Turbulence Modeling in CFD simulations

Scope of Turbulence Modeling

• In fluid mechanics, flow flow are classified as laminar and turbulent flow based on streamlines or velocity level.
• Turbulence refers to the chaotic and unpredictable fluid motion that occurs when a fluid flows rapidly over a surface or through a confined space. Turbulent flow is characterized by high levels of velocity fluctuations and vortices, which mix and transport mass, momentum, and energy in a highly irregular manner.
• Turbulent flows are encountered in many natural and industrial settings, including atmospheric flows, ocean currents, rivers, and airflows around vehicles and aircraft.
• Turbulence plays a crucial role in many physical and engineering processes, such as mixing, heat transfer, and combustion, and is an active area of research in fluid dynamics.

• One of the defining features of turbulence is its multi-scale nature. Turbulent flows consist of a hierarchy of eddies or vortices, ranging in size from the largest scales of motion, such as the size of an airplane wing, down to the smallest scales of motion, which are typically on the order of a few micrometers.
• The study of turbulence involves the development of mathematical models and experimental techniques to describe and understand the complex behavior of fluid flows in turbulent regimes. Turbulence can be modeled using a variety of approaches, including direct numerical simulation (DNS), large eddy simulation (LES), and Reynolds-averaged Navier-Stokes (RANS) modeling.
• Turbulence has a significant impact on many areas of science and engineering. For example, the prediction and control of turbulent flows is important for the design of more efficient and environmentally friendly engines and the accurate prediction of weather patterns.

Key Subjects for Turbulent Modeling

For CFD modeling of turbulent flow, engineers must understand fluid mechanics thoroughly. Governing equations, Stresses or forces acting on fluid must be understood well. The following subjects are important for turbulent analysis:

1. Advanced Fluid Mechanics: fluid mechanics with the viscous and non-viscous flow,   critical Reynolds numbers, and features of turbulent flow must be known to engineers
   • Navier stokes equations
   • Linear and Non linear viscosity
   • Laminar and turbulent boundary layers,
   • Features of turbulent flow
   • Compressible flow
2. Experimental Analysis of Turbulent Flow:
   • Statistical analysis of experimental data
   • Determination of averaged and fluctuating quantities
   • Turbulent stress
3. Numerical Modeling of Flow
   • Numerical modeling of governing equations
   • Numerical schemes
   • Pressure velocity coupling
4. Thermodynamics

 

Majors ways of Numerical Analysis

• Turbulence modeling is a field within fluid dynamics that aims to predict and analyze the behavior of turbulent flows. Turbulent flows are characterized by chaotic and random fluctuations in velocity, pressure, and other fluid properties. Due to the complex nature of turbulence, analytical solutions to the Navier-Stokes equations, which describe the behavior of fluid flow, are often difficult or impossible to obtain
• Turbulence modeling seeks to overcome this challenge by developing mathematical models and numerical methods that can accurately predict the behavior of turbulent flows. This is done by breaking down the turbulent flow into smaller, more manageable components and modeling their behavior. There are several different types of turbulence models, including:

1. 1. Complete modeling of Flow: Reynolds-Averaged Navier-Stokes (RANS) models, which are based on time-averaged equations and assume that the turbulent fluctuations can be modeled as a separate term.
   2. Partial Modeling of Flow: Large Eddy Simulation (LES) models, which simulate the larger turbulent structures directly and model the smaller scales.
   3. Complete Resolving of Flow (No Modeling): Direct Numerical Simulation (DNS) models, which simulate the entire range of scales of the turbulent flow and are the most accurate but computationally expensive.

• Turbulence modeling has numerous applications in engineering, including aerodynamics, combustion, heat transfer, and fluid-structure interaction. It is used to design and optimize aircraft, cars, and other vehicles, as well as industrial processes such as chemical reactions and energy production.

