Electromagnetic wave equation

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Electromagnetic_wave_equation 

The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It is a three-dimensional form of the wave equation. The homogeneous form of the equation, written in terms of either the electric field E or the magnetic field B, takes the form:

$${\displaystyle \begin{array}{rl}\left(v_{\mathrm{p}\mathrm{h}}^2{\mathrm{\nabla }}^2-\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{t}^2}\right)\mathbf{E}& =\mathbf{0}\\ \left(v_{\mathrm{p}\mathrm{h}}^2{\mathrm{\nabla }}^2-\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{t}^2}\right)\mathbf{B}& =\mathbf{0}\end{array}}$$

where

$${\displaystyle v_{\mathrm{p}\mathrm{h}}=\frac{1}{\sqrt{\mu \epsilon }}}$$

is the speed of light (i.e. phase velocity) in a medium with permeability μ, and permittivity ε, and ∇² is the Laplace operator. In a vacuum, v_ph = c₀ = 299792458 m/s, a fundamental physical constant. The electromagnetic wave equation derives from Maxwell's equations. In most older literature, B is called the magnetic flux density or magnetic induction. The following equations$${\displaystyle \begin{array}{rl}\mathrm{\nabla }\cdot \mathbf{E}& =0\\ \mathrm{\nabla }\cdot \mathbf{B}& =0\end{array}}$$predicate that any electromagnetic wave must be a transverse wave, where the electric field E and the magnetic field B are both perpendicular to the direction of wave propagation.

The origin of the electromagnetic wave equation

A postcard from Maxwell to Peter Tait.

In his 1865 paper titled A Dynamical Theory of the Electromagnetic Field, James Clerk Maxwell utilized the correction to Ampère's circuital law that he had made in part III of his 1861 paper On Physical Lines of Force. In Part VI of his 1864 paper titled Electromagnetic Theory of Light, Maxwell combined displacement current with some of the other equations of electromagnetism and he obtained a wave equation with a speed equal to the speed of light. He commented:

The agreement of the results seems to show that light and magnetism are affections of the same substance, and that light is an electromagnetic disturbance propagated through the field according to electromagnetic laws.

Maxwell's derivation of the electromagnetic wave equation has been replaced in modern physics education by a much less cumbersome method involving combining the corrected version of Ampère's circuital law with Faraday's law of induction.

To obtain the electromagnetic wave equation in a vacuum using the modern method, we begin with the modern 'Heaviside' form of Maxwell's equations. In a vacuum- and charge-free space, these equations are:

$${\displaystyle \begin{array}{rl}\mathrm{\nabla }\cdot \mathbf{E}& =0\\ \mathrm{\nabla }\times \mathbf{E}& =-\frac{\mathrm{\partial }\mathbf{B}}{\mathrm{\partial }t}\\ \mathrm{\nabla }\cdot \mathbf{B}& =0\\ \mathrm{\nabla }\times \mathbf{B}& ={\mu }_0{\epsilon }_0\frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t}\end{array}}$$

These are the general Maxwell's equations specialized to the case with charge and current both set to zero. Taking the curl of the curl equations gives:

$${\displaystyle \begin{array}{rl}\mathrm{\nabla }\times \left(\mathrm{\nabla }\times \mathbf{E}\right)& =\mathrm{\nabla }\times \left(-\frac{\mathrm{\partial }\mathbf{B}}{\mathrm{\partial }t}\right)=-\frac{\mathrm{\partial }}{\mathrm{\partial }t}\left(\mathrm{\nabla }\times \mathbf{B}\right)=-{\mu }_0{\epsilon }_0\frac{{\mathrm{\partial }}^2\mathbf{E}}{\mathrm{\partial }{t}^2}\\ \mathrm{\nabla }\times \left(\mathrm{\nabla }\times \mathbf{B}\right)& =\mathrm{\nabla }\times \left({\mu }_0{\epsilon }_0\frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t}\right)={\mu }_0{\epsilon }_0\frac{\mathrm{\partial }}{\mathrm{\partial }t}\left(\mathrm{\nabla }\times \mathbf{E}\right)=-{\mu }_0{\epsilon }_0\frac{{\mathrm{\partial }}^2\mathbf{B}}{\mathrm{\partial }{t}^2}\end{array}}$$

