Suhas V Patankar and SIMPLE Algorithm

“SIMPLE Algorithm which is not that Simple”


Author: Dr. Sandeep Mouvanal

Prof. Suhas V Patankar was born on 22 Feb 1941 in India (Pune, Maharashtra) and is known for his groundbreaking contribution to the field of Computational fluid dynamics through the introduction of the SIMPLE Algorithm along with his supervisor Prof. Brain Spalding [1]. SIMPLE algorithm revolutionized CFD, the numerical simulation of fluid flow, making it more accurate efficient, and useful for the industry. Moreover, his pioneering work in finite volume methods provided engineers and researchers with a robust framework for tackling complex fluid dynamics problems.

Patankar received his bachelor’s degree in mechanical engineering in 1962 from the College of Engineering, Pune, which is affiliated with the University of Pune, and his Master of Technology degree in mechanical engineering from the Indian Institute of Technology Bombay in 1964. In 1967 he received his Ph.D. in mechanical engineering from the Imperial College, University of London. He is currently a Professor Emeritus at the University of Minnesota. He is also president of Innovative Research, Inc.

Portrait of Prof Suhas V Patankar, Pen Sketch (pen on paper, 15 x 11″, Artist: Dr.Sandeep Mouvanal)

Some Important Contributions:

  • SIMPLE Algorithm:
    Patankar is widely recognized for his development of the SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) algorithm. This algorithm revolutionized the numerical solution of fluid flow problems by providing a robust and efficient method for coupling pressure and velocity fields in computational fluid dynamics simulations. The SIMPLE algorithm has become a cornerstone in CFD software packages and has enabled engineers to solve complex flow problems with greater accuracy and reliability.
  • Finite Volume Method:
    Patankar has made significant contributions to the development and popularization of the finite volume method for solving partial differential equations. His work laid the foundation for applying finite volume techniques to a wide range of engineering problems, including fluid dynamics, heat transfer, and chemical reaction kinetics. The finite volume method is highly regarded for its ability to accurately capture flow physics and conserve mass, momentum, and energy.
  • Books and Publications:
    Patankar has authored several influential books and research papers that have become standard references in the field of computational fluid dynamics and heat transfer. His book titled “Numerical Heat Transfer and Fluid Flow” is a seminal work that has guided countless engineers and researchers in understanding the fundamentals of numerical methods for fluid dynamics and heat transfer simulations.
  • Turbulence Modeling:
    Patankar has also contributed to the development of turbulence modeling techniques for turbulent flow simulations. His research has helped improve the accuracy and reliability of turbulence models used in CFD simulations, particularly in industrial applications where turbulent flows play a significant role.
  • Education and Mentorship:
    Beyond his technical contributions, Patankar has played a crucial role in educating and mentoring generations of engineers and researchers in the field of computational fluid dynamics. Through his teaching and supervision of graduate students, he has inspired numerous individuals to pursue careers in computational modeling and simulation.

Books Authored by Prof. Patankar

  1. Numerical Heat Transfer and Fluid flow

This book by Prof. Patankar is the most cited book in the field of CFD and the best to learn the finite volume method. As of today (15th Feb 2024), the total number of citations is 45,000. He explains the finite volume method, a powerful tool for simulating fluid flow, in easy-to-understand language. His book introduces groundbreaking methods like the SIMPLE technique, which made fluid simulations more accurate. It also covers tricky topics like turbulence and heat transfer, using real-world examples to make them relatable.

His writing style is friendly and approachable, making complex ideas accessible to everyone. By showing how these methods evolved, Patankar gives readers a deeper appreciation of the field. Even though it was written years ago, the book’s lessons are still relevant today. This was the book I used to learn FVM during my masters and I use this as the main reference to teach students at Flowthermolab.

Other Books by Prof. Patankar

SIMPLE Algorithm

The acronym SIMPLE stands for Semi-Implicit Method for Pressure Linked Equations. The algorithm was the contribution of Patankar and Spalding (1972) and is essentially a guess-and-correct procedure for calculating pressure on the staggered grid arrangement.

Why is it a significant contribution to the CFD world?
The answer is simple, after this contribution, CFD became more popular, especially in solving complex industrial problems, and hence the origin of several commercial CFD software and the CFD industry/job market.

Now let’s try to understand the actual problem solved by the SIMPLE Algorithm.
Let us consider the governing equation of fluid flow, the Navier-Stokes Equation. In the 2D case with a steady laminar flow, we have the x-momentum equation, the y-momentum equation, and also the continuity equation as shown below.

x-momentum equation

    \[   \frac{\partial}{\partial x}(\rho u u)+\frac{\partial}{\partial y}(\rho v u)=\frac{\partial}{\partial x}\left(\mu \frac{\partial u}{\partial x}\right)+\frac{\partial}{\partial y}\left(\mu \frac{\partial u}{\partial y}\right)-\frac{\partial p}{\partial x}+S_u     \qquad (1)\]


y-momentum equation

    \[   \frac{\partial}{\partial x}(\rho u v)+\frac{\partial}{\partial y}(\rho v v)=\frac{\partial}{\partial x}\left(\mu \frac{\partial v}{\partial x}\right)+\frac{\partial}{\partial y}\left(\mu \frac{\partial v}{\partial y}\right)-\frac{\partial p}{\partial y}+S_v     \qquad (2)\]


continuity equation

    \[   \frac{\partial}{\partial x}(\rho u)+\frac{\partial}{\partial y}(\rho v)=0     \qquad (3)\]



Every velocity component appears in each momentum equation, and the velocity field must also satisfy the continuity equation.
The convective terms of the momentum equations contain non-linear quantities: for example, the first term of equation (1) is the x derivative of \rho u^2.
All three equations are intricately coupled because every velocity component appears in each momentum equation and the continuity equation. The most complex issue to resolve is the role played by the pressure. It appears in both momentum equations, but there is no (transport or other) equation for the pressure.

If the pressure gradient is known, the process of obtaining discretized equations for velocities from the momentum equations is the same as that for any other scalar and is easy to discretize. But in reality or the real-world applications we have no clue about the pressure field or its gradient and we need to calculate it as part of the computation process.

If the flow is compressible the continuity equation may be used as the transport equation for density and, in addition to (1)–(3), the energy equation is the transport equation for temperature. The pressure may then be obtained from density and temperature by using the equation of state p = p(\rho, T ).
However, if the flow is incompressible the density is constant and hence by definition not linked to the pressure. In this case, coupling between pressure and velocity introduces a constraint in the solution of the flow field: if the correct pressure field is applied in the momentum equations the resulting velocity field should satisfy continuity. Both the problems associated with the non-linearities in the equation set and the pressure–velocity linkage can be resolved by adopting an iterative solution strategy such as the SIMPLE algorithm of Patankar and Spalding (1972).

In this algorithm, the convective fluxes per unit mass through cell faces are evaluated from so-called guessed velocity components. Furthermore, a guessed pressure field is used to solve the momentum equations, and a pressure correction equation, deduced from the continuity equation, is solved to obtain a pressure correction field, which is in turn used to update the velocity and pressure fields. To start the iteration process we use initial guesses for the velocity and pressure fields. As the algorithm proceeds our aim must be progressively to improve these guessed fields. The process is iterated until the convergence of the velocity and pressure fields. [2]

Flow chart of SIMPLE Algorithm [2]

SIMPLE Algorithm in steps:
(1) Guess the value of pressure, p^*
(2) Solve for momentum eqn and calculate the guessed velocities, u^*, v^* and w^*
(3) Solve for p^{\prime}, correction pressure (through the pressure correction equation)
(4) Using p^{\prime}, update u, v and w
(5) Correct p \rightarrow p^*+p^{\prime}
(6) Go to step 1 and iterate till converges \left(p^{\prime}=0\right)

Improved Versions:
Further enhancements to the SIMPLE Algorithm came as The SIMPLER (SIMPLE Revised) algorithm of Patankar (1980) as an improved version of SIMPLE. Then came the SIMPLEC (SIMPLE-Consistent) algorithm of Van Doormal and Raithby (1984); which follows the same steps as the SIMPLE algorithm, with the difference that the momentum equations are manipulated so that the SIMPLEC velocity correction equations omit terms that are less significant than those in SIMPLE. Then came the PISO algorithm, which stands for Pressure Implicit with Splitting of Operators, of Issa (1986) is a pressure–velocity calculation procedure developed originally for non-iterative computation of unsteady compressible flows. It has been adapted successfully for the iterative solution of steady-state problems. PISO involves one predictor step and two corrector steps and may be seen as an extension of SIMPLE, with a further corrector step to enhance it.

References:

[1] “Suhas Patankar” Wikipedia, Wikimedia Foundation, 15 Feb 2024, https://en.wikipedia.org/wiki/Brian_Spalding
[2] Versteeg, Henk Kaarle, and Weeratunge Malalasekera. An introduction to computational fluid dynamics: the finite volume method. Pearson education, 2007.
[3] Patankar, Suhas. Numerical heat transfer and fluid flow. Taylor & Francis, 2018.
[4] Suhas Patankar, research.com. 15 Feb 2024. https://research.com/u/suhas-v-patankar
[5] S.V. Patankar, D.B. Spalding, A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows, Int. J. Heat Mass Transfer 15 (10) (1972) 1787–1806.

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