Channel flow DNS
Theory and Physics
Overview
Professor, does DNS of channel flow hold a special position in the CFD world?
It's the most fundamental benchmark in turbulence research. The DNS at $Re_\tau = 180$ by Kim, Moin & Moser (1987) was groundbreaking and has since become the gold standard for turbulence model validation.
What is $Re_\tau$?
It's the friction Reynolds number, defined by the wall friction velocity $u_\tau$ and the channel half-width $\delta$.
Here, $\tau_w$ is the wall shear stress. $Re_\tau$ is a dimensionless number representing the ratio of the inner scale to the outer scale of wall turbulence.
Wall Law
Is the wall law also related?
The most important theoretical achievement for turbulent channel flow is the wall law. The distance from the wall is non-dimensionalized using inner variables.
In the viscous sublayer ($y^+ < 5$), there is a linear relationship:
In the logarithmic region ($y^+ > 30$), the log-law applies:
Here, $\kappa \approx 0.41$ (von Karman constant) and $B \approx 5.2$.
So DNS data is used to validate this law, right?
The DNS at $Re_\tau = 590$ by Moser, Kim & Mansour (1999) (commonly known as the MKM dataset) is an important dataset that confirmed the universality of the wall law with high precision. Currently, data up to $Re_\tau = 5200$ (Lee & Moser, 2015) exists.
Major DNS Databases
| Researchers | $Re_\tau$ | Grid Points | Year |
|---|---|---|---|
| Kim, Moin & Moser | 180 | $192 \times 129 \times 160$ | 1987 |
| Moser, Kim & Mansour | 180, 395, 590 | Up to $384 \times 257 \times 384$ | 1999 |
| Hoyas & Jimenez | 2003 | $6144 \times 633 \times 4608$ | 2006 |
| Lee & Moser | 5200 | $10240 \times 1536 \times 7680$ | 2015 |
10 billion grid points at $Re_\tau = 5200$? That's an incredible computational scale.
The story of how the world's first turbulent DNS in 1987 took "several weeks"
When Kim, Moin & Moser published their DNS (Direct Numerical Simulation) of channel flow in 1987, the computation took several weeks using the highest-performance supercomputer of the time. The Reynolds number was only 180 (based on friction velocity), with about 4 million grid points. Today, this calculation could be completed in a few days on a laboratory workstation. What made this paper groundbreaking was that it "numerically visualized the internal structure of turbulence for the first time." The moment streak structures and streamwise vortices near the wall appeared as predicted by theory, fluid dynamics researchers worldwide were reportedly thrilled. That paper is still cited thousands of times per year.
Physical Meaning of Each Term
- Temporal Term $\partial(\rho\phi)/\partial t$: Imagine the moment you turn on a faucet. At first, the 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, or the flow fluctuations each time an engine valve opens and closes—all are unsteady phenomena. So what is a steady-state analysis? It looks only at "after sufficient time has passed and the flow has settled down"—meaning this term is set to zero. Since computational cost drops significantly, starting with a steady-state solution is a basic CFD strategy.
- 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 corner 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 speed increases, 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 transport by flow, conduction is transfer by molecules. There's 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. That's molecular diffusion. Now, a question—honey and water, which flows more easily? Obviously water, right? Honey has high viscosity ($\mu$), so it flows poorly. Higher viscosity strengthens the diffusion term, making the fluid move in a "thick" manner. In low Reynolds number flows (slow, viscous), diffusion dominates. Conversely, in high Re number flows, convection overwhelmingly dominates, and diffusion plays a supporting role.
- Pressure Term $-\nabla p$: When you push the plunger of a syringe, the liquid shoots out forcefully from the needle tip, right? Why? Because the piston 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: "Pressure" in CFD is often gauge pressure, not absolute pressure. If results go wrong immediately after switching to compressible analysis, confusion between absolute/gauge pressure might be the cause.
- Source Term $S_\phi$: Warmed air rises—why? Because it becomes lighter (lower density) than its surroundings, so it's pushed up by buoyancy. This buoyancy is added to the equation as a source term. Other examples: chemical reaction heat from 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 natural convection analysis, forgetting to include buoyancy means 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: Linear relationship between shear stress and strain rate (viscosity model required for non-Newtonian fluids)
- Incompressibility Assumption (for Ma < 0.3): Density is treated as constant. For Mach number above 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 pressure 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 coefficient $\nu = \mu/\rho$ [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 Method
What kind of algorithm is used to solve DNS?
