---
title: "RANS equations and the closure problem"
subject: "Applied Computational Fluid Dynamics"
chapter: 5
key: "semester_7/applied_computational_fluid_dynamics:reynolds-averaging-closure-problem"
source: https://www.bctengineer.com/q/semester_7/applied_computational_fluid_dynamics/reynolds-averaging-closure-problem
site: BCT Engineering
---

# RANS equations and the closure problem

Applied Computational Fluid Dynamics · Chapter 5: Turbulence Modeling

- Asked in: year not given, 6 marks
- Video: [Closure of RANS Equations — Lesson 5](https://www.youtube.com/watch?v=aiZSQvjti84) — Ansys Learning
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/applied_computational_fluid_dynamics/reynolds-averaging-closure-problem
- Whole chapter: https://www.bctengineer.com/md/semester_7/applied_computational_fluid_dynamics/chapter-5

## Question

Explain Reynolds decomposition and time averaging. Derive the Reynolds-averaged continuity and momentum equations for incompressible flow, and explain the closure problem and the Boussinesq hypothesis.

## Short answer

Flow variables are split into mean and fluctuating parts; time-averaging the incompressible N-S equations yields RANS equations with Reynolds stresses -ρ⟨u'_i u'_j⟩. More unknowns than equations cause a closure problem. The Boussinesq hypothesis models the stresses via eddy viscosity: -ρ⟨u'_i u'_j⟩ = μ_t(∂ū_i/∂x_j+∂ū_j/∂x_i) - (2/3)ρk δ_{ij}.

## Answer

#### Reynolds decomposition and averaging
Each flow variable is split into a mean and a fluctuation:
$$
u=\bar u+u',\qquad p=\bar p+p',\qquad \bar u=\frac1T\int_0^Tu\,dt,\qquad \overline{u'}=0
$$
Rules: $\overline{\bar u}=\bar u$, $\overline{u'}=0$, but $\overline{u'v'}\neq0$ in general. For unsteady mean flow, ensemble averaging is used.

#### RANS equations
Substitute into the incompressible continuity and Navier-Stokes equations and average:

Continuity:
$$
\frac{\partial\bar u_i}{\partial x_i}=0
$$
Momentum:
$$
\rho\left(\frac{\partial\bar u_i}{\partial t}+\bar u_j\frac{\partial\bar u_i}{\partial x_j}\right)=-\frac{\partial\bar p}{\partial x_i}+\frac{\partial}{\partial x_j}\left(\mu\frac{\partial\bar u_i}{\partial x_j}-\rho\overline{u'_iu'_j}\right)
$$
The new term $-\rho\overline{u'_iu'_j}$ is the **Reynolds stress tensor**: 6 independent components (symmetric) which represent the extra momentum transport by turbulence. The term comes from the averaging of the non-linear convective term.

#### Closure problem
The averaged equations have 4 equations (continuity + 3 momentum) but 10 unknowns (4 mean variables + 6 Reynolds stresses). Writing equations for the stresses brings in triple correlations $\overline{u'u'u'}$, and so on: every level introduces more unknowns than equations. The set cannot be closed without **turbulence models** that relate the Reynolds stresses to mean quantities.

#### Boussinesq hypothesis
Reynolds stresses are assumed to act like viscous stresses, proportional to the mean strain rate, through an eddy viscosity $\mu_t$:
$$
-\rho\overline{u'_iu'_j}=\mu_t\left(\frac{\partial\bar u_i}{\partial x_j}+\frac{\partial\bar u_j}{\partial x_i}\right)-\frac23\rho k\delta_{ij},\qquad k=\tfrac12\overline{u'_iu'_i}
$$
The problem then becomes one of finding $\mu_t$: zero-equation (mixing length), one-equation (Spalart-Allmaras) and two-equation (k-$\varepsilon$, k-$\omega$) models. Alternatively, Reynolds stress models (RSM) solve transport equations for each stress, without the isotropy assumption.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. In Reynolds decomposition u = ubar + u', which statement is true?
   A. The time average of u' is zero, but the average of u'v' is generally nonzero
   B. The average of u'v' is always zero
   C. The average of u' is nonzero
   D. ubar is the instantaneous value
   Answer: A. Fluctuations average to zero but products of fluctuations do not, which leads to the Reynolds stresses. (Key point: Reynolds decomposition)

2. Where does the Reynolds stress term -rho avg(u_i' u_j') come from?
   A. Averaging the pressure gradient
   B. Averaging the viscous term
   C. Averaging the continuity equation
   D. Averaging the non-linear convective term
   Answer: D. Averaging the non-linear convective term leaves the avg of u_i'u_j', acting as an extra momentum transport. (Key point: Origin of Reynolds stresses)

3. Why is the closure problem present?
   A. There are 4 equations but 10 unknowns (4 mean variables + 6 Reynolds stresses)
   B. There are 10 equations but 4 unknowns
   C. Continuity is missing
   D. Pressure is unknown
   Answer: A. Extra equations for the stresses add triple correlations, so a turbulence model is needed to close the system. (Key point: Closure problem)

4. What does the Boussinesq hypothesis assume?
   A. Reynolds stresses are zero
   B. Reynolds stresses are proportional to the mean strain rate through an eddy viscosity mu_t
   C. Reynolds stresses are solved by their own transport equations
   D. Turbulence is anisotropic
   Answer: B. It treats turbulent stresses like viscous stresses with mu_t; the task becomes finding mu_t. RSM, by contrast, solves stress equations. (Key point: Boussinesq hypothesis)

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