7.3 Turbulence modeling

Turbulence modeling is crucial in fluid dynamics, helping engineers predict chaotic fluid behavior in various applications. It bridges the gap between complex physics and practical engineering solutions, enabling better designs in aerospace, automotive, and environmental fields.

Different approaches, from simple RANS models to complex DNS, offer varying levels of accuracy and computational cost. Choosing the right model involves balancing precision with available resources, making turbulence modeling both an art and a science in fluid dynamics.

Basics of turbulence

• Turbulence is a complex and chaotic state of fluid motion characterized by irregular fluctuations in velocity, pressure, and other flow properties
• Understanding and modeling turbulence is crucial in various engineering applications, such as aerodynamics, combustion, and heat transfer
• Turbulent flows exhibit enhanced mixing, increased drag, and dissipation of energy compared to laminar flows

Characteristics of turbulent flows

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• Turbulent flows are highly unsteady and irregular, with significant variations in velocity and pressure over time and space
• They exhibit a wide range of length and time scales, from large eddies to small-scale turbulent structures
• Turbulent flows are characterized by high Reynolds numbers, indicating the dominance of inertial forces over viscous forces
• They exhibit enhanced mixing and transport of momentum, heat, and mass due to the chaotic motion of fluid particles

Turbulence vs laminar flow

• Laminar flow is characterized by smooth, parallel layers of fluid with no mixing between layers
• Turbulent flow, on the other hand, exhibits chaotic and irregular motion with significant mixing between layers
• The transition from laminar to turbulent flow occurs when the Reynolds number exceeds a critical value, which depends on the geometry and flow conditions
• Turbulent flows are more common in practical engineering applications due to their prevalence at high Reynolds numbers

Importance of turbulence modeling

• Accurate modeling of turbulence is essential for predicting flow behavior, heat transfer, and other important phenomena in various engineering applications
• Turbulence modeling enables the design and optimization of devices such as aircraft wings, turbomachinery, and heat exchangers
• It helps in understanding and mitigating adverse effects of turbulence, such as increased drag, noise, and vibrations
• Turbulence modeling is crucial for the development of more efficient and sustainable engineering systems

Turbulence modeling approaches

• Various approaches have been developed to model turbulence, each with its own advantages and limitations
• The choice of turbulence modeling approach depends on the desired accuracy, computational cost, and the specific application

Direct numerical simulation (DNS)

• DNS involves solving the Navier-Stokes equations without any turbulence modeling assumptions
• It resolves all scales of turbulence, from the largest eddies to the smallest dissipative scales
• DNS requires extremely fine spatial and temporal resolution, making it computationally expensive and limited to low Reynolds number flows
• DNS is mainly used for fundamental research and validation of turbulence models

Reynolds-averaged Navier-Stokes (RANS) models

• RANS models are based on the Reynolds decomposition, which separates the flow variables into mean and fluctuating components
• The Navier-Stokes equations are averaged, resulting in additional terms called Reynolds stresses that represent the effects of turbulence
• RANS models introduce closure assumptions to model the Reynolds stresses, such as the Boussinesq hypothesis and eddy viscosity concept
• Examples of RANS models include zero-equation models (mixing length model), one-equation models (Spalart-Allmaras), and two-equation models (k-epsilon, k-omega)

Large eddy simulation (LES)

• LES resolves the large-scale turbulent structures directly while modeling the smaller scales using subgrid-scale (SGS) models
• The Navier-Stokes equations are filtered to separate the resolved and unresolved scales
• LES captures more flow details compared to RANS models but is computationally more expensive
• Examples of SGS models include the Smagorinsky model, dynamic Smagorinsky model, and wall-adapting local eddy-viscosity (WALE) model

Hybrid RANS-LES models

• Hybrid models combine the advantages of RANS and LES approaches to achieve a balance between accuracy and computational cost
• They use RANS modeling in near-wall regions where turbulence is anisotropic and LES in the outer regions where large-scale structures dominate
• Examples of hybrid models include detached eddy simulation (DES) and scale-adaptive simulation (SAS)
• Hybrid models provide a compromise between the computational efficiency of RANS and the accuracy of LES

RANS turbulence models

• RANS models are widely used in engineering applications due to their computational efficiency and robustness
• They are based on the Reynolds-averaged Navier-Stokes equations and introduce closure assumptions to model the Reynolds stresses

Boussinesq hypothesis

• The Boussinesq hypothesis assumes that the Reynolds stresses are proportional to the mean strain rate tensor
• It introduces the concept of eddy viscosity, which relates the Reynolds stresses to the mean velocity gradients
• The Boussinesq hypothesis is the foundation for many RANS turbulence models

Eddy viscosity concept

• Eddy viscosity represents the turbulent transport of momentum by eddies, analogous to molecular viscosity
• It is not a physical property of the fluid but rather a modeling concept to account for the effects of turbulence
• The eddy viscosity is typically modeled using algebraic expressions or transport equations

