Boundary Condition, Upper Boundary, Inlet Boundary & Outflow Boundary Notes | EduRev

Created by: Mahima Chaudhary

: Boundary Condition, Upper Boundary, Inlet Boundary & Outflow Boundary Notes | EduRev

 Page 1


Objectives_template
file:///D|/chitra/nptel_phase2/mechanical/cfd/lecture%2025/25_1.htm[6/20/2012 4:52:47 PM]
 Module 4: Vorticity Stream Function Approach for Solving Flow Problems
 Lecture 25:
 
The Lecture deals with:
Boundary Condition
Upper Boundary
Inlet Boundary
Outflow Boundary
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Page 2


Objectives_template
file:///D|/chitra/nptel_phase2/mechanical/cfd/lecture%2025/25_1.htm[6/20/2012 4:52:47 PM]
 Module 4: Vorticity Stream Function Approach for Solving Flow Problems
 Lecture 25:
 
The Lecture deals with:
Boundary Condition
Upper Boundary
Inlet Boundary
Outflow Boundary
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Objectives_template
file:///D|/chitra/nptel_phase2/mechanical/cfd/lecture%2025/25_2.htm[6/20/2012 4:52:47 PM]
 Module 4: Vorticity Stream Function Approach for Solving Flow Problems
 Lecture 25:
  
Boundary Condition
Now let us discuss about the boundary conditions.
Figure 25.1
Consider Fig. 25.1, we shall call B1 and B3 as bottom wall. Similar kind of boundary
conditions are aplicable on B1 and B3. At the nodal points which are coinciding with the
solid wall we can directly put  and  Since the line B1-B2-B3 is a
streamline, any constant value of  on it is acceptable. The usual choice is  The
wall vorticity is an extremely important evaluation. At no-slip boundaries,  is
produced. It is the diffusion and subsequent advection of the wall produced vorticity which
governs the physics. Using boundary B1 as an example, we expend  by a Taylor
series as
(25.1)
But  by no-slip condition and 
Again, 
Along the wall,  [because  constant  ].
Thus, 
 
 
 
 
 
 
 
 
 
 
 
 
 
Page 3


Objectives_template
file:///D|/chitra/nptel_phase2/mechanical/cfd/lecture%2025/25_1.htm[6/20/2012 4:52:47 PM]
 Module 4: Vorticity Stream Function Approach for Solving Flow Problems
 Lecture 25:
 
The Lecture deals with:
Boundary Condition
Upper Boundary
Inlet Boundary
Outflow Boundary
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Objectives_template
file:///D|/chitra/nptel_phase2/mechanical/cfd/lecture%2025/25_2.htm[6/20/2012 4:52:47 PM]
 Module 4: Vorticity Stream Function Approach for Solving Flow Problems
 Lecture 25:
  
Boundary Condition
Now let us discuss about the boundary conditions.
Figure 25.1
Consider Fig. 25.1, we shall call B1 and B3 as bottom wall. Similar kind of boundary
conditions are aplicable on B1 and B3. At the nodal points which are coinciding with the
solid wall we can directly put  and  Since the line B1-B2-B3 is a
streamline, any constant value of  on it is acceptable. The usual choice is  The
wall vorticity is an extremely important evaluation. At no-slip boundaries,  is
produced. It is the diffusion and subsequent advection of the wall produced vorticity which
governs the physics. Using boundary B1 as an example, we expend  by a Taylor
series as
(25.1)
But  by no-slip condition and 
Again, 
Along the wall,  [because  constant  ].
Thus, 
 
 
 
 
 
 
 
 
 
 
 
 
 
Objectives_template
file:///D|/chitra/nptel_phase2/mechanical/cfd/lecture%2025/25_2.htm[6/20/2012 4:52:47 PM]
Substituting this into (25.1) and solving for  with  gives
More general from regardless of the wall orientation or value of  at the boundary, it can
be written as
(25.2)
where  is the distance from  to ( ) in the normal direction [  denotes at the
wall].
 
 
Page 4


Objectives_template
file:///D|/chitra/nptel_phase2/mechanical/cfd/lecture%2025/25_1.htm[6/20/2012 4:52:47 PM]
 Module 4: Vorticity Stream Function Approach for Solving Flow Problems
 Lecture 25:
 
The Lecture deals with:
Boundary Condition
Upper Boundary
Inlet Boundary
Outflow Boundary
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Objectives_template
file:///D|/chitra/nptel_phase2/mechanical/cfd/lecture%2025/25_2.htm[6/20/2012 4:52:47 PM]
 Module 4: Vorticity Stream Function Approach for Solving Flow Problems
 Lecture 25:
  
Boundary Condition
Now let us discuss about the boundary conditions.
Figure 25.1
Consider Fig. 25.1, we shall call B1 and B3 as bottom wall. Similar kind of boundary
conditions are aplicable on B1 and B3. At the nodal points which are coinciding with the
solid wall we can directly put  and  Since the line B1-B2-B3 is a
streamline, any constant value of  on it is acceptable. The usual choice is  The
wall vorticity is an extremely important evaluation. At no-slip boundaries,  is
produced. It is the diffusion and subsequent advection of the wall produced vorticity which
governs the physics. Using boundary B1 as an example, we expend  by a Taylor
series as
(25.1)
But  by no-slip condition and 
Again, 
Along the wall,  [because  constant  ].
Thus, 
 
 
 
 
 
 
 
 
 
 
 
 
 
Objectives_template
file:///D|/chitra/nptel_phase2/mechanical/cfd/lecture%2025/25_2.htm[6/20/2012 4:52:47 PM]
Substituting this into (25.1) and solving for  with  gives
More general from regardless of the wall orientation or value of  at the boundary, it can
be written as
(25.2)
where  is the distance from  to ( ) in the normal direction [  denotes at the
wall].
 
