Navier-Stokes equations

The Navier-Stokes equations (/nævˈjeɪ ˈstoʊks/ nav-YAY STOHKS) describe the motion of viscous fluids. This system of partial differential equations was named after Claude-Louis Navier and George Gabriel Stokes, who developed them over a few decades of progressive work, from 1822 (Navier) to 1842-1850 (Stokes). Siméon Denis Poisson independently achieved the same results.

The Navier-Stokes equations mathematically express momentum balance for Newtonian fluids and make use of the conservation of mass. They are sometimes accompanied by an equation of state relating pressure, temperature and density. They arise from applying Newton's second law to fluid motion, together with the assumption that the stress in the fluid is the sum of a diffusing viscous term (proportional to the gradient of velocity) and a pressure term, hence describing viscous flow. The Navier-Stokes equations generalize the Euler equations in that the latter model only considers inviscid flow.

The Navier-Stokes equations are of great scientific and engineering interest because they may be used to model a wide variety of scenarios. In their full or simplified forms, they can assist in the design of aircraft and cars, the study of blood flow, the design of power stations, the analysis of pollution, and many other problems. Coupled with Maxwell's equations, they comprise the fundamentals of magnetohydrodynamics.

The Navier-Stokes equations are also of great interest in a purely mathematical sense. Despite their wide range of practical uses, the conjecture that they have smooth (meaning infinitely differentiable) or bounded solutions in three dimensions has not yet been proven. This is called the Navier-Stokes existence and smoothness problem. The Clay Mathematics Institute has called this one of the seven most important open problems in mathematics and has offered a $1 million prize for a solution or a counterexample.

The solution of the equations is a flow velocity. It is a vector field, to every point in a fluid, at any moment in a time interval, it gives a vector whose direction and magnitude are those of the velocity of the fluid at that point in space and at that moment in time. It is studied in three spatial dimensions and one time dimension, and higher-dimensional analogues are studied in both pure and applied mathematics. Once the velocity field is calculated, other quantities of interest, such as pressure or temperature, may be found using dynamical equations and relations. This is different from what one normally sees in classical mechanics, where solutions are typically trajectories of the position of a particle or deflection of a continuum. Studying velocity instead of position makes more sense for a fluid, although for visualization purposes, one can compute various trajectories. In particular, the streamlines of a vector field, interpreted as flow velocity, are the paths along which a massless fluid particle would travel. These paths are the integral curves whose derivative at each point is equal to the vector field, and they can represent visually the behavior of the vector field at a point in time.

The Navier-Stokes momentum equation can be derived as a particular form of the Cauchy momentum equation, whose general convective form is: \[\frac{\mathrm{D} \mathbf{u}}{\mathrm{D} t} = \frac 1 \rho \nabla \cdot \boldsymbol{\sigma} + \mathbf{a}.\] By setting the Cauchy stress tensor \(\boldsymbol{\sigma}\) to be the sum of a viscosity term \(\boldsymbol{\tau}\) (the deviatoric stress) and a pressure term \(-p \mathbf{I}\) (volumetric stress), we arrive at:

\(\rho\frac{\mathrm{D} \mathbf{u}}{\mathrm{D} t} = - \nabla p + \nabla \cdot \boldsymbol \tau + \rho\,\mathbf{a}\)

where

• \(\frac{\mathrm{D}}{\mathrm{D}t}\) is the material derivative, defined as \(\frac{\partial}{\partial t} + \mathbf{u} \cdot \nabla\),
• \(\rho\) is the (mass) density,
• \(\mathbf{u}\) is the flow velocity,
• \(\nabla \cdot \,\) is the divergence,
• \(p\) is the pressure,
• \(t\) is time,
• \(\boldsymbol{\tau}\) is the deviatoric stress tensor, which has order 2,
• \(\mathbf{a}\) represents body accelerations acting on the continuum, for example gravity, inertial accelerations, electrostatic accelerations, and so on.

In this form, it is apparent that in the assumption of an inviscid fluid, no deviatoric stress, Cauchy equations reduce to the Euler equations.

Assuming conservation of mass, with the known properties of divergence and gradient we can use the mass continuity equation, which represents the mass per unit volume of a homogenous fluid with respect to space and time (i.e., material derivative \(\frac{\mathbf{D}}{\mathbf{Dt}}\)) of any finite volume (\(\mathbf{V}\)) to represent the change of velocity in fluid media: \[\begin{aligned} & \frac{\mathbf{D}m}{\mathbf{Dt}} = \iiint\limits_V \left(\frac{ \mathbf{D}\rho}{\mathbf{Dt}} + \rho (\nabla \cdot \mathbf{u})\right) \, dV \\[5pt] & \frac{\mathbf{D}\rho}{\mathbf{Dt}} + \rho (\nabla \cdot \mathbf{u} )=\frac{\partial\rho}{\partial t} + (\nabla \rho) \cdot \mathbf{u} + \rho(\nabla \cdot \mathbf{u})= \frac{\partial\rho}{\partial t} + \nabla\cdot(\rho \mathbf{u})= 0 \end{aligned}\]where

