Turbulence modeling is a crucial aspect of simulating complex fluid flows in engineering applications. It tackles the challenges of predicting chaotic, unsteady behavior across multiple scales, from large eddies to small-scale structures.

Various approaches exist, including Reynolds-averaged Navier-Stokes (RANS) models, large eddy simulation (LES), and direct numerical simulation (DNS). Each method balances accuracy and computational cost, aiming to capture turbulent flow dynamics efficiently for practical applications.

Turbulence modeling overview

  • Turbulence modeling plays a crucial role in simulating and understanding complex fluid flows encountered in various engineering applications (aerospace, automotive, energy systems)
  • Turbulent flows exhibit chaotic, unsteady, and multiscale behavior, making them challenging to predict and control

Characteristics of turbulent flows

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  • Turbulent flows are characterized by irregular velocity fluctuations, enhanced mixing, and increased dissipation of energy
  • Exhibit a wide range of spatial and temporal scales, from large eddies to small-scale turbulent structures (Kolmogorov scales)
  • Display high sensitivity to initial and boundary conditions, leading to chaotic behavior and limited predictability

Challenges in turbulence modeling

  • Capturing the complex, multiscale nature of turbulent flows requires sophisticated mathematical and computational models
  • Direct numerical simulation (DNS) of turbulent flows is computationally expensive, often prohibitive for high Reynolds number flows
  • Turbulence models aim to provide accurate and efficient approximations of turbulent flow behavior while reducing computational costs

Reynolds-averaged Navier-Stokes (RANS) models

  • RANS models are based on the concept of decomposing flow variables into mean and fluctuating components
  • Introduce additional terms in the governing equations, representing the effects of turbulent fluctuations on the mean flow

Concept of Reynolds averaging

  • Reynolds averaging involves decomposing flow variables (velocity, pressure) into mean and fluctuating components
  • Resulting in the Reynolds-averaged Navier-Stokes (RANS) equations, which govern the mean flow behavior
  • Introduces the Reynolds stress tensor, representing the effects of turbulent fluctuations on the mean flow

Boussinesq eddy-viscosity hypothesis

  • The Boussinesq hypothesis relates the Reynolds stress tensor to the mean strain rate tensor through an eddy viscosity
  • Assumes that the effect of turbulence can be modeled as an increased viscosity, known as the turbulent or eddy viscosity
  • Simplifies the closure problem by reducing the number of unknown terms in the RANS equations

Zero-equation models

  • Zero-equation models, such as the mixing length model, estimate the eddy viscosity based on algebraic expressions
  • Rely on empirical relations and do not involve additional transport equations for turbulence quantities
  • Suitable for simple flows but have limited accuracy and generality

One-equation models

  • One-equation models, such as the Spalart-Allmaras model, solve a transport equation for a turbulence quantity (turbulent kinetic energy or eddy viscosity)
  • Provide improved accuracy compared to zero-equation models by accounting for the transport and history effects of turbulence
  • Widely used in aerospace applications due to their robustness and computational efficiency

Two-equation models

  • Two-equation models, such as the k-epsilon and k-omega models, solve transport equations for two turbulence quantities (turbulent kinetic energy and dissipation or specific dissipation rate)
  • Offer a balance between accuracy and computational cost, making them popular in industrial applications

k-epsilon model

  • The k-epsilon model solves transport equations for the turbulent kinetic energy (k) and its dissipation rate (epsilon)
  • Robust and widely used in various engineering applications due to its simplicity and numerical stability
  • Performs well for free-shear flows but has limitations in predicting near-wall behavior and adverse pressure gradients

k-omega model

  • The k-omega model solves transport equations for the turbulent kinetic energy (k) and the specific dissipation rate (omega)
  • Provides improved near-wall treatment compared to the k-epsilon model, making it suitable for wall-bounded flows
  • Sensitive to freestream turbulence levels and requires careful treatment of boundary conditions

SST model

  • The Shear Stress Transport (SST) model combines the advantages of the k-epsilon and k-omega models
  • Employs a blending function to switch between the k-omega model near the wall and the k-epsilon model in the freestream
  • Offers improved predictions of flow separation and adverse pressure gradients compared to the standard k-epsilon and k-omega models

Reynolds stress models

  • Reynolds stress models (RSM) solve transport equations for each component of the Reynolds stress tensor
  • Capture the anisotropic nature of turbulence and provide more accurate predictions for complex flows (swirling, rotating, or strongly separated flows)
  • Computationally expensive compared to eddy-viscosity models due to the increased number of equations solved

Large eddy simulation (LES)

  • LES is a turbulence modeling approach that directly resolves large-scale turbulent structures while modeling the effects of small-scale structures
  • Applies a spatial filtering operation to the Navier-Stokes equations, separating resolved and subgrid-scale (SGS) motions

Concept of spatial filtering

  • Spatial filtering in LES decomposes the flow field into resolved (large-scale) and subgrid-scale (small-scale) components
  • The resolved scales are directly computed, while the effects of the subgrid scales are modeled using SGS models
  • Filtering operation is typically based on the grid resolution, with the filter width related to the grid size

Subgrid-scale (SGS) modeling

  • SGS models aim to represent the effects of the unresolved, small-scale turbulent structures on the resolved flow
  • Account for the energy transfer between the resolved and subgrid scales and the dissipation of energy at small scales
  • Commonly used SGS models include the Smagorinsky model, dynamic models, and scale-similarity models

Smagorinsky model

  • The Smagorinsky model is a simple and widely used SGS model in LES
  • Relates the SGS stresses to the resolved strain rate tensor through an eddy-viscosity assumption
  • Employs a constant model coefficient, which requires calibration and can lead to excessive dissipation in certain flow regions

Dynamic SGS models

  • Dynamic SGS models, such as the dynamic Smagorinsky model, dynamically compute the model coefficients based on the resolved flow information
  • Adapt the model coefficients to the local flow conditions, reducing the need for a priori calibration
  • Provide improved predictions compared to the standard Smagorinsky model, especially in transitional and wall-bounded flows

Direct numerical simulation (DNS)

  • DNS involves directly solving the Navier-Stokes equations without any turbulence modeling assumptions
  • Resolves all spatial and temporal scales of turbulence, from the largest eddies to the Kolmogorov scales

Concept of DNS

  • DNS captures the full spectrum of turbulent motions by resolving all relevant scales
  • Requires extremely fine spatial and temporal resolution to accurately represent the smallest turbulent structures
  • Provides detailed information about turbulence dynamics, energy transfer, and dissipation mechanisms

Computational requirements for DNS

  • DNS is computationally expensive due to the high resolution required to capture all turbulent scales
  • Computational cost scales with the Reynolds number, making DNS feasible only for low to moderate Reynolds number flows
  • Requires massive computational resources and parallel computing techniques for large-scale simulations

Applications of DNS

  • DNS is primarily used for fundamental research and understanding of turbulence physics
  • Provides benchmark data for validating and improving turbulence models and numerical methods
  • Offers insights into complex flow phenomena, such as transition to turbulence, turbulent mixing, and heat transfer

Hybrid RANS-LES methods

  • Hybrid RANS-LES methods combine the advantages of RANS models and LES to efficiently simulate complex turbulent flows
  • Employ RANS modeling in near-wall regions and LES in detached or separated flow regions

Detached eddy simulation (DES)

  • DES is a hybrid RANS-LES approach that switches between RANS and LES based on a model-specific criterion
  • Uses RANS modeling in attached boundary layers and LES in detached or separated flow regions
  • Provides a computationally efficient alternative to full LES while capturing important unsteady flow features

Delayed detached eddy simulation (DDES)

  • DDES is an improved version of DES that addresses the issue of grid-induced separation in the original DES formulation
  • Employs a shielding function to delay the switch from RANS to LES, preventing premature activation of LES in attached boundary layers
  • Offers better control over the RANS-LES transition and improved predictions of flow separation

Improved delayed detached eddy simulation (IDDES)

  • IDDES further enhances the DDES formulation by combining the advantages of DDES and wall-modeled LES (WMLES)
  • Includes a modified subgrid length scale definition and a blending function for smooth transition between RANS and LES
  • Provides improved predictions of complex flows with both attached and separated regions, such as high-lift configurations and aeroacoustics

Wall modeling for LES

  • Wall modeling in LES aims to reduce the computational cost associated with resolving the near-wall turbulent structures
  • Allows for coarser grid resolution near the walls while still capturing the essential flow physics

Importance of near-wall resolution

  • Near-wall turbulent structures play a crucial role in determining wall shear stress, heat transfer, and flow separation
  • Resolving the near-wall region in LES requires extremely fine grid resolution, leading to high computational costs
  • Wall modeling approaches aim to relax the near-wall resolution requirements while maintaining accuracy

