PowerPointPrsentation

1
Boundary Layer Flows
by
Faisal Ahmed
M.Sc. (Student) Computational Engineering
Institute of Fluid Mechanics FAU
Erlangen-Nürnberg Cauerstrasse 4, D-91058 Erlangen
Topic of Presentation
2
Contents of the Presentation

• The presentation deals with
• Continuity and Navier - Stokes equation
• Derivation of the Boundary Layer Equation and
  explanation of the characteristics of the
  Boundary Layer Flow.
• Laminar Boundary Layer Flow
• -- Similarity solution of Blasius for flat
  plate boundary layer
• -- Boundary Layer solution for a free jet flow
• Considerations for Boundary Layer Flow with heat
  transfer
• Introduction for Turbulent Boundary Layer Flow
• Details of plain channel flow
• Concluding remarks

3
Continuity Equation
Navier-Stokes Equation
Where
Continuity and Navier-Stokes
Equation
4
For incompressible flow we have
Employing this in the expression for
we have
and
Also neglecting the effect of gravity we get the
Continuity and Navier-Stokes Equation as
following
Continuity Equation
Navier-Stokes Equation
Continuity and Navier-Stokes Equation for
Incompressible Flow
5
The concept of Boundary Layer Flow

• Flows at high Reynolds number can be divided up
  into two unequally large regions.
• Bulk of the flow region. Here viscosity can be
  neglected, the flow corresponds to inviscid
  limiting solution.
• A very thin region near the wall , where the
  viscosity must be taken into account. This region
  is called the Boundary Layer.

Experiment by L. Prandtl
Snapshot of the flow along a thin flat plate,
dragged through water.

• Observations
• Near the wall there is a thin layer where the
  velocity is considerably lower.
• Thickness of this layer is increases along the
  plate from the front to back.

Introduction to
Boundary Layer Flow
6
Velocity Profile

• Constant velocity profile perpendicular to the
  plate at the leading edge.
• The thickness of the boundary layer gets larger
  as the distance from the leading edge increases.
  Which indicates that more and more particles are
  caught up by the retardation. Therefore the
  boundary layer thickness is a monotonically
  increasing function of x.

The transition from boundary-layer flow to outer
flow takes place continuously, so that precise
boundary in principle cannot be given. Very
often the boundary is given at the point where
the velocity reaches a 99 of the outer velocity.
Fig Boundary layer on a flat plate
Characteristics of
Boundary Layer Flow
7
We consider the sketch on the right side
For a high Reynolds number
Convection time
Diffusion time
For the boundary point
Fig Boundary on a flat plate
So we get,
Derivation of Boundary Layer
Equation
8
For the figure shown, we have for a 2D case the
Continuity and Navier-Stokes Equation as follows
Continuity Equation
Navier-Stokes Equation
x momentum
y momentum
We apply the following non-dimensional parameters
to the continuity and Navier Stokes equation
Derivation of Boundary
Layer Equation (Contd.)
9
Applying the dimensionless parameters to the
Continuity equation we get
Dividing by we get
So, we get the Continuity equation for the
Boundary Layer Flow as following
Derivation of Boundary
Layer Equation (Contd.)
10
Applying the dimensionless parameters to the
x-momentum equation we get
x momentum
Dividing by we get
So we get the x-momentum equation for the
Boundary Layer Flow as below
Derivation of Boundary
Layer Equation (Contd.)
11
Applying the dimensionless parameters to the
y-momentum equation we get
y momentum
Dividing by we get
Now, as , we get the y-momentum
equation as follows
Derivation of Boundary Layer
Equation (Contd.)
12
So we can write the equations for the Boundary
Layer Flow as below
Continuity equation
x-Momentum equation
y-Momentum equation
Equations for Boundary Flow
13
We consider the flow over a flat plat as shown in
the figure

• As the velocity of the potential flow is constant
  , we have
• Steady Flow so,

So the Continuity and x-Momentum equation becomes
as below
Continuity equation
x-Momentum equation
With the boundary conditions
Boundary layer for a flow on flat plate
14
The similarity law of the velocity profile can be
written as
with
Where
So , we get
Fig Boundary layer on a flat plate
If is the stream function from definition
we know that ,
Upon integration we get,
Boundary layer for a flow on flat plate (contd.)
15
So we get
and
Therefore,
,
and
Blasius Equation
With boundary conditions
at
and
Boundary layer for a flow on flat plate (contd.)
16

• Remarks
• Two partial differential equations (Continuity
  and Momentum) are transformed into one Ordinary
  differential equation for the stream function
  using Similarity transformation.
• This 3rd order nonlinear equation can be solved
  by numerical methods, e.g. a Runge-Kutta Method
  known as Shooting Method.

• Velocity distribution
• ? Longitudinal velocity
  is shown in the figure. Curvature close to the
  wall is very little and close to the wall
  vanishes.

Boundary layer for a flow on flat plate (contd.)
17

• The transverse component of the velocity in the
  boundary layer is shown in the figure. At the
  outer edge of the boundary layer, i.e. for
  the transverse component is non-zero. We
  get,

Boundary layer for a flow on flat plate (contd.)
18
Here we have a 2D free laminar jet as shown in
the figure.