We can use the vector identity

$${\displaystyle \mathrm{\nabla }\times \left(\mathrm{\nabla }\times \mathbf{V}\right)=\mathrm{\nabla }\left(\mathrm{\nabla }\cdot \mathbf{V}\right)-{\mathrm{\nabla }}^2\mathbf{V}}$$

where V is any vector function of space. And

$${\displaystyle {\mathrm{\nabla }}^2\mathbf{V}=\mathrm{\nabla }\cdot \left(\mathrm{\nabla }\mathbf{V}\right)}$$

where ∇V is a dyadic which when operated on by the divergence operator ∇ ⋅ yields a vector. Since

$${\displaystyle \begin{array}{rl}\mathrm{\nabla }\cdot \mathbf{E}& =0\\ \mathrm{\nabla }\cdot \mathbf{B}& =0\end{array}}$$

then the first term on the right in the identity vanishes and we obtain the wave equations:

$${\displaystyle \begin{array}{rl}\frac{1}{c_0^2}\frac{{\mathrm{\partial }}^2\mathbf{E}}{\mathrm{\partial }{t}^2}-{\mathrm{\nabla }}^2\mathbf{E}& =\mathbf{0}\\ \frac{1}{c_0^2}\frac{{\mathrm{\partial }}^2\mathbf{B}}{\mathrm{\partial }{t}^2}-{\mathrm{\nabla }}^2\mathbf{B}& =\mathbf{0}\end{array}}$$

where

$${\displaystyle c_0=\frac{1}{\sqrt{{\mu }_0{\epsilon }_0}}=2.99792458\times {10}^8\text{m/s}}$$

is the speed of light in free space.

Covariant form of the homogeneous wave equation

Time dilation in transversal motion. The requirement that the speed of light is constant in every inertial reference frame leads to the theory of Special Relativity.

These relativistic equations can be written in contravariant form as

$${\displaystyle ◻A^{\mu }=0}$$

where the electromagnetic four-potential is

$${\displaystyle A^{\mu }=\left(\frac{\varphi }{c},\mathbf{A}\right)}$$

with the Lorenz gauge condition:

$${\displaystyle {\mathrm{\partial }}_{\mu }{A}^{\mu }=0,}$$

and where

$${\displaystyle ◻={\mathrm{\nabla }}^2-\frac{1}{c^2}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{t}^2}}$$

is the d'Alembert operator.

Homogeneous wave equation in curved spacetime

Main article: Maxwell's equations in curved spacetime

The electromagnetic wave equation is modified in two ways, the derivative is replaced with the covariant derivative and a new term that depends on the curvature appears.

$${\displaystyle -{A^{\alpha ;\beta }}_{;\beta }+{R^{\alpha }}_{\beta }{A}^{\beta }=0}$$

where ${\displaystyle {R^{\alpha }}_{\beta }}$ is the Ricci curvature tensor and the semicolon indicates covariant differentiation.

The generalization of the Lorenz gauge condition in curved spacetime is assumed:

$${\displaystyle {A^{\mu }}_{;\mu }=0.}$$

Inhomogeneous electromagnetic wave equation

Main article: Inhomogeneous electromagnetic wave equation

Localized time-varying charge and current densities can act as sources of electromagnetic waves in a vacuum. Maxwell's equations can be written in the form of a wave equation with sources. The addition of sources to the wave equations makes the partial differential equations inhomogeneous.

Solutions to the homogeneous electromagnetic wave equation

Main article: Wave equation

The general solution to the electromagnetic wave equation is a linear superposition of waves of the form

$${\displaystyle \begin{array}{rl}\mathbf{E}(\mathbf{r},t)& =g(\varphi (\mathbf{r},t))=g(\omega t-\mathbf{k}\cdot \mathbf{r})\\ \mathbf{B}(\mathbf{r},t)& =g(\varphi (\mathbf{r},t))=g(\omega t-\mathbf{k}\cdot \mathbf{r})\end{array}}$$

for virtually any well-behaved function g of dimensionless argument φ, where ω is the angular frequency (in radians per second), and k = (k_x, k_y, k_z) is the wave vector (in radians per meter).

Although the function g can be and often is a monochromatic sine wave, it does not have to be sinusoidal, or even periodic. In practice, g cannot have infinite periodicity because any real electromagnetic wave must always have a finite extent in time and space. As a result, and based on the theory of Fourier decomposition, a real wave must consist of the superposition of an infinite set of sinusoidal frequencies.

In addition, for a valid solution, the wave vector and the angular frequency are not independent; they must adhere to the dispersion relation:

$${\displaystyle k=|\mathbf{k}|=\frac{\omega }{c}=\frac{2\pi }{\lambda }}$$

where k is the wavenumber and λ is the wavelength. The variable c can only be used in this equation when the electromagnetic wave is in a vacuum.

