![[CFD] Rhie & Chow Interpolation (Part 1): Chequerboard Oscillations @fluidmechanics101](https://i.ytimg.com/vi/yqZ59Xn_aF8/maxresdefault.jpg)
Uploaded July 2021 | Updated September 2026, 2 weeks ago
An introduction to Momentum Weighted Interpolation (often referred to as Rhie & Chow Interpolation), a method which is used by many modern CFD codes to address chequerboard oscillations in pressure-based solvers which use a collocated variable arrangement. This talk is Part 1 of a 3 Part lecture series. The focus of Part 1 is understanding chequerboard oscillations and why they can occur in CFD solvers that use a collocated variable arrangement.
Timestamps:
1) 02:34 A recap of the finite volume method and the discretisation of the momentum equation
2) 23:44 What are chequerboard oscillations?
3) 41:50 What are the potential options for removing these oscillations?
#RhieChowInterpolation #fluidmechanics101 #cfd
********************************************************
TYPO: Note there is a small typo in Equation 26.
pe = 1/2 (pE + pP) pw = 1/2 (pP+pW)
The (-) should be (+)
********************************************************
Some useful references:
1. Rhie and Chow, 'Numerical Study of the Turbulent Flow Past an Airfoil with Trailing Edge Separation', AIAA Journal, 21(11), 1983.
https://www.academia.edu/24382500/Numerical_Study_of_the_Turbulent_Flow_Past_an_Airfoil_with_Trailing_Edge_Separation
2. Bartholomew et al. 'Unified formulation of the momentum-weighted interpolation for collocated
variable arrangements', Journal of Computational Physics, 2018, 375, 177 – 208.
researchgate.net/publication/327217521_Unified_formulation_of_the_momentum-weighted_interpolation_for_collocated_variable_arrangements
3. Versteeg & Malalasekeera, 'An Introduction to Computational Fluid Dynamics: The Finite Volume Method', Longman Scientific & Technical, 1995.
amazon.co.uk/Introduction-Computational-Fluid-Dynamics-Finite/dp/0131274988
4. Ferziger & Peric, 'Computational Methods for Fluid Dynamics', 3rd Edition, Springer-Verlag, Berlin, Heidelberg, New York, 2002.
amazon.co.uk/Computational-Methods-Fluid-Dynamics-Ferziger/dp/3540420746
============================================
Want to learn more?
============================================
Grab a copy of my CFD Fundamentals Course (for beginners):
udemy.com/course/computational-fluid-dynamics-fundamentals-course/?referralCode=F9CF2DFFCA224E93834E
Learn how to write your own CFD code in MATLAB and python (for intermediates):
dr-aidan-wimshurst-s-school.teachable.com/p/the-simple-algorithm
Learn how I draw my figures and diagrams in Inkscape (for everyone):
dr-aidan-wimshurst-s-school.teachable.com/p/inkscape-for-scientists-and-engineers
============================================
Did you like the video?
============================================
Download the lecture slides from my website:
fluidmechanics101.com/pages/lectures.html
Buy me a coffee to say thanks:
buymeacoffee.com/NKEZrhvg0
Support the channel on Patreon (and get useful extras for your CFD studies):
patreon.com/fluidmechanics101
============================================
Donations:
============================================
1) PayPal
paypal.me/fluidmechanics101
==================================
Disclaimer
==================================
The methods, algorithms, equations, formulae, diagrams and explanations in this talk are for educational and demonstrative purposes only. They should never be used to analyse, design, accredit or validate real scientific / engineering / mathematical structures and flow systems. For such applications, appropriate trained, qualified and accredited (SQEP) engineers / scientists should be consulted along with the appropriate documentation, procedures and engineering standards. Furthermore, the information contained within this talk has not been verified, peer reviewed or checked in any way and is likely to contain several errors. It is therefore not appropriate to use this talk itself (or any of the algorithms, equations, formulae, diagrams and explanations contained within this talk) as an academic or technical reference. The reader should consult the original references and follow the verification and validation processes adopted by your company / institution when carrying out engineering calculations and analyses. Fluid Mechanics 101 and Dr. Aidan Wimshurst are not accountable or liable in any form for the use or misuse of the information contained in this talk beyond the specific educational and demonstrative purposes for which it was intended.
An introduction to Momentum Weighted Interpolation (often referred to as Rhie & Chow Interpolation), a method which is used by many modern CFD codes to address chequerboard oscillations in pressure-based solvers which use a collocated variable arrangement. This talk is Part 1 of a 3 Part lecture series. The focus of Part 1 is understanding chequerboard oscillations and why they can occur in CFD solvers that use a collocated variable arrangement.