 

Classification of Turbulence Modeling

Here is a list of some commonly used turbulence models in computational fluid dynamics (CFD)

1. Reynolds-Averaged Navier – Stokes (RANS) models:
   • Standard k-epsilon model
   • RNG k-epsilon model
   • Realizable k-epsilon model
   • SST k-omega model
   • Reynolds Stress Model (RSM)
   • Spalart-Allmaras (SA) model
   • Explicit Algebraic Reynolds Stress Model (EARSM)

     

2. Large Eddy Simulation (LES) model
   • Smagorinsky model
   • Dynamic Smagorinsky model
   • Wall-Adapting Local Eddy-viscosity (WALE) model
   • Detached Eddy Simulation (DES)
   • Delayed Detached Eddy Simulation (DDES)
3. Hybrid models
   • Scale-Adaptive Simulation (SAS)
   • Zonal LES/RANS models
   • Detached-eddy Zonal LES/RANS (DZLES/RANS)
4. Direct Numerical Simulation (DNS) models
   • Full DNS
   • Pseudo-spectral DNS
   • Lattice Boltzmann Method (LBM)

• Each turbulence model has its advantages and disadvantages, and the choice of model depends on the specific flow conditions, the desired accuracy, and the available computational resources. The selection of an appropriate turbulence model is an important factor in obtaining accurate CFD simulations.

RANS (Reynolds-Averaged Navier-Stokes) Models

Reynolds Decomposition 

• Reynolds decomposition is a mathematical technique used to separate the mean and fluctuating components of a variable in a turbulent flow. It is named after Osborne Reynolds, who pioneered the study of fluid dynamics in the late 19th century.

 

• In Reynolds decomposition, any variable in a turbulent flow, such as velocity or pressure, is expressed as the sum of a mean value and a fluctuating value. The mean value is obtained by time-averaging the variable over a sufficiently long period of time, while the fluctuating component represents the deviation from the mean at any given instant.

• For example, the velocity of a fluid can be expressed as:

   u = U + u’

where U is the mean velocity and u’ is the fluctuating component. Similarly, pressure can be expressed as:

p = P + p’

where P is the mean pressure and p’ is the fluctuating pressure.

• The Reynolds decomposition allows us to separate the effects of the mean flow and the turbulent fluctuations, and it is a key concept in the mathematical models used to describe turbulent flows, such as the Reynolds-averaged Navier-Stokes (RANS) equations.
• Reynolds decomposition can be extended to multiple dimensions, where each velocity component or scalar quantity can be decomposed into a mean and fluctuating component. This leads to a more complete description of the turbulent flow and enables us to better understand the transport phenomena that occur in turbulent flows.

 

• Substitute instantaneous values as sum of two parts and take average of equations on both sides

 

 Reynolds  Averaged  Navier Stokes (RANS)  Equations

• The Reynolds-Averaged Navier-Stokes (RANS) equations are a set of equations that describe the motion of fluid in a turbulent flow. They are derived by averaging the Navier-Stokes equations over time and/or space to account for the turbulent fluctuations in the flow. The RANS equations are given by:

• The additional term in the averaged momentum equations is called as Reynolds or turbulent Stress. This is due to fluctuating velocities in turbulent flows, which enhance the momentum of fluid flows. The additional term in RANS equations is calculated based on turbulent models.
• The RANS equations are used extensively in computational fluid dynamics (CFD) to simulate turbulent flows in engineering and scientific applications.
• The equations are often solved numerically using computational methods such as finite difference, finite element, or spectral methods. However, the RANS equations are still an active area of research, and there are ongoing efforts to develop more accurate and efficient models for simulating turbulent flows.

Reynolds or Turbulent Stress

• Reynolds stress is a type of turbulence-related stress that arises in fluid flows. It is named after Osborne Reynolds, a pioneering physicist who studied fluid dynamics in the late 19th century.
• Reynolds stress is a measure of the fluctuating component of the momentum transport in a fluid, which arises due to turbulent fluctuations. It represents the interaction between the turbulent eddies and the mean flow of the fluid.

• In general, the Reynolds stress is expressed in terms of the fluctuating velocity components in a fluid and can be written as a tensor with six components, corresponding to the various combinations of velocity components. The Reynolds stress tensor plays a crucial role in the mathematical models used to describe turbulent flows, such as the Reynolds-averaged Navier-Stokes (RANS) equations.
• Reynolds stress has important practical applications in many fields, including aerospace engineering, meteorology, and oceanography. For example, it is used to model the behavior of turbulent fluid flows around aircraft wings, as well as in the prediction of ocean currents and the spread of pollutants in the environment.
• The turbulence in the flow is modeled using additional equations that provide closure for the Reynolds stresses, which represent the turbulent fluctuations in the flow. The most commonly used closure models for the Reynolds stresses are the k-epsilon model and the Reynolds stress model.
• The Reynolds stress model is a more complex closure model that directly solves the six components of the Reynolds stress tensor, which represents the anisotropic nature of turbulence. The Reynolds stress model requires additional transport equations for the anisotropic components of turbulence.