Zero-equation models

• Zero-equation models, such as the mixing length model, are the simplest RANS models
• They rely on algebraic expressions to estimate the eddy viscosity based on the local mean flow properties
• Zero-equation models are computationally inexpensive but have limited accuracy and applicability

One-equation models

• One-equation models, such as the Spalart-Allmaras model, solve a transport equation for a turbulence quantity related to the eddy viscosity
• The transport equation accounts for the production, dissipation, and transport of turbulence
• One-equation models provide improved accuracy compared to zero-equation models but still have limitations in complex flows

Two-equation models

• Two-equation models, such as k-epsilon and k-omega models, solve transport equations for two turbulence quantities: turbulent kinetic energy (k) and dissipation rate (epsilon) or specific dissipation rate (omega)
• These models provide a more complete description of turbulence by accounting for the production, dissipation, and transport of turbulence quantities
• Two-equation models are widely used in engineering applications due to their robustness and reasonable accuracy

k-epsilon model

• The k-epsilon model solves transport equations for the turbulent kinetic energy (k) and its dissipation rate (epsilon)
• It is one of the most widely used RANS models due to its simplicity and good performance in many flows
• The standard k-epsilon model has limitations in predicting flows with strong adverse pressure gradients and separation

k-omega model

• The k-omega model solves transport equations for the turbulent kinetic energy (k) and the specific dissipation rate (omega)
• It performs better than the k-epsilon model in near-wall regions and flows with adverse pressure gradients
• The standard k-omega model is sensitive to freestream turbulence levels and may require careful boundary condition treatment

SST k-omega model

• The shear stress transport (SST) k-omega model combines the advantages of the k-epsilon model in the freestream and the k-omega model near the walls
• It uses a blending function to switch between the two models based on the distance from the wall
• The SST k-omega model provides improved predictions for flows with separation and adverse pressure gradients

Reynolds stress models (RSM)

• Reynolds stress models solve transport equations for each component of the Reynolds stress tensor
• They account for the anisotropy of turbulence and can capture complex turbulence-driven secondary flows
• RSMs are more computationally expensive than eddy viscosity models due to the additional transport equations
• They require careful numerical treatment and may suffer from stability issues in some cases

LES turbulence modeling

• Large eddy simulation (LES) resolves the large-scale turbulent structures directly while modeling the smaller scales using subgrid-scale (SGS) models
• LES provides a more accurate representation of turbulence compared to RANS models but is computationally more expensive

Filtering of Navier-Stokes equations

• In LES, the Navier-Stokes equations are filtered to separate the resolved scales from the unresolved scales
• The filtering operation is typically performed in space using a low-pass filter
• The filtered equations govern the evolution of the resolved scales while the effects of the unresolved scales are modeled by the SGS models

Subgrid-scale (SGS) models

• SGS models represent the effects of the unresolved small-scale turbulent structures on the resolved scales
• They introduce a subgrid-scale stress tensor that accounts for the momentum transfer between the resolved and unresolved scales
• SGS models are based on the eddy viscosity concept and assume that the subgrid-scale stresses are proportional to the resolved strain rate tensor

Smagorinsky model

• The Smagorinsky model is a simple and widely used SGS model
• It expresses the subgrid-scale eddy viscosity as a function of the resolved strain rate tensor and a model coefficient
• The model coefficient is often determined based on theoretical considerations or by dynamic procedures

Dynamic Smagorinsky model

• The dynamic Smagorinsky model improves upon the standard Smagorinsky model by dynamically computing the model coefficient based on the resolved flow field
• It uses a test filter to determine the local value of the model coefficient, adapting it to the local flow conditions
• The dynamic procedure helps to overcome some of the limitations of the standard Smagorinsky model, such as excessive dissipation in laminar regions

Wall-adapting local eddy-viscosity (WALE) model

• The WALE model is designed to improve the near-wall behavior of the SGS model
• It accounts for the correct scaling of the eddy viscosity near the walls and reproduces the proper near-wall asymptotic behavior
• The WALE model is less sensitive to the choice of model coefficients compared to the Smagorinsky model and provides improved predictions in wall-bounded flows

Boundary conditions for turbulence

• Proper treatment of boundary conditions is crucial for accurate turbulence modeling, especially near solid walls where turbulence is strongly influenced by the presence of the wall

Wall functions

• Wall functions are used to bridge the near-wall region, where turbulence is anisotropic and the grid resolution is typically insufficient to resolve the viscous sublayer
• They provide algebraic expressions for the velocity and turbulence quantities near the wall based on the law of the wall and the logarithmic law
• Wall functions allow for a coarser grid resolution near the walls, reducing the computational cost