Timestamps:
1) 02:34 A recap of the finite volume method and the discretisation of the momentum equation
2) 23:44 What are chequerboard oscillations?
3) 41:50 What are the potential options for removing these oscillations?
#RhieChowInterpolation #fluidmechanics101 #cfd
********************************************************
TYPO: Note there is a small typo in Equation 26.
pe = 1/2 (pE + pP) pw = 1/2 (pP+pW)
The (-) should be (+)
********************************************************
Some useful references:
1. Rhie and Chow, 'Numerical Study of the Turbulent Flow Past an Airfoil with Trailing Edge Separation', AIAA Journal, 21(11), 1983.
https://www.academia.edu/24382500/Numerical_Study_of_the_Turbulent_Flow_Past_an_Airfoil_with_Trailing_Edge_Separation
2. Bartholomew et al. 'Unified formulation of the momentum-weighted interpolation for collocated
variable arrangements', Journal of Computational Physics, 2018, 375, 177 – 208.
researchgate.net/publication/327217521_Unified_formulation_of_the_momentum-weighted_interpolation_for_collocated_variable_arrangements
3. Versteeg & Malalasekeera, 'An Introduction to Computational Fluid Dynamics: The Finite Volume Method', Longman Scientific & Technical, 1995.
amazon.co.uk/Introduction-Computational-Fluid-Dynamics-Finite/dp/0131274988
4. Ferziger & Peric, 'Computational Methods for Fluid Dynamics', 3rd Edition, Springer-Verlag, Berlin, Heidelberg, New York, 2002.
amazon.co.uk/Computational-Methods-Fluid-Dynamics-Ferziger/dp/3540420746
============================================
Want to learn more?
============================================
Grab a copy of my CFD Fundamentals Course (for beginners):
udemy.com/course/computational-fluid-dynamics-fundamentals-course/?referralCode=F9CF2DFFCA224E93834E
Learn how to write your own CFD code in MATLAB and python (for intermediates):
dr-aidan-wimshurst-s-school.teachable.com/p/the-simple-algorithm
Learn how I draw my figures and diagrams in Inkscape (for everyone):
dr-aidan-wimshurst-s-school.teachable.com/p/inkscape-for-scientists-and-engineers
============================================
Did you like the video?
============================================
Download the lecture slides from my website:
fluidmechanics101.com/pages/lectures.html
Buy me a coffee to say thanks:
buymeacoffee.com/NKEZrhvg0
Support the channel on Patreon (and get useful extras for your CFD studies):
patreon.com/fluidmechanics101
============================================
Donations:
============================================
1) PayPal
paypal.me/fluidmechanics101
==================================
Disclaimer
==================================
The methods, algorithms, equations, formulae, diagrams and explanations in this talk are for educational and demonstrative purposes only. They should never be used to analyse, design, accredit or validate real scientific / engineering / mathematical structures and flow systems. For such applications, appropriate trained, qualified and accredited (SQEP) engineers / scientists should be consulted along with the appropriate documentation, procedures and engineering standards. Furthermore, the information contained within this talk has not been verified, peer reviewed or checked in any way and is likely to contain several errors. It is therefore not appropriate to use this talk itself (or any of the algorithms, equations, formulae, diagrams and explanations contained within this talk) as an academic or technical reference. The reader should consult the original references and follow the verification and validation processes adopted by your company / institution when carrying out engineering calculations and analyses. Fluid Mechanics 101 and Dr. Aidan Wimshurst are not accountable or liable in any form for the use or misuse of the information contained in this talk beyond the specific educational and demonstrative purposes for which it was intended.
![[CFD] Rhie & Chow Interpolation (Part 1): Chequerboard Oscillations](https://i.ytimg.com/vi/yqZ59Xn_aF8/mqdefault.jpg "[CFD] Rhie & Chow Interpolation (Part 1): Chequerboard Oscillations
An introduction to Momentum Weighted Interpolation (often referred to as Rhie & Chow Interpolation), a method which is used by many modern CFD codes to address chequerboard oscillations in pressure-based solvers which use a collocated variable arrangement. This talk is Part 1 of a 3 Part lecture series. The focus of Part 1 is understanding chequerboard oscillations and why they can occur in CFD solvers that use a collocated variable arrangement.
Timestamps:
1) 02:34 A recap of the finite volume method and the discretisation of the momentum equation
2) 23:44 What are chequerboard oscillations?
