Pressure-velocity coupling is achieved by using Equation 23–46 to derive an additional condition for pressure by reformatting the continuity equation (Equation 23–45). The pressure-based solver allows you to solve your flow problem in either a segregated or coupled manner. Ansys Fluent provides the option to choose among five pressure-velocity coupling algorithms: SIMPLE, SIMPLEC, PISO, Coupled, and (for unsteady flows using the non-iterative time advancement scheme (NITA)) Fractional Step (FSM). All the aforementioned schemes, except the "coupled" scheme, are based on the predictor-corrector approach. For instructions on how to select these algorithms, see Choosing the Pressure-Velocity Coupling Method in the User’s Guide.
Note that SIMPLE, SIMPLEC, PISO, and Fractional Step use the pressure-based segregated algorithm, while Coupled uses the pressure-based coupled solver.
Important: The pressure-velocity coupling schemes that are applicable when using the Eulerian multiphase model are Phase Coupled SIMPLE, Multiphase Coupled, and Full Multiphase Coupled. These are discussed in detail in Selecting the Pressure-Velocity Coupling Method in the User's Guide.
The SIMPLE algorithm uses a relationship between velocity and pressure corrections to enforce mass conservation and to obtain the pressure field.
If the momentum equation is solved with a guessed pressure field 
, the resulting face flux, 
, computed from Equation 23–46
 | (23–50) |
does not satisfy the continuity equation. Consequently, a correction 
is added to the face flux 
so that the corrected face
flux, 
 | (23–51) |
satisfies the continuity equation. The SIMPLE algorithm postulates
that 
be written as
 | (23–52) |
where 
is the cell pressure
correction.
The SIMPLE algorithm substitutes the flux correction equations
(Equation 23–51 and Equation 23–52)
into the discrete continuity equation (Equation 23–45) to obtain a discrete equation for the pressure correction 
in
the cell:
 | (23–53) |
where the source term 
is the net flow rate into the cell:
 | (23–54) |
The pressure-correction equation (Equation 23–53) may be solved using the algebraic multigrid (AMG) method described in Algebraic Multigrid (AMG). Once a solution is obtained, the cell pressure and the face flux are corrected using
 | (23–55) |
 | (23–56) |
Here 
is the under-relaxation
factor for pressure (see Under-Relaxation of Variables for information about under-relaxation). The corrected face flux, 
, satisfies
the discrete continuity equation identically during each iteration.
A number of variants of the basic SIMPLE algorithm are available in the literature. In addition to SIMPLE, Ansys Fluent offers the SIMPLEC (SIMPLE-Consistent) algorithm [667]. SIMPLE is the default for transient simulations, but many problems will benefit from the use of SIMPLEC, as described in SIMPLE vs. SIMPLEC in the User’s Guide.
The SIMPLEC procedure is similar to the SIMPLE procedure outlined
above. The only difference lies in the expression used for the face
flux correction, 
. As in SIMPLE, the
correction equation may be written as
 | (23–57) |
However, the coefficient 
is redefined as a function
of 
. The use of this modified correction
equation has been shown to accelerate convergence in problems where
pressure-velocity coupling is the main deterrent to obtaining a solution.
For meshes with some degree of skewness, the approximate relationship between the correction of mass flux at the cell face and the difference of the pressure corrections at the adjacent cells is very rough. Since the components of the pressure-correction gradient along the cell faces are not known in advance, an iterative process similar to the PISO neighbor correction described below is desirable. After the initial solution of the pressure-correction equation, the pressure-correction gradient is recalculated and used to update the mass flux corrections. This process, which is referred to as "skewness correction", significantly reduces convergence difficulties associated with highly distorted meshes. The SIMPLEC skewness correction allows Ansys Fluent to obtain a solution on a highly skewed mesh in approximately the same number of iterations as required for a more orthogonal mesh. Note that the skewness correction can use a default or an enhanced formulation.
The Pressure-Implicit with Splitting of Operators (PISO) pressure-velocity coupling scheme, part of the SIMPLE family of algorithms, is based on the higher degree of the approximate relation between the corrections for pressure and velocity. One of the limitations of the SIMPLE and SIMPLEC algorithms is that new velocities and corresponding fluxes do not satisfy the momentum balance after the pressure-correction equation is solved. As a result, the calculation must be repeated until the balance is satisfied. To improve the efficiency of this calculation, the PISO algorithm performs two additional corrections: neighbor correction and skewness correction.