• \(\frac{\mathrm{D}m}{\mathrm{D}t}\) is the material derivative of mass per unit volume (density, \(\rho\)),
• \(\iiint \limits_V \bigl(F(x_1, x_2, x_3 ,t)\bigr) \, dV\) is the mathematical operation for the integration throughout the volume (\(V\)),
• \(\frac{\partial }{\partial t}\) is the partial derivative mathematical operator,
• \(\nabla \cdot \mathbf{u}\,\) is the divergence of the flow velocity (\(\mathbf{u}\)), which is a scalar field,
• \(\nabla \rho \,\) is the gradient of density (\(\rho\)), which is the vector derivative of a scalar field,

to arrive at the conservation form of the equations of motion. This is often written:

\(\frac {\partial}{\partial t} (\rho\,\mathbf{u}) + \nabla \cdot (\rho\,\mathbf{u} \otimes \mathbf{u}) = - \nabla p + \nabla \cdot \boldsymbol \tau + \rho\,\mathbf{a}\)

where \(\otimes\) is the outer product of the flow velocity (\(\mathbf{u}\)): \[\mathbf u \otimes \mathbf u = \mathbf u \mathbf u^{\mathsf T}\]

Condensed: the full section is in Wikipedia.

A significant feature of the Cauchy equation and consequently all other continuum equations (including Euler and Navier-Stokes) is the presence of convective acceleration: the effect of acceleration of a flow with respect to space. While individual fluid particles indeed experience time-dependent acceleration, the convective acceleration of the flow field is a spatial effect, one example being fluid speeding up in a nozzle.

Remark: here, the deviatoric stress tensor is denoted \(\boldsymbol{\tau}\) as it was in the general continuum equations and in the incompressible flow section.

The compressible momentum Navier-Stokes equation results from the following assumptions on the Cauchy stress tensor:

• the stress is Galilean invariant: it does not depend directly on the flow velocity, but only on spatial derivatives of the flow velocity. So the stress variable is the tensor gradient \(\nabla \mathbf{u}\), or more simply the rate-of-strain tensor: \(\boldsymbol{\varepsilon}\left(\nabla \mathbf{u}\right) \equiv \frac{1}{2}\nabla \mathbf{u} + \frac{1}{2} \left(\nabla \mathbf{u}\right)^{\mathsf T}\)
• the deviatoric stress is linear in this variable: \(\boldsymbol{\sigma}(\boldsymbol \varepsilon) = -p \mathbf I + \mathbf{C} : \boldsymbol \varepsilon\), where \(p\) is independent on the strain rate tensor, \(\mathbf{C}\) is the fourth-order tensor representing the constant of proportionality, called the viscosity or elasticity tensor, and : is the double-dot product.
• the fluid is assumed to be isotropic, as with gases and simple liquids, and consequently \(\mathbf{C}\) is an isotropic tensor; furthermore, since the deviatoric stress tensor is symmetric, by Helmholtz decomposition it can be expressed in terms of two scalar Lamé parameters, the second viscosity \(\lambda\) and the dynamic viscosity \(\mu\), as it is usual in linear elasticity: Linear stress constitutive equation  (expression similar to the one for elastic solid)

  \(\boldsymbol \sigma(\boldsymbol \varepsilon) = - p \mathbf I + \lambda \operatorname{tr} (\boldsymbol \varepsilon) \mathbf I + 2 \mu \boldsymbol \varepsilon\)

  where \(\mathbf{I}\) is the identity tensor, and \(\operatorname{tr} (\boldsymbol \varepsilon)\) is the trace of the rate-of-strain tensor. So this decomposition can be explicitly defined as: \[\boldsymbol \sigma = -p \mathbf I + \lambda (\nabla\cdot\mathbf{u}) \mathbf I + \mu \left(\nabla\mathbf{u} + ( \nabla\mathbf{u} )^\mathsf{T}\right).\]

Since the trace of the rate-of-strain tensor in three dimensions is the divergence (i.e. rate of expansion) of the flow: \[\operatorname{tr} (\boldsymbol \varepsilon) = \nabla\cdot\mathbf{u}.\]

Given this relation, and since the trace of the identity tensor in three dimensions is three: \[\operatorname{tr} (\boldsymbol I) = 3.\]

the trace of the stress tensor in three dimensions becomes: \[\operatorname{tr} (\boldsymbol \sigma ) = -3p + (3 \lambda + 2 \mu )\nabla\cdot\mathbf{u}.\]

