Question

# Lattice Boltzmann equation vs Navier-Stokes equations?

What are the limitations of these 2 equations? Can LB be used to model and simulate in aerodynamics problems?

2nd Dec, 2020
Maxim Solovchuk
National Health Research Institutes
@ J. Blair Perot , very nice answer. Small comment to your question mark: Navier Stokes equation is valid at small Knudsen numbers, Kn<<1 (for example )

23rd Jul, 2014
J. Blair Perot
University of Massachusetts Amherst
LBM solves the Boltzmann equation which in theory is more general than N.S.  But in practice LBM combines some physics with numerical errors in clever ways that can be good and bad.  Some things to think about...  All of which are probably fixable with some or a lot of effort.
Classic LBM doesn't have enough particle speeds to represent the energy equation (it only captures conservation of mass and momentum).  So without significant alteration (and larger stencils with more speeds) and careful stability analysis the classic LBM methods are not appropriate for compressible flow or heat transfer.
Classic LBM also has trouble with truly incompressible flow.  It solves a low Mach number approximation with a speed of sound that is not the right speed of sound.  While the method is usually very  'fast' per timestep, any attempt to get properly close to the correct speed of sound causes very small timesteps and makes the method 'slow' per second of computed flow.
The method uses a uniform mesh spacing everywhere in the domain.  Since fluids tend to have thin layers, such as boundary layers and shear layers, and trailing vortices, the resolution required to capture these thin layers can cause massive over-resolution everywhere else.  Again, the method can be 'fast' per grid cell, but requires massive numbers of grid cells (due to uniform resolution) causing 'slow' overall solution times in practice (for real flow problems).
Variable viscosity is tricky.  The method uses clever tricks to turn numerical viscosity into physical viscosity.  This is one reason the method can be so fast.  But  these tricks don't work when the effective viscosity changes due to temperature, concentration, material properties (2-phase), or turbulence (eddy viscosity).
Its good points are that  the LBM is very clever numerically, very parallel, and very fast per cell timestep.  I gloss over these advantages because every LBM paper will tell you all this in more detail.  But these advantages are real and are potentially decisive (depending on your application).
Another big plus, is that the issue of upwinding and how to properly discretize advection is completely removed.  The advection part is, in fact, performed perfectly, by LBM.  Any attempt to remove the uniform mesh restriction on the LBM will destroy the exact advection property.  You get one or the other ...
Some people claim that boundary conditions are an issue with LBM.  But I don't see them as any harder than BC issues for N.S. (which are actually quite complex to pose and implement properly).
The claim is frequently made that LBM are great for multi-phase flows.  I have never seen a justification of this claim and see no real fundamental advantage to LBM for those flows.
N.S.  is limited by the continuum hypothesis.  Less than Mach 3 roughly.  And a Knudsen number over 100?
23 Recommendations

