32
2 Model of Quasi-Steady Weld Pool Dynamics and Numerical Simulation
a n v n =
a nb v nb + b + ( p P − p N )A n
(2.33)
where: (p P –p N ) A n —corresponding pressure resultant force.
The shadows in the figure indicate the control volume, the short arrows indicate
the positions of u, v and w, and the main grid points are indicated by small black dots.
Relative to the main grid point, the position of u is only misplaced in the x direction.
In other words, whether the position of u is exactly at the midpoint between two
grids is related to the method of defining the control volume. The position of u must
fall on the surface of the control volume, independent of whether the surface of the
control volume is at the midpoint between the two grids.
Figure 2.5c shows the control volume of the z-direction momentum equation. It
has a half step displacement in the z direction. The discretization equation for velocity
component w is as follows:
a n w n =
a nb w nb + b + ( p P − p T )A t
(2.34)
Figure 2.5d is the control volume of pressure. Obviously, velocity u exists at the
east–west interface of the control volume of pressure, velocity v exists at the southnorth interface of the control volume of pressure, and velocity w exists at the upper
and lower interfaces of the control volume of pressure.
2.3.3 Pressure Correction Method for Solving Navier-Stokers
Equation
2.3.3.1 Basic Idea of Pressure Correction Method
In essence, the pressure correction technique is an iterative method, in which only
some new physical reasoning methods are used. The purpose is to build a bridge to
solve the next iteration from the results of the last iteration.
This method is suitable for solving incompressible viscous flow problems. It
specifically includes the following contents:
(1) Assume a pressure field, denote it as p*, and start the iteration.
(2) Substitute the already known p* into the momentum equation to solve the
values of the velocity u, v, and w. Since the values of these velocities are
calculated according to p*, they are denoted as u*, v*, and w*.
(3) u*, v*, and w* are obtained according to the predicted pressure, so they do not
necessarily satisfy the continuity equation. Form a pressure correction value
p
by using the continuity equation. (p* + p’) may be more consistent with the
continuity equation. The pressure correction formula is:
2 Model of Quasi-Steady Weld Pool Dynamics and Numerical Simulation
a n v n =
a nb v nb + b + ( p P − p N )A n
(2.33)
where: (p P –p N ) A n —corresponding pressure resultant force.
The shadows in the figure indicate the control volume, the short arrows indicate
the positions of u, v and w, and the main grid points are indicated by small black dots.
Relative to the main grid point, the position of u is only misplaced in the x direction.
In other words, whether the position of u is exactly at the midpoint between two
grids is related to the method of defining the control volume. The position of u must
fall on the surface of the control volume, independent of whether the surface of the
control volume is at the midpoint between the two grids.
Figure 2.5c shows the control volume of the z-direction momentum equation. It
has a half step displacement in the z direction. The discretization equation for velocity
component w is as follows:
a n w n =
a nb w nb + b + ( p P − p T )A t
(2.34)
Figure 2.5d is the control volume of pressure. Obviously, velocity u exists at the
east–west interface of the control volume of pressure, velocity v exists at the southnorth interface of the control volume of pressure, and velocity w exists at the upper
and lower interfaces of the control volume of pressure.
2.3.3 Pressure Correction Method for Solving Navier-Stokers
Equation
2.3.3.1 Basic Idea of Pressure Correction Method
In essence, the pressure correction technique is an iterative method, in which only
some new physical reasoning methods are used. The purpose is to build a bridge to
solve the next iteration from the results of the last iteration.
This method is suitable for solving incompressible viscous flow problems. It
specifically includes the following contents:
(1) Assume a pressure field, denote it as p*, and start the iteration.
(2) Substitute the already known p* into the momentum equation to solve the
values of the velocity u, v, and w. Since the values of these velocities are
calculated according to p*, they are denoted as u*, v*, and w*.
(3) u*, v*, and w* are obtained according to the predicted pressure, so they do not
necessarily satisfy the continuity equation. Form a pressure correction value
p
by using the continuity equation. (p* + p’) may be more consistent with the
continuity equation. The pressure correction formula is: