Bansilal Ramnath Agarwal Charitable Trust’s

VISHWAKARMA INSTITUTE OF TECHNOLOGY, PUNE

(An Autonomous Institute affiliated to University of Pune)

 

  

DEPARTMENT OF MECHANICAL ENGINEERING

PRESSURE-VELOCITY COUPLING ALGORITHM

(2021-22)

CFD-Home Assignment (Blog)

GUIDED BY:

Prof. (Dr.) S. S. Shinde

                                                 

SUBMITTED BY:

Prathmesh Ganesh Amle

Roll no- A-7 (11920048)

 

 

INTRODUCTION

The convection of a scalar variable φ depends on the magnitude and direction of the local velocity field. To develop our methods in the previous chapter we assumed that the velocity field was somehow known. In general, the velocity field is, however, not known and emerges as part of the overall solution process along with all other flow variables. In this chapter we look at the most popular strategies for computing the entire flow field. Transport equations for each velocity component – momentum equations – can be derived from the general transport equation (2.39) by replacing the variable φ by u, v and w respectively. Every velocity component appears in each momentum equation, and the velocity field must also satisfy the continuity equation. This can be clearly shown by considering the equations governing a two-dimensional laminar steady flow:

-There are three governing equations for three momentum (velocity components) and one continuity equation.

*Convection terms are non-linear in nature.

     *There are four variables- three velocity components and one pressure. No. equations and no. of variables are satisfied.

*  *There is no separate equation for pressure, though it plays an important role in each velocity component equation.

*Velocity variables are very intimately coupled, as each velocity variable term appears in every equation including continuity equation.

*Correct pressure field should be known and to be used in momentum equation, to obtain correct velocity field, which in turn satisfies both momentum equation as well as continuity equation.

*The linkage is set through a procedure called pressure-velocity coupling.

 

 Arrangement of Variables

After defining domain and mesh, one needs to select points where variables have to be computed.

·    It is critical as it decides the way the governing equations are solved, the programming, the memory size and time requirement.

·      There are three approaches - Staggered, Collocated and Semi-staggered.

·   It is clear that, if the velocities are defined at the scalar grid nodes, the influence of pressure is not properly represented in the discretized momentum equations. A remedy for this problem is to use a staggered grid for velocity components (Harlow and Welch, 1965).

·  The idea is to evaluate scalar variables, such as pressure, density, temperature etc., at ordinary nodal points but to calculate velocity components on staggered grids cantered around the cell faces.

 

STAGGERED GRID

The natural choice of the grid for a finite volume method is a staggered grid. It is called staggered because the velocity components are staggered with respect to the pressure, which is placed in the centre of the cells. The staggered grid ensures that the resulting discrete system will not be singular.

                                                                    Fig: - Staggered Grid

COLLOCATED GRID

·      All the variables are stored at the same nodal location.

·      Approximation for the derivative terms are straight forward.

·      Programming is easier and memory requirement is minimum.

·      Used when the boundary conditions and its slope are discontinuous.

·      Disadvantage: May result in Checker-board problem.

Fig: - Collocated Grid

 

The Marker-And-Cell (MAC) method

The Marker-And-Cell (MAC) method was developed by Francis H. Harlow, J. Eddie Welch, and the T-3 team at the Los Alamos National Laboratory in 1965. This was the first successful technique that allowed incompressible fluids to flow properly without too much distortion requiring the calculations to be “reset” by hand as with the Particle-In-Cell method. In the MAC method, particles are used as markers that locate the material in a mesh which in turn defines the location of the fluid’s free surfaces.

~Start with guess value for pressure and velocity components.

~Intermediate velocity for the X-direction velocity component can be written as,

Similarly for v and w velocity component in y and z directions

~Calculate error in continuity D*

~If the magnitude of D* greater than some prescribed value ε, correction pressure is calculated as δp= -βD*,

~The velocity components and pressure are corrected as,

Similarly for v and w velocity component

~Repeat above steps for all cells until no cells has the magnitude of D* greater than the prescribed value.

