7.2 Reynolds-averaged Navier-Stokes equations

Reynolds decomposition

Reynolds decomposition is the starting point for deriving the RANS equations. The core idea is straightforward: split every flow variable into its time-averaged (mean) part and the leftover fluctuation. This separation lets you write governing equations for the mean flow alone, with turbulence effects appearing as extra terms you then need to model.

Mean and fluctuating components

Any instantaneous flow variable (velocity, pressure, temperature, etc.) is written as the sum of a mean component and a fluctuating component:

$u_i = \bar{u}_i + u'_i$

Here $\bar{u}_i$ is the time-averaged velocity and $u'_i$ is the instantaneous deviation from that average. By definition, the time average of the fluctuation is zero: $\overline{u'_i} = 0$. This property is used repeatedly when deriving the RANS equations.

Time averaging

The mean value is defined by averaging over a time interval $T$ that is long compared to the turbulent fluctuations but short compared to any slow unsteadiness in the mean flow:

$\bar{u}_i = \frac{1}{T} \int_{t}^{t+T} u_i \, dt$

A few key properties of the time-averaging operator that you'll use constantly:

• $\overline{\bar{u}_i} = \bar{u}_i$ (averaging an already-averaged quantity gives itself)
• $\overline{u'_i} = 0$ (mean of the fluctuation is zero)
• $\overline{\bar{u}_i \, u'_j} = 0$ (cross-terms between mean and fluctuation vanish)
• $\overline{u'_i \, u'_j} \neq 0$ in general (products of fluctuations do not average to zero)

That last point is exactly where the Reynolds stresses come from. Applying these averaging rules to the Navier-Stokes equations produces the RANS equations.

RANS equations

The RANS equations govern the mean flow field. They look almost identical to the original Navier-Stokes equations, except for one critical addition: the Reynolds stress term, which captures how turbulent fluctuations transfer momentum.

Continuity equation for mean flow

For incompressible flow, time-averaging the continuity equation gives:

$\frac{\partial \bar{u}_i}{\partial x_i} = 0$

This has the same form as the original continuity equation. The mean flow satisfies mass conservation on its own, with no extra turbulence terms.

Momentum equation for mean flow

Time-averaging the momentum equation yields:

$\frac{\partial \bar{u}_i}{\partial t} + \bar{u}_j \frac{\partial \bar{u}_i}{\partial x_j} = -\frac{1}{\rho} \frac{\partial \bar{p}}{\partial x_i} + \nu \frac{\partial^2 \bar{u}_i}{\partial x_j \partial x_j} - \frac{\partial \overline{u'_i u'_j}}{\partial x_j}$

Every term here mirrors the standard Navier-Stokes equation except the last one. That final term, $-\frac{\partial \overline{u'_i u'_j}}{\partial x_j}$, is the divergence of the Reynolds stress tensor. It acts like an additional stress on the mean flow caused by turbulent mixing.

Reynolds stress tensor

The Reynolds stress tensor is defined as:

$\tau_{ij} = -\rho \overline{u'_i u'_j}$

It's a symmetric second-order tensor (so $\overline{u'_i u'_j} = \overline{u'_j u'_i}$), which means it has six independent components in 3D. Physically, each component represents a rate of momentum transfer by turbulent fluctuations. The diagonal components ($\overline{u'^2}$, $\overline{v'^2}$, $\overline{w'^2}$) are normal stresses related to the turbulent kinetic energy, while the off-diagonal components are turbulent shear stresses.

Closure problem

The RANS momentum equation introduced six new unknowns (the independent Reynolds stress components), but no new equations came along with them. The system is now underdetermined. This is the closure problem: you need a turbulence model to express the Reynolds stresses in terms of known mean-flow quantities so the equations can actually be solved.

Turbulence modeling

Turbulence models provide the missing relationships needed to close the RANS equations. They range from simple algebraic expressions to full transport equations for each Reynolds stress component. The choice of model always involves a trade-off between accuracy and computational cost.

Boussinesq hypothesis

The Boussinesq hypothesis draws an analogy between turbulent and viscous stresses. Just as viscous stresses are proportional to the strain rate through molecular viscosity, the Reynolds stresses are assumed proportional to the mean strain rate through an eddy viscosity $\mu_t$:

$\tau_{ij} = 2\mu_t \bar{S}_{ij} - \frac{2}{3} \rho k \delta_{ij}$

where $\bar{S}_{ij} = \frac{1}{2}\left(\frac{\partial \bar{u}_i}{\partial x_j} + \frac{\partial \bar{u}_j}{\partial x_i}\right)$ is the mean strain rate tensor, $k = \frac{1}{2}\overline{u'_i u'_i}$ is the turbulent kinetic energy, and $\delta_{ij}$ is the Kronecker delta. The second term ensures the correct trace of the tensor (the normal stresses sum to $2\rho k$).

This hypothesis reduces the closure problem from modeling six stress components to modeling a single scalar, $\mu_t$. It works well for many shear-dominated flows but struggles when turbulence is strongly anisotropic.