Wall functions

  • Wall functions are algebraic models that relate the near-wall velocity to the wall shear stress using empirical laws (logarithmic law of the wall)
  • Provide a computationally efficient way to represent the near-wall flow behavior without resolving the viscous sublayer
  • Suitable for high Reynolds number flows with attached boundary layers but have limitations in predicting flow separation and heat transfer

Two-layer models

  • Two-layer models divide the near-wall region into a viscous sublayer and a logarithmic layer
  • Employ simplified RANS-like equations in the viscous sublayer and LES in the outer layer
  • Provide improved predictions compared to wall functions by capturing the essential near-wall physics
  • Require additional grid resolution compared to wall functions but still offer significant computational savings compared to fully-resolved LES

Turbulence model validation

  • Validation of turbulence models is essential to assess their accuracy, reliability, and range of applicability
  • Involves comparing model predictions with experimental data or high-fidelity simulations (DNS or resolved LES)

Benchmark test cases

  • Benchmark test cases are well-documented experimental or numerical datasets that serve as reference for turbulence model validation
  • Cover a range of flow configurations, such as channel flows, boundary layers, jets, wakes, and separated flows
  • Provide detailed measurements of velocity fields, Reynolds stresses, and other turbulence statistics

Comparison with experimental data

  • Comparing turbulence model predictions with experimental data is a critical step in model validation
  • Experimental data can include mean velocity profiles, turbulence intensities, Reynolds stresses, and flow visualizations
  • Assesses the ability of turbulence models to capture key flow features and turbulence characteristics

Uncertainty quantification in turbulence modeling

  • Uncertainty quantification (UQ) aims to quantify the impact of model assumptions, input uncertainties, and numerical errors on the simulation results
  • Employs techniques such as sensitivity analysis, Bayesian inference, and stochastic collocation methods
  • Provides a framework for assessing the reliability and robustness of turbulence model predictions and guiding model improvement efforts

Turbulence modeling in complex geometries

  • Turbulence modeling in complex geometries poses additional challenges related to mesh generation and near-wall treatment
  • Requires careful consideration of grid resolution, quality, and adaptivity to accurately capture the flow physics

Challenges in mesh generation

  • Complex geometries often involve irregular shapes, sharp edges, and small-scale features that require special attention during mesh generation
  • Ensuring adequate grid resolution in critical flow regions while maintaining computational efficiency is a key challenge
  • Unstructured or hybrid grids are commonly used to handle complex geometries, but they can introduce numerical errors and increase computational costs

Near-wall treatment

  • Near-wall treatment in complex geometries is crucial for accurate prediction of wall shear stress, heat transfer, and flow separation
  • Requires careful grid design to capture the steep gradients and anisotropic turbulence in the near-wall region
  • Wall functions and two-layer models need to be adapted to handle non-planar or curved walls and to account for the local flow conditions

Adaptive mesh refinement

  • Adaptive mesh refinement (AMR) is a technique that dynamically adapts the grid resolution based on the local flow features and turbulence characteristics
  • Allows for efficient use of computational resources by refining the mesh in regions of interest while coarsening it in less critical areas
  • Helps capture the multiscale nature of turbulence and improves the accuracy of turbulence model predictions in complex geometries
  • Turbulence modeling continues to evolve, driven by advances in computational resources, numerical methods, and data-driven approaches
  • Emerging trends aim to improve the accuracy, efficiency, and generality of turbulence models

Machine learning-based models

  • Machine learning (ML) techniques, such as neural networks and Gaussian processes, are being explored for turbulence modeling
  • ML-based models can learn complex relationships between flow variables and turbulence quantities from high-fidelity simulation data or experimental measurements
  • Offer the potential for improved accuracy and reduced computational cost compared to traditional physics-based models

Data-driven modeling approaches

  • Data-driven modeling approaches leverage large datasets from high-fidelity simulations or experiments to inform turbulence model development
  • Employ techniques such as dimensionality reduction, sparse regression, and Bayesian inference to extract relevant turbulence features and model coefficients
  • Aim to develop turbulence models that are tailored to specific flow configurations or operating conditions, improving their predictive capabilities

Multiscale modeling strategies

  • Multiscale modeling strategies aim to bridge the gap between different levels of turbulence modeling (DNS, LES, RANS) and capture the multiscale nature of turbulence
  • Employ techniques such as hybrid RANS-LES methods, scale-resolving simulations, and concurrent multiscale approaches
  • Allow for efficient simulation of complex turbulent flows by combining the strengths of different modeling approaches and resolving the relevant scales of turbulence
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