8.3 Turbulence Models and Closure Problem
4 min read•august 16, 2024
Turbulence models are crucial for solving the in . They allow us to simulate complex turbulent flows without resolving all scales, making computational fluid dynamics practical for engineering applications.
Different types of turbulence models offer trade-offs between accuracy and computational cost. From simple to advanced Reynolds stress models, the choice depends on the specific flow problem, available resources, and required accuracy level.
Turbulence Models for RANS Equations
Closure Problem and Need for Turbulence Models
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- Reynolds-Averaged Navier-Stokes (RANS) equations introduce additional unknown terms () resulting in more unknowns than equations
- Closure problem arises from this discrepancy between unknowns and equations necessitating turbulence models to provide additional equations or approximations
- Turbulence models capture effects of turbulent fluctuations on mean flow properties without resolving all scales of turbulent motion
- Primary goal relates Reynolds stresses to mean flow quantities or introduces additional transport equations
- Essential for practical engineering applications allows simulation of complex turbulent flows without computational expense of resolving all turbulent scales
- Choice of turbulence model significantly impacts accuracy and reliability of flow predictions
Purpose and Impact of Turbulence Models
- Provide mathematical descriptions to capture turbulent fluctuation effects
- Allow closure of RANS equations by relating Reynolds stresses to mean flow quantities
- Enable simulation of complex flows without resolving all turbulent scales
- Reduce computational costs compared to fully resolved simulations
- Impact accuracy and reliability of flow predictions based on model choice
- Facilitate practical engineering applications of computational fluid dynamics
Turbulence Model Classification and Comparison
Categories of Turbulence Models
- Broadly classified into three main categories algebraic models, , and
- Algebraic models (zero-equation models) use algebraic expressions to relate Reynolds stresses directly to mean flow quantities
- One-equation models introduce a single additional transport equation typically for
- Two-equation models (k-ε and k-ω) solve two additional transport equations usually for turbulent kinetic energy and or specific dissipation rate
- Advanced models include Reynolds stress models (RSM) and providing higher fidelity at increased computational cost
Comparison and Selection of Turbulence Models
- Each class offers trade-off between computational efficiency and accuracy
- More complex models generally provide better results for wider range of flow conditions
- Selection depends on specific flow problem, available computational resources, and required accuracy level
- Algebraic models offer simplicity and low computational cost but limited accuracy for complex flows
- One-equation models provide improved accuracy over algebraic models with moderate computational cost
- Two-equation models balance accuracy and computational cost for many engineering applications
- Advanced models (RSM, LES) offer highest fidelity but require significant computational resources
Implementing Turbulence Models
Common Turbulence Model Implementations
- Algebraic models (mixing length model) use simple expressions to relate eddy viscosity to mean flow gradient and characteristic length scale
- (one-equation) solves transport equation for modified eddy viscosity suitable for aerodynamic applications
- k-ε model (two-equation) solves transport equations for turbulent kinetic energy (k) and dissipation rate (ε) to determine eddy viscosity
- k-ω model (two-equation) uses specific dissipation rate (ω) instead of ε offering improved performance near solid boundaries
Implementation Considerations
- Careful consideration of boundary conditions, wall treatment, and numerical stability issues required
- Proper discretization and solution techniques crucial for accurate and stable solutions
- Validation and verification of implementations essential to ensure reliability of simulation results
- Boundary layer resolution and near-wall treatment critical for accurate predictions
- Numerical schemes must be selected to ensure stability and convergence of turbulence model equations
- Implementation of source terms and coupling with mean flow equations requires attention to detail
Limitations of Turbulence Models
Model-Specific Limitations
- Algebraic models struggle with complex flows involving separation, recirculation, or strong pressure gradients
- One-equation models (Spalart-Allmaras) perform well for attached boundary layers and mildly separated flows but less accurate for complex 3D flows
- Two-equation models offer good balance for many applications but struggle with strong streamline curvature
- k-ε model overpredicts turbulence levels in adverse pressure gradients and regions of flow separation
- k-ω model performs better near solid boundaries and in adverse pressure gradients but sensitive to freestream turbulence levels
General Limitations and Applicability
- All RANS-based models have limitations in predicting highly unsteady flows, transition to turbulence, and flows with significant anisotropy
- Applicability depends on flow regime, geometry complexity, required accuracy, and available computational resources
- Model selection and validation crucial for specific applications
- Limited ability to capture non-equilibrium effects and rapid changes in mean strain rate
- Struggle with flows involving strong rotation, buoyancy, or compressibility effects
- May require tuning of model constants for specific flow classes reducing universality
Key Terms to Review (20)
A. M. B. S. P. D. G. R. Spalding: A. M. B. S. P. D. G. R. Spalding refers to a turbulence model that provides a means for approximating the behavior of turbulent flows, particularly in the context of fluid dynamics. This model uses a set of equations to describe the statistical properties of turbulence, aiming to close the governing equations by introducing empirical correlations for unresolved scales of motion, making it crucial in computational fluid dynamics simulations.
