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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:
uiuj=uiuj⇒−ui′uj′
(Reynolds stresses)
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)
Main steps to derive the RANS momentum equations?
1st Decompose:
u=u+u′
2nd Time-average Navier-Stokes
3rd Apply time-averaging rules
4th Introduce Reynolds stresses:
−ρui′uj′
Where do Reynolds stresses come from?
They arise from time-averaging nonlinear convective terms.
Physically, they represent turbulent momentum transfer and must be modeled.
Why must CFD results be validated?
Model assumptions (e.g., turbulence closure) can deviate from reality. Validation ensures reliability.
Prove the time averaging rule:
a′b′=ab−ab
Let
a=a+a′,b=b+b′
Then
ab=ab+ab′+ba′+a′b′
Time Average:
ab=ab+a′b′
—>
a′b′=ab−ab
Reynolds averaging of:
∂(ujui)/∂xj ?
∂(ujui)/∂xj=∂/∂xj(ujui+uj′ui′) Turbulence stress arises as:
∂uj′ui′/∂xj
What is the Boussinesq Hypothesis (1877)?
It models Reynolds stresses as:
−ui′uj′=νt(∂xj∂ui+∂xi∂uj)
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
How is each term in the k equation treated?
Production: Modeled
Dissipation: Modeled
Convection: Exact
Diffusion: Modeled
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.
Why is CFD validation important?
Simulations use assumptions and models that may not reflect real flows. Validation ensures accuracy.
Eddy viscosity based on k and turbulence timescale τ ?
νt=Cμkτ
How are model constants determined?
By fitting to experimental data or DNS in canonical cases like flat plates, pipe flows, etc.
Mixing length model (Nikuradse form)?
(νt=lm2∂y∂Ulm/R=0.14−0.08(1−y/R)2−0.06(1−y/R)4)
What is an adverse pressure gradient (APG)?
Region where pressure increases in the flow direction. Causes deceleration and flow separation.
How to determine the reattachment point?
The point where wall shear stress τw becomes zero and changes sign (negative to positive).
Describe the flow before the step in a backward-facing step setup.
Flow develops as a boundary layer, increasing momentum thickness until the step.
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
Useful for defining upstream flow state.
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:
(ui′uj′)
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
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.
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
Why aren’t Reynolds Stress Models (RSMs) widely used?
High computational cost
Complex implementation and stability issues
Requires fine mesh and detailed boundary conditions
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/uτ
y+=yuτ/ν
where
uτ=τw/ρ
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.
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
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.
Mesh requirement for Enhanced Wall Treatment in Fluent?
First node should be at y+ = 1 Requires fine mesh near walls.
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.
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)