7.2 Reynolds-averaged Navier-Stokes equations

Reynolds-averaged Navier-Stokes equations are a key tool for modeling turbulent flows. They separate flow variables into mean and fluctuating components, allowing us to focus on average behavior while accounting for turbulence effects.

RANS equations introduce the Reynolds stress tensor, which represents turbulent momentum transport. Closure models like eddy viscosity approaches are used to relate these stresses to mean flow variables, enabling practical simulations of complex turbulent flows in engineering applications.

Reynolds decomposition

• Fundamental concept in turbulence modeling that separates flow variables into mean and fluctuating components
• Enables the derivation of Reynolds-averaged Navier-Stokes (RANS) equations, which govern the mean flow behavior
• Allows for the statistical description of turbulent flows and the modeling of turbulence effects on the mean flow

Mean and fluctuating components

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• Flow variables (velocity, pressure, etc.) are decomposed into a mean component and a fluctuating component
• Mean component represents the time-averaged value of the variable
• Fluctuating component represents the instantaneous deviation from the mean
• Decomposition is expressed as: $u_i = \bar{u}_i + u'_i$, where $\bar{u}_i$ is the mean and $u'_i$ is the fluctuating component

Time averaging

• Process of averaging flow variables over a sufficiently long time interval to obtain the mean values
• Time-averaging operator is defined as: $\bar{u}_i = \frac{1}{T} \int_{t}^{t+T} u_i dt$
• Time interval $T$ should be much larger than the characteristic time scales of turbulent fluctuations
• Time averaging of the Navier-Stokes equations leads to the RANS equations

RANS equations

• Governing equations for the mean flow in turbulent flows, obtained by applying Reynolds decomposition and time averaging to the Navier-Stokes equations
• Represent the conservation of mass and momentum for the mean flow
• Introduce additional terms, such as the Reynolds stress tensor, that account for the effects of turbulent fluctuations on the mean flow

Continuity equation for mean flow

• Represents the conservation of mass for the mean flow
• Obtained by time-averaging the continuity equation: $\frac{\partial \bar{u}_i}{\partial x_i} = 0$
• Ensures that the mean flow satisfies the mass conservation principle

Momentum equation for mean flow

• Represents the conservation of momentum for the mean flow
• Obtained by time-averaging the Navier-Stokes equations: $\frac{\partial \bar{u}_i}{\partial t} + \bar{u}_j \frac{\partial \bar{u}_i}{\partial x_j} = -\frac{1}{\rho} \frac{\partial \bar{p}}{\partial x_i} + \nu \frac{\partial^2 \bar{u}_i}{\partial x_j \partial x_j} - \frac{\partial \overline{u'_i u'_j}}{\partial x_j}$
• Introduces the Reynolds stress tensor $\overline{u'_i u'_j}$, which represents the effects of turbulent fluctuations on the mean flow

Reynolds stress tensor

• Symmetric second-order tensor that arises from the time averaging of the nonlinear convective terms in the Navier-Stokes equations
• Represents the transport of momentum due to turbulent fluctuations
• Defined as: $\tau_{ij} = -\rho \overline{u'_i u'_j}$
• Requires modeling to close the RANS equations and solve for the mean flow

Closure problem

• Challenge in turbulence modeling that arises from the introduction of the Reynolds stress tensor in the RANS equations
• RANS equations contain more unknowns (Reynolds stresses) than equations, making the system underdetermined
• Closure models are needed to relate the Reynolds stresses to the mean flow variables and close the RANS equations
• Turbulence models, such as eddy viscosity models and Reynolds stress models, are used to provide closure

Turbulence modeling

• Approach to model the effects of turbulent fluctuations on the mean flow in RANS simulations
• Aims to provide closure for the RANS equations by relating the Reynolds stresses to the mean flow variables
• Includes a wide range of models with varying levels of complexity and accuracy (eddy viscosity models, Reynolds stress models, and large eddy simulation)

Boussinesq hypothesis

• Assumption that the Reynolds stresses can be modeled using an eddy viscosity, analogous to the molecular viscosity in laminar flows
• Relates the Reynolds stresses to the mean strain rate tensor: $\tau_{ij} = 2\mu_t \bar{S}_{ij} - \frac{2}{3} \rho k \delta_{ij}$
• Introduces the concept of eddy viscosity $\mu_t$, which represents the turbulent transport of momentum
• Forms the basis for many turbulence models, such as algebraic models, one-equation models, and two-equation models

