Key Turbulence Models

Why This Matters

Turbulence modeling sits at the heart of computational fluid dynamics—it's the mathematical bridge between the elegant but unsolvable full Navier-Stokes equations and practical engineering predictions. You're being tested on your understanding of the closure problem: turbulence creates more unknowns than equations, so every model represents a different philosophical and mathematical approach to closing that gap. The concepts here connect directly to scale separation, Reynolds decomposition, eddy viscosity assumptions, and the fundamental trade-off between computational cost and physical fidelity.

Don't just memorize model names and their acronyms. Know why each approach exists, what physical assumptions it makes, and when those assumptions break down. Exam questions will ask you to select appropriate models for specific flow scenarios, compare the mathematical foundations of different approaches, and explain why a model that works beautifully for one application fails spectacularly for another. Understanding the underlying principles—not just the formulas—is what separates strong answers from mediocre ones.

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Reynolds-Averaged Approaches (RANS-Based Models)

These models apply Reynolds decomposition to split flow variables into mean and fluctuating components, then time-average the governing equations. The closure problem emerges because averaging the nonlinear convective terms produces unknown Reynolds stress terms $-\rho \overline{u'_i u'_j}$. Each RANS variant offers a different strategy for modeling these stresses.

Reynolds-Averaged Navier-Stokes (RANS) Equations

• Foundation of industrial CFD—derived by decomposing velocity into mean $\bar{u}$ and fluctuating $u'$ components, then time-averaging the Navier-Stokes equations
• Closure problem requires additional turbulence models because averaging introduces six unknown Reynolds stress components $\overline{u'_i u'_j}$
• Computational efficiency makes RANS the workhorse for engineering design, though it struggles with unsteady separation, strong streamline curvature, and highly anisotropic turbulence

k-ε Model

• Two-equation model solving transport equations for turbulent kinetic energy $k = \frac{1}{2}\overline{u'_i u'_i}$ and its dissipation rate $\varepsilon$
• Eddy viscosity computed as $\nu_t = C_\mu \frac{k^2}{\varepsilon}$, assuming turbulence is isotropic and scales with these two quantities
• Industrial standard for wall-bounded flows, though it overpredicts spreading rates in jets and performs poorly near separation points

k-ω Model

• Alternative two-equation formulation using specific dissipation rate $\omega = \varepsilon / k$ instead of $\varepsilon$ directly
• Superior near-wall performance—integrates smoothly to the wall without requiring damping functions, better for adverse pressure gradients and boundary layer flows
• Freestream sensitivity is the major drawback; results depend strongly on the arbitrary freestream value of $\omega$

Spalart-Allmaras Model

• One-equation simplicity—solves a single transport equation for modified eddy viscosity $\tilde{\nu}$, avoiding the $k$ and $\varepsilon$ equations entirely
• Aerospace heritage designed specifically for external aerodynamics; calibrated for attached boundary layers and mild separation
• Computational savings with reasonable accuracy for streamlined bodies, but lacks the physics to handle complex recirculating flows or strong pressure gradients

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Reynolds Stress and Algebraic Approaches

When the eddy viscosity assumption fails—particularly for flows with strong rotation, curvature, or anisotropy—these models solve directly for the Reynolds stress tensor or use algebraic relations to capture directional effects.

Reynolds Stress Model (RSM)

• Six transport equations for each independent component of the Reynolds stress tensor $\overline{u'_i u'_j}$, abandoning the isotropic eddy viscosity assumption
• Anisotropy captured naturally, making RSM essential for swirling flows, rotating systems, and flows with strong streamline curvature
• Computational cost and stability challenges—more equations mean more numerical stiffness and sensitivity to boundary conditions

Algebraic Stress Model

• Implicit algebraic relations derived from the RSM transport equations by assuming local equilibrium of turbulence production and dissipation
• Anisotropy without full RSM cost—captures directional effects while maintaining computational tractability similar to two-equation models
• Compromise solution that works well for mildly complex turbulence but loses accuracy when convection and diffusion of Reynolds stresses matter significantly

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Scale-Resolving Simulations (LES and DNS)

Rather than modeling all turbulent fluctuations, these approaches resolve turbulent structures directly at some or all scales. The governing philosophy shifts from statistical averaging to spatial filtering or complete resolution.

Large Eddy Simulation (LES)

• Spatial filtering separates large energy-containing eddies (resolved) from small dissipative scales (modeled via subgrid-scale closure)
• Transient accuracy captures vortex shedding, mixing, and unsteady phenomena that RANS fundamentally cannot predict
• Grid requirements scale as $Re^{9/5}$ for wall-bounded flows, making LES computationally demanding but increasingly feasible for complex industrial problems

Smagorinsky Model

• Subgrid-scale closure for LES that models unresolved stresses as $\tau_{ij}^{sgs} = -2(C_s \Delta)^2 |\bar{S}| \bar{S}_{ij}$
• Eddy viscosity at subgrid scales—assumes small eddies behave like enhanced molecular viscosity, with magnitude proportional to local strain rate $|\bar{S}|$
• Constant $C_s$ limitation—the Smagorinsky constant requires tuning for different flows; dynamic variants compute $C_s$ locally to improve generality

Direct Numerical Simulation (DNS)

• No modeling whatsoever—resolves every scale of turbulent motion from the largest energy-containing eddies down to the Kolmogorov scale $\eta$
• Grid requirements scale as $Re^3$, restricting DNS to relatively low Reynolds numbers (typically $Re < 10^4$ for channel flows)
• Benchmark standard for validating turbulence models and understanding fundamental physics; not a design tool but essential for research

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Hybrid RANS-LES Methods

These approaches combine RANS efficiency near walls with LES accuracy in separated regions, addressing the prohibitive cost of wall-resolved LES while capturing unsteady large-scale structures.

Detached Eddy Simulation (DES)

• Automatic switching between RANS (attached boundary layers) and LES (separated/wake regions) based on local grid spacing relative to turbulent length scale
• Length scale modification replaces the RANS turbulent length scale with the grid spacing $\Delta$ when $\Delta < \delta$, triggering LES behavior
• Gray area problem—the RANS-to-LES transition zone can produce spurious behavior; improved variants (DDES, IDDES) address this with shielding functions

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Quick Reference Table

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Self-Check Questions

1. Both k-ε and k-ω are two-equation models—what fundamental quantity do they share, and how do their second equations differ in physical interpretation?

2. You're simulating flow around a rotating turbine blade with strong streamline curvature. Which model family would you choose over standard k-ε, and why does the eddy viscosity assumption fail here?

3. Compare and contrast LES and RANS in terms of: (a) what they resolve vs. model, (b) computational cost scaling with Reynolds number, and (c) ability to predict vortex shedding.

4. A colleague proposes using DNS for a production automotive aerodynamics simulation at $Re = 10^7$. Explain why this is impractical and identify two alternative approaches that could capture unsteady effects.

5. What distinguishes DES from simply running LES everywhere, and what numerical artifact can occur in the transition region between RANS and LES modes?