Eddy viscosity models

These models all use the Boussinesq hypothesis and differ in how they determine $\mu_t$.

Algebraic models

Also called zero-equation models, these compute $\mu_t$ directly from local mean-flow quantities with no additional transport equations. The mixing length model is the classic example: $\mu_t = \rho \, l_m^2 \left|\frac{\partial \bar{u}}{\partial y}\right|$, where $l_m$ is a prescribed mixing length. The Baldwin-Lomax model is another common algebraic model used in aerodynamics.

These are cheap to run but limited. You have to specify the length scale yourself, which makes them unreliable for flows where the turbulence structure isn't known in advance.

One-equation models

These solve a single transport equation for a turbulence quantity, typically the turbulent kinetic energy $k$ or a modified viscosity. The Spalart-Allmaras model is the most widely used one-equation model; it solves a transport equation directly for a variable related to $\mu_t$ and was designed specifically for aerospace applications.

One-equation models improve on algebraic models by letting turbulence quantities be transported by the flow, but they still rely on an algebraically prescribed length scale.

Two-equation models

These are the workhorses of industrial CFD. By solving transport equations for two turbulence quantities, they can determine both a turbulence velocity scale and a length scale, making them self-contained.

• $k$-$\varepsilon$ model: Solves for turbulent kinetic energy $k$ and its dissipation rate $\varepsilon$. Eddy viscosity: $\mu_t = C_\mu \rho \frac{k^2}{\varepsilon}$. Robust and well-validated for free shear flows, but performs poorly near walls without modifications.
• $k$-$\omega$ model: Solves for $k$ and the specific dissipation rate $\omega$. Eddy viscosity: $\mu_t = \rho \frac{k}{\omega}$. Better near-wall behavior than standard $k$-$\varepsilon$, but sensitive to freestream values of $\omega$.
• SST $k$-$\omega$ model (Menter): Blends $k$-$\omega$ near walls with $k$-$\varepsilon$ in the freestream, combining the strengths of both. This is one of the most popular choices for general-purpose engineering simulations.

Reynolds stress models

Reynolds stress models (RSMs) abandon the Boussinesq hypothesis entirely. Instead, they solve individual transport equations for all six independent components of $\overline{u'_i u'_j}$, plus an equation for $\varepsilon$ (or $\omega$). That's seven extra PDEs compared to two for a $k$-$\varepsilon$ model.

The payoff is that RSMs can capture anisotropic turbulence effects that eddy viscosity models miss, such as stress-driven secondary flows in ducts or strong swirling flows. The cost is higher computational expense and more difficulty achieving numerical convergence.

Large eddy simulation (LES)

LES takes a fundamentally different approach from RANS. Rather than averaging out all turbulence, it directly resolves the large, energy-carrying eddies and only models the small-scale (subgrid-scale) motions.

The Navier-Stokes equations are spatially filtered rather than time-averaged, producing filtered equations with a subgrid-scale (SGS) stress term. SGS models like the Smagorinsky model or the dynamic Smagorinsky model provide closure for these small-scale effects.

LES produces time-resolved, three-dimensional flow fields and captures far more turbulence physics than RANS. However, it requires much finer grids (especially near walls) and time-accurate simulations, making it significantly more expensive. Note that LES is not strictly a RANS method; it's included here for comparison since it represents an alternative closure strategy.

Boundary conditions

Boundary conditions in RANS simulations must specify not only the mean-flow variables but also the turbulence quantities ($k$, $\varepsilon$, $\omega$, or Reynolds stresses, depending on the model). The treatment of walls is particularly important because turbulence behavior changes dramatically in the near-wall region.

Wall functions

Wall functions bridge the viscous sublayer without resolving it on the grid. They use the known logarithmic velocity profile (the "law of the wall") to relate the wall shear stress to the velocity at the first grid point off the wall.

• The first cell center is typically placed in the log-law region, around $y^+ \approx 30\text{–}300$
• This allows much coarser near-wall meshes, saving significant computational cost
• Wall functions work well for high-Reynolds-number flows with attached boundary layers
• They become unreliable when the log-law assumptions break down (e.g., separation, strong pressure gradients, low-Re flows)

Near-wall resolution

The alternative to wall functions is resolving the viscous sublayer directly. This requires placing the first grid point at $y^+ < 1$ (often $y^+ \approx 1$) and using enough cells through the boundary layer to capture the viscous sublayer, buffer layer, and log-law region.

• Provides more accurate wall shear stress and heat transfer predictions
• Necessary for flows with separation, strong pressure gradients, or low Reynolds numbers
• Requires turbulence models formulated to be valid all the way to the wall (e.g., $k$-$\omega$ or low-Re $k$-$\varepsilon$ variants)
• Substantially increases mesh count and computational cost compared to wall functions

Numerical methods for RANS

Solving the RANS equations numerically involves discretizing the computational domain, approximating the governing equations on that discrete domain, and iterating to a converged solution. For incompressible flows, the coupling between pressure and velocity requires special algorithms.