Eddy viscosity models

• Class of turbulence models that employ the Boussinesq hypothesis to relate the Reynolds stresses to the mean strain rate tensor
• Aim to model the eddy viscosity $\mu_t$ using additional transport equations for turbulence quantities
• Vary in complexity, from simple algebraic models to more advanced one-equation and two-equation models

Algebraic models

• Simplest type of eddy viscosity models, also known as zero-equation models
• Express the eddy viscosity as a function of local mean flow variables and empirical constants
• Examples include the mixing length model and the Baldwin-Lomax model
• Computationally inexpensive but limited in accuracy and generality

One-equation models

• Eddy viscosity models that solve a single transport equation for a turbulence quantity, typically the turbulent kinetic energy $k$
• Relate the eddy viscosity to the turbulent kinetic energy and a turbulent length scale
• Examples include the Spalart-Allmaras model and the $k$-$\sqrt{k}L$ model
• Offer improved accuracy compared to algebraic models but still rely on empirical relations for the turbulent length scale

Two-equation models

• Widely used eddy viscosity models that solve two transport equations for turbulence quantities
• Most common examples are the $k$-$\varepsilon$ and $k$-$\omega$ models, which solve equations for the turbulent kinetic energy $k$ and the turbulent dissipation rate $\varepsilon$ or specific dissipation rate $\omega$
• Relate the eddy viscosity to $k$ and $\varepsilon$ or $\omega$: $\mu_t = C_\mu \rho \frac{k^2}{\varepsilon}$ or $\mu_t = \rho \frac{k}{\omega}$
• Provide a good balance between accuracy and computational cost, making them popular choices for industrial applications

Reynolds stress models

• Higher-level turbulence models that directly solve transport equations for the individual components of the Reynolds stress tensor
• Avoid the assumption of isotropic eddy viscosity by accounting for the anisotropic nature of turbulence
• Require solving six additional transport equations for the Reynolds stresses, along with an equation for the turbulent dissipation rate
• Offer improved accuracy for complex flows with strong anisotropy but are computationally more expensive than eddy viscosity models

Large eddy simulation (LES)

• Turbulence modeling approach that directly resolves large-scale turbulent structures while modeling the effects of smaller-scale structures
• Applies a spatial filtering operation to the Navier-Stokes equations, separating the resolved scales from the subgrid scales
• Resolves the large-scale turbulent motions using the filtered Navier-Stokes equations
• Models the effects of subgrid-scale motions using subgrid-scale (SGS) models, such as the Smagorinsky model or the dynamic Smagorinsky model
• Provides more accurate and detailed turbulence predictions compared to RANS models but is computationally more expensive

Boundary conditions

• Specify the flow behavior at the boundaries of the computational domain in RANS simulations
• Crucial for accurately representing the physical flow conditions and ensuring the well-posedness of the numerical problem
• Include wall boundary conditions, inlet/outlet conditions, and symmetry conditions

Wall functions

• Approach to model the near-wall flow behavior in RANS simulations without fully resolving the viscous sublayer
• Based on the assumption of a logarithmic velocity profile near the wall
• Relate the wall shear stress and the mean velocity at the first grid point away from the wall using empirical formulas
• Reduce the computational cost by allowing for coarser grid resolution near the walls
• Suitable for high-Reynolds-number flows with attached boundary layers

Near-wall resolution

• Alternative approach to wall functions that involves fully resolving the viscous sublayer and the buffer layer in RANS simulations
• Requires a highly refined grid near the walls, with the first grid point typically located at $y^+ < 1$
• Captures the detailed flow behavior in the near-wall region, including the viscous sublayer and the buffer layer
• Provides more accurate predictions of wall shear stress and heat transfer compared to wall functions
• Necessary for low-Reynolds-number flows, flows with strong pressure gradients, and flows with separation

Numerical methods for RANS

• Discretization and solution techniques used to solve the RANS equations numerically
• Involve the spatial discretization of the computational domain, the discretization of the governing equations, and the solution of the resulting algebraic equations
• Common methods include the finite volume method, the finite element method, and pressure-velocity coupling algorithms

Finite volume method

• Widely used discretization method for RANS simulations in computational fluid dynamics (CFD)
• Divides the computational domain into a set of control volumes (cells) and applies the conservation principles to each cell
• Integrates the RANS equations over each control volume and approximates the fluxes at the cell faces using interpolation schemes
• Results in a set of algebraic equations for the mean flow variables at the cell centers
• Ensures local and global conservation of mass, momentum, and energy