3) 41:50 What are the potential options for removing these oscillations?
#RhieChowInterpolation #fluidmechanics101 #cfd
********************************************************
TYPO: Note there is a small typo in Equation 26.
pe = 1/2 (pE + pP) pw = 1/2 (pP+pW)
The (-) should be (+)
********************************************************
Some useful references:
1. Rhie and Chow, Numerical Study of the Turbulent Flow Past an Airfoil with Trailing Edge Separation, AIAA Journal, 21(11), 1983.
https://www.academia.edu/24382500/Numerical_Study_of_the_Turbulent_Flow_Past_an_Airfoil_with_Trailing_Edge_Separation
2. Bartholomew et al. Unified formulation of the momentum-weighted interpolation for collocated
variable arrangements, Journal of Computational Physics, 2018, 375, 177 – 208.
https://www.researchgate.net/publication/327217521_Unified_formulation_of_the_momentum-weighted_interpolation_for_collocated_variable_arrangements
3. Versteeg & Malalasekeera, An Introduction to Computational Fluid Dynamics: The Finite Volume Method, Longman Scientific & Technical, 1995.
https://www.amazon.co.uk/Introduction-Computational-Fluid-Dynamics-Finite/dp/0131274988
4. Ferziger & Peric, Computational Methods for Fluid Dynamics, 3rd Edition, Springer-Verlag, Berlin, Heidelberg, New York, 2002.
https://www.amazon.co.uk/Computational-Methods-Fluid-Dynamics-Ferziger/dp/3540420746
Want to learn more?
Grab a copy of my CFD Fundamentals Course (for beginners):
https://www.udemy.com/course/computational-fluid-dynamics-fundamentals-course/?referralCode=F9CF2DFFCA224E93834E
Learn how to write your own CFD code in MATLAB and python (for intermediates):
https://dr-aidan-wimshurst-s-school.teachable.com/p/the-simple-algorithm
Learn how I draw my figures and diagrams in Inkscape (for everyone):
https://dr-aidan-wimshurst-s-school.teachable.com/p/inkscape-for-scientists-and-engineers
Did you like the video?
Download the lecture slides from my website:
https://www.fluidmechanics101.com/pages/lectures.html
Buy me a coffee to say thanks:
https://www.buymeacoffee.com/NKEZrhvg0
Support the channel on Patreon (and get useful extras for your CFD studies):
https://www.patreon.com/fluidmechanics101
Donations:
1) PayPal
https://www.paypal.me/fluidmechanics101
Disclaimer
The methods, algorithms, equations, formulae, diagrams and explanations in this talk are for educational and demonstrative purposes only. They should never be used to analyse, design, accredit or validate real scientific / engineering / mathematical structures and flow systems. For such applications, appropriate trained, qualified and accredited (SQEP) engineers / scientists should be consulted along with the appropriate documentation, procedures and engineering standards. Furthermore, the information contained within this talk has not been verified, peer reviewed or checked in any way and is likely to contain several errors. It is therefore not appropriate to use this talk itself (or any of the algorithms, equations, formulae, diagrams and explanations contained within this talk) as an academic or technical reference. The reader should consult the original references and follow the verification and validation processes adopted by your company / institution when carrying out engineering calculations and analyses. Fluid Mechanics 101 and Dr. Aidan Wimshurst are not accountable or liable in any form for the use or misuse of the information contained in this talk beyond the specific educational and demonstrative purposes for which it was intended.")
![[CFD] The Energy Equation for Solids and Fluids in CFD](https://i.ytimg.com/vi/z8dZHze_EPo/mqdefault.jpg "[CFD] The Energy Equation for Solids and Fluids in CFD
An introduction to the differential form of the energy conservation equation for fluid flows in CFD. The presented equation is valid for both incompressible and compressible flows, as well as stationary and moving solid regions and hence can be used in conjugate heat transfer applications. The following topics are covered:
1) 2:11 What form of the differential energy equation is valid for solids and which form is valid for fluids?
2) 4:31 What is the thermal/internal energy equation?
3) 9:18 Why do we need to consider the total energy equation?
4) 21:41 What about enthalpy?
#cfd #energyequation #heattransfer
Some useful references:
1) F. Incropera, D. Dewitt, T. Bergman, A. Lavine
Fundamentals of Heat and Mass Transfer
John Wiley and Sons, 6th Edition
2) OpenFOAM V2.2.0 Release Notes
The Energy Equation in OpenFOAM
https://cfd.direct/openfoam/energy-equation/
3) ANSYS FLUENT User Manual
13.2.1 Heat Transfer Theory
https://www.sharcnet.ca/Software/Fluent6/html/ug/node568.htm
Want to learn more?