• The Boussinesq hypothesis is a widely used assumption in fluid mechanics that relates the eddy viscosity, a measure of the turbulent transport of momentum, to the rate of strain tensor in the Navier-Stokes equations. The hypothesis was first proposed by French physicist Joseph Boussinesq in 1877.
• The Boussinesq hypothesis assumes that the turbulent flow can be modeled as a Newtonian fluid with an effective viscosity that varies with the local flow conditions. The eddy viscosity, denoted as μ_t, is related to the rate of strain tensor S, which describes the deformation of the fluid elements, by the equation:

• where ρ is the fluid density, Cu is a dimensionless constant, l is the characteristic length scale of the turbulence, and |S| is the magnitude of the rate of the strain tensor.
• The Boussinesq hypothesis assumes that the eddy viscosity is proportional to the magnitude of the rate of strain tensor and the characteristic length scale of the turbulence.
• The Boussinesq hypothesis is commonly used in turbulence models, such as the Reynolds-averaged Navier-Stokes (RANS) equations, which are used to simulate turbulent flows in practical engineering applications. The hypothesis provides a simple and computationally efficient way to incorporate the effects of turbulence into the Navier-Stokes equations, allowing for the prediction of the mean flow behavior and turbulence statistics.
• However, the Boussinesq hypothesis is an approximation and is valid only under certain assumptions, such as the assumption of isotropy and homogeneity of the turbulence. In more complex turbulent flows, the Boussinesq hypothesis may need to be modified or replaced by more advanced turbulence models that take into account the anisotropic and non-homogeneous nature of the turbulence.

Turbulent Viscosity

• In turbulent flows, the flow velocity varies randomly in space and time. As a result, the viscous stresses in the flow are not sufficient to describe the turbulent motion. To account for the effects of turbulence, an additional “eddy viscosity” is introduced, which is analogous to the molecular viscosity that arises from the random molecular motion in a fluid
• The turbulent viscosity or eddy viscosity represents the rate of diffusion of momentum due to the effects of turbulence. It is a proportionality constant that relates the Reynolds stress to the mean strain rate of the fluid. In RANS (Reynolds-averaged Navier-Stokes) models, the turbulent viscosity is modeled based on various assumptions, such as the assumption that it is proportional to the local strain rate or some function of it.
• The eddy viscosity is often used to close the set of equations in the Reynolds-averaged Navier-Stokes equations, allowing the calculation of the mean flow field and turbulence quantities, such as the Reynolds stresses. The choice of turbulence model and the modeling of the turbulent viscosity depends on the specific application and the desired level of accuracy and computational resources.

The k-epsilon model

• The k-epsilon model is a two-equation model that solves for two additional variables: the turbulence kinetic energy (k) and the rate of dissipation of turbulence kinetic energy (epsilon).
• The model assumes that the turbulent eddies decay due to a combination of viscous diffusion and turbulent transport, and the rate of decay is proportional to the dissipation rate of kinetic energy.
• Transport equations in k-e models: Turbulent kinetic energy equations

   

• Turbulent Dissipation Rate (ε)

             

• Turbulent Viscosity is calculated after numerically solving the above two equations

SST k-ω turbulence model

• The SST k-ω turbulence model, also known as the Menter model, is a widely used turbulence model in computational fluid dynamics (CFD) that combines the k-ω and k-epsilon models to provide accurate predictions of complex turbulent flows.

• The SST stands for “Shear Stress Transport,” which refers to the transport of shear stresses due to turbulence
• The SST k-ω model is a two-equation model that solves for two variables: the turbulence kinetic energy (k) and the specific dissipation rate (ω).
• The model combines the advantages of the k-epsilon model in the near-wall region and the k-ω model in the far-field region to provide accurate predictions of both the near-wall and far-field turbulence effects
• In the near-wall region, the model uses the k-epsilon model to accurately capture the turbulence structure and provide accurate predictions of the wall shear stress. In the far-field region, the model uses the k-ω model to accurately predict the turbulent kinetic energy and the specific dissipation rate.
• The SST k-ω model also includes a modification to the turbulence eddy viscosity that accounts for the effects of rotation and curvature on the turbulence. This modification improves the accuracy of the model in flows with strong streamlined curvature or rotation, such as flows in turbomachinery.
• The SST k-ω model is widely used in a range of engineering applications, such as aerospace, automotive, and marine engineering. It is particularly useful in predicting flows with strong shear and pressure gradients, such as boundary layers, wakes, and separated flows.

The Reynolds Stress Model (RSM)

• The Reynolds Stress Model (RSM) is a type of turbulence model used in computational fluid dynamics (CFD) that directly solves for the six components of the Reynolds stress tensor, which represents the anisotropic nature of turbulence. The RSM is a more advanced model compared to the standard k-epsilon and k-omega models, as it provides a more detailed description of the turbulence characteristics.
• The RSM solves a set of partial differential equations that relate the mean velocity, the Reynolds stresses, and the turbulence kinetic energy to the mean pressure and the turbulent viscosity. The model uses additional transport equations for the anisotropic components of the Reynolds stress tensor, which allows for a more accurate representation of the complex turbulence structure.
• The RSM is particularly useful in predicting complex flows, such as those with strong streamline curvature or separation, as well as flows with highly anisotropic turbulence, such as flows in swirling or rotating systems. The model can also capture the effects of secondary flows and turbulent mixing more accurately than simpler turbulence models.
• However, the RSM has some limitations, including the need for extensive data input and computational resources. The model requires accurate boundary conditions and input data, which can be difficult to obtain in practical applications. The complexity of the model also makes it computationally expensive, which limits its use in large-scale simulations.
• Overall, the RSM is a powerful tool in predicting complex turbulent flows, but it requires careful consideration of the limitations and the trade-offs between accuracy and computational cost.

Large Eddy Simulation (LES)

• Large eddy simulation (LES) is a computational technique used in fluid dynamics to simulate turbulent flows. In LES, the large-scale eddies of a turbulent flow are directly resolved, while the small-scale eddies are modeled based on a subgrid-scale (SGS) model.
• LES is particularly useful for simulating complex turbulent flows where the scales of the turbulent motion span a wide range, from large-scale structures to the smallest eddies. These flows cannot be accurately modeled using traditional computational fluid dynamics (CFD) approaches that rely on turbulence models to account for the effects of the small-scale eddies.

Filtering of Large Scales

• In LES, the large-scale eddies are directly resolved by dividing the computational domain into a grid of cells and solving the Navier-Stokes equations at each grid point. However, the small-scale eddies, which are not resolved by the grid, are modeled using a SGS model that accounts for their effects on the resolved scales.

• The SGS model in LES is based on filtering the Navier-Stokes equations using a low-pass filter to separate the large-scale motions from the small-scale motions. The filtered equations are then supplemented with a closure model that represents the effects of the small-scale motions on the resolved scales.
• LES provides more accurate and detailed information on the turbulent flow field compared to traditional turbulence models, such as Reynolds-averaged Navier-Stokes (RANS) modeling. However, LES requires significantly more computational resources due to the higher resolution required to resolve the large-scale eddies, making it more computationally expensive than RANS.
• LES is widely used in engineering and scientific applications where accurate predictions of turbulent flows are critical, such as in aircraft design, combustion engines, and atmospheric and oceanic simulations.

 

Subgrid-Scale (SGS) model

• Large Eddy Simulation (LES) is a computational fluid dynamics technique used to simulate turbulent flows. Unlike traditional Reynolds-averaged Navier-Stokes (RANS) models, LES resolves the larger turbulent structures while modeling the smaller scales using a subgrid-scale (SGS) model. This approach allows for more accurate predictions of turbulent flows and is particularly useful in applications where turbulence plays a significant role.
• In LES, the flow domain is divided into a grid of computational cells, and the governing equations of fluid dynamics are solved for each cell. However, due to the range of scales involved in turbulent flows, it is not computationally feasible to resolve all the scales in a practical simulation. Therefore, the smaller scales are modeled using an SGS model, which provides a closure for the governing equations of the large-scale motion.

