8.5 Turbulence modeling in CFD

Turbulence in fluid dynamics

Turbulence is a chaotic state of fluid motion where velocity, pressure, and other flow properties fluctuate irregularly across a wide range of scales. Accurately modeling this behavior is central to CFD predictions in aerodynamics, combustion, and heat transfer. The multiscale nature of turbulence forces a fundamental tradeoff: resolving more detail costs more computation, so different modeling strategies exist depending on what you can afford and what you need.

Characteristics of turbulent flows

Randomness and irregularity

Turbulent flows show random, chaotic fluctuations in velocity and pressure that span a wide range of length and time scales. This makes turbulence inherently unpredictable in a pointwise sense, though its statistical properties (means, variances, spectra) can be described and modeled. The mathematical challenge is that no closed-form solution exists for the Navier-Stokes equations in turbulent regimes.

Diffusivity and mixing

Turbulence dramatically enhances the transport of momentum, heat, and mass compared to laminar flow. Turbulent eddies stir the fluid, leading to rapid mixing and homogenization of properties. This is why turbulent mixing matters so much in combustion chambers, chemical reactors, and heat exchangers: the mixing rate often controls the overall process performance.

Dissipation and energy cascade

Energy in turbulent flow moves from large eddies to progressively smaller ones through what's called the energy cascade. At the smallest scales (the Kolmogorov microscales), viscosity converts kinetic energy into heat. The rate at which turbulent kinetic energy dissipates, $\epsilon$, is one of the most important quantities in turbulence modeling because it sets the scale at which energy is ultimately removed from the flow.

Approaches to turbulence modeling

Three main strategies exist, each resolving a different portion of the turbulent spectrum:

Direct numerical simulation (DNS)

DNS solves the full Navier-Stokes equations without any modeling assumptions. Every scale of motion is resolved, from the largest energy-containing eddies down to the Kolmogorov microscales. The grid must be fine enough to capture the smallest dissipative structures, and the time step must resolve the fastest fluctuations.

The computational cost scales roughly as $Re^3$ (in 3D), which makes DNS feasible only for relatively low Reynolds numbers. It serves primarily as a research tool and a benchmark for validating cheaper models.

Large eddy simulation (LES)

LES directly resolves the large, energy-containing eddies and models only the small-scale (subgrid-scale) motions using SGS models. Since the large eddies carry most of the energy and are geometry-dependent, resolving them captures the dominant physics. The small scales tend to be more universal and easier to model.

LES is far cheaper than DNS but still requires substantially finer grids than RANS, especially near walls. It's well suited for flows with large-scale unsteadiness, separation, or mixing where RANS struggles.

Reynolds-averaged Navier-Stokes (RANS)

RANS decomposes every flow variable into a time-averaged mean and a fluctuating component, then solves equations for the mean flow. The effect of turbulent fluctuations appears as the Reynolds stress tensor, which must be modeled through turbulence closure models. This introduces additional transport equations for quantities like turbulent kinetic energy ($k$) and dissipation rate ($\epsilon$ or $\omega$).

RANS is by far the cheapest approach and dominates industrial CFD. The tradeoff is that all unsteady turbulent content is modeled rather than resolved, which limits accuracy in flows with strong separation, swirl, or transient behavior.

RANS turbulence models

Spalart-Allmaras model

This is a one-equation model that solves a single transport equation for a modified turbulent viscosity, $\tilde{\nu}$. It was developed specifically for aerodynamic applications and performs well in attached and mildly separated wall-bounded flows. Its simplicity and robustness make it popular in the aerospace industry, though it lacks the generality of two-equation models for complex internal flows.

k-epsilon models

These two-equation models solve transport equations for the turbulent kinetic energy $k$ and its dissipation rate $\epsilon$. The turbulent viscosity is then computed as:

$\nu_t = C_\mu \frac{k^2}{\epsilon}$

The standard k-epsilon model is the workhorse of industrial CFD due to its simplicity and reasonable accuracy in many shear flows. However, it performs poorly in the near-wall region (requiring wall functions) and in flows with strong streamline curvature or rotation. Two common variants address specific weaknesses:

• Realizable k-epsilon: enforces a physical constraint on the normal stresses, improving behavior in flows with strong strain rates
• RNG k-epsilon: derived from renormalization group theory, provides better performance in rapidly strained and swirling flows

k-omega models

These two-equation models solve for $k$ and the specific dissipation rate $\omega$ (which can be thought of as $\omega \sim \epsilon / k$). The standard k-omega model handles near-wall regions and adverse pressure gradients better than k-epsilon, but it's sensitive to freestream values of $\omega$.

The SST (Shear Stress Transport) k-omega model, developed by Menter, blends k-omega behavior near walls with k-epsilon behavior in the freestream. It also includes a limiter on the eddy viscosity to improve prediction of flow separation. The SST model is one of the most widely recommended general-purpose RANS models.

Reynolds stress models

Reynolds stress models (RSM) solve individual transport equations for all six independent components of the Reynolds stress tensor $\overline{u_i' u_j'}$, plus an equation for $\epsilon$. This means RSM can capture anisotropic turbulence, which eddy-viscosity models (Spalart-Allmaras, k-epsilon, k-omega) cannot, since those assume the Reynolds stresses are proportional to the mean strain rate.

RSM is more accurate for flows with strong swirl, rotation, or secondary motions. The cost is higher (seven equations instead of two) and the models are harder to converge numerically, so they're used selectively rather than as a default choice.

LES turbulence models

Smagorinsky model

The Smagorinsky model is the simplest and most widely used SGS model. It computes a subgrid-scale eddy viscosity as:

$\nu_{SGS} = (C_s \Delta)^2 |\bar{S}|$

where $C_s$ is the Smagorinsky constant (typically around 0.1–0.2), $\Delta$ is the filter width (related to grid spacing), and $|\bar{S}|$ is the magnitude of the resolved strain-rate tensor. The constant $C_s$ is set empirically and may need tuning for different flows, which is a notable limitation.

Dynamic Smagorinsky model

The dynamic model, proposed by Germano, removes the need to prescribe $C_s$ by computing it on the fly from the resolved field. It applies a test filter (coarser than the grid filter) and uses the Germano identity to extract the coefficient from the energy content between the two filter levels.

This makes the model self-adjusting: $C_s$ varies in space and time, naturally going to zero near walls and in laminar regions. The dynamic model significantly improves accuracy and robustness compared to the constant-coefficient version.

Wall-adapting local eddy-viscosity (WALE) model

The WALE model is designed to give correct near-wall scaling of the SGS viscosity without requiring damping functions. It uses both the strain rate and the rotation rate of the resolved field to compute $\nu_{SGS}$, which naturally produces the correct $y^3$ behavior of eddy viscosity near walls (where $y$ is the wall distance). This makes WALE particularly attractive for wall-bounded flows and complex geometries where defining a wall distance for damping functions would be awkward.

Hybrid RANS-LES methods

Full LES of wall-bounded flows at high Reynolds numbers is extremely expensive because the near-wall eddies become very small. Hybrid methods address this by using RANS in the boundary layer and LES in separated or free-shear regions.

Detached eddy simulation (DES)

DES uses a RANS model (typically Spalart-Allmaras) in attached boundary layers and switches to LES where the local grid spacing is fine enough to resolve turbulent structures. The switch is controlled by comparing the turbulent length scale from the RANS model to the local grid size. When the grid is fine relative to the modeled length scale, the model behaves as LES.

A known problem with original DES is grid-induced separation: if the grid in the boundary layer is ambiguously sized (too fine for RANS, too coarse for LES), the model can prematurely switch to LES mode and produce nonphysical separation.

Delayed detached eddy simulation (DDES)

DDES addresses grid-induced separation by adding a shielding function that keeps the boundary layer in RANS mode regardless of grid refinement. The switch to LES is "delayed" until the flow is clearly in a separated or detached region. This makes DDES more robust and less sensitive to grid design than original DES.

Improved delayed detached eddy simulation (IDDES)

IDDES extends DDES by incorporating wall-modeled LES (WMLES) capability. Depending on local flow conditions and grid resolution, IDDES can operate in RANS mode, LES mode, or WMLES mode, transitioning seamlessly between them. This gives IDDES the flexibility to handle situations where the grid near the wall is fine enough to support LES content, while still falling back to RANS where it isn't.

Boundary conditions for turbulence

Wall functions

Wall functions bridge the gap between the wall and the first grid point by applying analytical or semi-empirical profiles for velocity and turbulence quantities in the near-wall region (viscous sublayer and buffer layer). They allow the first cell center to sit in the log-law region (typically $y^+ \approx 30\text{–}300$), which dramatically reduces the number of cells needed near walls. Wall functions work well for attached, equilibrium boundary layers but can lose accuracy in flows with separation, strong pressure gradients, or heat transfer.

Low-Reynolds number approach

The low-Reynolds number (or "resolve-to-wall") approach places enough grid points within the viscous sublayer to directly solve the turbulence equations all the way to the wall surface. This requires the first cell center at $y^+ \approx 1$, which means many more cells in the near-wall region. The payoff is higher accuracy for wall-bounded flows, especially where wall functions break down. Most modern two-equation models have low-Re formulations available.

Turbulence model validation

Comparison with experimental data

Simulation results should be compared against experimental measurements to assess model accuracy. Common experimental techniques used for validation include:

• Particle image velocimetry (PIV): provides full-field velocity measurements
• Laser Doppler anemometry (LDA): gives point-wise velocity with high temporal resolution
• Hot-wire anemometry: measures velocity fluctuations at high frequency
• Wind tunnel force/pressure measurements: provide integrated quantities like drag and pressure distributions

Matching both mean flow quantities and turbulence statistics (Reynolds stresses, spectra) gives a more complete picture of model performance.

Comparison with DNS results

DNS provides a complete, high-fidelity dataset with access to every flow variable at every point in space and time. This makes it invaluable for validating turbulence models in detail, including quantities that are difficult to measure experimentally (pressure-strain correlations, dissipation rates, budget terms in transport equations).

DNS validation databases exist for canonical flows like channel flow, pipe flow, boundary layers, and homogeneous isotropic turbulence. These are particularly useful for testing whether a model captures the correct physics in well-controlled conditions.

Limitations of turbulence models

Assumptions and simplifications

Every turbulence model introduces closure assumptions that limit its generality:

• Eddy-viscosity RANS models assume the Reynolds stresses are aligned with the mean strain rate (the Boussinesq hypothesis). This fails in flows with strong anisotropy, swirl, or secondary motions.
• RANS empirical constants are calibrated against specific benchmark flows and may not transfer well to very different configurations.
• LES SGS models assume the unresolved scales behave in a relatively simple, universal manner. In underresolved LES (where the filter cuts into the energy-containing range), the SGS model carries too much responsibility and accuracy degrades.

Accuracy vs. computational cost

The hierarchy from cheapest to most expensive, and correspondingly least to most resolved, is:

Hybrid RANS-LES methods (DES, DDES, IDDES) sit between RANS and LES in both cost and fidelity. Choosing the right approach depends on the flow physics you need to capture, the accuracy required, and the computational budget available.