Grab a copy of my CFD Fundamentals Course (for beginners):
https://www.udemy.com/course/computational-fluid-dynamics-fundamentals-course/?referralCode=F9CF2DFFCA224E93834E
Learn how to write your own CFD code in MATLAB and python (for intermediates):
https://dr-aidan-wimshurst-s-school.teachable.com/p/the-simple-algorithm
Learn how I draw my figures and diagrams in Inkscape (for everyone):
https://dr-aidan-wimshurst-s-school.teachable.com/p/inkscape-for-scientists-and-engineers
Did you like the video?
Download the lecture slides from my website:
https://www.fluidmechanics101.com/pages/lectures.html
Buy me a coffee to say thanks:
https://www.buymeacoffee.com/NKEZrhvg0
Support the channel on Patreon (and get useful extras for your CFD studies):
https://www.patreon.com/fluidmechanics101
Donations:
1) PayPal
https://www.paypal.me/fluidmechanics101
Disclaimer
The methods, algorithms, equations, formulae, diagrams and explanations in this talk are for educational and demonstrative purposes only. They should never be used to analyse, design, accredit or validate real scientific / engineering / mathematical structures and flow systems. For such applications, appropriate trained, qualified and accredited (SQEP) engineers / scientists should be consulted along with the appropriate documentation, procedures and engineering standards. Furthermore, the information contained within this talk has not been verified, peer reviewed or checked in any way and is likely to contain several errors. It is therefore not appropriate to use this talk itself (or any of the algorithms, equations, formulae, diagrams and explanations contained within this talk) as an academic or technical reference. The reader should consult the original references and follow the verification and validation processes adopted by your company / institution when carrying out engineering calculations and analyses. Fluid Mechanics 101 and Dr. Aidan Wimshurst are not accountable or liable in any form for the use or misuse of the information contained in this talk beyond the specific educational and demonstrative purposes for which it was intended.")
![[CFD] When and Why do I need Operating Pressure, Temperature and Density?](https://i.ytimg.com/vi/zFDDhXEsIZ4/mqdefault.jpg "[CFD] When and Why do I need Operating Pressure, Temperature and Density?
An introduction to the concepts of operating pressure, temperature and density, as they are used in CFD. Short, concise arguments are presented for why CFD codes use them, and when you should apply them. The following topics are covered:
1) 0:42 When and why do I need an operating pressure?
2) 12:59 When and why do I need an operating density?
3) 17:31 When and why do I need an operating temperature?
#cfd #operatingpressure #idealgaslaw
Some useful references:
1) ANSYS FLUENT User Manual
8.14.3 Operating Pressure
https://www.sharcnet.ca/Software/Fluent6/html/ug/node374.htm
2) ANSYS CFX User Manual
1.2.8 Setting a reference pressure
https://www.sharcnet.ca/Software/Ansys/16.2.3/en-us/help/cfx_mod/i1298632.html
Want to learn more?
Grab a copy of my CFD Fundamentals Course (for beginners):
https://www.udemy.com/course/computational-fluid-dynamics-fundamentals-course/?referralCode=F9CF2DFFCA224E93834E
Learn how to write your own CFD code in MATLAB and python (for intermediates):
https://dr-aidan-wimshurst-s-school.teachable.com/p/the-simple-algorithm
Learn how I draw my figures and diagrams in Inkscape (for everyone):
https://dr-aidan-wimshurst-s-school.teachable.com/p/inkscape-for-scientists-and-engineers
Did you like the video?
Download the lecture slides from my website:
https://www.fluidmechanics101.com/pages/lectures.html
Buy me a coffee to say thanks:
https://www.buymeacoffee.com/NKEZrhvg0
Support the channel on Patreon (and get useful extras for your CFD studies):
https://www.patreon.com/fluidmechanics101
Donations:
1) PayPal
https://www.paypal.me/fluidmechanics101
Disclaimer
The methods, algorithms, equations, formulae, diagrams and explanations in this talk are for educational and demonstrative purposes only. They should never be used to analyse, design, accredit or validate real scientific / engineering / mathematical structures and flow systems. For such applications, appropriate trained, qualified and accredited (SQEP) engineers / scientists should be consulted along with the appropriate documentation, procedures and engineering standards. Furthermore, the information contained within this talk has not been verified, peer reviewed or checked in any way and is likely to contain several errors. It is therefore not appropriate to use this talk itself (or any of the algorithms, equations, formulae, diagrams and explanations contained within this talk) as an academic or technical reference. The reader should consult the original references and follow the verification and validation processes adopted by your company / institution when carrying out engineering calculations and analyses. Fluid Mechanics 101 and Dr. Aidan Wimshurst are not accountable or liable in any form for the use or misuse of the information contained in this talk beyond the specific educational and demonstrative purposes for which it was intended.")