21st Jul, 2014
Moritz von Stosch
DataHow AG
I worked with LB during my Masters project, where I simulated central drop collisions. It appeared to me that the LB models are particularly suitable for the modeling of problems that comprise different phases, e.g. a liquid and a gas phase. It also seemed that the LB approach is computationally more efficient than the Navier Stockes approach, but I never made a rigorous comparison. Further, I know that there was a PhD-student working on LBs in the Aerodynamics Department at the RWTH-Aachen university.
1 Recommendation
21st Jul, 2014
Ivan V. Kazachkov
Ніжинський державний університет ім. Миколи Гоголя
In short, LB is based on statistical approach, therefore it is applied in case when the NSE cannot give more, e.g. multiphase and multicomponent flows, where you need to follow individual particles, etc. If you have homogeneous flow, better to use NSE as more precise eqs.
1 Recommendation
22nd Jul, 2014
Erlend Magnus Viggen
Norwegian University of Science and Technology
The Navier-Stokes equations describe the behaviour of fluid flow in the continuum approximation. Then you have the Boltzmann equation, which is a more detailed description of the behaviour of a gas. However, the Navier-Stokes equations follow from the Boltzmann equation in a certain limit.
The thing about the Lattice Boltzmann method is that it is a discretisation of the Boltzmann equation rather than the Navier-Stokes equations. Somewhat paradoxically, the former tends to give a simpler numerical scheme than the latter. And since a numerical solution to the Navier-Stokes equations follow from a numerical solution to the Boltzmann equation (in a certain limit, as I said), it is in principle possible to use LB to simulate anything related to Navier-Stokes, such as aerodynamics.
2 Recommendations
22nd Jul, 2014
James M Buick
University of Portsmouth
The LBM can be shown to satisfy the incompressible NS equation. Son anything which can be simulated with the incompressible NS equation can be simulated using the LBM. The question is then which is the better approach. There is not a definite answer to this, but  generally the LBM has advantages over the NS solver for multi-phase flows, non-Newtonian flows and small-scale flows where the continuity approximation in the NS equation does not hold. If your aerodynamics problem has a pressure/density difference then you can still use the LBM. For compressibility of a few percent the standard LBM has been shown to e appropriate.  Variants of the LBM model have been used in the literature for strongly compressible flows.
13 Recommendations
23rd Jul, 2014
J. Blair Perot
University of Massachusetts Amherst
LBM solves the Boltzmann equation which in theory is more general than N.S.  But in practice LBM combines some physics with numerical errors in clever ways that can be good and bad.  Some things to think about...  All of which are probably fixable with some or a lot of effort.
Classic LBM doesn't have enough particle speeds to represent the energy equation (it only captures conservation of mass and momentum).  So without significant alteration (and larger stencils with more speeds) and careful stability analysis the classic LBM methods are not appropriate for compressible flow or heat transfer.
Classic LBM also has trouble with truly incompressible flow.  It solves a low Mach number approximation with a speed of sound that is not the right speed of sound.  While the method is usually very  'fast' per timestep, any attempt to get properly close to the correct speed of sound causes very small timesteps and makes the method 'slow' per second of computed flow.
The method uses a uniform mesh spacing everywhere in the domain.  Since fluids tend to have thin layers, such as boundary layers and shear layers, and trailing vortices, the resolution required to capture these thin layers can cause massive over-resolution everywhere else.  Again, the method can be 'fast' per grid cell, but requires massive numbers of grid cells (due to uniform resolution) causing 'slow' overall solution times in practice (for real flow problems).
Variable viscosity is tricky.  The method uses clever tricks to turn numerical viscosity into physical viscosity.  This is one reason the method can be so fast.  But  these tricks don't work when the effective viscosity changes due to temperature, concentration, material properties (2-phase), or turbulence (eddy viscosity).
Its good points are that  the LBM is very clever numerically, very parallel, and very fast per cell timestep.  I gloss over these advantages because every LBM paper will tell you all this in more detail.  But these advantages are real and are potentially decisive (depending on your application).
Another big plus, is that the issue of upwinding and how to properly discretize advection is completely removed.  The advection part is, in fact, performed perfectly, by LBM.  Any attempt to remove the uniform mesh restriction on the LBM will destroy the exact advection property.  You get one or the other ...
Some people claim that boundary conditions are an issue with LBM.  But I don't see them as any harder than BC issues for N.S. (which are actually quite complex to pose and implement properly).
The claim is frequently made that LBM are great for multi-phase flows.  I have never seen a justification of this claim and see no real fundamental advantage to LBM for those flows.
N.S.  is limited by the continuum hypothesis.  Less than Mach 3 roughly.  And a Knudsen number over 100?
23 Recommendations
21st Sep, 2014
Andreas Lintermann
Forschungszentrum Jülich
After reading your comments I have to come back to the answer J. Blair Perot has given which is quiet detailed and explains well the differences between NS and LBM. However, LBM is not limited to a fixed mesh resolution although the grid distance is usually equal in all directions for a certain resolution (there have been approaches with non Cartesian meshes, i.e., boundary fitted meshes). In fact by using a transformation between different meshes of resolution it is possible to obtain stable results. You can find more information on this in the work "Depuis, A. and Chopard, B., Theory and applications of an alternative lattice Boltzmann grid refinement algorithm, Physical Review E, Vol. 67 (6), p. 1-7, (2003), doi:10.1103/PhysRevE.67.066707".
3 Recommendations
26th Sep, 2014
Swinburne University of Technology
Thank you so much for all of the answers. I'm actually looking for a PhD studentship. I'm interested to learn and applied LBM for my PhD. Now, I'm just doing some reading and I have some questions regarding it.
1 Recommendation
11th Mar, 2015
Instituto Tecnológico de Aragón
Just to add a different point of view: the lattice Boltzmann method can be viewed as an efficient numerical method to solve the Navier-Stokes equations. Have a look to (and references 12, 14 and 17 herein):
2 Recommendations
11th Jan, 2017
Shabnam Ghahremanian
Amirkabir University of Technology
What is the advantage of molecular dynamics method to LBM and particle tracing?
24th Nov, 2018
CFD Open Series
LB is the general molecular dynamic version of NS equation. It is a integral-differential equation. So, simply said, LB is encompasses the NS equation. To answer the second part of your question , Yes.
2nd Dec, 2020
Maxim Solovchuk
National Health Research Institutes
@ J. Blair Perot , very nice answer. Small comment to your question mark: Navier Stokes equation is valid at small Knudsen numbers, Kn<<1 (for example )

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