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)Read More

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