The SIMPLE algorithm

The SIMPLE algorithm gives a method of calculating pressure and velocities. The method is iterative, and when other scalars are coupled to the momentum equations the calculation needs to be done sequentially. The sequence of operations in a CFD procedure which employs the SIMPLE algorithm is given as

Fig:- Flow chart SIMPLE algorithm

 

 

The SIMPLER (SIMPLE Revised) algorithm

The SIMPLER (SIMPLE Revised) algorithm of Patankar (1980) is an improved version of SIMPLE. In this algorithm the discretized continuity equation is used to derive a discretized equation for pressure, instead of a pressure correction equation as in SIMPLE. Thus, the intermediate pressure field is obtained directly without the use of a correction. Velocities are, however, still obtained through the velocity corrections of SIMPLE.

Fig:- Flow Chart SIMPLER algorithm

 

The SIMPLEC algorithm

The SIMPLEC (SIMPLE-Consistent) algorithm of Van Doormal and Raithby (1984) 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.

The u-velocity correction equation of SIMPLEC is given by

 

Similarly, the modified v-velocity correction equation is

The discretized pressure correction equation is the same as in SIMPLE, except that the d-terms are calculated from equations. The sequence of operations of SIMPLEC is identical to that of SIMPLE.

 

 

The PIMPLE Algorithm

The PIMPLE Algorithm is a combination of PISO (Pressure Implicit with Splitting of Operator) and SIMPLE (Semi-Implicit Method for Pressure-Linked Equations). All these algorithms are iterative solvers but PISO and PIMPLE are both used for transient cases whereas SIMPLE is used for steady-state cases. The best way to think about the PIMPLE algorithm is to imagine it as a SIMPLE algorithm for every time step, where outer correctors are the iterations, and once converged will move on to the next time step until the solution is complete.

 

 Fig:- Flow chart PIMPLE algorithm

 

The PISO algorithm

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.

Fig:- Flow chart PISO algorithm

Comparison between SIMPLE, ITS VARIENTS AND PISO

v The SIMPLE algorithm is relatively straightforward and has been successfully implemented in numerous CFD procedures.

 

v The other variations of SIMPLE can produce savings in computational effort due to improved convergence. In SIMPLE, the pressure correction p′ is satisfactory for correcting velocities but not so good for correcting pressure.

 

v Hence the improved procedure SIMPLER uses the pressure corrections to obtain velocity corrections only. A separate, more effective, pressure equation is solved to yield the correct pressure field.

 

v Since no terms are omitted to derive the discretized pressure equation in SIMPLER, the resulting pressure field corresponds to the velocity field. Therefore, in SIMPLER the application of the correct velocity field results in the correct pressure field, whereas it does not in SIMPLE.

 

v Consequently, the method is highly effective in calculating the pressure field correctly. This has significant advantages when solving the momentum equations. Although the number of calculations involved in SIMPLER is about 30% larger than that for SIMPLE, the fast convergence rate reportedly reduces the computer time by 30–50% (Anderson et al., 1984).

 

v Further details of SIMPLE and its variants may be found in Patankar (1980). SIMPLEC and PISO have proved to be as efficient as SIMPLER in certain types of flows but it is not clear whether it can be categorically stated that they are better than SIMPLER.

 

v Comparisons have shown that the performance of each algorithm depends on the flow conditions, the degree of coupling between the momentum equation and scalar equations (in combusting flows, for example, due to the dependence of the local density on concentration and temperature), the amount of under-relaxation used, and sometimes even on the details of the numerical technique used for solving the algebraic equations.

 

A comprehensive comparison of PISO, SIMPLER and SIMPLEC methods for a variety of steady flow problems by Jang et al. (1986) showed that, for problems in which momentum equations are not coupled to a scalar variable, PISO showed robust convergence behavior and required less computational effort than SIMPLER and SIMPLEC.

