I have this idea in my mind that Direct Numerical Simulation (DNS) is only used in research since the Reynolds numbers of industrial applications are too high and require computer horsepower we don't have yet. Is that still true nowadays?

Any reference on the subject?

  • 1
    $\begingroup$ To the best of my knowledge, LES and VLES are the two mostly used in the industry right now. Note that LES mesh requirements are almost the same as that of DNS except near the walls. $\endgroup$
    – Naghi
    Commented Aug 6, 2020 at 9:21
  • 3
    $\begingroup$ @Alish DNS is at least two to three orders of magnitude more computationally expensive than LES. $\endgroup$
    – Algo
    Commented Aug 6, 2020 at 10:26
  • $\begingroup$ @Algo And is it heavily due to mesh requirements? $\endgroup$
    – Naghi
    Commented Aug 6, 2020 at 10:34
  • $\begingroup$ @Alish in LES you can model small scale eddies and capture the rest, which will result in a less expensive mesh requirements than DNS. $\endgroup$
    – Algo
    Commented Aug 6, 2020 at 11:06

1 Answer 1


I believe this is too broad to be answered here, but DNS has already been employed for high Reynolds number simulations in the range of 12,000 < Re < 33,500 - for swirling jets, but still "out of reach for large systems".

In order to visualize the size of scales required to capture turbulent flow parameters for a high Reynolds number flow, let's take for example a simple air flow in a pipe with $\text{Re} \approx 10^5$ having fairly moderate turbulence intensity of 3%.

Now, the most important factor to take into consideration for DNS and LES is the smallest eddies (Kolmogorov eddy) scale, those are the smallest scales of eddies that can exist in a turbulent flow before being converted into heat through viscous dissipation and your DNS simulation should capture such eddies.

(I am not going to include all the calculations to make the answer concise, but you can refer to Wilcox (2006) and Rodriguez (2019) for more).

Now, back to our flow situation, you are expecting to have Kolmogorov scales with the following estimations:

Kolmogorov eddy length scale     0.000152879 [meters]
Kolmogorov eddy time scale       0.00148359  [seconds]
Kolmogorov eddy velocity scale   0.103046    [meter/second]

So, you basically have an idea about the vast size of the mesh and the time step required to have a Courant number less than one. And higher Reynolds number will have even smaller scales (imagine having a DNS setup for a supersonic flow over a 3D wing).

However, I believe this paragraph by Rodriguez (2019), might give you an insight about the future of DNS (which is very very limited right now):

In any case, as computational power increases, DNS will not only be used for turbulence research and small systems but for larger engineering designs as well; this trend is inevitable and was predicted long ago. This optimistic premise is supported by the strong potential from recent advances in quantum computers, topological quantum materials, and quantum algorithms [...] quantum algorithms already solve linear systems of equations, which are essential for CFD solvers.

Furthermore, it is expected that quantum algorithms will result in an exponential decrease of the time required to solve systems of linear equations. Indeed, the literature as of 2019 indicates the potential for computational speed increases of at least a factor of 1000! And of course, the detailed DNS calculations will uncover fluid functionality that can be leveraged onto vastly improved engineered system behavior and performance.

I really can't recommend this book enough (specifically for your question): Applied Computational Fluid Dynamics and Turbulence Modeling - Rodriguez (2019)


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