DNS (Direct Numerical Simulation) resolves all scales of turbulence. It solves the Navier-Stokes equations directly without models. For channel flow DNS, the pseudo-spectral method is mainstream.
Pseudo-Spectral Method
For the streamwise ($x$) and spanwise ($z$) directions, periodic boundary conditions allow Fourier series expansion; for the wall-normal direction ($y$), Chebyshev polynomial expansion is used.
The nonlinear term (convection term) is calculated in physical space and transformed to wavenumber space via FFT. This is why it's called the "pseudo" spectral method.
So the nonlinear term isn't calculated directly in wavenumber space.
Calculating the convection term in wavenumber space results in a convolution, increasing computational cost to $O(N^2)$. Calculating in physical space and using FFT keeps it at $O(N \log N)$. However, aliasing errors occur, so the 3/2 rule (de-aliasing) is applied.
Grid Resolution Requirements
DNS requires grids smaller than the Kolmogorov scale $\eta$. For channel flow, this is expressed in inner variables.
| Direction | Recommended Resolution | Remarks |
|---|---|---|
| Streamwise $\Delta x^+$ | 5〜10 | Resolves streak structures |
| Spanwise $\Delta z^+$ | 3〜5 | Streak structure width $\lambda_z^+ \approx 100$ |
| Wall-normal $\Delta y^+_{wall}$ | < 1 | Resolves viscous sublayer |
| Wall-normal $\Delta y^+_{center}$ | 5〜10 | Channel center |
Time Integration
What about time integration?
A hybrid approach is standard: implicit method (Crank-Nicolson) for the viscous term and explicit method (3rd-order Adams-Bashforth or 3-stage Runge-Kutta) for the nonlinear term. The CFL condition is determined by the explicit part, typically $\Delta t^+ \approx 0.1$〜$0.5$.
How long do you need to integrate to obtain statistics?
A wash-out time of $T u_\tau / \delta > 10$ is needed to reach statistical stationarity, followed by sampling requiring $T u_\tau / \delta > 20$〜$50$. Convergence of statistics is slower for higher-order moments.
Why the spectral method became the "standard" for channel DNS
The spectral method (Fourier expansion + Chebyshev polynomials) is widely used in channel flow DNS because it can "achieve the same accuracy at lower cost" compared to finite difference methods. Resolving all scales of turbulence requires extremely high accuracy, and finite difference methods would require an extreme increase in grid points to keep up. On the other hand, the spectral method can solve smooth, periodic flows with "mathematically maximum efficiency." However, it also has drawbacks, such as difficulty with complex geometries. That's why benchmark DNS calculations tend to concentrate on simple shapes like channels and pipes, which in turn has led to the enrichment of databases.
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 shapes and multiphysics. Mesh-free methods like SPH are also developing.
CFL Condition (Courant Number)
Explicit method: CFL ≤ 1 is the stability condition. Implicit method: 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 residuals for the continuity equation, momentum, and energy drop by 3-4 orders of magnitude. The mass conservation residual is particularly important.
Relaxation Factor
Typical initial values: Pressure: 0.2〜0.3, Velocity: 0.5〜0.7. Reduce the factor if divergence occurs. Increase after convergence to accelerate.
Internal Iterations for Unsteady Calculations
Iterate within each time step until a steady solution converges. Internal iteration count: 5〜20 times as 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 determined (predictor step), then pressure is corrected so that mass conservation is satisfied with that velocity (corrector step), and the velocity is revised using the corrected pressure—this back-and-forth is repeated to approach the correct solution. It resembles two people leveling a shelf: one adjusts the height, the other balances it, and they repeat this alternately.
Analogy for the Upwind Scheme
The upwind scheme is a method that "stands in the river flow and prioritizes upstream information." A person in the river cannot tell where the water comes from by looking downstream—it's a discretization method reflecting the physics that upstream information determines downstream conditions. Although it's first-order accurate, it is highly stable because it correctly captures the flow direction.
Practical Guide
Practical Guide
How can I use DNS data to validate RANS models?
Performing DNS itself requires large-scale computation, but in practice, it's important to utilize publicly available data to validate the accuracy of RANS or LES.
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