 

 NEW RESEARCH/NEW METHOD

ESIMPLE, a new pressure–velocity coupling algorithm for built-environment CFD simulator.

Highlights

•We present an extension of the SIMPLE algorithm (ESIMPLE) and its performance is compared with SIMPLE, PISO, SIMPLEC and LIMPO

 

•For that, the four algorithms are tested in three built-environment scenarios: a cubical room (1 m3) with heated floor; a lab-scale room, with an occupant, with assessment of the algorithm’s accuracy against experimental data; and a real-scale office room with an occupant.

 

•For the best scenario, a three times faster convergence rate was attained by the new algorithm.

 

Built environments are major energy consumers and, therefore, tools supporting their efficient design and guaranteeing thermal comfort and indoor air quality are a key factor for energy, environmental sustainability and healthiness. This has been particularly stressed recently by the need to understand the phenomenon of transport of pollutants and/or pathogens leading to exhalation of droplets and aerosols in built environments from people potentially contaminated with the new coronaviruses. In the pursuance of such objectives, CFD procedures have been widely used as prediction tools due to its ability and flexibility in capturing the main features of built environment flows. On the other hand, CFD methods are supported by complex and time-consuming calculation procedures, especially when used to predict heat and mass transport phenomena in built environment. A possible strategy to reduce computational time is the optimization of the pressure-velocity coupling. An extension of the SIMPLE algorithm (ESIMPLE) is proposed, and its performance compared with the well stablished algorithms, SIMPLE, PISO, SIMPLEC and LIMPO. For that, three test case scenarios are simulated: i) a cubical room (1 m3) with heated floor; ii) a small-scale room, with an occupant, mimicking an office room; and iii) a real-scale office room with an occupant. For the worst scenario, ESIMPLE yielded similar CPU-time required for convergence, and for the best scenario, a three times faster convergence rate was attained. Simultaneously, this newly proposed coupling scheme algorithm, yielded a lower number of iteration steps required for convergence, in 6 of the 9 simulated cases.

 

CONCLUSION

~It was also observed that when the scalar variables were closely linked to velocities, PISO had no significant advantage over the other methods.

~Iterative methods using SIMPLER and SIMPLEC have robust convergence characteristics in strongly coupled problems, and it could not be ascertained which of SIMPLER or SIMPLEC was superior.

~ESIMPLE is a new method which has more convergence rate than the other variants of SIMPLE.

 

 

 

 REFERENCES

[1] Author open overlay panel Nuno Serraab Viriato Semiaoa ESIMPLE, a new pressure–velocity coupling algorithm for built-environment CFD simulations https://doi.org/10.1016/j.buildenv.2021.108170Get rights and content.

[2] H. Versteeg and M. Malalasekra, An Introduction to Computational Fluid Dynamics: The Finite Volume Method.

[3] Handbook of Computational Fluid Mechanics http/doi.org/10.1016/B978-0-12-553010-1.X5000-2.

[4] Patankar and Spalding, SIMPLE Algorithm, 1972.

[5] Patankar, SIMPLER Algorithm, 1980.

[6] Vandoormal and Raithby, SIMPLEC Algorithm, 1984.

[7] Dr. S. Vengadesan Computational Fluid Mechanics Videos.

Acknowledgement

 

We are profoundly grateful to Prof. (Dr.) S. S. Shinde for his guidance and continuous encouragement throughout to see that this home assignment rights its target since its commencement to its completion. His continuous support and advice have contributed tremendously towards bringing this home assignment to fruition.

We would like to express deepest appreciation towards Prof. (Dr.) R. M. Jalnekar, Director Vishwakarma Institute of Technology. Prof. (Dr.) M. B. Chaudhari, Head of the Mechanical Engineering Department.

I would like to take this opportunity to express our gratitude and sincere thanks to all those who directly or indirectly have contributed in matters related to this home assignment.

Date: 03/01/2022

Prathmesh Amle