4.5 Turbulence modeling

Turbulence modeling is crucial in aerodynamics, impacting drag, heat transfer, and noise generation. It involves predicting chaotic fluid motion with irregular fluctuations in velocity and pressure. Understanding turbulence is key to optimizing aircraft design and performance.

Various approaches exist, from direct numerical simulation to Reynolds-averaged Navier-Stokes models. Each method balances accuracy and computational cost differently. RANS models, like k-epsilon and k-omega, are widely used in industry for their efficiency in predicting averaged flow quantities.

Fundamentals of turbulence

• Turbulence is a complex and chaotic state of fluid motion characterized by irregular fluctuations in velocity, pressure, and other flow properties
• Understanding turbulence is crucial in aerodynamics as it significantly impacts drag, heat transfer, mixing, and noise generation
• Turbulent flows exhibit a wide range of scales, from large eddies that contain most of the energy to small dissipative scales where viscosity dominates

Characteristics of turbulent flows

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• Turbulent flows are highly unsteady and irregular, with significant variations in velocity and pressure fields
• They exhibit enhanced mixing and increased rates of momentum, heat, and mass transfer compared to laminar flows
• Turbulent flows are characterized by the presence of eddies of various sizes that interact and transfer energy across different scales
• Turbulence is a three-dimensional phenomenon, with fluctuations occurring in all spatial directions

Turbulent vs laminar flow

• Laminar flow is characterized by smooth, parallel layers of fluid with no mixing between layers
• Turbulent flow, on the other hand, is characterized by chaotic and irregular motion with significant mixing between fluid layers
• The transition from laminar to turbulent flow depends on the Reynolds number (ratio of inertial forces to viscous forces) and the geometry of the flow
• In aerodynamics, both laminar and turbulent flows can occur, and understanding their characteristics is essential for accurate predictions and design optimization

Importance in aerodynamics

• Turbulence plays a crucial role in determining the aerodynamic performance of vehicles, including lift, drag, and heat transfer
• Accurate modeling of turbulence is necessary for the design and optimization of aircraft wings, turbomachinery, and other aerodynamic components
• Turbulence affects the behavior of boundary layers, which are thin regions near solid surfaces where viscous effects are significant
• Turbulent flows can lead to increased drag, noise generation, and structural vibrations, making their understanding and control critical in aerodynamic applications

Turbulence modeling approaches

• Turbulence modeling aims to develop mathematical and computational methods to predict the behavior of turbulent flows
• Different approaches have been developed to balance accuracy, computational cost, and the level of detail captured in the simulations
• The choice of turbulence modeling approach depends on the specific application, available computational resources, and the required level of accuracy

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 down to the smallest dissipative scales
• DNS requires extremely fine spatial and temporal resolution, making it computationally expensive and limited to low Reynolds number flows and simple geometries
• DNS is mainly used for fundamental research and validation of turbulence models

Large eddy simulation (LES)

• LES resolves the large-scale turbulent motions directly while modeling the effects of smaller scales using subgrid-scale (SGS) models
• It captures more flow details than RANS models but is less computationally expensive than DNS
• LES is suitable for flows with complex geometries and moderate Reynolds numbers
• The accuracy of LES depends on the quality of the SGS model and the resolution of the computational grid

Reynolds-averaged Navier-Stokes (RANS)

• RANS models are based on the Reynolds-averaged Navier-Stokes equations, which decompose the flow variables into mean and fluctuating components
• The effects of turbulence are modeled using additional transport equations for turbulence quantities, such as turbulent kinetic energy and dissipation rate
• RANS models are computationally efficient and widely used in industrial applications
• They provide averaged flow quantities but do not capture the detailed unsteady behavior of turbulent flows

Hybrid RANS-LES methods

• Hybrid methods combine RANS and LES approaches to balance accuracy and computational cost
• They use RANS models in regions near walls where the grid resolution is insufficient for LES, and switch to LES in regions away from walls where the grid is fine enough
• Examples of hybrid methods include Detached Eddy Simulation (DES) and Scale-Adaptive Simulation (SAS)
• Hybrid methods aim to provide improved accuracy compared to RANS while being more computationally efficient than full LES

