Direct Numerical Simulation Part III – The Reynolds Number Limitation

“To mechanical progress there is apparently no end: for as in the past so in the future, each step in any direction will remove limits and bring in past barriers which have till then blocked the way in other directions; and so what for the time may appear to be a visible or practical limit will turn out to be but a bend in the road…” – Osborne Reynolds

osborne reynolds

My first actual encounter with DNS was while researching for my thesis relating to the role of hairpins in transition and turbulence(specifically originating from bypass transition mechanism). ChannelFlow code as simple as it was made me feel ever so powerful in my direct confrontation with turbulence… 😉

Turbulence phenomena is very precisely described by a seemingly simple set of equations, the Navier-Stokes equations, their nature is such that analytic solutions to even the most simple turbulent flows can not be obtained and resorting to numerical solutions seems like the only hope.

But the resourcefulness of the plea to a direct numerical description of the equations is a mixed blessing as it seems the availability of such a description is directly matched to the power of a dimensionless number reflecting on how well momentum is diffused relative to the flow velocity (in the cross-stream direction) and on the thickness of a boundary layer relative to the body – The Reynolds Number.


It is found that the computational effort in Direct Numerical Simulation (DNS) of the Navier-Stokes equations rises as Reynolds number in the power of 9/4 which renders such calculations as prohibitive for most engineering applications of practical interest and it shall remain so for the foreseeable future, its use confined to simple geometries and a limited range of Reynolds numbers  in the aim of supplying significant insight into turbulence physics that can not be attained in the laboratory.

Turbulent Boundary Layer (P. Schlatter and D. Henningson of KTH)

Saying all that, it is not expected that DNS will take on vital role in the engineering design process, where many designs are to be evaluated working through a repetitive cycle of obtaining a CAD geometry–> grid generation–>Solving the equation–>post-processing the results–>optimization decisions.
Nonetheless, DNS shall find its place in the industrial CFD community for specialized research as it does in the academy, where on the line of an academic study which lasts up to approximately 5 years only a few high-fidelity simulations are conducted.

So how high a Reynolds Number is Enough?

Understanding the limitation raised in the above paragraph leads to skepticism on the relevance of DNS for computing large scale engineering problems.


The enormous Reynolds number involved for real-life engineering problems and the assumption that in order to get valuable results such problems should be solved for similar Reynolds number does preclude serious hope for such DNS. However when approaching an engineering problem one should remember to make aid of qualitative universality apparent in turbulence flows and ask himself if there is not a high enough Reynolds to acquire valuable results about the simulated phenomena, i.e. ask one self what is the objectives to be drawn of such DNS.


As an example we may take a glimpse at the canonical problem of zero-pressure-gradient boundary layer (See P. Moin and X. Wu 2011). As one checks the turbulence statistics to understand the behavior of the Reynolds number dependence, scaling it with inner or outer variables, it can be seen that it does not collapse at different Reynolds number and the claim of strong dependence upon the Reynolds number is somewhat misleading. The fact that there is a dependence is a consequence of the scaling in use without significant change in the qualitative nature of the physics involved.


DNS of a zero-pressure-gradient flat plate boundary layer (P. Moin and X. Wu)

The fact of the matter is that accurate computation of mean flow and second order statistics often do not require significant scale separation between energetic and dissipative scales. The dependency impacts only if small scale features such as intermittency or high order statistics are to be explored. In this case indeed a wide separation is to be held in order for results to be congruent at higher Reynolds numbers.

All of the above leads to the conclusion that the Reynolds number should be as high as the objective of the calculation demands. DNS should be thought of as a research tool, and is still not a brute force solution for the NS equations.

Nevertheless the objective of using DNS to allow better insight in controlled studies of specific qualitative nature of the physics of the flow, improvement of scaling laws and last but not least the development of strongly improved and validated turbulence models (subject of Part IV) to specifically tuned for a branch of CFD for real-life engineering large scale problems could be fulfilled.

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