Direct Numerical Simulation (DNS) — why it stays a research tool

A napkin calculation showing why the cost of solving the Navier–Stokes equations without any turbulence model explodes as the 9/4 power of the Reynolds number.

Opening

In Chapter 9 we saw that the largest eddy LL (capital L, the integral scale) and the smallest eddy η\eta (eta, the Kolmogorov scale, where viscosity finally turns kinetic energy into heat) coexist in the same flow. This chapter asks what happens to a computer the moment you try to fit both ends onto a single grid. By the end you can answer “why is DNS not used in industry?” with a one-line napkin calculation.

Main 1 — Definition of DNS: no model

DNS (Direct Numerical Simulation) means solving the Navier–Stokes equations directly, with no turbulence model at all. The RANS approach of Chapter 5 splits off a mean field and closes the unknown −ui′uj′‾-\overline{u'_i u'_j} term with a model; the LES approach of Chapter 11 resolves only the large eddies and models the small ones. DNS has no such closure. Instead it refines the grid all the way down to η\eta and resolves every scale of the energy cascade numerically and directly.

For that reason DNS is often called “a numerical experiment without approximation.” It produces new turbulence statistics with the same standing as a wind-tunnel measurement — a research instrument, not a design tool.

Main 2 — Grid count scales as Re9/4\mathrm{Re}^{9/4}

Only one estimate is needed. Let the side length of the domain be LL and the smallest cell size be η\eta. The number of grid points along one direction is simply

Lη\frac{L}{\eta}

From Chapter 9, with the integral-scale Reynolds number Re=UL/ν\mathrm{Re} = UL/\nu (where ν\nu, nu, is the kinematic viscosity),

Lη∼Re3/4\frac{L}{\eta} \sim \mathrm{Re}^{3/4}

In three dimensions all three directions have to be resolved, so the total grid count NN is

N∼(Lη)3∼Re9/4N \sim \left(\frac{L}{\eta}\right)^3 \sim \mathrm{Re}^{9/4}

That exponent 9/4 equals 2.25. A tenfold increase in Reynolds number multiplies the grid by about 178; a hundredfold increase multiplies it by about 31,623.

Add the CFL stability condition on the time step, Δt∝Δx/U∝η/U\Delta t \propto \Delta x / U \propto \eta / U, and the combined space-and-time cost scales as Re3\mathrm{Re}^3. The table below sticks to the spatial grid alone.

Main 3 — Napkin numbers: at what Re does a supercomputer become mandatory

The single relation N∼Re9/4N \sim \mathrm{Re}^{9/4} already decides whether DNS is practical.

Re\mathrm{Re}N∼Re9/4N \sim \mathrm{Re}^{9/4}What it really means
10210^2∼104.5≈3×104\sim 10^{4.5} \approx 3 \times 10^4Minutes on a laptop
10310^3∼106.75≈6×106\sim 10^{6.75} \approx 6 \times 10^6An overnight run on a workstation
10410^4∼109≈109\sim 10^{9} \approx 10^9One run on a national supercomputer
10510^5∼1011.25≈2×1011\sim 10^{11.25} \approx 2 \times 10^{11}A multi-year project on a top-tier system
10610^6∼1013.5≈3×1013\sim 10^{13.5} \approx 3 \times 10^{13}Out of reach on current hardware

External automotive aerodynamics sits at Re∼107\mathrm{Re} \sim 10^7, an airliner wing at Re∼108\mathrm{Re} \sim 10^8. Real industrial problems live two or three rows below the bottom of this table. So DNS is reserved for validating industrial codes and producing reference data for new models, while RANS and LES occupy the design loop.

In Python

Let us print the table for ourselves. numpy only, just Re9/4\mathrm{Re}^{9/4}.

import numpy as np

# DNS grid-point estimate: N ~ Re^(9/4)
# 9/4 = 2.25, lifting Chapter 9's L/eta ~ Re^(3/4) into three dimensions

Re_values = [1e2, 1e3, 1e4, 1e5, 1e6]
exponent = 9 / 4  # 2.25

print(f"{'Re':>10} | {'N ~ Re^(9/4)':>15}")
print("-" * 30)
for Re in Re_values:
    N = Re ** exponent
    print(f"{Re:>10.1e} | {N:>15.2e}")

# Note: adding the CFL condition on the time step pushes the total cost
# to Re^3. The table above counts only the spatial grid; wall-clock cost
# rises even faster.

Running it shows N≈3.16×109N \approx 3.16 \times 10^9 at Re=104\mathrm{Re} = 10^4. Chapter 1’s claim that “the Reynolds number alone fixes the qualitative state of the flow” gets a computational rephrasing: a single variable also fixes the boundary of what is computable.

To the next chapter

If the DNS grid cost is unacceptable, two paths remain. Average out every fluctuation and model what is left (RANS, Chapters 5–6), or resolve only the large eddies and model the small ones (LES). Chapter 11: Large Eddy Simulation (LES) takes the second path quantitatively — the central question is how DNS’s Re9/4\mathrm{Re}^{9/4} shrinks under LES.