Monochromatic, sinusoidal steady-state

The simplest set of solutions to the wave equation result from assuming sinusoidal waveforms of a single frequency in separable form:

$${\displaystyle \mathbf{E}(\mathbf{r},t)=\mathrm{\Re }\left\{\mathbf{E}(\mathbf{r})e^{i\omega t}\right\}}$$

where

• i is the imaginary unit,
• ω = 2π f  is the angular frequency in radians per second,
•  f  is the frequency in hertz, and
• ${\displaystyle e^{i\omega t}=\cos (\omega t)+i\sin (\omega t)}$ is Euler's formula.

Plane wave solutions

Main article: Sinusoidal plane-wave solutions of the electromagnetic wave equation

Consider a plane defined by a unit normal vector

$${\displaystyle \mathbf{n}=\frac{\mathbf{k}}{k}.}$$

Then planar traveling wave solutions of the wave equations are

$${\displaystyle \begin{array}{rl}\mathbf{E}(\mathbf{r})& ={\mathbf{E}}_0{e}^{-i\mathbf{k}\cdot \mathbf{r}}\\ \mathbf{B}(\mathbf{r})& ={\mathbf{B}}_0{e}^{-i\mathbf{k}\cdot \mathbf{r}}\end{array}}$$

where r = (x, y, z) is the position vector (in meters).

These solutions represent planar waves traveling in the direction of the normal vector n. If we define the z direction as the direction of n, and the x direction as the direction of E, then by Faraday's Law the magnetic field lies in the y direction and is related to the electric field by the relation

$${\displaystyle c^2\frac{\mathrm{\partial }B}{\mathrm{\partial }z}=\frac{\mathrm{\partial }E}{\mathrm{\partial }t}.}$$

Because the divergence of the electric and magnetic fields are zero, there are no fields in the direction of propagation.

This solution is the linearly polarized solution of the wave equations. There are also circularly polarized solutions in which the fields rotate about the normal vector.

Spectral decomposition

Because of the linearity of Maxwell's equations in a vacuum, solutions can be decomposed into a superposition of sinusoids. This is the basis for the Fourier transform method for the solution of differential equations. The sinusoidal solution to the electromagnetic wave equation takes the form

$${\displaystyle \begin{array}{rl}\mathbf{E}(\mathbf{r},t)& ={\mathbf{E}}_0\cos (\omega t-\mathbf{k}\cdot \mathbf{r}+{\varphi }_0)\\ \mathbf{B}(\mathbf{r},t)& ={\mathbf{B}}_0\cos (\omega t-\mathbf{k}\cdot \mathbf{r}+{\varphi }_0)\end{array}}$$

where

• t is time (in seconds),
• ω is the angular frequency (in radians per second),
• k = (k_x, k_y, k_z) is the wave vector (in radians per meter), and
• ${\displaystyle {\varphi }_0}$ is the phase angle (in radians).

The wave vector is related to the angular frequency by

$${\displaystyle k=|\mathbf{k}|=\frac{\omega }{c}=\frac{2\pi }{\lambda }}$$

where k is the wavenumber and λ is the wavelength.

The electromagnetic spectrum is a plot of the field magnitudes (or energies) as a function of wavelength.

Multipole expansion

Assuming monochromatic fields varying in time as ${\displaystyle e^{-i\omega t}}$, if one uses Maxwell's Equations to eliminate B, the electromagnetic wave equation reduces to the Helmholtz equation for E:

$${\displaystyle ({\mathrm{\nabla }}^2+k^2)\mathbf{E}=0,\mathbf{B}=-\frac{i}{k}\mathrm{\nabla }\times \mathbf{E},}$$

with k = ω/c as given above. Alternatively, one can eliminate E in favor of B to obtain:

$${\displaystyle ({\mathrm{\nabla }}^2+k^2)\mathbf{B}=0,\mathbf{E}=-\frac{i}{k}\mathrm{\nabla }\times \mathbf{B}.}$$

A generic electromagnetic field with frequency ω can be written as a sum of solutions to these two equations. The three-dimensional solutions of the Helmholtz Equation can be expressed as expansions in spherical harmonics with coefficients proportional to the spherical Bessel functions. However, applying this expansion to each vector component of E or B will give solutions that are not generically divergence-free (∇ ⋅ E = ∇ ⋅ B = 0), and therefore require additional restrictions on the coefficients.

