact with the still fluid suddenly starts to move, the layers of fluid close to the wall are dragged along while the layers farther away from the wall move with a lower velocity.
• The viscous layer develops as a result of the no-slip boundary condition at the wall.
![Page 11: Boundary Layer Flow](https://reader033.vdocument.in/reader033/viewer/2022061612/56816261550346895dd2bddb/html5/thumbnails/11.jpg)
Flow separation Flow separation occurs when (a) the velocity at the wall is zero or negative and an inflection point exists in the velocity profile, and a positive or adverse pressure gradient occurs in the direction of flow.
![Page 12: Boundary Layer Flow](https://reader033.vdocument.in/reader033/viewer/2022061612/56816261550346895dd2bddb/html5/thumbnails/12.jpg)
Boundary layer theory• Consider a flow over a semi-infinite flat plate (and also
for a finite flat plate), under steady state conditions:
Fluid Velocity, v Away from plate,
inviscid flow assumption is valid.
Near the plate, viscosity effects are significant.
![Page 13: Boundary Layer Flow](https://reader033.vdocument.in/reader033/viewer/2022061612/56816261550346895dd2bddb/html5/thumbnails/13.jpg)
Boundary layer theorySolid Boundary
No SlipVelocity 0
Velocity v
Velocity v
0
INF
0
INF
INVISCID FLOWASSUMPTION OK HERE
FRICTION CANNOT BENEGLECTED HERE
![Page 14: Boundary Layer Flow](https://reader033.vdocument.in/reader033/viewer/2022061612/56816261550346895dd2bddb/html5/thumbnails/14.jpg)
Boundary layer theorySolid Boundary
0
INF
0
INF
x
dd
BL thickness99% Free stream velocity
BL thicknessincreases with x
What happens to d when you move in x?
![Page 15: Boundary Layer Flow](https://reader033.vdocument.in/reader033/viewer/2022061612/56816261550346895dd2bddb/html5/thumbnails/15.jpg)
Laminar boundary layer
![Page 16: Boundary Layer Flow](https://reader033.vdocument.in/reader033/viewer/2022061612/56816261550346895dd2bddb/html5/thumbnails/16.jpg)
Laminar boundary layer
Re,xxvN
Re,
5.0 5.0x
x xvN
d
BL Reynolds number:
Blasius approximation of d:
![Page 17: Boundary Layer Flow](https://reader033.vdocument.in/reader033/viewer/2022061612/56816261550346895dd2bddb/html5/thumbnails/17.jpg)
Turbulent boundary layer
![Page 18: Boundary Layer Flow](https://reader033.vdocument.in/reader033/viewer/2022061612/56816261550346895dd2bddb/html5/thumbnails/18.jpg)
Turbulent boundary layerAt high enough fluid velocity, inertial forces dominateo Viscous forces cannot prevent a wayward particle from motion
o Chaotic flow ensuesTurbulence near the wall For wall-bounded flows, turbulence initiates near the wall
![Page 19: Boundary Layer Flow](https://reader033.vdocument.in/reader033/viewer/2022061612/56816261550346895dd2bddb/html5/thumbnails/19.jpg)
Turbulent boundary layer• In turbulent flow, the velocity component normal
to the surface is much smaller than the velocity parallel to the surface
• The gradients of the flow across the layer are much greater than the gradients in the flow direction.
![Page 20: Boundary Layer Flow](https://reader033.vdocument.in/reader033/viewer/2022061612/56816261550346895dd2bddb/html5/thumbnails/20.jpg)
Turbulent boundary layer
![Page 21: Boundary Layer Flow](https://reader033.vdocument.in/reader033/viewer/2022061612/56816261550346895dd2bddb/html5/thumbnails/21.jpg)
Turbulent boundary layer
Eddies and Vorticity
• An eddy is a particle of vorticity that typically forms within regions of velocity gradient
• An eddy begins as a disturbance near the wall, followed by the formation of a vortex filament that later stretches into a horseshoe or hairpin vortex
![Page 22: Boundary Layer Flow](https://reader033.vdocument.in/reader033/viewer/2022061612/56816261550346895dd2bddb/html5/thumbnails/22.jpg)
Turbulent boundary layer Turbulence is comprised of irregular, chaotic, three-
dimensional fluid motion, but containing coherent structures.
Turbulence occurs at high Reynolds numbers, where instabilities give way to chaotic motion.
Turbulence is comprised of many scales of eddies, which dissipate energy and momentum through a series of scale ranges. The largest eddies contain the bulk of the kinetic energy, and break up by inertial forces. The smallest eddies contain the bulk of the vorticity, and dissipate by viscosity into heat.
Turbulent flows are not only dissipative, but also dispersive through the advection mechanism.
![Page 23: Boundary Layer Flow](https://reader033.vdocument.in/reader033/viewer/2022061612/56816261550346895dd2bddb/html5/thumbnails/23.jpg)
Dimensional Analysis
![Page 24: Boundary Layer Flow](https://reader033.vdocument.in/reader033/viewer/2022061612/56816261550346895dd2bddb/html5/thumbnails/24.jpg)
Buckingham Pi Theorem• Tells how many dimensionless groups (p)
may define a system.• Theorem:
If n variables are involved in a problem and these are expressed using k base dimensions, then (n – k) dimensionless groups are required to characterize the system/problem.