Algebraic Models: Algebraic models are mathematical representations that use algebraic equations to describe complex physical phenomena, particularly in fluid dynamics. These models simplify the behavior of turbulent flows by representing relationships among different variables, allowing for predictions and analyses that are computationally less intensive than fully resolved numerical simulations. They are crucial in addressing the closure problem, where equations governing turbulent flows need additional relations to solve for all unknowns.
Boussinesq Approximation: The Boussinesq approximation is an approach used in fluid dynamics that simplifies the equations of motion for buoyant flows by assuming that density variations are small and primarily affect the buoyancy forces. This approximation enables the modeling of thermal convection in fluids, making it particularly useful in studying turbulence, geophysical flows, and other scenarios where temperature differences influence fluid behavior without significantly changing the overall density.
Closure problem: The closure problem refers to the challenge in turbulence modeling where the governing equations produce more unknown quantities than equations available to solve them. This issue arises particularly in the context of averaged equations, leading to complications in accurately predicting turbulent flow behaviors. It highlights the necessity of developing additional relationships or models to relate these unknown quantities to measurable parameters.
Dissipation rate: The dissipation rate is a measure of the rate at which turbulent kinetic energy is converted into thermal energy due to viscous forces in a fluid. This concept is crucial for understanding turbulence dynamics, as it directly influences the energy distribution in turbulent flows and helps characterize the scales of motion within the fluid. By quantifying how energy dissipates, it plays a vital role in turbulence models that aim to predict and describe fluid behavior under various conditions.
Finite Volume Method: The finite volume method is a numerical technique used for solving partial differential equations that arise in fluid dynamics by dividing the computational domain into small control volumes. This method focuses on the conservation laws, ensuring that the flow of mass, momentum, and energy are accurately represented across the boundaries of these control volumes, making it especially effective for problems involving shock waves, turbulence, and complex geometries.
Homogeneity assumption: The homogeneity assumption refers to the idea that a fluid's properties, such as density and viscosity, are uniform throughout the entire flow field. This simplification is often used in turbulence modeling to analyze complex flow patterns without considering variations in these properties that can complicate calculations. By assuming homogeneity, researchers can focus on the overall behavior of turbulent flows, leading to more manageable mathematical models.
K-epsilon model: The k-epsilon model is a popular turbulence modeling approach used in fluid dynamics that employs two transport equations: one for the turbulent kinetic energy (k) and another for the turbulent dissipation rate (epsilon). This model provides a practical way to close the Reynolds-Averaged Navier-Stokes (RANS) equations, allowing for more accurate predictions of turbulent flows by capturing the effects of turbulence on mean flow characteristics.
K-omega model: The k-omega model is a two-equation turbulence model used to predict the behavior of turbulent flows by solving two transport equations, one for the turbulent kinetic energy (k) and one for the specific dissipation rate (ω). This model effectively addresses the closure problem associated with the Reynolds-Averaged Navier-Stokes equations by providing a more accurate representation of turbulence, especially in boundary layers and near-wall flows.