The main idea of the PISO algorithm is to move the repeated calculations required by SIMPLE and SIMPLEC inside the solution stage of the pressure-correction equation [274]. After one or more additional PISO loops, the corrected velocities satisfy the continuity and momentum equations more closely. This iterative process is called a momentum correction or "neighbor correction". The PISO algorithm takes a little more CPU time per solver iteration, but it can dramatically decrease the number of iterations required for convergence, especially for transient problems.
For meshes with some degree of skewness, the approximate relationship between the correction of mass flux at the cell face and the difference of the pressure corrections at the adjacent cells is very rough. Since the components of the pressure-correction gradient along the cell faces are not known in advance, an iterative process similar to the PISO neighbor correction described above is desirable [177]. After the initial solution of the pressure-correction equation, the pressure-correction gradient is recalculated and used to update the mass flux corrections. This process, which is referred to as "skewness correction", significantly reduces convergence difficulties associated with highly distorted meshes. The PISO skewness correction allows Ansys Fluent to obtain a solution on a highly skewed mesh in approximately the same number of iterations as required for a more orthogonal mesh. Note that the skewness correction can use a default formulation or an enhanced formulation.
For meshes with a high degree of skewness, the simultaneous coupling of the neighbor and skewness corrections at the same pressure correction equation source may cause divergence or a lack of robustness. An alternate, although more expensive, method for handling the neighbor and skewness corrections inside the PISO algorithm is to apply one or more iterations of skewness correction for each separate iteration of neighbor correction. For each individual iteration of the classical PISO algorithm from [274], this technique allows a more accurate adjustment of the face mass flux correction according to the normal pressure correction gradient.
In the FSM, the momentum equations are decoupled from the continuity equation using a mathematical technique called operator-splitting or approximate factorization. The resulting solution algorithm is similar to the segregated solution algorithms described earlier. The formalism used in the approximate factorization allows you to control the order of splitting error. Because of this, the FSM is adopted in Ansys Fluent as a velocity-coupling scheme in a non-iterative time-advancement (NITA) algorithm (Non-Iterative Time-Advancement Scheme).
As previously mentioned, the pressure-based solver allows you to solve your flow problem in either a segregated or coupled manner. Using the coupled approach offers some advantages over the non-coupled or segregated approach. The coupled scheme obtains a robust and efficient single phase implementation for steady-state flows, with superior performance compared to the segregated solution schemes. This pressure-based coupled algorithm offers an alternative to the density-based and pressure-based segregated algorithm with SIMPLE-type pressure-velocity coupling. For transient flows, using the coupled algorithm is necessary when the quality of the mesh is poor, or if large time steps are used.
The pressure-based segregated algorithm solves the momentum equation and pressure correction equations separately. This semi-implicit solution method results in slow convergence.
The coupled algorithm solves the momentum and pressure-based continuity equations together. The full implicit coupling is achieved through an implicit discretization of pressure gradient terms in the momentum equations, and an implicit discretization of the face mass flux, including the Rhie-Chow pressure dissipation terms.
In the momentum equations (Equation 23–41),
the pressure gradient for component 
is of the form
 | (23–58) |
Where 
is the coefficient
derived from the Gauss divergence theorem and coefficients of the
pressure interpolation schemes (Equation 23–42). Finally, for any 
th cell, the discretized form of the momentum equation
for component 
is defined as
 | (23–59) |
In the continuity equation, Equation 23–45, the balance of fluxes is replaced using the flux expression in Equation 23–46, resulting in the discretized form
 | (23–60) |
As a result, the overall system of equations (Equation 23–59 and Equation 23–60), after being transformed to the
-form, is presented as
 | (23–61) |
where the influence of a cell 
on a cell 
has the form
 | (23–62) |
and the unknown and residual vectors have the form
 | (23–63) |
 | (23–64) |
Note that Equation 23–61 is solved using the coupled AMG, which is detailed in The Coupled and Scalar AMG Solvers.