RANS turbulence models

• RANS turbulence models are based on the Boussinesq hypothesis and the concept of eddy viscosity
• They introduce additional transport equations to model the effects of turbulence on the mean flow
• RANS models vary in complexity, from zero-equation models to full Reynolds stress models

Boussinesq hypothesis

• The Boussinesq hypothesis assumes that the turbulent stresses are proportional to the mean strain rate tensor
• It introduces the concept of eddy viscosity, which relates the turbulent stresses to the mean velocity gradients
• The Boussinesq hypothesis simplifies the modeling of turbulence by reducing the number of unknowns in the RANS equations

Eddy viscosity concept

• Eddy viscosity is a hypothetical viscosity used to model the effects of turbulence on the mean flow
• It represents the enhanced mixing and transport caused by turbulent eddies
• The eddy viscosity is not a physical property of the fluid but a modeling parameter that depends on the flow conditions and the turbulence model used

Zero-equation models

• Zero-equation models, also known as algebraic models, do not solve any additional transport equations for turbulence quantities
• They estimate the eddy viscosity based on local flow properties and empirical correlations
• Examples include the Baldwin-Lomax model and the Cebeci-Smith model
• Zero-equation models are simple and computationally efficient but have limited accuracy and generality

One-equation models

• One-equation models solve a single transport equation for a turbulence quantity, usually the turbulent kinetic energy ($k$)
• The eddy viscosity is then calculated based on $k$ and a length scale determined by empirical relations or the local flow properties
• Examples include the Spalart-Allmaras model and the Baldwin-Barth model
• One-equation models offer improved accuracy compared to zero-equation models but still rely on empirical relations for the length scale

Two-equation models

• Two-equation models solve two additional transport equations for turbulence quantities, typically the turbulent kinetic energy ($k$) and either the turbulent dissipation rate ($\epsilon$) or the specific dissipation rate ($\omega$)
• The eddy viscosity is calculated based on $k$ and $\epsilon$ or $\omega$, providing a more complete description of the turbulence scales
• Two-equation models are widely used in industrial applications due to their good balance between accuracy and computational cost

k-epsilon model

• The $k$-$\epsilon$ model solves transport equations for the turbulent kinetic energy ($k$) and the turbulent dissipation rate ($\epsilon$)
• It is one of the most widely used turbulence models due to its robustness and reasonable accuracy for a wide range of flows
• The standard $k$-$\epsilon$ model has limitations in handling flows with strong adverse pressure gradients and separation
• Variants of the $k$-$\epsilon$ model, such as the RNG $k$-$\epsilon$ and the realizable $k$-$\epsilon$, have been developed to address some of these limitations

k-omega model

• The $k$-$\omega$ model solves transport equations for the turbulent kinetic energy ($k$) and the specific dissipation rate ($\omega$)
• It is known for its good performance in near-wall regions and its ability to handle low-Reynolds-number effects
• The standard $k$-$\omega$ model is sensitive to the freestream values of $\omega$, which can affect its accuracy in some cases
• The Wilcox $k$-$\omega$ model is a widely used variant that addresses some of the limitations of the standard model

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 and the local flow conditions
• The SST $k$-$\omega$ model is known for its good performance in flows with adverse pressure gradients and separation
• It has become a popular choice for aerodynamic applications due to its accuracy and robustness

Reynolds stress models

• Reynolds stress models (RSM) solve transport equations for the individual components of the Reynolds stress tensor instead of using the Boussinesq hypothesis
• They provide a more accurate representation of the turbulence anisotropy and the effects of rotation and curvature
• RSMs require solving additional transport equations (up to 7) compared to eddy viscosity models, making them computationally more expensive
• RSMs are useful for flows with strong anisotropy, such as swirling flows and flows with significant streamline curvature

