5.9 Systems of equations

Most CFD calculations involve solving a system of equations that represent the physics of the problem. For example, laminar flow by natural convection can be represented by the equations introduced in Sec. 2.20 , reproduced below.

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The system provides 3 equations (1 vector, 2 scalar) which can be solved for 3 unknowns, eqn, eqn and eqn.

As discussed in Sec. 3.4 , the finite volume method solves an individual matrix equation for each variable, e.g. eqn for eqn. The vector equation for eqn is decoupled into 3 matrix equations for individual components, i.e. eqn, eqn and eqn.

Each individual matrix equation for one solution variable, e.g. eqn, incorporates terms from other variables, e.g. eqn, into the source vector eqn. The contribution to eqn is calculated using current values of the respective variables. Systems of equations are thereby solved by successive substitution of solved variables into the source vectors of subsequent equations.

An iterative solution for a single equation, like the one in Sec. 5.7 , can be extended to a system of equations. Time eqn is incremented by eqn and equations are solved in sequence, before returning to start the next time step with the increment of eqn.

The substitutions in the momentum and pressure equations are particularly important, culminating in corrections to eqn and the advective flux eqn, discussed in Sec. 5.10 .

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At the start of any time step the current eqn becomes eqn for the discretisation of the eqn term in the momentum equation. The advection term eqn is discretised by Eq. (3.8 ), treating one eqn as flux eqn and the other as the advected quantity.

The equation is solved for eqn. The new solution for eqn is substituted into the eqn equation which is solved for eqn. The new solution for eqn is then used to correct eqn in order to help enforce the mass conservation constraint eqn (eqn).

Before the current solution step is completed, eqn is also corrected to reduce the error in the discretisation of eqn when it then becomes eqn in the following solution step. The correction also provides a better “initial guess” eqn for the matrix solution of the next momentum equation, which reduces the solution time.

Notes on CFD: General Principles - 5.9 Systems of equations