 
Objectives_template
file:///D|/chitra/nptel_phase2/mechanical/cfd/lecture%2025/25_3.htm[6/20/2012 4:52:47 PM]
 Module 4: Vorticity Stream Function Approach for Solving Flow Problems
 Lecture 25:
 
Upper Boundary
The upper boundary B5 in Fig 25.1 is having the usual no-slip and impervious conditions
for velocity components. i.e.,  For vorticity  (25.2) will apply.
But how to evaluate  at the upper wall?
The value of  at the upper wall is constant and may be evaluated by integrating the 
velocity profile at the inlet. Integration may be performed through Simpson's rule to get
 (25.3)
If we want to model the condition of no boundary at B5, or, in other words, in y-direction,
fluid at infinite extent is assumed, the problem is little more difficult. However, Thoman
and Szewczyk (1966) used a treatment which specifies this far-field condition of 
with  and  Thus  was applied through a Neumann
condition at the boundary along B5 as
(25.4)
where B5 is considered at 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Page 5


Objectives_template
file:///D|/chitra/nptel_phase2/mechanical/cfd/lecture%2025/25_1.htm[6/20/2012 4:52:47 PM]
 Module 4: Vorticity Stream Function Approach for Solving Flow Problems
 Lecture 25:
 
The Lecture deals with:
Boundary Condition
Upper Boundary
Inlet Boundary
Outflow Boundary
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Objectives_template
file:///D|/chitra/nptel_phase2/mechanical/cfd/lecture%2025/25_2.htm[6/20/2012 4:52:47 PM]
 Module 4: Vorticity Stream Function Approach for Solving Flow Problems
 Lecture 25:
  
Boundary Condition
Now let us discuss about the boundary conditions.
Figure 25.1
Consider Fig. 25.1, we shall call B1 and B3 as bottom wall. Similar kind of boundary
conditions are aplicable on B1 and B3. At the nodal points which are coinciding with the
solid wall we can directly put  and  Since the line B1-B2-B3 is a
streamline, any constant value of  on it is acceptable. The usual choice is  The
wall vorticity is an extremely important evaluation. At no-slip boundaries,  is
produced. It is the diffusion and subsequent advection of the wall produced vorticity which
governs the physics. Using boundary B1 as an example, we expend  by a Taylor
series as
(25.1)
But  by no-slip condition and 
Again, 
Along the wall,  [because  constant  ].
Thus, 
 
 
 
 
 
 
 
 
 
 
 
 
 
Objectives_template
file:///D|/chitra/nptel_phase2/mechanical/cfd/lecture%2025/25_2.htm[6/20/2012 4:52:47 PM]
Substituting this into (25.1) and solving for  with  gives
More general from regardless of the wall orientation or value of  at the boundary, it can
be written as
(25.2)
where  is the distance from  to ( ) in the normal direction [  denotes at the
wall].
 
 
Objectives_template
file:///D|/chitra/nptel_phase2/mechanical/cfd/lecture%2025/25_3.htm[6/20/2012 4:52:47 PM]
 Module 4: Vorticity Stream Function Approach for Solving Flow Problems
 Lecture 25:
 
Upper Boundary
The upper boundary B5 in Fig 25.1 is having the usual no-slip and impervious conditions
for velocity components. i.e.,  For vorticity  (25.2) will apply.
But how to evaluate  at the upper wall?
The value of  at the upper wall is constant and may be evaluated by integrating the 
velocity profile at the inlet. Integration may be performed through Simpson's rule to get
 (25.3)
If we want to model the condition of no boundary at B5, or, in other words, in y-direction,
fluid at infinite extent is assumed, the problem is little more difficult. However, Thoman
and Szewczyk (1966) used a treatment which specifies this far-field condition of 
with  and  Thus  was applied through a Neumann
condition at the boundary along B5 as
(25.4)
where B5 is considered at 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Objectives_template
file:///D|/chitra/nptel_phase2/mechanical/cfd/lecture%2025/25_4.htm[6/20/2012 4:52:47 PM]
 Module 4: Vorticity Stream Function Approach for Solving Flow Problems
 Lecture 25:
 
Inlet Boundary
Inlet boundary in Fig. 25.1 cannot have any unique prescription. It will depend on the
physical situation. For the axial velocity  uniform or parabolic or any possible profile
can be taken. Most widely used conditions are:
 for  to JMAX (25.5)
or,
(25.6)
For normal velocity  Formm and Harlow (1963) set
 for  to JMAX (25.7)
The stream function  can be obtained from the axial velocity profile at the inlet as
(25.8)
Vorticity  also depends on inlet velocity profile. Pao and Daugherty (1969) used
uniform axial velocity profile,  and then specified  Greenspan (1969) fixed up
 from axial velocity profile and assumed  which result in
(25.9)
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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