Fundamentals of DNS (Direct Numerical Simulation)
Fundamentals of DNS (Direct Numerical Simulation): Theoretical Foundations
Overview
Professor, DNS directly resolves all scales of turbulence, right? How does it work?
DNS (Direct Numerical Simulation) is a method that directly solves the Navier-Stokes equations without any turbulence models. It resolves all vortices from the Kolmogorov microscale $\eta$ to the integral scale $L$ using a mesh. It reproduces turbulence physics most faithfully, but requires computational resources of a different order of magnitude.
Required Resolution
How many mesh points are needed for DNS?
The number of mesh points in each direction is proportional to $L/\eta$. The Kolmogorov scale is $\eta = (\nu^3/\varepsilon)^{1/4}$, and its ratio to the integral scale is determined by the Reynolds number.
In three dimensions, this many grid points are needed in each direction, so the total number of grid points is,
How much is that for $Re = 10^4$?
$N \sim (10^4)^{9/4} \approx 10^9$. That's about 1 billion grid points. Furthermore, the number of time steps is proportional to $Re_L^{1/2}$, so the total computational cost is proportional to $Re_L^{11/4}$. For industrial-level $Re \sim 10^6$, $N \sim 10^{13.5}$, which is currently impossible with today's computers.
Governing Equations
What equations are solved in DNS?
For incompressible flow, it's the standard Navier-Stokes equations themselves.
There are no model terms whatsoever. This is DNS's greatest strength, and the results can be considered the "exact solution" of the N-S equations. It is the most reliable validation data for RANS and LES models.
Kolmogorov Scaling
Could you tell me more about the Kolmogorov scale?
It's the smallest scale in the turbulent energy cascade.
| Scale | Expression | Physical Meaning |
|---|---|---|
| Length scale $\eta$ | $(\nu^3/\varepsilon)^{1/4}$ | Size of the smallest vortex dominated by viscous dissipation |
| Velocity scale $u_\eta$ | $(\nu\varepsilon)^{1/4}$ | Velocity of the smallest vortex |
| Time scale $\tau_\eta$ | $(\nu/\varepsilon)^{1/2}$ | Turnover time of the smallest vortex |
DNS requires $\Delta x \leq \pi\eta$ (as a guideline, $\Delta x \approx 2\eta$) and $\Delta t \leq \tau_\eta$. If the mesh does not satisfy this condition, small-scale vortices will cause aliasing and make the computation unstable.
DNS's "9/4 Power Wall" โ How much does the computational cost increase when $Re$ is doubled?
The number of grid points required for DNS increases on the order of $N \sim Re^{9/4}$ to resolve down to the Kolmogorov scale. For example, doubling $Re$ increases the computational cost by a factor of about $2^{9/4} \approx 4.76$. There are estimates that DNS for a practical aircraft wing ($Re \sim 10^7$) would take decades even with current supercomputers. DNS is often spoken of as a symbol of "the researcher's ideal versus the wall of reality" because this ruthless exponential story tells it all.
Computational Methods for Fundamentals of DNS (Direct Numerical Simulation)
DNS Numerical Methods
What kind of numerical schemes are used in DNS?
In DNS, high-precision schemes are essential so that numerical errors do not contaminate the physics of turbulence.
| Method | Spatial Accuracy | Application | Representative Codes |
|---|---|---|---|
| Spectral Method | Exponential (spectral accuracy) | Periodic simple geometries | Channelflow, SIMSON |
| Compact Finite Difference (6th order) | 6th order | Slightly complex geometries | Incompact3d |
| Standard Finite Difference (2nd order central) | 2nd order | General geometries | OpenFOAM, Nek5000 |
| Spectral Element Method | High order (p-order) | Complex geometries | Nek5000, Nektar++ |
Is the spectral method the most accurate?
Yes. For simple geometries with periodic boundary conditions (channel flow, isotropic turbulence box, etc.), FFT-based spectral methods are the most efficient. Aliasing of high-wavenumber components is removed by the 3/2 rule or Phase Shift. However, they cannot be applied to complex geometries.
Pressure Poisson Equation
How is pressure calculated in incompressible DNS?
To satisfy the continuity equation, the pressure Poisson equation is solved.
In the spectral method, it can be solved directly in wavenumber space, making it very efficient. In finite difference methods, iterative methods (PCG, FFT-based direct solver, etc.) are used.
Time Integration
What about the scheme in the time direction?
The standard approach is to combine an implicit method (Crank-Nicolson) for the viscous term with an explicit method (3rd order Adams-Bashforth or low-storage Runge-Kutta) for the convection term.
| Scheme | Accuracy | Stability | Typical Use |
|---|---|---|---|
| AB3 + CN | 2nd~3rd order | Conditionally stable (CFL constraint) | Channel flow DNS |
| RK3 + CN | 3rd order | Good | High-precision DNS |
| RK4 | 4th order | Strict CFL constraint | Spectral method DNS |
How small is the time step in DNS?
Both the CFL condition $\Delta t \leq \Delta x / U_{\max}$ and the viscous stability condition $\Delta t \leq \Delta x^2 / (2\nu)$ must be satisfied. For channel flow DNS ($Re_\tau = 590$), $\Delta t^+ = \Delta t u_\tau^2/\nu \approx 0.02$. Computations require tens to hundreds of thousands of time steps in physical time.
Why Spectral Methods Are Chosen for DNS โ The Temptation of "Zero Differentiation Error"
The reason Fourier spectral methods are preferred as a numerical method for DNS is that "the accuracy of differentiation is theoretically infinite." In physical space difference methods, differentiation error depends on grid spacing, but in Fourier spectral methods, each wavenumber mode is treated independently, yielding accuracy equivalent to analytical differentiation. However, the drawbacks are clear: "only periodic boundary conditions can be handled" and "cannot be extended to non-uniform grids." That's why problems handled in DNS are concentrated on "simple geometry problems" like channel flow and isotropic turbulenceโa rational choice born from the pursuit of accuracy.
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