5.1 RANS, LES, and DNS Methodologies

Turbulent combustion modeling is complex, but there are three main approaches: RANS, LES, and DNS. Each method offers a different balance between accuracy and computational cost, tackling the challenge of resolving turbulent motions across various scales.

RANS averages flow variables, LES filters out small scales, and DNS resolves everything. Understanding these methods is crucial for predicting turbulent combustion behavior in real-world applications, from jet engines to power plants.

Turbulence Modeling Approaches

Reynolds-Averaged Navier-Stokes (RANS) and Large Eddy Simulation (LES)

• Reynolds-Averaged Navier-Stokes (RANS) decomposes flow variables into mean and fluctuating components
• RANS equations solve for time-averaged quantities, reducing computational cost
• Large Eddy Simulation (LES) resolves larger turbulent scales directly while modeling smaller scales
• LES applies spatial filtering to separate large and small-scale motions
• Both RANS and LES require closure models to account for unresolved turbulence

Direct Numerical Simulation (DNS) and Turbulence Modeling

• Direct Numerical Simulation (DNS) resolves all scales of turbulent motion without modeling
• DNS provides highest fidelity results but demands extreme computational resources
• Turbulence modeling bridges the gap between resolved and unresolved scales
• Models range from simple algebraic relations to complex transport equations
• Subgrid-scale modeling in LES addresses effects of small-scale motions on resolved scales
• Common subgrid-scale models include Smagorinsky model and dynamic procedures

Computational Considerations

Computational Cost and Resolution Scales

• Computational cost increases dramatically from RANS to LES to DNS
• RANS requires least resources, suitable for industrial applications (aircraft design)
• LES balances accuracy and cost, used in complex flows (combustion chambers)
• DNS demands massive computing power, limited to fundamental research (channel flow)
• Resolution scales differ significantly between approaches
• RANS resolves mean flow, models all turbulence scales
• LES resolves energy-containing eddies, models dissipative scales
• DNS resolves all scales down to Kolmogorov microscales

Spatial Filtering and Grid Requirements

• Spatial filtering in LES separates resolved and subgrid scales
• Filter width typically linked to computational grid spacing
• Grid resolution determines the portion of turbulent spectrum directly computed
• RANS grids can be relatively coarse, focused on mean flow features
• LES grids must capture energy-containing eddies (inertial subrange)
• DNS grids require extreme refinement to resolve Kolmogorov scales
• Grid design impacts accuracy, stability, and computational efficiency

RANS and LES Techniques

Time-averaging and Spatial Filtering

• Time-averaging in RANS decomposes variables into mean and fluctuating parts
• RANS equations derived by applying Reynolds decomposition to Navier-Stokes equations
• Spatial filtering in LES separates resolved and subgrid-scale motions
• LES filter operation defines cutoff between directly computed and modeled scales
• Both techniques introduce unclosed terms requiring modeling (Reynolds stresses, subgrid stresses)

Turbulent Kinetic Energy and Energy Spectrum

• Turbulent kinetic energy represents intensity of velocity fluctuations
• RANS models often solve transport equation for turbulent kinetic energy
• LES resolves part of turbulent kinetic energy spectrum directly
• Energy spectrum describes distribution of turbulent energy across scales
• RANS models entire spectrum, LES resolves large scales and models small scales
• Proper resolution of energy-containing scales crucial for LES accuracy