Low-Reynolds-number models

• Low-Reynolds-number models are designed to resolve the near-wall region directly without the use of wall functions
• They employ damping functions to modify the turbulence model equations and account for the viscous effects near the wall
• Low-Reynolds-number models require a fine grid resolution near the walls to capture the steep gradients in velocity and turbulence quantities

Near-wall resolution requirements

• The choice between wall functions and low-Reynolds-number models depends on the desired accuracy and the available computational resources
• Wall functions are suitable for high-Reynolds-number flows where the near-wall region is not of primary interest
• Low-Reynolds-number models are necessary for accurate predictions of wall-bounded flows, such as boundary layers and flow separation
• The near-wall grid resolution should be chosen based on the turbulence model and the desired level of accuracy

Turbulence model validation

• Validation of turbulence models is essential to assess their accuracy and reliability in predicting turbulent flows
• Validation involves comparing the model predictions with experimental data or high-fidelity simulations

Benchmark test cases

• Benchmark test cases are well-documented experiments or simulations that provide detailed data for turbulence model validation
• Examples of benchmark cases include turbulent channel flow, backward-facing step flow, and flow over a circular cylinder
• These cases cover a range of flow conditions and geometries to assess the performance of turbulence models in different scenarios

Comparison with experimental data

• Experimental data, such as velocity profiles, turbulence statistics, and wall shear stress, are used to validate turbulence models
• The model predictions are compared with the experimental data to evaluate the accuracy of the model
• Statistical measures, such as mean error and correlation coefficients, are used to quantify the agreement between the model and the experiments

Sensitivity analysis of model parameters

• Turbulence models often involve empirical constants or model parameters that need to be calibrated
• Sensitivity analysis is performed to assess the influence of these parameters on the model predictions
• The sensitivity analysis helps to identify the key parameters and their optimal values for a given flow configuration
• It also provides insights into the robustness and reliability of the turbulence model

Applications of turbulence modeling

• Turbulence modeling is widely used in various engineering fields to predict and analyze turbulent flows
• Some of the key applications of turbulence modeling include:

Aerodynamics and aerospace engineering

• Turbulence modeling is essential for the design and optimization of aircraft wings, propulsion systems, and other aerospace components
• It helps in predicting lift, drag, and heat transfer characteristics of aircraft and spacecraft
• Examples include the analysis of flow over airfoils, wings, and turbomachinery components

Automotive and transportation engineering

• Turbulence modeling is used in the development of vehicles, from cars to trains and ships
• It assists in optimizing the aerodynamic performance, reducing drag, and improving fuel efficiency
• Applications include the analysis of flow around vehicles, in engine combustion chambers, and in cooling systems

Environmental and atmospheric flows

• Turbulence modeling is crucial for understanding and predicting environmental and atmospheric flows, such as wind, ocean currents, and pollutant dispersion
• It helps in the assessment of wind loads on structures, the design of wind turbines, and the study of atmospheric boundary layers
• Turbulence modeling is also used in weather forecasting and climate modeling

Industrial and process engineering

• Turbulence modeling is applied in various industrial processes, such as mixing, heat transfer, and chemical reactions
• It aids in the design and optimization of industrial equipment, such as heat exchangers, reactors, and turbomachinery
• Examples include the analysis of flow in pipelines, mixing tanks, and combustion systems

Limitations and challenges

• Despite the significant progress in turbulence modeling, there are still limitations and challenges that need to be addressed

Accuracy vs computational cost

• There is a trade-off between the accuracy of turbulence models and their computational cost
• Higher-fidelity models, such as DNS and LES, provide more accurate predictions but are computationally expensive
• RANS models are computationally efficient but may not capture all the relevant turbulence phenomena accurately
• Balancing accuracy and computational cost is a major challenge in turbulence modeling

Model selection and calibration

• Selecting the appropriate turbulence model for a given flow problem is not always straightforward
• Different models have their strengths and weaknesses, and their performance may vary depending on the flow conditions and geometry
• Calibrating the model parameters to match experimental data or high-fidelity simulations can be time-consuming and requires expertise

Complex geometry and flow conditions

• Turbulence modeling becomes more challenging in complex geometries and flow conditions, such as flow separation, recirculation, and high-speed flows
• The presence of strong gradients, curvature, and rotational effects can strain the assumptions of turbulence models
• Addressing these complexities often requires advanced modeling techniques, such as adaptive mesh refinement and higher-order numerical schemes

Future developments in turbulence modeling

• There is an ongoing research effort to improve turbulence modeling techniques and address their limitations
• Some of the future developments include:
  • Hybrid RANS-LES models that combine the strengths of both approaches
  • Data-driven turbulence models that leverage machine learning techniques to improve predictions
  • Adaptive turbulence models that automatically adjust to the local flow conditions
  • High-fidelity simulations, such as DNS and LES, becoming more feasible with the advancements in computational resources
• These developments aim to provide more accurate and reliable turbulence modeling tools for a wide range of engineering applications