The SIMPLE Algorithm

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Computational methodologies for forced convection
  1. One-Dimensional Steady-State Convection and Diffusion
    1. Central Difference Scheme
    2. Upwind Scheme
    3. Hybrid Scheme
    4. Exponential and Power Law Schemes
    5. A Generalized Expression of Discretization Schemes
  2. Multidimensional Convection and Diffusion Problems
  3. Numerical Solution of Flow Field
    1. Special Difficulties
    2. Staggered grid
    3. Pressure Correction Equation
    4. The SIMPLE Algorithm
  4. Numerical Simulation of Interfaces and Free Surfaces
  5. Application of Computational Methods

The above approach for the solution of the incompressible flow field was named SIMPLE, which stands for Semi-Implicit Method for Pressure-Linked Equations. It was originally proposed by Patankar and Spalding (1972) and summarized in Patankar (1980). This algorithm is a semi-implicit method because the first terms on the right-hand side of eqs. (4.300) – (4.302) are neglected. If these terms are retained, we will have to solve for the velocity corrections ( ) for the entire flow field simultaneously and the algorithm will become fully-implicit. The procedure for SIMPLE algorithm can be summarized as following:
1. Guess a pressure field, p*.
2. Obtain the starred velocity field, (u*, v*, and w*) from eqs. (4.295) – (4.297).
3. Solve the pressure correction equation (4.309) to get .
4. Obtain a new pressure field from eq. (4.298).
5. Improve the velocity field using eqs. (4.305) – (4.307).
6. Obtain solutions for other ’s from eq. (4.274). If the flow field is not affected by a particular , it should be solved after a converged solution for flow field is obtained.
7. Treat the corrected pressure in Step 4 as a new starred pressure and go back to step 2. The iteration procedure is repeated until a converged solution is obtained.
Since the invention of the SIMPLE algorithm in 1972, it has evolved into the classic approach for computational fluid dynamics and heat transfer. Although there are several improved versions in the literature, SIMPLE still remains one of basic the most powerful algorithms. The objective of this section is to introduce the readers to the world of computational fluid dynamics and heat transfer without the overwhelming numerical details. Additional information can be found in the seminal books of Patankar (1980) and Tao (2001), as well as the Handbook of Numerical Heat Transfer (Minkowycz et al., 2006).