MEC320 - Tutorial 2 - Turbulence Modelling

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31 Terms

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What are the differences between Navier-Stokes and Reynolds Averaged Navier-Stokes (RANS) equations?

Navier-Stokes equations describe instantaneous flow variables, while RANS equations describe time-averaged variables and include Reynolds stresses due to turbulence:

uiujuiujuiuj\overline{u_{i}u_{j}}\neq\overline{u}_{i}\,\overline{u}_{j}\Rightarrow-\overline{u_{i}^{\prime}u_{j}^{\prime}}

(Reynolds stresses)

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What do we mean by linear and non-linear terms in the momentum equations?

Linear terms: involve single velocity components (e.g., ∂u/∂x)
Non-linear terms: involve products of velocities (e.g., u ∂u/∂x)

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Main steps to derive the RANS momentum equations?

1st Decompose:

u=u+uu = \overline{u} + u'

2nd Time-average Navier-Stokes

3rd Apply time-averaging rules

4th Introduce Reynolds stresses:

ρuiuj- \rho \overline{u_i' u_j'}

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Where do Reynolds stresses come from?

They arise from time-averaging nonlinear convective terms.
Physically, they represent turbulent momentum transfer and must be modeled.

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Why must CFD results be validated?

Model assumptions (e.g., turbulence closure) can deviate from reality. Validation ensures reliability.

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Prove the time averaging rule:

ab=abab\overline{a'b'} = \overline{ab} - \overline{a} \,\overline{b}

Let

a=a+a,b=b+ba=\overline{a}+a^{\prime},b=\overline{b}+b^{\prime}

Then

ab=ab+ab+ba+abab=\overline{a}\,\overline{b}+\overline{a}b^{\prime}+\overline{b}a^{\prime}+a^{\prime}b^{\prime}

Time Average:

ab=ab+ab\overline{ab} = \overline{a}\,\overline{b} + \overline{a' b'}

—>

ab=abab\overline{a' b'} = \overline{ab} - \overline{a} \,\overline{b}

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Reynolds averaging of:

(ujui)/xj∂(u_j u_i)/∂x_j ?

(ujui)/xj=/xj(ujui+ujui)\overline{\partial (u_j u_i)/\partial x_j} = \partial/\partial x_j (\overline{u}_j \, \overline{u}_i + \overline{u_j' u_i'}) Turbulence stress arises as:

ujui/xj∂ \overline{u_j' u_i'} / ∂x_j

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What is the Boussinesq Hypothesis (1877)?

It models Reynolds stresses as:

uiuj=νt(uixj+ujxi)- \overline{u_i' u_j'} = \nu_t \left( \frac{\partial \overline{u}_i}{\partial x_j} + \frac{\partial \overline{u}_j}{\partial x_i} \right)

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Differences between Navier-Stokes, RANS, and EVM RANS?

  • Navier-Stokes: Instantaneous, no modeling

  • RANS: Time-averaged, includes Reynolds stresses

  • EVM-RANS: Uses eddy viscosity (νt) to model Reynolds stresses

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How is each term in the k equation treated?

  • Production: Modeled

  • Dissipation: Modeled

  • Convection: Exact

  • Diffusion: Modeled

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Physical meaning of k and ε equation terms?

  • k : Turbulent kinetic energy

  • ε : Rate of dissipation of k
    Terms model generation, transport, and loss of turbulence energy.

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Why is CFD validation important?

Simulations use assumptions and models that may not reflect real flows. Validation ensures accuracy.

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Eddy viscosity based on k and turbulence timescale τ ?

νt=Cμkτ\nu_t = C_\mu k \tau

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How are model constants determined?

By fitting to experimental data or DNS in canonical cases like flat plates, pipe flows, etc.

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Mixing length model (Nikuradse form)?

(νt=lm2Uylm/R=0.140.08(1y/R)20.06(1y/R)4)\left(\nu_{t}=l_{m}^2\left|\frac{\partial U}{\partial y}\right|l_{m}/R=0.14-0.08(1-y/R\right)^2-0.06\left(1-y/R)^4\right)

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What is an adverse pressure gradient (APG)?

Region where pressure increases in the flow direction. Causes deceleration and flow separation.

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How to determine the reattachment point?

The point where wall shear stress τw becomes zero and changes sign (negative to positive).

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Describe the flow before the step in a backward-facing step setup.

Flow develops as a boundary layer, increasing momentum thickness until the step.

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What does this mean:
"The wall boundary-layer thickness was 1.9 cm, and the Reynolds number (based on momentum thickness) was 5000 at a location 4 step-heights upstream of the step"

Gives inflow conditions:

θ=1.9cm,Reθ=5000\theta = 1.9\, \text{cm}, Re_\theta = 5000

Useful for defining upstream flow state.

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What are the differences between eddy viscosity models (EVMs) and Reynolds stress models (RSMs)?

  • EVMs assume isotropic turbulence and use scalar eddy viscosity (νt)

  • RSMs directly solve transport equations for each component of Reynolds stress tensor capturing anisotropy:

(uiuj)\left(\overline{u_{i}^{\prime}u_{j}^{\prime}}\right)

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What are the differences between linear and non-linear eddy viscosity models?

  • Linear EVMs relate Reynolds stresses linearly to strain rate

  • Non-linear EVMs include higher-order terms, capturing anisotropy and curvature effects more accurately

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Why is knowing key flow physics important for CFD modeling?

Because turbulence models perform differently under different conditions (e.g., separation, rotation, anisotropy). Selecting a suitable model depends on dominant flow features.

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Compare k-e and k-w models.

  • k-e: better for free-shear flows, uses wall functions

  • k-w: better near walls and in adverse pressure gradients

  • SST k-w: blends both, offering accuracy and robustness

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Why aren’t Reynolds Stress Models (RSMs) widely used?

  • High computational cost

  • Complex implementation and stability issues

  • Requires fine mesh and detailed boundary conditions

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What is the law of the wall and what are u+ and y+ ?

The law of the wall describes velocity profiles near a wall

u+=u/uτu^+ = u / u_\tau

y+=yuτ/νy^{+}=yu_{\tau}/\nu

where

uτ=τw/ρu_\tau = \sqrt{\tau_w / \rho}

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Why should the first node be placed at y+ < 1 or y+ > 30 ?

To ensure proper use of low-Re models (y+ < 1) or wall functions (y+ > 30). Intermediate values yield inaccurate results.

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Differences between standard, non-equilibrium wall functions and enhanced wall treatment?

  • Standard: assume equilibrium boundary layer

  • Non-equilibrium: account for pressure gradients and separation

  • Enhanced: resolves viscous sublayer and works with y+ < 1

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Why aren’t low-Re models only for low Reynolds number flows?

"Low-Re" refers to resolving near-wall regions, not global flow Re. These models are used when resolving viscous sublayers.

29
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Mesh requirement for Enhanced Wall Treatment in Fluent?

First node should be at y+ = 1 Requires fine mesh near walls.

30
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Summarize near-wall modelling strategies

Use wall functions when mesh can't resolve viscous sublayer.

Use enhanced treatments or low-Re models when accuracy near walls is critical.

Mesh strategy must match model.

31
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Compare standard k-e and SST k-w

Equations: Both solve 2-equation models

Near-wall: k-e uses wall functions, SST k-w resolves near-wall

Mesh: k-e coarse (y+ > 30), SST: fine (y+ < 1)