168
7. Solution of the Navier-Stokes Equations
For the case of constant density and viscosity, this equation simplifies further;
the viscous and unsteady terms disappear by virtue of the continuity equation
leaving:
The pressure equation can be solved by one of the numerical methods for
elliptic equations described in Chaps. 3 and 4. It is important to note that
the right hand side of the pressure equation is a sum of derivatives of terms in
the momentum equations; these must be approximated in a manner consistent
with their treatment in the equations they are derived from.
It is also important to note that the Laplacian operator in the pressure
equation is the product of the divergence operator originating from the continuity equation and the gradient operator that comes from the momentum
equations. In a numerical approximation, it is essential that the consistency of
these operators be maintained i.e. the approximation of the Poisson equations
must be defined as the product of the divergence and gradient approximations used in the basic equations. Violation of this constraint leads to lack
of satisfaction of the continuity equation. To emphasize the importance of
this issue, the two derivatives of the pressure in the above equations were
separated: the outer derivative stems from the continuity equation while the
inner derivative arises from the momentum equations. The outer and inner
derivatives may be discretized using different schemes - they have to be those
used in the momentum and continuity equations.
A pressure equation of this kind is used to calculate the pressure in both
explicit and implicit solution methods. To maintain consistency among the
approximations used, it is best to derive the equation for the pressure from the
discretized momentum and continuity equations rather than by approximating the Poisson equation. The pressure equation can also be used to calculate
the pressure from a velocity field obtained by solving vorticity/streamfunction
equations, see Sect. 7.4.2.
7.3.2 A Simple Explicit Time Advance Scheme
Before considering commonly used methods for solving the steady state
Navier-Stokes equations, let us look at a method for the unsteady equations that illustrates how the numerical Poisson equation for the pressure is
constructed and the role it plays in enforcing continuity. The choice of the
approximations to the spatial derivatives is not important here so the semidiscretized (discrete in space but not time) momentum equations are written
symbolically as:
7. Solution of the Navier-Stokes Equations
For the case of constant density and viscosity, this equation simplifies further;
the viscous and unsteady terms disappear by virtue of the continuity equation
leaving:
The pressure equation can be solved by one of the numerical methods for
elliptic equations described in Chaps. 3 and 4. It is important to note that
the right hand side of the pressure equation is a sum of derivatives of terms in
the momentum equations; these must be approximated in a manner consistent
with their treatment in the equations they are derived from.
It is also important to note that the Laplacian operator in the pressure
equation is the product of the divergence operator originating from the continuity equation and the gradient operator that comes from the momentum
equations. In a numerical approximation, it is essential that the consistency of
these operators be maintained i.e. the approximation of the Poisson equations
must be defined as the product of the divergence and gradient approximations used in the basic equations. Violation of this constraint leads to lack
of satisfaction of the continuity equation. To emphasize the importance of
this issue, the two derivatives of the pressure in the above equations were
separated: the outer derivative stems from the continuity equation while the
inner derivative arises from the momentum equations. The outer and inner
derivatives may be discretized using different schemes - they have to be those
used in the momentum and continuity equations.
A pressure equation of this kind is used to calculate the pressure in both
explicit and implicit solution methods. To maintain consistency among the
approximations used, it is best to derive the equation for the pressure from the
discretized momentum and continuity equations rather than by approximating the Poisson equation. The pressure equation can also be used to calculate
the pressure from a velocity field obtained by solving vorticity/streamfunction
equations, see Sect. 7.4.2.
7.3.2 A Simple Explicit Time Advance Scheme
Before considering commonly used methods for solving the steady state
Navier-Stokes equations, let us look at a method for the unsteady equations that illustrates how the numerical Poisson equation for the pressure is
constructed and the role it plays in enforcing continuity. The choice of the
approximations to the spatial derivatives is not important here so the semidiscretized (discrete in space but not time) momentum equations are written
symbolically as: