4.5 Turbulence modeling

Fundamentals of turbulence

Turbulence is a chaotic state of fluid motion where velocity, pressure, and other flow properties fluctuate irregularly across a wide range of scales. In aerodynamics, turbulence directly affects drag, heat transfer, mixing, and noise generation, so modeling it accurately is central to predicting and optimizing aircraft performance.

Turbulent flows contain eddies spanning from large energy-carrying structures down to tiny dissipative scales where viscosity converts kinetic energy into heat. Energy transfers from large eddies to progressively smaller ones in what's known as the energy cascade, until viscosity dissipates it at the smallest scales (the Kolmogorov scales).

Characteristics of turbulent flows

• Turbulent flows are highly unsteady and irregular, with significant variations in velocity and pressure fields.
• They exhibit enhanced mixing and increased rates of momentum, heat, and mass transfer compared to laminar flows.
• Eddies of various sizes interact and transfer energy across scales, making turbulence inherently three-dimensional with fluctuations in all spatial directions.

Turbulent vs laminar flow

Laminar flow moves in smooth, parallel layers with no mixing between them. Turbulent flow, by contrast, involves chaotic motion with vigorous mixing across fluid layers.

The transition between the two depends primarily on the Reynolds number, which is the ratio of inertial forces to viscous forces ($Re = \frac{\rho U L}{\mu}$). Higher Reynolds numbers favor turbulence. The geometry and surface conditions of the body also influence where and how transition occurs.

In aerodynamics, both flow regimes commonly appear on the same surface. A wing's boundary layer, for example, typically starts laminar near the leading edge and transitions to turbulent further downstream.

Importance in aerodynamics

• Turbulence determines key performance metrics: lift, drag, and heat transfer rates on aerodynamic surfaces.
• It governs boundary layer behavior, the thin region near solid surfaces where viscous effects dominate.
• Turbulent boundary layers produce higher skin friction drag than laminar ones, but they also resist flow separation better.
• Turbulence-driven noise and structural vibrations are major concerns in aircraft design, making accurate turbulence prediction essential.

Turbulence modeling approaches

Because turbulence involves motions across an enormous range of length and time scales, no single modeling approach works best for every situation. The main approaches differ in how much of the turbulence spectrum they resolve directly versus how much they model.

Direct numerical simulation (DNS)

DNS solves the full Navier-Stokes equations without any turbulence modeling assumptions, resolving every scale of motion from the largest eddies down to the Kolmogorov scale.

• Requires extremely fine spatial and temporal resolution. Grid points scale roughly as $Re^{9/4}$, so computational cost grows very rapidly with Reynolds number.
• Currently limited to relatively low Reynolds numbers and simple geometries.
• Primarily used for fundamental turbulence research and as benchmark data for validating other models.

Large eddy simulation (LES)

LES directly resolves the large, energy-carrying eddies while modeling the effect of smaller eddies using subgrid-scale (SGS) models. The idea is that small-scale turbulence tends to be more universal and isotropic, making it easier to model than the large scales.

• Captures significantly more flow detail than RANS, including unsteady vortex dynamics.
• Less expensive than DNS, but still demands fine grids, especially near walls.
• Accuracy depends heavily on the quality of the SGS model and the grid resolution.
• Well-suited for flows with complex unsteady features at moderate Reynolds numbers.

Reynolds-averaged Navier-Stokes (RANS)

RANS models decompose every flow variable into a time-averaged (mean) component and a fluctuating component using Reynolds decomposition. Averaging the Navier-Stokes equations produces extra unknown terms (the Reynolds stresses), which must be modeled.

• Computationally efficient, making RANS the workhorse of industrial CFD.
• Provides time-averaged flow quantities (mean velocity, pressure, skin friction) but does not capture unsteady turbulent structures.
• Accuracy depends on how well the turbulence model represents the Reynolds stresses for the flow in question.

Hybrid RANS-LES methods

Hybrid methods use RANS in regions where LES would be too expensive (typically near walls) and switch to LES in regions where the grid can support it (typically away from walls).

• Detached Eddy Simulation (DES) is the most well-known hybrid approach. It uses a RANS model in attached boundary layers and transitions to LES in separated regions.
• Scale-Adaptive Simulation (SAS) adjusts its resolved scale based on the local flow, allowing LES-like behavior where the grid permits.
• These methods aim for RANS-level cost near walls with LES-level accuracy in separated and wake regions, though the transition between RANS and LES zones remains a modeling challenge.

RANS turbulence models

RANS models range from simple algebraic expressions to full transport equations for every component of the Reynolds stress tensor. Most industrial models rely on the eddy viscosity concept to close the RANS equations.

Boussinesq hypothesis

The Boussinesq hypothesis assumes that the Reynolds stress tensor is proportional to the mean strain rate tensor, analogous to how viscous stresses relate to strain in a Newtonian fluid:

$-\overline{u_i' u_j'} = 2 \nu_t S_{ij} - \frac{2}{3} k \delta_{ij}$

This reduces the problem from modeling six independent Reynolds stress components to determining a single scalar quantity: the eddy viscosity $\nu_t$.

The hypothesis works well for many shear-dominated flows but breaks down when turbulence is strongly anisotropic (e.g., swirling flows, flows with strong streamline curvature).

Eddy viscosity concept

Eddy viscosity ($\nu_t$) is not a physical fluid property. It's a modeling quantity that represents the enhanced mixing and momentum transport caused by turbulent eddies. Its value depends on the local flow conditions and varies throughout the flow field.

Different RANS models differ primarily in how they calculate $\nu_t$.

Zero-equation models

Zero-equation (algebraic) models don't solve any transport equations for turbulence quantities. Instead, they estimate $\nu_t$ directly from local mean flow properties and empirical correlations.

• Examples: Baldwin-Lomax model, Cebeci-Smith model.
• Very fast and simple to implement.
• Limited accuracy and poor generality. They can't account for turbulence history effects (e.g., how upstream conditions affect downstream turbulence).
• Mostly used for quick estimates or as part of more complex methods.

One-equation models

One-equation models solve a single transport equation, usually for a quantity related to the eddy viscosity or the turbulent kinetic energy $k$.

• The Spalart-Allmaras model solves one equation directly for a modified eddy viscosity $\tilde{\nu}$. It was designed specifically for aerospace boundary layer flows and performs well for attached and mildly separated flows.
• One-equation models improve on algebraic models by accounting for transport effects, but they still rely on empirical relations for the turbulence length scale.

Two-equation models

Two-equation models solve transport equations for two turbulence quantities, which together determine both the velocity scale and the length scale of the turbulence. This makes them more general than one-equation models.

The eddy viscosity is typically computed as:

$\nu_t = C_\mu \frac{k^2}{\epsilon} \quad \text{or} \quad \nu_t = \frac{k}{\omega}$

depending on the model.

k-epsilon model

The $k$-$\epsilon$ model solves transport equations for turbulent kinetic energy ($k$) and its dissipation rate ($\epsilon$).

• One of the most widely used models due to its robustness across a broad range of flows.
• Performs well in fully turbulent, free-shear flows (jets, wakes, mixing layers).
• Struggles with strong adverse pressure gradients, separation, and near-wall flows (the standard version requires wall functions).
• Variants like the RNG $k$-$\epsilon$ and realizable $k$-$\epsilon$ address some of these weaknesses through modified coefficients and constraints.

k-omega model

The $k$-$\omega$ model solves transport equations for $k$ and the specific dissipation rate $\omega$ (which can be thought of as $\omega \sim \epsilon / k$).

• Performs well in near-wall regions and can be integrated directly to the wall without wall functions.
• Handles low-Reynolds-number effects and transitional flows better than $k$-$\epsilon$.
• The standard version is sensitive to freestream values of $\omega$, which can affect results in the outer part of boundary layers and in free-shear flows.
• The Wilcox $k$-$\omega$ model is a widely used variant that mitigates some of this freestream sensitivity.

SST k-omega model

The Shear Stress Transport (SST) $k$-$\omega$ model, developed by Menter, blends the strengths of both two-equation models:

1. It uses the $k$-$\omega$ formulation in the near-wall region, where that model excels.
2. It transitions to a $k$-$\epsilon$-like formulation in the freestream, avoiding the freestream sensitivity of $k$-$\omega$.
3. A blending function based on wall distance and local flow variables controls the switch between formulations.
4. It also includes a limiter on the eddy viscosity in adverse pressure gradient regions, improving separation prediction.

The SST $k$-$\omega$ model has become one of the most popular choices for external aerodynamic simulations because of its accuracy across a wide range of flow conditions, particularly flows involving adverse pressure gradients and separation.

Reynolds stress models

Reynolds stress models (RSM) abandon the Boussinesq hypothesis entirely. Instead, they solve individual transport equations for each of the six independent components of the Reynolds stress tensor, plus an equation for $\epsilon$ (or equivalent), totaling up to seven additional equations.

• RSMs capture turbulence anisotropy directly, making them more accurate for swirling flows, flows with strong streamline curvature, and three-dimensional separation.
• The added computational cost and complexity (more equations, more difficult convergence) mean they're used less frequently than eddy viscosity models in routine industrial work.
• They remain valuable when the Boussinesq hypothesis is known to be inadequate for the flow physics at hand.

Turbulent boundary layers

Turbulent boundary layers form near solid surfaces when the Reynolds number is high enough for turbulence to develop. Their structure and behavior directly control skin friction drag, heat transfer, and susceptibility to separation.

Characteristics of turbulent boundary layers

Compared to laminar boundary layers, turbulent ones are:

• Thicker, because turbulent mixing spreads momentum further from the wall.
• Fuller in their velocity profile, meaning the velocity rises more steeply near the wall and then levels off more gradually.
• Higher in skin friction, due to the increased momentum transfer toward the wall.
• More resistant to separation, because the energetic mixing brings high-momentum fluid from the outer flow down toward the surface.

The near-wall region has a well-defined layered structure:

• Viscous sublayer ($y^+ < 5$): Viscous stresses dominate; velocity varies linearly with wall distance.
• Buffer layer ($5 < y^+ < 30$): Transition zone where both viscous and turbulent stresses are significant.
• Logarithmic layer ($y^+ > 30$, up to roughly 10-20% of the boundary layer thickness): Turbulent stresses dominate; velocity follows a log-law profile.

Law of the wall

The law of the wall describes the mean velocity profile in the inner region of a turbulent boundary layer using non-dimensional wall coordinates:

$u^+ = \frac{u}{u_\tau}, \quad y^+ = \frac{y \, u_\tau}{\nu}$

where $u_\tau = \sqrt{\tau_w / \rho}$ is the friction velocity.

• In the viscous sublayer: $u^+ = y^+$
• In the log layer: $u^+ = \frac{1}{\kappa} \ln y^+ + C$, where $\kappa \approx 0.41$ (von Kármán constant) and $C \approx 5.0$ for smooth walls.

This universal profile is the foundation for wall functions used in RANS models. Wall functions bridge the near-wall region analytically, allowing the first grid point to be placed in the log layer rather than deep in the viscous sublayer. Getting the near-wall treatment right is critical for accurate skin friction and heat transfer predictions.

Boundary layer separation

Separation occurs when the flow near the wall reverses direction, typically caused by an adverse pressure gradient (pressure increasing in the flow direction) or abrupt geometric changes.

• Separation creates recirculation zones, increases pressure drag, and can cause loss of lift on airfoils (stall).
• Turbulent boundary layers resist separation better than laminar ones because turbulent mixing continuously replenishes momentum near the wall.
• Accurately predicting where and how separation occurs is one of the most demanding tests for any turbulence model. The SST $k$-$\omega$ model and RSMs generally handle separation better than the standard $k$-$\epsilon$ model.

Turbulent boundary layer control

Boundary layer control techniques aim to reduce drag, delay separation, or enhance mixing.

Passive methods (no external energy input):

• Vortex generators: Small vanes that create streamwise vortices, energizing the boundary layer and delaying separation.
• Riblets: Tiny streamwise grooves on the surface that reduce turbulent skin friction by a few percent.
• Surface roughness: Can be used strategically to trip the boundary layer to turbulent where desired.

Active methods (require external energy):

• Suction: Removes low-momentum fluid near the wall, thinning the boundary layer and delaying separation.
• Blowing/injection: Adds momentum to the near-wall region.
• Synthetic jets: Oscillating jets that energize the boundary layer without net mass addition.

These techniques are applied in high-lift systems, laminar flow control on wings, and drag reduction on fuselages and nacelles.

Numerical considerations

Implementing turbulence models in CFD requires careful attention to meshing, near-wall treatment, solver settings, and validation. Poor numerical choices can undermine even the best turbulence model.

Mesh resolution requirements

Different modeling approaches have vastly different mesh requirements:

For RANS simulations, you still need adequate refinement in regions with high gradients: boundary layers, wakes, shear layers, and near shocks. Always perform a mesh sensitivity study (run the same case on progressively finer grids) to confirm that your results are grid-independent.

Near-wall treatment

The choice of near-wall treatment must be consistent with your turbulence model and your mesh:

• Low-Reynolds-number (resolve-to-wall) approach: The first cell center is placed at $y^+ \approx 1$, and the model resolves the viscous sublayer directly. Required for models like the standard $k$-$\omega$ and SST $k$-$\omega$.
• Wall function approach: The first cell center is placed at $30 < y^+ < 300$, and algebraic wall functions bridge the viscous-affected region. Commonly used with the $k$-$\epsilon$ model.

Placing your first cell at an intermediate $y^+$ value (say 5-20) that falls in the buffer layer is a common mistake. Neither the resolved approach nor wall functions are accurate in this range.

Convergence and stability issues

Turbulence equations are nonlinear and tightly coupled to the mean flow equations, which can cause convergence difficulties.

Practical strategies to improve convergence:

1. Initialize carefully. Start from a reasonable initial condition (e.g., a converged inviscid or laminar solution, or uniform freestream values for $k$ and $\omega$).
2. Use appropriate under-relaxation factors. Turbulence equations often need lower relaxation factors than the momentum equations.
3. Monitor multiple indicators. Track residuals, but also monitor integrated quantities (lift, drag, mass flow) and point values to confirm the solution has truly converged.
4. Consider multigrid or implicit solvers to accelerate convergence for stiff systems.

Validation and verification

Verification checks that the equations are solved correctly: Are the discretization errors small enough? Does the solution converge with grid refinement?

Validation checks that the right equations are being solved: Do the results match experimental data or higher-fidelity simulations (DNS/LES)?

Both steps are necessary. A verified solution can still be wrong if the turbulence model is inappropriate for the flow. Validate against experimental data for flow conditions and geometries relevant to your application, not just a single benchmark case.

Applications in aerodynamics

Airfoil and wing design

Turbulence models predict lift, drag, and pitching moment on airfoils and wings. They're used to optimize wing shapes for maximum lift-to-drag ratio and to predict the onset and extent of flow separation, which directly determines stall behavior. The SST $k$-$\omega$ model is a common choice for these applications.

High-lift configurations

Flaps, slats, and other high-lift devices create complex flow fields with multiple boundary layers, wakes, and their interactions. Turbulence models must capture confluent boundary layers (where a wake merges with a boundary layer on the next element), separation on deflected flap surfaces, and unsteady wake effects. Getting these interactions right is critical for predicting maximum lift and stall margins during takeoff and landing.

Turbomachinery flows

Compressors and turbines involve high Reynolds numbers, strong pressure gradients, blade wakes, tip clearance flows, and secondary flows. Turbulence models predict stage efficiency, loss distributions, and heat transfer on blades. The confined, rotating environment makes turbulence anisotropy important, and RSMs or transition-sensitive models are sometimes needed for accurate predictions.

Hypersonic flows

At high Mach numbers, turbulence modeling faces additional complications:

• Compressibility effects alter turbulence structure and must be accounted for in the model (e.g., through compressibility corrections to $k$-$\epsilon$ or $k$-$\omega$).
• Shock-boundary layer interactions create regions of intense adverse pressure gradients and possible separation.
• High temperatures can cause chemical reactions and real-gas effects that couple with the turbulence.

Accurate turbulence prediction is essential for thermal protection system design on re-entry vehicles and for inlet performance prediction on scramjets.

Limitations and challenges

Accuracy vs computational cost

The fundamental tension in turbulence modeling is that more accurate methods cost more. DNS and LES provide detailed, time-resolved predictions but remain impractical for full aircraft configurations at flight Reynolds numbers ($Re \sim 10^7$ or higher). RANS models are affordable but rely on assumptions that limit their accuracy. Hybrid RANS-LES methods offer a middle ground but introduce their own challenges around the RANS-to-LES transition zone.

Complex geometry and flow conditions

RANS models are calibrated against relatively simple canonical flows (flat plates, pipes, simple shear layers). When applied to flows with strong curvature, massive separation, or three-dimensional vortex structures, the underlying assumptions (especially the Boussinesq hypothesis) can break down. No single RANS model performs best across all flow types, which is why model selection should always be guided by the specific flow physics involved.

Transition prediction

Most RANS turbulence models assume the flow is fully turbulent everywhere. In reality, boundary layers start laminar and transition to turbulent at some point. Predicting this transition location matters greatly for drag estimation (laminar skin friction is much lower than turbulent).

Transition models, such as the $\gamma$-$Re_\theta$ correlation-based model, add transport equations for intermittency and transition onset criteria. These increase complexity and computational cost but are essential for applications like natural laminar flow wing design, where maintaining laminar flow over a large portion of the wing is the primary design goal.

Future developments in turbulence modeling

• Machine learning approaches are being explored to improve RANS closure models by training on DNS and experimental data. These data-driven models aim to correct known deficiencies in eddy viscosity predictions for specific flow classes.
• Multiscale methods like the Variational Multiscale (VMS) approach provide a more rigorous framework for separating resolved and modeled scales in LES.
• Exascale computing is gradually making wall-resolved LES feasible for increasingly complex geometries, which will expand the range of flows where high-fidelity turbulence data is available for model development and validation.