The multipole expansion circumvents this difficulty by expanding not E or B, but r ⋅ E or r ⋅ B into spherical harmonics. These expansions still solve the original Helmholtz equations for E and B because for a divergence-free field F, ∇² (r ⋅ F) = r ⋅ (∇² F). The resulting expressions for a generic electromagnetic field are:

$${\displaystyle \begin{array}{rl}\mathbf{E}& =e^{-i\omega t}\sum _{l,m}\sqrt{l(l+1)}\left[a_E(l,m){\mathbf{E}}_{l,m}^{(E)}+a_M(l,m){\mathbf{E}}_{l,m}^{(M)}\right]\\ \mathbf{B}& =e^{-i\omega t}\sum _{l,m}\sqrt{l(l+1)}\left[a_E(l,m){\mathbf{B}}_{l,m}^{(E)}+a_M(l,m){\mathbf{B}}_{l,m}^{(M)}\right],\end{array}}$$

where ${\displaystyle {\mathbf{E}}_{l,m}^{(E)}}$ and ${\displaystyle {\mathbf{B}}_{l,m}^{(E)}}$ are the electric multipole fields of order (l, m), and ${\displaystyle {\mathbf{E}}_{l,m}^{(M)}}$ and ${\displaystyle {\mathbf{B}}_{l,m}^{(M)}}$ are the corresponding magnetic multipole fields, and a_E(l, m) and a_M(l, m) are the coefficients of the expansion. The multipole fields are given by

$${\displaystyle \begin{array}{rl}{\mathbf{B}}_{l,m}^{(E)}& =\sqrt{l(l+1)}\left[B_l^{(1)}{h}_l^{(1)}(kr)+B_l^{(2)}{h}_l^{(2)}(kr)\right]{\mathbf{\Phi }}_{l,m}\\ {\mathbf{E}}_{l,m}^{(E)}& =\frac{i}{k}\mathrm{\nabla }\times {\mathbf{B}}_{l,m}^{(E)}\\ {\mathbf{E}}_{l,m}^{(M)}& =\sqrt{l(l+1)}\left[E_l^{(1)}{h}_l^{(1)}(kr)+E_l^{(2)}{h}_l^{(2)}(kr)\right]{\mathbf{\Phi }}_{l,m}\\ {\mathbf{B}}_{l,m}^{(M)}& =-\frac{i}{k}\mathrm{\nabla }\times {\mathbf{E}}_{l,m}^{(M)},\end{array}}$$

where h_l^(1,2)(x) are the spherical Hankel functions, E_l^(1,2) and B_l^(1,2) are determined by boundary conditions, and

$${\displaystyle {\mathbf{\Phi }}_{l,m}=\frac{1}{\sqrt{l(l+1)}}(\mathbf{r}\times \mathrm{\nabla })Y_{l,m}}$$

are vector spherical harmonics normalized so that

$${\displaystyle \int {\mathbf{\Phi }}_{l,m}^{\ast }\cdot {\mathbf{\Phi }}_{l^{\prime },m^{\prime }}d\mathrm{\Omega }={\delta }_{l,l^{\prime }}{\delta }_{m,m^{\prime }}.}$$

The multipole expansion of the electromagnetic field finds application in a number of problems involving spherical symmetry, for example antennae radiation patterns, or nuclear gamma decay. In these applications, one is often interested in the power radiated in the far-field. In this regions, the E and B fields asymptotically approach

$${\displaystyle \begin{array}{rl}\mathbf{B}& \approx \frac{e^{i(kr-\omega t)}}{kr}\sum _{l,m}(-i{)}^{l+1}\left[a_E(l,m){\mathbf{\Phi }}_{l,m}+a_M(l,m)\hat{\mathbf{r}}\times {\mathbf{\Phi }}_{l,m}\right]\\ \mathbf{E}& \approx \mathbf{B}\times \hat{\mathbf{r}}.\end{array}}$$

The angular distribution of the time-averaged radiated power is then given by

$${\displaystyle \frac{dP}{d\mathrm{\Omega }}\approx \frac{1}{2k^2}{\left|\sum _{l,m}(-i{)}^{l+1}\left[a_E(l,m){\mathbf{\Phi }}_{l,m}\times \hat{\mathbf{r}}+a_M(l,m){\mathbf{\Phi }}_{l,m}\right]\right|}^2.}$$

Turbulence modeling

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Turbulence_modeling 

A simulation of a physical wind tunnel airplane model

In fluid dynamics, turbulence modeling is the construction and use of a mathematical model to predict the effects of turbulence. Turbulent flows are commonplace in most real-life scenarios. In spite of decades of research, there is no analytical theory to predict the evolution of these turbulent flows. The equations governing turbulent flows can only be solved directly for simple cases of flow. For most real-life turbulent flows, CFD simulations use turbulent models to predict the evolution of turbulence. These turbulence models are simplified constitutive equations that predict the statistical evolution of turbulent flows.

