Turbulence — CAE Glossary
What is Turbulence
"Turbulence" comes up repeatedly in textbooks, but what exactly is it? My understanding is limited to it being the opposite of laminar flow….
Put simply, it is a flow state where velocity and pressure fluctuate randomly both in time and space. In laminar flow, fluid particles move in orderly, parallel layers, but in turbulence, vortices of various sizes intertwine in three dimensions, causing intense mixing. For example, smoke rising from a chimney flows straight at first but becomes chaotic partway up—that's the transition to turbulence.
Under what conditions does turbulence occur?
Transition occurs when Reynolds number $\mathrm{Re} = \dfrac{\rho U L}{\mu}$ exceeds a critical value. For pipe flow, Re ≈ 2300 is the threshold; for a flat plate boundary layer, Rex ≈ 5×105. Physically, inertial forces (which disturb the flow) overcome viscous forces (which stabilize it), triggering turbulence. Most industrial flows—air around cars, water in pipes, combustor flames in jet engines—operate at Re = 105–107, which is fully turbulent.
Since turbulence is "random," doesn't that mean it's unpredictable? Wouldn't that make simulation impossible?
Good question. The instantaneous behavior of individual eddies is indeed chaotic and difficult to predict long-term. However, statistical properties (mean velocity, variance, energy spectrum, etc.) are reproducible. That's why in CFD we adopt the strategy of "predicting statistical quantities correctly." It's similar to weather forecasting—we can predict tomorrow's temperature but not the exact shape of a specific cloud.
Kolmogorov Cascade and Energy Spectrum
I often hear about the "Kolmogorov cascade." How does it work?
Proposed by Kolmogorov in 1941, this theory explains how turbulent energy moves across eddy size scales. There are three stages:
1. Energy-Containing Range (Large Scale): From boundary conditions (walls, obstacles, shear layers), main-flow energy is injected into large eddies. The eddy size is on the order of a characteristic length $L$.
2. Inertial Subrange (Intermediate Scale): Large eddies become unstable and split into slightly smaller eddies, which split further. In this range, viscosity barely matters; energy cascades "like a waterfall" to smaller scales—hence the term "cascade."
3. Dissipation Range (Smallest Scale): At the Kolmogorov scale $\eta = (\nu^3/\varepsilon)^{1/4}$, viscous forces dominate and energy converts to heat. This is the endpoint.
How large is the Kolmogorov scale $\eta$ in practice?
The scale ratio is $L/\eta \sim \mathrm{Re}^{3/4}$. For instance, flow around a car (L ≈ 1 m, Re ≈ 106) has $\eta$ ≈ 0.03 mm—half the thickness of a human hair. This explains why resolving all scales with DNS demands an enormous mesh.
The famous $E(\kappa) \propto \kappa^{-5/3}$ formula in energy spectrum—which part of the cascade does it correspond to?
It corresponds to the inertial subrange. Kolmogorov argued via dimensional analysis that in this region, the energy spectrum depends only on dissipation rate $\varepsilon$ and wavenumber $\kappa$:
where $C_K \approx 1.5$ is the Kolmogorov constant. This $-5/3$ law has been confirmed repeatedly in experiments and is regarded as one of the most beautiful results in turbulence theory. When you plot energy density $E(\kappa)$ against wavenumber $\kappa$ on log-log axes, a clean straight line appears in the inertial subrange.
Reynolds Decomposition and Turbulence Modeling
When simulating turbulence, I've heard we first do "Reynolds decomposition." What does that accomplish?
It means splitting instantaneous flow quantities into time-averaged and fluctuating components. For velocity:
where $\overline{u}_i$ is time-averaged velocity and $u'_i$ is the fluctuation (which averages to zero). Pressure decomposes similarly: $p = \bar{p} + p'$. When you substitute this decomposition into the Navier-Stokes equation and time-average, you obtain the Reynolds-Averaged Navier-Stokes (RANS) equation.
Does decomposing alone simplify the equation?
Actually, the opposite occurs—new terms are introduced, not eliminated. After averaging, a term $-\rho\overline{u'_i u'_j}$ appears, called Reynolds stress, which captures the correlation between velocity fluctuations. We have more unknowns than equations—this is the famous turbulence closure problem.
If it doesn't close, how do you solve it?
We introduce a turbulence model to approximate the Reynolds stress in terms of known quantities, closing the system. The most widely used is the eddy viscosity model (Boussinesq hypothesis), which assumes Reynolds stress is proportional to mean strain rate:
Here, $\nu_t$ is eddy viscosity and $k = \frac{1}{2}\overline{u'_i u'_i}$ is turbulent kinetic energy. To find $\nu_t$, we solve additional transport equations like the $k$-$\varepsilon$ or $k$-$\omega$ SST model.