![[CFD] What Wall Functions Do I Need for Turbulent Kinetic Energy?](https://i.ytimg.com/vi/zSIgKsQSa9k/mqdefault.jpg "[CFD] What Wall Functions Do I Need for Turbulent Kinetic Energy?
[CFD] What Wall Functions Do I Need for Turbulent Kinetic Energy?
An introduction to the wall functions that are used to capture the near wall behaviour of the turbulent kinetic energy (k) in all mainstream CFD codes (OpenFOAM, Fluent, CFX, Star). The following topics are covered:
1) 2:11 How do we modify the production and dissipation of k in the wall adjacent cell?
2) 13:33 Why do some CFD codes specify dk/dn at the wall?
3) 22:12 Why do some CFD codes specify k directly at the cell centroid?
#wallfunctions #tke #kwallfunction
Some useful references:
1) ANSYS FLUENT User Manual
12.10.2 Standard Wall Functions
http://www.afs.enea.it/project/neptunius/docs/fluent/html/th/node99.htm
2) G. Kalitzin, G. Medic, G. Iaccarino and P. Durbin
Near-wall behaviour of RANS turbulene models and implications for wall functions
Journal of Computational Physics 204 (2005) 265-291
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.584.2507&rep=rep1&type=pdf
3) Moser, Kim & Mansour, Direct numerical simulation of turbulent channel flow up to Re_tau = 590. Physics of Fluids, 11(4), 1999.
https://cfd.spbstu.ru/agarbaruk/doc/1999_Moser-Kim-Mansour_Direct-numerical-simulation-of-turbulent-channel-flow-up-to-Re-590.pdf
4) The complete set of statistical DNS data for turbulent channel flow:
http://turbulence.ices.utexas.edu/MKM_1999.html
5) J. Bredberg. On the wall boundary condition for turbulence models. Internal report 00/4, Chalmers University of Technology, Sweden, 2000.
http://www.tfd.chalmers.se/~lada/postscript_files/jonas_report_WF.pdf
Want to learn more?
Grab a copy of my CFD Fundamentals Course (for beginners):
https://www.udemy.com/course/computational-fluid-dynamics-fundamentals-course/?referralCode=F9CF2DFFCA224E93834E
Learn how to write your own CFD code in MATLAB and python (for intermediates):
https://dr-aidan-wimshurst-s-school.teachable.com/p/the-simple-algorithm
Learn how I draw my figures and diagrams in Inkscape (for everyone):
https://dr-aidan-wimshurst-s-school.teachable.com/p/inkscape-for-scientists-and-engineers
Did you like the video?
Download the lecture slides from my website:
https://www.fluidmechanics101.com/pages/lectures.html
Buy me a coffee to say thanks:
https://www.buymeacoffee.com/NKEZrhvg0
Support the channel on Patreon (and get useful extras for your CFD studies):
https://www.patreon.com/fluidmechanics101
Donations:
1) PayPal
https://www.paypal.me/fluidmechanics101
Disclaimer
The methods, algorithms, equations, formulae, diagrams and explanations in this talk are for educational and demonstrative purposes only. They should never be used to analyse, design, accredit or validate real scientific / engineering / mathematical structures and flow systems. For such applications, appropriate trained, qualified and accredited (SQEP) engineers / scientists should be consulted along with the appropriate documentation, procedures and engineering standards. Furthermore, the information contained within this talk has not been verified, peer reviewed or checked in any way and is likely to contain several errors. It is therefore not appropriate to use this talk itself (or any of the algorithms, equations, formulae, diagrams and explanations contained within this talk) as an academic or technical reference. The reader should consult the original references and follow the verification and validation processes adopted by your company / institution when carrying out engineering calculations and analyses. Fluid Mechanics 101 and Dr. Aidan Wimshurst are not accountable or liable in any form for the use or misuse of the information contained in this talk beyond the specific educational and demonstrative purposes for which it was intended.")