Turbulent boundary layers

• Turbulent boundary layers are regions near solid surfaces where the flow transitions from laminar to turbulent and the effects of viscosity are significant
• Understanding the characteristics and behavior of turbulent boundary layers is crucial for accurate predictions of drag, heat transfer, and flow separation

Characteristics of turbulent boundary layers

• Turbulent boundary layers exhibit a highly unsteady and irregular flow structure with significant velocity fluctuations
• They have a higher momentum transfer and increased mixing compared to laminar boundary layers, resulting in higher skin friction and heat transfer rates
• Turbulent boundary layers are thicker than laminar boundary layers and have a more gradual velocity profile
• The near-wall region of a turbulent boundary layer consists of a viscous sublayer, a buffer layer, and a logarithmic layer

Law of the wall

• The law of the wall describes the velocity profile in the near-wall region of a turbulent boundary layer
• It consists of a linear relation in the viscous sublayer ($u^+ = y^+$) and a logarithmic relation in the logarithmic layer ($u^+ = \frac{1}{\kappa} \ln y^+ + C$)
• The law of the wall is used to develop wall functions for RANS models, which provide a bridge between the near-wall region and the fully turbulent outer layer
• Proper treatment of the near-wall region is crucial for accurate predictions of skin friction, heat transfer, and flow separation

Boundary layer separation

• Boundary layer separation occurs when the flow near the wall reverses direction due to adverse pressure gradients or geometrical features
• Separation leads to increased drag, loss of lift, and the formation of recirculation zones
• Turbulent boundary layers are more resistant to separation than laminar boundary layers due to their higher momentum transfer
• Accurate prediction of boundary layer separation is essential for the design and optimization of aerodynamic surfaces

Turbulent boundary layer control

• Turbulent boundary layer control aims to manipulate the flow to reduce drag, delay separation, or enhance mixing
• Passive control methods include surface roughness, vortex generators, and riblets, which modify the boundary layer structure without external energy input
• Active control methods involve the application of external energy, such as suction, blowing, or wall motion, to influence the boundary layer behavior
• Boundary layer control techniques are used in various aerodynamic applications, such as high-lift systems, flow control on wings, and drag reduction on aircraft surfaces

Numerical considerations

• Numerical considerations are important when implementing turbulence models in computational fluid dynamics (CFD) simulations
• Proper treatment of mesh resolution, near-wall regions, convergence, and validation is crucial for accurate and reliable results

Mesh resolution requirements

• Turbulence models have different mesh resolution requirements depending on their formulation and the level of detail they aim to capture
• DNS and LES require extremely fine meshes to resolve the small-scale turbulent structures, leading to high computational costs
• RANS models have lower mesh resolution requirements but still need proper refinement in regions with high gradients and near walls
• Mesh sensitivity studies should be conducted to ensure that the solution is grid-independent and captures the relevant flow features

Near-wall treatment

• The near-wall region in turbulent flows requires special treatment due to the high gradients and the importance of viscous effects
• Low-Reynolds-number models resolve the near-wall region down to the viscous sublayer, requiring a very fine mesh near the wall ($y^+ \approx 1$)
• Wall functions are used with high-Reynolds-number models to bridge the near-wall region and the fully turbulent outer layer, allowing for coarser meshes near the wall ($y^+ > 30$)
• The choice of near-wall treatment depends on the turbulence model, the flow conditions, and the available computational resources

Convergence and stability issues

• Turbulence models can introduce additional numerical challenges related to convergence and stability
• The non-linearity and coupling of the turbulence equations with the mean flow equations can lead to stiffness and slow convergence
• Proper initialization, relaxation factors, and solution strategies (such as multigrid methods) can help improve convergence and stability
• Monitoring residuals, solution variables, and integral quantities is essential to ensure that the solution has converged to a steady state or a statistically stable condition

Validation and verification

• Validation and verification (V&V) are essential processes to assess the accuracy and reliability of turbulence model simulations
• Verification involves ensuring that the mathematical model is correctly implemented and that the numerical solution is converging to the exact solution of the model equations
• Validation compares the simulation results with experimental data or high-fidelity simulations to assess the accuracy and predictive capability of the turbulence model
• V&V should be performed for a range of flow conditions and geometries relevant to the intended application of the turbulence model

Applications in aerodynamics

• Turbulence modeling is widely used in various aerodynamic applications to predict and optimize the performance of vehicles and components
• Accurate modeling of turbulence is crucial for the design and analysis of aircraft, turbomachinery, and high-speed vehicles

Airfoil and wing design

• Turbulence models are used to predict the lift, drag, and moment characteristics of airfoils and wings
• They help in optimizing the shape and configuration of wings to maximize lift-to-drag ratio and improve aerodynamic efficiency
• Turbulence models are also used to predict the onset and extent of flow separation, which can significantly impact the performance of wings

High-lift configurations

• High-lift devices, such as flaps and slats, are used to increase lift during takeoff and landing
• Turbulence models are employed to predict the complex flow physics associated with high-lift configurations, including boundary layer separation, wake interactions, and unsteady flow phenomena
• Accurate modeling of turbulence is essential for the design and optimization of high-lift systems to ensure safe and efficient operation of aircraft

Turbomachinery flows

• Turbomachinery, such as compressors and turbines, involves complex turbulent flows with high Reynolds numbers and strong pressure gradients
• Turbulence models are used to predict the performance, efficiency, and heat transfer characteristics of turbomachinery components
• Accurate modeling of turbulence is crucial for the design and optimization of turbomachinery to improve efficiency, reduce losses, and ensure reliable operation

Hypersonic flows

• Hypersonic flows, characterized by high Mach numbers and strong shock waves, pose unique challenges for turbulence modeling
• Turbulence models need to account for compressibility effects, shock-boundary layer interactions, and chemical reactions in hypersonic flows
• Accurate prediction of turbulence is essential for the design and analysis of hypersonic vehicles, such as scramjets and re-entry vehicles

Limitations and challenges

• Despite significant advances in turbulence modeling, there are still limitations and challenges that need to be addressed for more accurate and reliable predictions
• These limitations and challenges drive the development of new turbulence models and the improvement of existing ones

Accuracy vs computational cost

• There is a trade-off between accuracy and computational cost in turbulence modeling
• High-fidelity approaches like DNS and LES provide detailed and accurate predictions but are computationally expensive and limited to simple geometries and low Reynolds numbers
• RANS models are computationally efficient but rely on assumptions and empirical relations that limit their accuracy and generality
• Hybrid RANS-LES methods aim to balance accuracy and cost but still face challenges in terms of model formulation, grid resolution, and transition between RANS and LES regions

Complex geometry and flow conditions

• Turbulence models often struggle with complex geometries and flow conditions, such as high curvature, strong pressure gradients, and flow separation
• The assumptions and empirical relations used in turbulence models may break down in these situations, leading to inaccurate predictions
• Adaptive mesh refinement and advanced numerical methods can help capture the complex flow features but increase the computational cost and complexity of the simulations

Transition prediction

• Predicting the transition from laminar to turbulent flow is a significant challenge in turbulence modeling
• Most turbulence models assume fully turbulent flow and do not accurately capture the transition process
• Transition modeling requires additional equations or modifications to the turbulence models, increasing the complexity and computational cost
• Accurate prediction of transition is crucial for applications such as laminar flow control, where maintaining laminar flow can significantly reduce drag

Future developments in turbulence modeling

• Research efforts are ongoing to develop more accurate, reliable, and efficient turbulence models
• Machine learning and data-driven approaches are being explored to improve turbulence models by leveraging large datasets and advanced algorithms
• Multiscale methods, such as the Variational Multiscale (VMS) method, aim to better capture the interaction between large and small scales of turbulence