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Does the Navier-Stokes equation model (experimentally observed) turbulence accurately? I was under the impression that this is an open question. Turbulence in

by quanto 4y ago

Does the Navier-Stokes equation model (experimentally observed) turbulence accurately? I was under the impression that this is an open question. Turbulence indeed is high Kolmogorov complexity (w.r.t some reasonable grammar of numerical mathematics) and thus high computational complexity. But the question of N-S equation's adequacy comes before the question of computing. I haven't worked on computational fluid dynamics in a long time -- happy to get insights from specialists.

btrettel 4y ago
> Does the Navier-Stokes equation model (experimentally observed) turbulence accurately? I was under the impression that this is an open question. No one I know working in turbulence regards the Navier-Stokes equations as a bad model for turbulence. They typically regard the equations as a source of truth about turbulence. Focusing on turbulence in incompressible fluids [0], almost every time I've seen a comparison of "direct numerical simulations" using the Navier-Stokes equations against experiments measuring turbulence, the two matched well. Off the top of my head the only time I've encountered a discrepancy was deemed not due to the Navier-Stokes equations themselves but due to not modeling the initial or boundary conditions correctly [1]. Initial conditions refer to the flow field at the start of the simulation. Boundary conditions refer to conditions at a boundary, like a wall. Initial condition modeling is a big issue in certain flow instabilities like the Rayleigh-Taylor instability to my knowledge. The Navier-Stokes equations are chaotic, so they are sensitive to small changes in the initial conditions. I'm not aware of anyone working on flow instabilities like the Rayleigh-Taylor instability who attributes a discrepancy between the experiments and simulations to an inadequacy of the Navier-Stokes equations. And to my knowledge I don't think any of the mathematicians working on the blow-up problem regard flow instabilities as clues indicating where to focus their attention to make the Euler/Navier-Stokes equations blow-up. [0] You could, of course, pick a situation where the Navier-Stokes equations are known to not work well independent of turbulence considerations. [1] Having worked a bit on initial and boundary condition modeling, though not in the context of the Rayleigh-Taylor instability, I can say that initial condition and boundary condition models are often bad, and they are understudied.
- andi999 4y ago
  Which situations are this in [0]?
  - btrettel 4y ago
    For high Knudsen numbers [0] the Navier-Stokes equations will be inaccurate. You can use a different approach based on the Boltzmann equation [1] like DSMC [2] for example. As I wrote elsewhere there are good reasons to not use the Boltzmann equation everywhere [3]. Navier-Stokes is simpler and often completely adequate. There are surely other situations as well; this is just what comes immediately to mind as I had a class covering some of this. [0] https://en.wikipedia.org/wiki/Knudsen_number https://en.wikipedia.org/wiki/Knudsen_number [1] https://en.wikipedia.org/wiki/Boltzmann_equation https://en.wikipedia.org/wiki/Boltzmann_equation [2] https://en.wikipedia.org/wiki/Direct_simulation_Monte_Carlo https://en.wikipedia.org/wiki/Direct_simulation_Monte_Carlo [3] https://news.ycombinator.com/item?id=31054992 https://news.ycombinator.com/item?id=31054992

by quanto 4y ago

btrettel 4y ago

> Does the Navier-Stokes equation model (experimentally observed) turbulence accurately? I was under the impression that this is an open question. No one I know working in turbulence regards the Navier-Stokes equations as a bad model for turbulence. They typically regard the equations as a source of truth about turbulence. Focusing on turbulence in incompressible fluids [0], almost every time I've seen a comparison of "direct numerical simulations" using the Navier-Stokes equations against experiments measuring turbulence, the two matched well. Off the top of my head the only time I've encountered a discrepancy was deemed not due to the Navier-Stokes equations themselves but due to not modeling the initial or boundary conditions correctly [1]. Initial conditions refer to the flow field at the start of the simulation. Boundary conditions refer to conditions at a boundary, like a wall. Initial condition modeling is a big issue in certain flow instabilities like the Rayleigh-Taylor instability to my knowledge. The Navier-Stokes equations are chaotic, so they are sensitive to small changes in the initial conditions. I'm not aware of anyone working on flow instabilities like the Rayleigh-Taylor instability who attributes a discrepancy between the experiments and simulations to an inadequacy of the Navier-Stokes equations. And to my knowledge I don't think any of the mathematicians working on the blow-up problem regard flow instabilities as clues indicating where to focus their attention to make the Euler/Navier-Stokes equations blow-up. [0] You could, of course, pick a situation where the Navier-Stokes equations are known to not work well independent of turbulence considerations. [1] Having worked a bit on initial and boundary condition modeling, though not in the context of the Rayleigh-Taylor instability, I can say that initial condition and boundary condition models are often bad, and they are understudied.

andi999 4y ago
Which situations are this in [0]?
- btrettel 4y ago
  For high Knudsen numbers [0] the Navier-Stokes equations will be inaccurate. You can use a different approach based on the Boltzmann equation [1] like DSMC [2] for example. As I wrote elsewhere there are good reasons to not use the Boltzmann equation everywhere [3]. Navier-Stokes is simpler and often completely adequate. There are surely other situations as well; this is just what comes immediately to mind as I had a class covering some of this. [0] https://en.wikipedia.org/wiki/Knudsen_number https://en.wikipedia.org/wiki/Knudsen_number [1] https://en.wikipedia.org/wiki/Boltzmann_equation https://en.wikipedia.org/wiki/Boltzmann_equation [2] https://en.wikipedia.org/wiki/Direct_simulation_Monte_Carlo https://en.wikipedia.org/wiki/Direct_simulation_Monte_Carlo [3] https://news.ycombinator.com/item?id=31054992 https://news.ycombinator.com/item?id=31054992