Closure problem

The Navier–Stokes equations govern the velocity and pressure of a fluid flow. In a turbulent flow, each of these quantities may be decomposed into a mean part and a fluctuating part. Averaging the equations gives the Reynolds-averaged Navier–Stokes (RANS) equations, which govern the mean flow. However, the nonlinearity of the Navier–Stokes equations means that the velocity fluctuations still appear in the RANS equations, in the nonlinear term ${\displaystyle -\rho \overline{v_i^{\mathrm{\prime }}{v}_j^{\mathrm{\prime }}}}$ from the convective acceleration. This term is known as the Reynolds stress, ${\displaystyle R_{ij}}$. Its effect on the mean flow is like that of a stress term, such as from pressure or viscosity.

To obtain equations containing only the mean velocity and pressure, we need to close the RANS equations by modelling the Reynolds stress term ${\displaystyle R_{ij}}$ as a function of the mean flow, removing any reference to the fluctuating part of the velocity. This is the closure problem.

Eddy viscosity

Joseph Valentin Boussinesq was the first to attack the closure problem, by introducing the concept of eddy viscosity. In 1877 Boussinesq proposed relating the turbulence stresses to the mean flow to close the system of equations. Here the Boussinesq hypothesis is applied to model the Reynolds stress term. Note that a new proportionality constant ${\displaystyle {\nu }_t>0}$, the (kinematic) turbulence eddy viscosity, has been introduced. Models of this type are known as eddy viscosity models (EVMs).

$${\displaystyle -\overline{v_i^{\mathrm{\prime }}{v}_j^{\mathrm{\prime }}}={\nu }_t\left(\frac{\mathrm{\partial }\overline{v_i}}{\mathrm{\partial }{x}_j}+\frac{\mathrm{\partial }\overline{v_j}}{\mathrm{\partial }{x}_i}\right)-\frac{2}{3}k{\delta }_{ij}}$$ which can be written in shorthand as $${\displaystyle -\overline{v_i^{\mathrm{\prime }}{v}_j^{\mathrm{\prime }}}=2{\nu }_t{S}_{ij}-\frac{2}{3}k{\delta }_{ij}}$$ where

• ${\displaystyle S_{ij}}$ is the mean rate of strain tensor
• ${\displaystyle {\nu }_t}$ is the (kinematic) turbulence eddy viscosity
• ${\displaystyle k=\frac{1}{2}\overline{v_i^{\prime }{v}_i^{\prime }}}$ is the turbulence kinetic energy
• and ${\displaystyle {\delta }_{ij}}$ is the Kronecker delta.

In this model, the additional turbulence stresses are given by augmenting the molecular viscosity with an eddy viscosity. This can be a simple constant eddy viscosity (which works well for some free shear flows such as axisymmetric jets, 2-D jets, and mixing layers).

The Boussinesq hypothesis – although not explicitly stated by Boussinesq at the time – effectively consists of the assumption that the Reynolds stress tensor is aligned with the strain tensor of the mean flow (i.e.: that the shear stresses due to turbulence act in the same direction as the shear stresses produced by the averaged flow). It has since been found to be significantly less accurate than most practitioners would assume. Still, turbulence models which employ the Boussinesq hypothesis have demonstrated significant practical value. In cases with well-defined shear layers, this is likely due the dominance of streamwise shear components, so that considerable relative errors in flow-normal components are still negligible in absolute terms. Beyond this, most eddy viscosity turbulence models contain coefficients which are calibrated against measurements, and thus produce reasonably accurate overall outcomes for flow fields of similar type as used for calibration.

Prandtl's mixing-length concept

Later, Ludwig Prandtl introduced the additional concept of the mixing length, along with the idea of a boundary layer. For wall-bounded turbulent flows, the eddy viscosity must vary with distance from the wall, hence the addition of the concept of a 'mixing length'. In the simplest wall-bounded flow model, the eddy viscosity is given by the equation: $${\displaystyle {\nu }_t=\left|\frac{\mathrm{\partial }u}{\mathrm{\partial }y}\right|l_m^2}$$ where

• ${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }y}}$ is the partial derivative of the streamwise velocity (u) with respect to the wall normal direction (y)
• ${\displaystyle l_m}$ is the mixing length.