Large eddy simulation (LES): Large eddy simulation (LES) is a mathematical approach used to simulate turbulent fluid flows by resolving the larger, energy-containing eddies while modeling the smaller, less significant ones. This method bridges the gap between direct numerical simulation (DNS), which resolves all scales of motion, and traditional Reynolds-averaged Navier-Stokes (RANS) models that apply turbulence averaging. LES captures the unsteady and complex characteristics of turbulent flows, making it particularly useful for understanding the dynamics of such systems.
One-equation models: One-equation models are turbulence modeling approaches that use a single transport equation to predict the turbulent kinetic energy in fluid flows. These models simplify the complexity of turbulence by providing a practical way to estimate flow behavior without requiring extensive computational resources, making them widely applicable in various engineering and scientific contexts.
Reynolds Number: Reynolds number is a dimensionless quantity that helps predict flow patterns in different fluid flow situations. It is defined as the ratio of inertial forces to viscous forces and is calculated using the formula $$Re = \frac{\rho v L}{\mu}$$, where $$\rho$$ is fluid density, $$v$$ is flow velocity, $$L$$ is characteristic length, and $$\mu$$ is dynamic viscosity. This number indicates whether a flow is laminar or turbulent, providing insight into the behavior of fluids in various scenarios.
Reynolds Stress Tensor: The Reynolds stress tensor is a mathematical representation that quantifies the influence of turbulent fluctuations in a fluid on its mean flow characteristics. This tensor captures the additional stresses that arise due to the chaotic nature of turbulence, which are not accounted for in the standard viscous stress terms of the Navier-Stokes equations. Understanding the Reynolds stress tensor is crucial for modeling turbulent flows, as it plays a key role in turbulence models and the closure problem associated with Reynolds-Averaged Navier-Stokes equations.
Reynolds-Averaged Navier-Stokes Equations: The Reynolds-Averaged Navier-Stokes (RANS) equations are a set of equations used to describe the motion of fluid substances by averaging the effects of turbulence over time. They incorporate the influence of turbulent fluctuations on the mean flow, allowing for more manageable calculations in turbulent flow scenarios. This approach addresses the complexity of turbulence by introducing additional terms that represent the averaged effects of turbulence, leading to the necessity of turbulence models to close the equations.
Spalart-Allmaras Model: The Spalart-Allmaras model is a one-equation turbulence model that is used to simulate the effects of turbulence in fluid dynamics. This model simplifies the calculations by solving a single transport equation for the turbulent viscosity, making it computationally efficient while still providing reliable results, especially for aerospace applications where boundary layer flow is important.
Spectral Method: The spectral method is a numerical technique used for solving differential equations by representing the solution as a sum of basis functions, typically chosen from orthogonal polynomials or Fourier series. This method transforms the problem into a spectral space, which often leads to high accuracy, especially for smooth problems, making it particularly valuable in analyzing turbulent flows and addressing the closure problem in turbulence modeling.
Strouhal Number: The Strouhal number is a dimensionless quantity used to describe oscillating flow mechanisms, defined as the ratio of inertial forces to viscous forces in a fluid system. It is particularly useful in characterizing unsteady flows where periodic phenomena occur, such as vortex shedding, and can help predict the frequency of these oscillations relative to the flow velocity and characteristic length scale of the object involved.
T. J. Sullivan: T. J. Sullivan is known for his contributions to turbulence modeling and the closure problem in fluid dynamics. His work often focuses on developing mathematical frameworks that help in understanding and predicting turbulent flows, which are critical in both engineering applications and environmental studies.
Turbulent kinetic energy: Turbulent kinetic energy (TKE) refers to the energy associated with the chaotic and fluctuating motion of fluid particles in a turbulent flow. It plays a vital role in understanding how turbulence affects the transport of momentum, heat, and mass in various fluid dynamics scenarios, and it serves as a key parameter in turbulence modeling and analysis.
Two-equation models: Two-equation models are mathematical formulations used in turbulence modeling that rely on two transport equations to describe the behavior of turbulent flows. These models help to provide a closure to the system of equations governing fluid motion by calculating important turbulence quantities, such as kinetic energy and its dissipation rate, enabling better predictions of flow characteristics.