• The assumptions are as follows
• The flow is symmetric around X-axis
• Velocity changes only in X-direction
• There is no external pressure source

Boundary layer for a 2D laminar free jet
19
So we have the Boundary Layer equation for this
case
With the following boundary conditions
and
at
And for ,
For the jet the total Momentum is,
Boundary layer for a 2D laminar free jet (contd.)
20
We assume that,
and
So then we get,
,
,
and
Putting them in the equation for the boundary
layer, we get
Substituting the value of in the expression
for momentum we get
Which leads to
and
So,
and
Boundary layer for a 2D laminar free jet
(contd.)
21
So, by applying the above conditions the
differential equation takes the following form
with the boundary conditions at
and

and
In order to eliminate the constant from
the differential equation we use the following
substitutions
and
So the equation takes the form
and
with boundary conditions at
and
Integrating the differential equation one time we
get
Applying the above b.c. s we get the equation as
Boundary layer for a 2D laminar free jet
(contd.)
22
For solving the above equation we use the
following expression
Which gives
So the expression for total momentum becomes
Which leads to
and
And mass flow rate
The velocity profile for the free jet with
varying is shown in the figure
Boundary layer for a 2D laminar free jet
(contd.)
23
We have the energy equation as
Where,
and
, we get the energy equation as below
Assuming,
and
The momentum equation is written as
Here,
viscous diffusion co-efficient and
Thermal viscous co-efficient
So we introduce a new dimensionless no. called
Prandtl no. which is expressed as
Treatment of boundary layer with heat transfer
24
Prandtl no. has been found to be the parameter
which relates the relative thickness of the
hydrodynamic and thermal boundary layers.
Below we show the velocity profile and boundary
layer thickness of hydrodynamic and thermal for
different Pr. no. range.
Treatment of boundary layer with heat transfer
(contd.)
25
Dimensionless form of the Momentum equation
where,
Here we used,
and
Dimensionless form of the Energy equation
Characteristics temperature difference
and heat transport
Treatment of boundary layer with heat transfer
(contd.)
26
We imagine the case of the flow over a flat
plate. The figure is indicative of the flow. The
boundary layer on the plate remains laminar close
to the leading edge.
When we move further downstream the flow becomes
turbulent. This transition of the boundary layer
from laminar to turbulent takes place in a region
of finite length. This fact is seen from the
above figure. For simplicity, we assume that the
transition takes place at a point, which called
the critical point. The Reynolds no. for this
critical point is defined from experiment as,
. But this value of critical
Reynolds no. depends strongly on how free the
flow is from perturbation. The treatment of
turbulent boundary needs to done by modeling .
Turbulent boundary layer
27
We do the following substitutions, which was
suggested by Reynolds, in Navier-Stokes equation,
and
So for the turbulent flow we get the Continuity
and momentum equation as follows
Continuity equation
RANS equation
Extra unknown term Reynolds stress
tensor with 6 unknowns.

• For Reynolds Stress Closure Modeling the
  following models are quite often used
• Zero-Equation models (Constant eddy viscosity
  models, Mixing length models)
• One equation models
• Two equation models ( models)

Turbulent boundary layer (contd.)
28
We consider a 2D, fully developed, plain
turbulent channel flow as shown in the figure
The momentum equations for the given case in all
the 2 direction is given as
Treatment of Channel Flow
29
Due to the wall boundary condition we have at the
wall,
Integration of the momentum equation for
gives
Here as, and
, also the change of
pressure is negligible in direction for
most of the practical application , so we can
write

Substituting this in the equation of momentum for
direction we get
Velocity scale
and Length scale
Treatment of Channel Flow (contd.)
30
We use the following non-dimensional parameters,
We get,
A plot for different parameters for 2D channel
flow is shown here. It is noticeable that for
most of the wide range of the plot
is the shortest term.
In FAU from experiment it was found that, the
relation can be expressed as
Figure Terms of the momentum equation for the 2D
channel flow
Treatment of Channel Flow (contd.)
31
For a high Reynolds no. flow it can be expressed
as
And later we get,
where, and
Here, e is know as the wall roughness
The results of the hotwire and LD anemometer was
plotted in a double logarithmic curve and the
slope was obtained to be -1.
In standard literature a wide range of value for
and B is found. Which due to the fact that,
suitable measuring device was not available.
Fig Plot of the experimental results
Treatment of Channel Flow (contd.)
32
The equation
is plotted for determining
Figure Normalized turbulent stress transport for
plane channel flow
The turbulent intensity is plotted against the
distance from the wall. The profile remains
linear at the beginning. We also notice the
upward shift of the curves as the Reynolds number
increases.
Treatment of Channel Flow (contd.)
33
All the research and experiment value are valid
for
For rough surface we can write
Which can also be written as
For the near wall region of the channel flow, the
flow is laminar, and this layer is called the
viscous sub layer. In this region the velocity
profile is linear .
For this region the non-dimensional velocity
Far from the wall or in turbulent zone
The log and the liner profile meet in the buffer
zone.
Treatment of Channel Flow (contd.)
34

• In our presentation we dealt with the analytical
  treatment of Boundary Layer Flows.
• We dealt with
• Continuity and Navier Stokes equations
• Derivation of Boundary Layer equation and two
  special cases of Laminar Boundary Layer flow
• Boundary Layer flow with heat transfer
• Turbulent boundary layer, and details treatment
  of plain channel flow.
• Nevertheless Boundary Layer concept has a wide
  application range in the filed of engineering
  starting from simple channel flow to Aerodynamics
  and Weather modeling. Still a huge amount of
  research is going on regarding the application of
  Boundary Layer in these fields.

Concluding Remarks