![Page 25: Boundary Layer Flow](https://reader033.vdocument.in/reader033/viewer/2022061612/56816261550346895dd2bddb/html5/thumbnails/25.jpg)
Buckingham Pi Theorem Example: In describing the motion of a pendulum, the variables are time [T], length [L], gravity [L/T2], mass [M]. Therefore, n = 4 k = 3. So, only one (4 – 3) dimensionless group is required to describe the system.
But how do we derive this?
1 2 3 1( , , , ) n kgf tL
p p p p p
![Page 26: Boundary Layer Flow](https://reader033.vdocument.in/reader033/viewer/2022061612/56816261550346895dd2bddb/html5/thumbnails/26.jpg)
Buckingham Pi TheoremHow to find the dimensional groups:
For the pendulum example: let a, b, c and d be the coefficients of t, L, g and m in the group, respectively.
In terms of dimensions:
1a b c dt L g mp
( 2 ) ( )2
ca b d a c b c dLT L M T L M
T
![Page 27: Boundary Layer Flow](https://reader033.vdocument.in/reader033/viewer/2022061612/56816261550346895dd2bddb/html5/thumbnails/27.jpg)
Buckingham Pi TheoremHow to find the dimensional groups:
For the pendulum example: let a, b, c and d be the coefficients of t, L, g and m in the group, respectively.
Since the group is dimensionless:
Therefore:
( 2 ) ( ) 0 0 0a c b c dT L M T L M
2 00
0
a cb cd
![Page 28: Boundary Layer Flow](https://reader033.vdocument.in/reader033/viewer/2022061612/56816261550346895dd2bddb/html5/thumbnails/28.jpg)
Buckingham Pi TheoremHow to find the dimensional groups:
For the pendulum example: let a, b, c and d be the coefficients of t, L, g and m in the group, respectively.
Arbitrarily choosing a = 1:
Therefore:
1 1; ; 02 2
c b d
1 11 02 21 1 gt L g m t
Lp p
![Page 29: Boundary Layer Flow](https://reader033.vdocument.in/reader033/viewer/2022061612/56816261550346895dd2bddb/html5/thumbnails/29.jpg)
Buckingham Pi TheoremExample:
Drag on a sphere
Drag depends on FOUR parameters: sphere size (D); fluid speed (v); fluid density (); fluid viscosity (m)
Difficult to know how to set up experiments to determine dependencies and how to present results (four graphs?)
, , ,F f D v
![Page 30: Boundary Layer Flow](https://reader033.vdocument.in/reader033/viewer/2022061612/56816261550346895dd2bddb/html5/thumbnails/30.jpg)
Buckingham Pi TheoremStep 1: List all the parameters involved
Let n be the number of parametersExample: For drag on a sphere: F, v, D, ,
and (n = 5)
Step 2: Select a set of primary dimensionsLet k be the number of primary dimensions
For this example: M (kg), L (m), t (sec); thus k = 3
2 3
F v D
ML L M MLt t L Lt
![Page 31: Boundary Layer Flow](https://reader033.vdocument.in/reader033/viewer/2022061612/56816261550346895dd2bddb/html5/thumbnails/31.jpg)
Buckingham Pi TheoremStep 3: Determine the number of dimensionless
groups required to define the system
Step 4: Select a set of k dimensional parameters that includes all the primary dimensions
For example: select , v, D
1 2
5 3 2( )
n kfp p
31 3
a ba b c c a b c a bM Lv D L M L t
L tp
3 0 0 0 0; 0; 0 a b c a bM L t M L t a b c
![Page 32: Boundary Layer Flow](https://reader033.vdocument.in/reader033/viewer/2022061612/56816261550346895dd2bddb/html5/thumbnails/32.jpg)
Buckingham Pi TheoremStep 4: Select a set of k dimensional
parameters that includes all the primary dimensions
For example: select , v, D AND F
3 2 0 0 0a d b c d a b dM L t M L t
1 3 2 a b d
a b c d cM L MLv D F LL t t
p
03 0
2 0
a db c d a
b d
![Page 33: Boundary Layer Flow](https://reader033.vdocument.in/reader033/viewer/2022061612/56816261550346895dd2bddb/html5/thumbnails/33.jpg)
Buckingham Pi TheoremStep 4: Select a set of k dimensional
parameters that includes all the primary dimensions
For example: select , v, D AND F
Let d = 1:
Therefore:1 2 2 1
1 2 2
Fv D Fv D
p
1 0 12 0 2
2 1 3 0 2
a ab b
c c
![Page 34: Boundary Layer Flow](https://reader033.vdocument.in/reader033/viewer/2022061612/56816261550346895dd2bddb/html5/thumbnails/34.jpg)
Buckingham Pi TheoremStep 4: Select a set of k dimensional
parameters that includes all the primary dimensions
Next group: select , v, D and
3 0 0 03
a b dc a d b c a d b dM L ML M L t M L t
L t Lt
2a b c dv Dp
03 0
0
a db c a d
b d
![Page 35: Boundary Layer Flow](https://reader033.vdocument.in/reader033/viewer/2022061612/56816261550346895dd2bddb/html5/thumbnails/35.jpg)
Buckingham Pi TheoremStep 4: Select a set of k dimensional
parameters that includes all the primary dimensions
Next group: select , v, D and Let a = 1:
Therefore:
1 0 11 0 1
1 3 1 0 1
d db b
c c
1 1 1 12 Re
vDv D Np
Re2 2and F f Nv D
![Page 36: Boundary Layer Flow](https://reader033.vdocument.in/reader033/viewer/2022061612/56816261550346895dd2bddb/html5/thumbnails/36.jpg)
Buckingham Pi Theorem