RANS / LES / DNS Hierarchy
I hear about LES and DNS besides RANS. What are the differences, and how do you choose between them?
The difference is how many eddy scales you directly compute. Recall the Kolmogorov cascade:
RANS (Reynolds-Averaged Navier-Stokes): All eddy scales are approximated by models. Cheapest computationally. Produces steady solutions, ideal for design parameter studies. However, large-scale unsteady behavior is missed.
LES (Large Eddy Simulation): Directly computes large eddies that the mesh can resolve; only smaller eddies are modeled via SGS models. Requires unsteady computation and costs 10–100 times more than RANS, but captures eddy structure and acoustic phenomena directly.
DNS (Direct Numerical Simulation): Directly resolves all eddies down to the Kolmogorov scale $\eta$. No turbulence models needed—most accurate—but required mesh count scales as $N \sim \mathrm{Re}^{9/4}$, limiting practical Re to about 104. Mainly used in academic research and model validation.
So in practical design work, you mostly use RANS?
Yes, RANS accounts for over 80% of industrial CFD today. In particular, the $k$-$\omega$ SST model is so versatile (from external aerodynamics to internal flows) that there's an implicit rule: "when in doubt, use SST." However, GPU availability is making LES more affordable, and it's increasingly used in automotive aerodynamics and turbine combustors. Hybrid methods like DES (Detached Eddy Simulation) and SAS (Scale-Adaptive Simulation) are also gaining traction.
How much does computational cost differ concretely?
Here's a rough benchmark for external flow around a 5 m car at Re = 106:
RANS: 10–50 million cells, steady-state convergence in hours to a day (100 cores).
LES: 100 million–1 billion cells, unsteady integration for days to weeks (1000+ cores).
DNS: 1013–1014 cells—currently infeasible even on supercomputers.
Understanding this scaling is key to selecting the right approach.
Turbulence Intensity and Boundary Conditions
Turbulence intensity always shows up in CFD inlet boundary conditions. What does it represent?
Turbulence intensity $TI$ is a dimensionless indicator of how "disturbed" a flow is. It is defined as:
where $u'_{\mathrm{rms}}$ is the root-mean-square of velocity fluctuations and $\overline{U}$ is mean velocity. Typical ranges:
- High-quality wind tunnel: TI = 0.1–0.5% (very clean flow)
- Pipe exit: TI = 1–5%
- Atmospheric environment: TI = 5–20%
- Gas turbine combustor exit: TI = 10–25%
Does setting inlet turbulence intensity carelessly affect results?
It does—especially for predicting boundary layer transition position and separation. Higher TI triggers transition earlier, forming a turbulent boundary layer that delays separation. Conversely, underestimated TI can falsely predict laminar separation where none exists. Best practice: use measured data if available; otherwise, use guidelines (internal flows: TI ≈ 5%, external: TI ≈ 1%), then perform sensitivity studies.
Besides inlet TI, I also see turbulence length scale and turbulence viscosity ratio. Do those need setting too?
Yes. For $k$-$\varepsilon$, you need inlet $k$ and $\varepsilon$; compute $k = \frac{3}{2}(TI \cdot \overline{U})^2$ from TI, then estimate $\varepsilon = C_\mu^{3/4} k^{3/2}/\ell$ using turbulence length scale $\ell$ (e.g., 7–10% of duct height). For $k$-$\omega$ SST, $\omega = k^{1/2}/(C_\mu^{1/4}\ell)$. It seems tedious, but sloppy inlet conditions distort the flow development for 5–10 diameters downstream. Don't skip this step.
Practical Points
Finally, what's the most common pitfall in turbulence CFD?
Three stand out:
1. Inadequate wall meshing: Near walls, turbulent velocity gradients change sharply. Neglecting $y^+$ management distorts wall shear and heat transfer predictions. Use wall functions at $y^+ = 30$–100, or resolve the wall region at $y^+ < 1$ with low-Re models.
2. Turbulence model selection errors: Standard $k$-$\varepsilon$ underestimates separation in strong adverse pressure gradients; even SST can struggle with swirl. Know each model's strengths and weaknesses.
3. Numerical dissipation in LES: Since LES directly computes eddies, excessive numerical dissipation from the discretization scheme damps eddies artificially. Second-order upwind schemes are too dissipative; use central difference or high-order schemes.
So turbulence is quite deep. You need to understand Kolmogorov physics and model limitations, then apply CFD mindfully.
Exactly. Maintain humility—"we cannot fully solve turbulence" (except via DNS)—while strategically selecting models and resolution to match your prediction goal. Nobel laureate Feynman once called turbulence "the last unsolved problem in classical physics." That wisdom guides good practice.
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