Finite element method

• Alternative discretization method for RANS simulations, particularly in solid mechanics and heat transfer applications
• Divides the computational domain into a set of elements (typically triangles or tetrahedra) and approximates the solution using a weighted residual formulation
• Expresses the mean flow variables as a linear combination of basis functions defined on the elements
• Results in a set of algebraic equations for the coefficients of the basis functions
• Offers flexibility in handling complex geometries and allows for higher-order approximations

Pressure-velocity coupling

• Numerical challenge in RANS simulations arising from the lack of an explicit equation for pressure in the incompressible RANS equations
• Requires special treatment to ensure the coupling between pressure and velocity and to satisfy the continuity equation
• Common algorithms include the SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) and PISO (Pressure-Implicit with Splitting of Operators) methods
• Involve an iterative process of solving the momentum equations, updating the pressure field, and correcting the velocity field to satisfy continuity

Applications of RANS

• RANS simulations are widely used in various engineering and scientific fields to predict and analyze turbulent flows
• Provide a computationally efficient approach to model complex turbulent flows in industrial applications
• Examples include aerodynamics, turbomachinery, combustion, and environmental flows

Aerodynamics

• RANS simulations are extensively used in the aerospace industry to predict the aerodynamic performance of aircraft, missiles, and other vehicles
• Applications include the design and optimization of wings, fuselages, and control surfaces
• RANS models are used to predict lift, drag, and moment coefficients, as well as the flow field around the vehicle
• Examples: airfoil design, wing optimization, high-lift configurations, and aircraft stability analysis

Turbomachinery

• RANS simulations are employed in the design and analysis of turbomachinery components, such as compressors, turbines, and pumps
• Used to predict the flow field, pressure distribution, and performance characteristics of turbomachinery stages
• Help in optimizing blade shapes, reducing losses, and improving efficiency
• Examples: gas turbine engines, steam turbines, centrifugal compressors, and hydraulic turbines

Combustion

• RANS simulations are applied to model turbulent reacting flows in combustion systems, such as internal combustion engines, gas turbines, and furnaces
• Coupled with combustion models to predict the interaction between turbulence and chemical reactions
• Used to analyze mixing, flame stability, pollutant formation, and combustion efficiency
• Examples: diesel engines, gas turbine combustors, industrial furnaces, and fire safety applications

Environmental flows

• RANS simulations are used to model turbulent flows in environmental and geophysical applications, such as atmospheric boundary layers, rivers, and coastal regions
• Help in understanding the transport and dispersion of pollutants, sediments, and heat in the environment
• Used to predict wind fields, turbulent mixing, and flow patterns in complex terrains and urban areas
• Examples: weather forecasting, air pollution modeling, river and coastal engineering, and wind energy assessment

Limitations and challenges

• Despite their widespread use, RANS simulations have several limitations and challenges that need to be considered when interpreting the results and applying the models to practical problems
• These limitations arise from the assumptions made in the turbulence models, the computational cost of resolving complex flows, and the difficulty in modeling certain flow phenomena

Assumption of isotropic turbulence

• Many RANS turbulence models, particularly eddy viscosity models, assume that the turbulence is isotropic (i.e., the turbulent fluctuations are the same in all directions)
• This assumption is not valid for many complex flows, such as flows with strong streamline curvature, swirl, or separation
• Leads to inaccuracies in the prediction of turbulence quantities and the mean flow field in such cases
• Reynolds stress models and LES can capture anisotropic turbulence effects but are computationally more expensive

Accuracy vs computational cost

• RANS simulations involve a trade-off between accuracy and computational cost
• Higher-fidelity turbulence models, such as Reynolds stress models and LES, provide more accurate predictions but require more computational resources
• Lower-fidelity models, such as eddy viscosity models, are computationally cheaper but may not capture all the relevant flow physics
• Choice of turbulence model depends on the specific application, the required accuracy, and the available computational resources

Modeling complex flows

• RANS simulations face challenges in modeling certain complex flow phenomena, such as:
  • Flow separation and reattachment
  • Transition from laminar to turbulent flow
  • Unsteady and time-dependent flows
  • Multiphase flows and fluid-structure interaction
• These phenomena involve complex interactions between turbulence, mean flow, and other physical processes that are difficult to model accurately with RANS approaches
• Advanced turbulence models, such as LES and hybrid RANS-LES methods, may be needed to capture these flow features
• Experimental validation and uncertainty quantification are important to assess the reliability of RANS predictions in complex flows