Fundamentals of DNS (Direct Numerical Simulation)
Theory and Physics
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" — 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.
Physical Meaning of Each Term
- Temporal term $\partial(\rho\phi)/\partial t$: Imagine the moment you turn on a faucet. At first, water comes out in an unstable, spluttering manner, but after a while, it becomes a steady flow, right? This "period of change" is described by the temporal term. The pulsation of blood flow due to heartbeats, and the flow fluctuations each time an engine valve opens and closes are all unsteady phenomena. So what is steady-state analysis? It looks only at "after sufficient time has passed and the flow has settled down"—in other words, setting this term to zero. Since computational cost drops significantly, the basic CFD strategy is to first try solving it as steady-state.
- Convection term $\nabla \cdot (\rho \mathbf{u} \phi)$: What happens if you drop a leaf into a river? It gets carried downstream by the flow, right? This is "convection"—the effect where fluid motion transports things. Warm air from a heater reaching the far end of a room is also because the "carrier," air, transports heat via convection. Here's the interesting part—this term contains "velocity × velocity," making it nonlinear. That is, as the flow becomes faster, this term rapidly strengthens, making control difficult. This is the root cause of turbulence. A common misconception: "Convection and conduction are similar things" → They are completely different! Convection is carried by flow, conduction is transmitted by molecules. There is an order of magnitude difference in efficiency.
- Diffusion term $\nabla \cdot (\Gamma \nabla \phi)$: Have you ever put milk in coffee and left it? Even without stirring, it naturally mixes after a while, right? That's molecular diffusion. Now, next question—honey and water, which flows more easily? Obviously water, right? Honey has high viscosity ($\mu$), so it flows poorly. When viscosity is large, the diffusion term becomes strong, and the fluid moves in a "thick" manner. In low Reynolds number flow (slow, viscous), diffusion dominates. Conversely, in high Re number flow, convection overwhelmingly dominates, and diffusion plays a supporting role.
- Pressure term $-\nabla p$: When you push the plunger of a syringe, liquid shoots out forcefully from the needle tip, right? Why? Because the plunger side is high pressure, and the needle tip is low pressure—this pressure difference provides the force that pushes the fluid. Dam discharge works on the same principle. On a weather map, where isobars are tightly packed? That's right, strong winds blow. "Flow is generated where there is a pressure difference"—this is the physical meaning of the pressure term in the Navier-Stokes equations. A point of confusion here: The "pressure" in CFD is often gauge pressure, not absolute pressure. When you switch to compressible analysis and suddenly get strange results, it might be due to confusing absolute/gauge pressure.
- Source term $S_\phi$: Warmed air rises—why? Because it becomes lighter (lower density) than its surroundings, so it is pushed upward by buoyancy. This buoyancy is added to the equation as a source term. Other examples: chemical reaction heat generated by a gas stove flame, Lorentz force acting on molten metal in a factory's electromagnetic pump... These are all actions that "inject energy or force into the fluid from the outside," expressed by the source term. What happens if you forget the source term? In a natural convection analysis, if you forget to include buoyancy, the fluid doesn't move at all—a physically impossible result where warm air doesn't rise in a room with the heater on in winter.
Assumptions and Applicability Limits
- Continuum assumption: Valid for Knudsen number Kn < 0.01 (mean free path ≪ characteristic length)
- Newtonian fluid assumption: Shear stress and strain rate have a linear relationship (non-Newtonian fluids require viscosity models)
- Incompressibility assumption (for Ma < 0.3): Density is treated as constant. For Mach number ≥ 0.3, compressibility effects must be considered
- Boussinesq approximation (Natural convection): Density variation is considered only in the buoyancy term; constant density is used in other terms
- Non-applicable cases: Rarefied gas (Kn > 0.1), supersonic/hypersonic flow (shock capturing required), free surface flow (VOF/Level Set, etc. required)
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Velocity $u$ | m/s | When converting from volumetric flow rate for inlet conditions, pay attention to cross-sectional area units |
| Pressure $p$ | Pa | Distinguish between gauge and absolute pressure. Use absolute pressure for compressible analysis |
| Density $\rho$ | kg/m³ | Air: approx. 1.225 kg/m³ @20°C, Water: approx. 998 kg/m³ @20°C |
| Viscosity coefficient $\mu$ | Pa·s | Be careful not to confuse with kinematic viscosity $\nu = \mu/\rho$ [m²/s] |
| Reynolds number $Re$ | Dimensionless | $Re = \rho u L / \mu$. Criterion for laminar/turbulent transition |
| CFL number | Dimensionless | $CFL = u \Delta t / \Delta x$. Directly related to time step stability |
Numerical Methods and Implementation
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 — 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 turbulence—a rational choice born from the pursuit of accuracy.
Upwind Scheme
1st order upwind: Large numerical diffusion but stable. 2nd order upwind: Improved accuracy but risk of oscillations. Essential for high Reynolds number flows.
Central Differencing
2nd order accurate, but numerical oscillations occur for Pe number > 2. Suitable for low Reynolds number diffusion-dominated flows.
TVD Schemes (MUSCL, QUICK, etc.)
Maintain high accuracy while suppressing numerical oscillations via limiter functions. Effective for capturing shocks and steep gradients.
Finite Volume Method vs Finite Element Method
FVM: Naturally satisfies conservation laws. Mainstream in CFD. FEM: Advantageous for complex geometries and multiphysics. Mesh-free methods like SPH are also developing.
CFL Condition (Courant Number)
Explicit methods: CFL ≤ 1 is the stability condition. Implicit methods: Stable even for CFL > 1, but affects accuracy and iteration count. LES: CFL ≈ 1 recommended. Physical meaning: Information should not travel more than one cell per time step.
Residual Monitoring
Convergence is judged when the residuals for the continuity equation, momentum, and energy each drop by 3-4 orders of magnitude. The mass conservation residual is particularly important.
Relaxation Factor
Pressure: 0.2~0.3, Velocity: 0.5~0.7 are typical initial values. If diverging, lower the relaxation factor. After convergence, increase to accelerate.
Internal Iterations for Unsteady Calculations
Iterate within each time step until a steady solution converges. Internal iteration count: 5~20 iterations is a guideline. If residuals fluctuate between time steps, review the time step size.
Analogy for the SIMPLE Method
The SIMPLE method is an "alternating adjustment" technique. First, velocity is tentatively obtained (predictor step), then pressure is corrected so that mass conservation is satisfied with that velocity (corrector step), and velocity is revised using the corrected pressure—this catchball is repeated to approach the correct solution. It resembles two people leveling a shelf: one adjusts the height, the other...
Related Topics
なった
詳しく
報告