Finite volume method

The finite volume method (FVM) is the dominant discretization approach in CFD for RANS. The procedure is:

1. Divide the domain into non-overlapping control volumes (cells)
2. Integrate the RANS equations over each control volume
3. Convert volume integrals of divergence terms to surface integrals using the divergence theorem
4. Approximate the fluxes at cell faces using interpolation schemes (upwind, central, blended, etc.)
5. Assemble the resulting algebraic equations for the unknowns at cell centers

A key advantage of FVM is that it guarantees conservation of mass, momentum, and energy both locally (per cell) and globally, by construction.

Finite element method

The finite element method (FEM) divides the domain into elements (triangles, quadrilaterals, tetrahedra, hexahedra) and approximates the solution as a weighted sum of basis functions defined on those elements. A weighted residual (typically Galerkin) formulation produces the algebraic system.

FEM offers greater flexibility for complex geometries and naturally accommodates higher-order polynomial approximations. It's more common in structural and multiphysics applications than in mainstream CFD, though stabilized FEM formulations (like SUPG) are used for flow problems.

Pressure-velocity coupling

In incompressible RANS, there's no explicit equation for pressure. Pressure appears in the momentum equations, and the continuity equation acts as a constraint on the velocity field. Special iterative algorithms enforce this coupling:

• SIMPLE (Semi-Implicit Method for Pressure-Linked Equations): Solves the momentum equations with a guessed pressure, derives a pressure correction from the continuity equation, and updates both pressure and velocity. Repeats until convergence.
• PISO (Pressure-Implicit with Splitting of Operators): Similar to SIMPLE but includes additional corrector steps within each time step, making it better suited for transient simulations.
• SIMPLEC and PIMPLE are common variants that improve convergence behavior.

Applications of RANS

RANS remains the most widely used turbulence modeling approach in engineering practice because it provides reasonable accuracy at manageable computational cost for a broad range of flows.

Aerodynamics

RANS simulations are standard in aerospace design for predicting lift, drag, and moment coefficients on airfoils, wings, and full aircraft configurations. The SST $k$-$\omega$ model is particularly popular here. Typical applications include airfoil shape optimization, high-lift system design, and stability analysis. RANS handles attached and mildly separated flows well, though massive separation (e.g., deep stall) pushes it beyond its reliable range.

Turbomachinery

Compressors, turbines, and pumps all involve complex rotating flows where RANS is used to predict pressure distributions, efficiency, and loss mechanisms. Blade shape optimization and stage matching rely heavily on RANS-based CFD. The periodic geometry of blade rows makes RANS especially efficient here, since you can often simulate a single blade passage.

Combustion

Turbulent reacting flows in engines, gas turbine combustors, and furnaces are modeled by coupling RANS with combustion models (flamelet models, eddy dissipation models, etc.). RANS predicts the turbulent mixing field, which controls flame behavior, pollutant formation ($NO_x$, soot), and combustion efficiency.

Environmental flows

Atmospheric boundary layers, pollutant dispersion in urban areas, river hydraulics, and wind farm siting all use RANS simulations. These applications often involve complex terrain and thermal stratification, where modified $k$-$\varepsilon$ models with buoyancy terms are common.

Limitations and challenges

Assumption of isotropic turbulence

Eddy viscosity models assume that the Reynolds stress tensor is aligned with the mean strain rate tensor, which implicitly treats turbulence as isotropic. This fails for flows with strong streamline curvature, rotation, swirl, or secondary motions driven by Reynolds stress anisotropy. A classic example: eddy viscosity models cannot predict turbulence-driven secondary flows in non-circular ducts because those flows are entirely caused by anisotropy in the normal Reynolds stresses.

Accuracy vs. computational cost

The hierarchy from algebraic models through two-equation models, RSMs, and LES represents increasing fidelity at increasing cost. For a typical industrial problem, a two-equation RANS simulation might take hours on a workstation, while a wall-resolved LES of the same geometry could take weeks on a cluster. Choosing the right level of modeling for a given application is one of the most important practical decisions in CFD.

Modeling complex flows

Several flow phenomena remain genuinely difficult for RANS:

• Separation and reattachment: RANS tends to predict separation too late and reattachment too early, especially with eddy viscosity models
• Laminar-turbulent transition: Standard RANS models assume fully turbulent flow. Transition models (e.g., the $\gamma$-$Re_\theta$ model) exist but add complexity and are not universally reliable
• Unsteady flows: RANS is inherently a steady-state framework. Unsteady RANS (URANS) can capture large-scale unsteadiness but misses broadband turbulent fluctuations
• Multiphase flows and fluid-structure interaction: Coupling RANS with additional physics introduces further modeling assumptions and numerical challenges

Hybrid RANS-LES methods (such as Detached Eddy Simulation, DES) attempt to combine the efficiency of RANS in attached boundary layers with the accuracy of LES in separated regions. These are increasingly used for flows where pure RANS is insufficient but full LES is too expensive.

Experimental validation remains essential for building confidence in RANS predictions, particularly for flows outside the calibration range of the turbulence model being used.