The SGS model is based on the idea that the effects of small-scale turbulence can be represented by a set of averaged quantities that are related to the resolved flow variables. These averaged quantities are referred to as the subgrid-scale stress tensor, and their calculation is the main task of the SGS model. The SGS stress tensor can be calculated using various closure models, such as the Smagorinsky model, Dynamic Smagorinsky model, WALE model, and Vreman model, among others.

Smagorinsky Model

• The Smagorinsky model is the most commonly used SGS model and is based on the assumption that the subgrid-scale stresses are proportional to the local strain rate of the resolved flow.
• The proportionality constant is determined by the eddy viscosity concept, which relates the turbulent viscosity to the resolved flow variables.
• The eddy viscosity is calculated based on the grid-scale turbulent kinetic energy, which is resolved by LES.

Dynamic Smagorinsky model

• The Dynamic Smagorinsky model is an extension of the Smagorinsky model that dynamically adjusts the model coefficient based on the local flow properties.
• This makes the model more adaptable to complex flows and improves its accuracy.

WALE (Wall-Adapting Local Eddy-viscosity) model

• The WALE (Wall-Adapting Local Eddy-viscosity) model is a more advanced SGS model that incorporates wall effects in turbulence modeling.
• This is important in applications such as boundary layer flows, where the proximity to the wall can significantly affect the turbulence characteristics.

Vreman model

• The Vreman model is another SGS model that uses a Lagrangian approach to model the subgrid-scale stresses.
• This model has been shown to be effective in predicting turbulent mixing in stratified flows.
• The accuracy of the SGS model depends on its ability to capture the relevant flow physics of the application being studied. Therefore, the choice of SGS model should be made based on the specific application and the desired balance between accuracy and computational cost. The accuracy of the SGS model can be validated through comparison with experimental data or direct numerical simulation (DNS) if available.
• Overall, SGS modeling is a critical component of LES and is used to represent the effects of small-scale turbulence on the resolved large-scale motion. The choice of SGS model should be based on the specific application and the desired balance between accuracy and computational cost.

Accuracy of SGS Model

• The accuracy of the SGS model can be validated through comparison with experimental data or DNS.
• SGS modeling is a powerful tool for predicting turbulent flows in a wide range of applications, including aerospace, automotive, and energy systems.

Direct numerical simulation (DNS)

• Direct numerical simulation (DNS) is a computational technique used in fluid dynamics to simulate turbulent flows. DNS solves the Navier-Stokes equations numerically, without any turbulence modeling, to accurately simulate the behavior of the flow at all scales, from the large-scale structures to the smallest eddies.
• In DNS, the entire range of turbulence scales is resolved, with no need for subgrid-scale models. This makes DNS the most accurate approach to simulate turbulent flows, as it captures all the details of the flow field, including small-scale features that cannot be resolved by traditional turbulence models.
• However, DNS is computationally very expensive and is limited to relatively simple geometries and low Reynolds numbers. The computational cost of DNS scales with the Reynolds number, which is a measure of the turbulence intensity and scales with the characteristic length and velocity of the flow. As the Reynolds number increases, so does the range of scales that must be resolved, making DNS increasingly computationally demanding.
• DNS is typically used as a research tool to study fundamental aspects of turbulence, such as the behavior of small-scale structures and the statistics of turbulence. It is also used to provide high-fidelity data for validating turbulence models used in engineering applications.
• DNS has been used to study a wide range of turbulent flows, such as boundary layers, jet flows, and wake flows. It has also been used to study complex phenomena such as turbulence in reacting flows, multiphase flows, and fluid-structure interactions.
• Overall, DNS is a powerful tool for understanding the physics of turbulence and developing accurate turbulence models, but its high computational cost limits its practical use in many engineering and scientific applications.

Summary

• RANS models are widely used in engineering applications, as they provide a good balance between accuracy and computational cost.
• However, they have limitations in accurately predicting complex flow phenomena, such as flow separation, vortex shedding, and turbulent mixing, which are important in many industrial applications.
• For these types of flows, more advanced turbulence models, such as Large Eddy Simulation (LES) or Direct Numerical Simulation (DNS), may be required.
• In summary, turbulence modeling is a crucial component of CFD simulations, and the choice of turbulence model depends on the specific flow conditions, the desired accuracy, and the available computational resources.