Can you briefly explain the differences between DNS, LES, and RANS?

DNS resolves all scales, LES resolves only large eddies and approximates small ones with SGS models, and RANS takes time averages and models all eddies. Accuracy is DNS > LES > RANS, but computational cost follows the same order. In practice, RANS dominates, LES is increasingly common, but DNS is mostly research.

In what situations can DNS actually be used?

Primarily low-Reynolds-number fundamental flows: channel flow, backward-facing step, mixing layers, jets, etc. Up to Re of several thousand to tens of thousands, modern supercomputers can handle it. The results are extremely valuable as validation data for turbulence models.

Direct Numerical Simulation (DNS) — CAE Glossary

Q: DNS means 'solving the Navier-Stokes equation directly without turbulence modeling,' right? Why is that special?

Turbulence contains a huge range of scales from large eddies to tiny ones, and trying to resolve all of them with a grid causes the grid points to scale as Re^(9/4), leading to explosive growth. That's why typical CFD uses turbulence models to approximate the effect of small eddies, but DNS doesn't do that at all. The major strength is that you get the 'true solution' with zero model error.

Q: I heard computational cost scales as Re^3. How difficult is that in practice?

For instance, a channel flow at Re=1,000 can be handled with a few million grid points, but for flow around an automobile at Re=10^7, the grid points would be on the order of 10^16, which is completely beyond current supercomputers. That's why industrial high-Reynolds-number problems are practically impossible with DNS; it's mostly used for academic benchmarking.

Category: Glossary | 2026-03-28

CAE visualization for dns - technical simulation diagram

What is DNS

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DNS means "solving the Navier-Stokes equation directly without turbulence modeling," right? Why is that special?

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Great question. Turbulence contains an enormous range of scales from large eddies to tiny ones, and energy cascades from large eddies to small ones. The smallest eddy scale is called the Kolmogorov scale $\eta$, and DNS resolves all eddies down to $\eta$ using the grid. Since it doesn't approximate with models, model error is zero—you get the "true solution," which is the greatest strength.

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How large is the Kolmogorov scale?

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The Kolmogorov scale is expressed by this equation:

$$\eta = \left(\frac{\nu^3}{\varepsilon}\right)^{1/4}$$

where $\nu$ is kinematic viscosity and $\varepsilon$ is the turbulent kinetic energy dissipation rate. For example, in pipe flow (Re≈10,000), $\eta$ is around 0.1 mm. For flow around an automobile (Re=10⁷), it becomes microns. To resolve this across the entire domain requires an astronomically large mesh.

DNS Computational Cost

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I heard computational cost scales as $Re^3$. How difficult is that in practice?

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Let's look at the scale ratio first. The ratio of the largest scale $L$ to the smallest scale $\eta$ is:

$$\frac{L}{\eta} \sim Re^{3/4}$$

In three dimensions, the number of grid points $N$ becomes:

$$N \sim \left(\frac{L}{\eta}\right)^3 \sim Re^{9/4}$$

Furthermore, you also need smaller time steps, so the total computational cost scales as $Re^3$. A channel flow at Re=1,000 can be handled with several million grid points, but for flow around an automobile at Re=10⁷, grid points reach 10¹⁶, which is completely beyond current supercomputers.

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10¹⁶ points? That's 10 quadrillion. That's impossible, right?

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Exactly. That's why applying DNS to industrial high-Reynolds number problems is practically impossible. Current supercomputers can achieve DNS at Re of a few thousand to tens of thousands; in 2024, the University of Tokyo's DNS on Fugaku for channel flow reached Re_τ≈8,000 or so, using trillions of grid points.

When DNS Can Be Used

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So where is DNS actually used? If it can't be used for much, what's the point?

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Actually, DNS has tremendous value. Here are the main applications:

Validation database for turbulence models: Compare the RANS or LES turbulence models against DNS "true solutions" to verify their validity. For example, Johns Hopkins Turbulence Database (JHTDB) makes DNS results publicly available, and it's referenced in countless studies worldwide.
Understanding turbulent physics: DNS can fully visualize 3D flow features that are difficult to measure experimentally, like vortex structure near walls and transition mechanisms.
Practical problems at low Reynolds numbers: Microfluidic devices or blood flow in biology (Re=hundreds to thousands) can sometimes be directly computed with DNS.

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So the idea is "create the exact solution with DNS, then use it to intelligently develop cheaper methods"?

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Exactly that. Recently, feeding DNS data into machine learning to develop better turbulence models is also booming. Fields called Physics-Informed Neural Networks (PINNs) and Data-Driven Turbulence Modeling use DNS data as essential "training data."

DNS vs LES vs RANS

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Can you summarize the differences between DNS, LES, and RANS?

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A rough summary looks like this:

Method	Eddy Resolution	Grid Points (Re=10⁴)	Application
DNS	All scales	~10⁷	Academic research & validation
LES	Large eddies only (SGS model)	~10⁵–10⁶	Aeroacoustics & unsteady flow
RANS	None (all modeled)	~10⁴–10⁵	Industrial design mainstream

Accuracy is DNS > LES > RANS, but computational cost increases in the same order. In practice, cost-accuracy tradeoff matters: RANS suffices for car air duct design, LES is needed for wing-tip vortex noise prediction, and DNS is referenced to validate those turbulence models.

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As supercomputers get faster, will we eventually be able to use DNS for flow around cars?

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Theoretically yes. But to solve DNS at Re=10⁷, you'd need 10⁶ to 10⁸ times more computing power than today, which is decades away even if Moore's Law continues. More realistically, hybrid approaches like WMLES (Wall-Modeled LES) and RANS/LES hybrids (DES, IDDES) are developing; those will have bigger practical impact.

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So DNS is like "ultimate accuracy but limited applicability—a special tool"?

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That's perfect. DNS is the "gold standard" of turbulence research and the foundation supporting other methods. In the CFD world, DNS results are always referenced as the trusted benchmark. Not used directly in practice, but indirectly it underpins the credibility of all CFD analysis—a vital technique.

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