This simple model is the basis for the "law of the wall", which is a surprisingly accurate model for wall-bounded, attached (not separated) flow fields with small pressure gradients.

More general turbulence models have evolved over time, with most modern turbulence models given by field equations similar to the Navier–Stokes equations.

Smagorinsky model for the sub-grid scale eddy viscosity

Joseph Smagorinsky was the first who proposed a formula for the eddy viscosity in Large Eddy Simulation models, based on the local derivatives of the velocity field and the local grid size:

${\displaystyle {\nu }_t=\mathrm{\Delta }x\mathrm{\Delta }y\sqrt{{\left(\frac{\mathrm{\partial }u}{\mathrm{\partial }x}\right)}^2+{\left(\frac{\mathrm{\partial }v}{\mathrm{\partial }y}\right)}^2+\frac{1}{2}{\left(\frac{\mathrm{\partial }u}{\mathrm{\partial }y}+\frac{\mathrm{\partial }v}{\mathrm{\partial }x}\right)}^2}}$

In the context of Large Eddy Simulation, turbulence modeling refers to the need to parameterize the subgrid scale stress in terms of features of the filtered velocity field. This field is called subgrid-scale modeling.

Spalart–Allmaras, k–ε and k–ω models

The Boussinesq hypothesis is employed in the Spalart–Allmaras (S–A), k–ε (k–epsilon), and k–ω (k–omega) models and offers a relatively low cost computation for the turbulence viscosity ${\displaystyle {\nu }_t}$. The S–A model uses only one additional equation to model turbulence viscosity transport, while the k–ε and k–ω models use two.

Common models

The following is a brief overview of commonly employed models in modern engineering applications.

• Spalart–Allmaras (S–A)
  The Spalart–Allmaras model is a one-equation model that solves a modelled transport equation for the kinematic eddy turbulent viscosity. The Spalart–Allmaras model was designed specifically for aerospace applications involving wall-bounded flows and has been shown to give good results for boundary layers subjected to adverse pressure gradients. It is also gaining popularity in turbomachinery applications.
• k–ε (k–epsilon)
  K-epsilon (k-ε) turbulence model is the most common model used in computational fluid dynamics (CFD) to simulate mean flow characteristics for turbulent flow conditions. It is a two-equation model which gives a general description of turbulence by means of two transport equations (PDEs). The original impetus for the K-epsilon model was to improve the mixing-length model, as well as to find an alternative to algebraically prescribing turbulent length scales in moderate to high complexity flows.
• k–ω (k–omega)
  In computational fluid dynamics, the k–omega (k–ω) turbulence model is a common two-equation turbulence model that is used as a closure for the Reynolds-averaged Navier–Stokes equations (RANS equations). The model attempts to predict turbulence by two partial differential equations for two variables, k and ω, with the first variable being the turbulence kinetic energy (k) while the second (ω) is the specific rate of dissipation (of the turbulence kinetic energy k into internal thermal energy).
• SST (Menter’s Shear Stress Transport)
  SST (Menter's shear stress transport) turbulence model is a widely used and robust two-equation eddy-viscosity turbulence model used in computational fluid dynamics. The model combines the k-omega turbulence model and K-epsilon turbulence model such that the k-omega is used in the inner region of the boundary layer and switches to the k-epsilon in the free shear flow.
• Reynolds stress equation model

  The Reynolds stress equation model (RSM), also referred to as second moment closure model, is the most complete classical turbulence modelling approach. Popular eddy-viscosity based models like the k–ε (k–epsilon) model and the k–ω (k–omega) models have significant shortcomings in complex engineering flows. This arises due to the use of the eddy-viscosity hypothesis in their formulation. For instance, in flows with high degrees of anisotropy, significant streamline curvature, flow separation, zones of recirculating flow or flows influenced by rotational effects, the performance of such models is unsatisfactory. In such flows, Reynolds stress equation models offer much better accuracy.

  Eddy viscosity based closures cannot account for the return to isotropy of turbulence, observed in decaying turbulent flows. Eddy-viscosity based models cannot replicate the behaviour of turbulent flows in the Rapid Distortion limit, where the turbulent flow essentially behaves like an elastic medium.