So by alternatively decomposing the stress tensor into isotropic and deviatoric parts, as usual in fluid dynamics: \[\boldsymbol \sigma = - \left[ p - \left(\lambda + \tfrac23 \mu\right) \left(\nabla\cdot\mathbf{u}\right) \right] \mathbf I + \mu \left(\nabla\mathbf{u} + \left( \nabla\mathbf{u} \right)^\mathsf{T} - \tfrac23 \left(\nabla\cdot\mathbf{u}\right)\mathbf I\right)\]

Introducing the bulk viscosity \(\zeta\), \[\zeta \equiv \lambda + \tfrac23 \mu ,\]

\(\boldsymbol \sigma = -\bigl[ p - \zeta (\nabla\cdot\mathbf{u})\bigr] \mathbf I + \mu \left[\nabla\mathbf{u} + ( \nabla\mathbf{u} )^\mathsf{T} - \tfrac23 (\nabla\cdot\mathbf{u})\mathbf I\right]\)

Condensed: the full section is in Wikipedia.

The incompressible momentum Navier-Stokes equation results from the following assumptions on the Cauchy stress tensor:

• the stress is Galilean invariant: it does not depend directly on the flow velocity, but only on spatial derivatives of the flow velocity. So the stress variable is the tensor gradient \(\nabla \mathbf{u}\).
• the fluid is assumed to be isotropic, as with gases and simple liquids, and consequently \(\boldsymbol{\tau}\) is an isotropic tensor; furthermore, since the deviatoric stress tensor can be expressed in terms of the dynamic viscosity \(\mu\): Stokes' stress constitutive equation  (expression used for incompressible elastic solids)

  \(\boldsymbol \tau = 2 \mu \boldsymbol \varepsilon\)

  where \[\boldsymbol{\varepsilon} = \tfrac{1}{2} \left( \mathbf{\nabla u} + \mathbf{\nabla u}^\mathsf{T} \right)\] is the rate-of-strain tensor. So this decomposition can be made explicit as:

  Stokes's stress constitutive equation  (expression used for incompressible viscous fluids)

  \(\boldsymbol \tau = \mu \left[\nabla\mathbf{u} + (\nabla\mathbf{u}) ^\mathsf{T}\right]\)

This is constitutive equation is also called the Newtonian law of viscosity. Dynamic viscosity μ need not be constant, in incompressible flows it can depend on density and on pressure. Any equation that makes explicit one of these transport coefficient in the conservative variables is called an equation of state.

The divergence of the deviatoric stress in case of uniform viscosity is given by: \[\nabla \cdot \boldsymbol \tau = 2 \mu \nabla \cdot \boldsymbol \varepsilon = \mu \nabla \cdot \left( \nabla\mathbf{u} + \nabla\mathbf{u} ^\mathsf{T} \right) = \mu \, \nabla^2 \mathbf{u}\] because \(\nabla \cdot \mathbf{u} = 0\) for an incompressible fluid.

Incompressibility rules out density and pressure waves like sound or shock waves, so this simplification is not useful if these phenomena are of interest. The incompressible flow assumption typically holds well with all fluids at low Mach numbers (say up to about Mach 0.3), such as for modelling air winds at normal temperatures. the incompressible Navier-Stokes equations are best visualized by dividing for the density:

\(\frac{D \mathbf{u}}{D t} = \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} = \nu \,\nabla^2 \mathbf{u} - \frac{1}{\rho}\nabla p + \frac{1}{\rho} \mathbf{f}\)

where \(\nu = \frac{\mu}{\rho}\) is called the kinematic viscosity. By isolating the fluid velocity, one can also state:

\(\left(\frac{\partial}{\partial t} + \mathbf{u} \cdot \nabla - \nu \,\nabla^2 \right) \mathbf{u} = - \frac{1}{\rho}\nabla p + \frac{1}{\rho} \mathbf{f}.\)

If the density is constant throughout the fluid domain, or, in other words, if all fluid elements have the same density, \(\rho\), then we have

\(\frac{D \mathbf{u}}{D t} = \nu \,\nabla^2 \mathbf{u} - \nabla \frac{p}{\rho} + \frac{1}{\rho} \mathbf{f},\)

where \(\frac{p}{\rho}\) is called the unit pressure head.

Condensed: the full section is in Wikipedia.

With partitioning of the problem domain and defining basis functions on the partitioned domain, the discrete form of the governing equation is \[\left(\mathbf{w}_i, \frac{\partial\mathbf{u}_j}{\partial t}\right) = -\bigl(\mathbf{w}_i, \left(\mathbf{u}\cdot\nabla\right)\mathbf{u}_j\bigr) - \nu\left(\nabla\mathbf{w}_i: \nabla\mathbf{u}_j\right) + \left(\mathbf{w}_i, \mathbf{f}^S\right).\]

It is desirable to choose basis functions that reflect the essential feature of incompressible flow. The elements must be divergence-free. While the velocity is the variable of interest, the existence of the stream function or vector potential is necessary by the Helmholtz theorem. Further, to determine fluid flow in the absence of a pressure gradient, one can specify the difference of stream function values across a 2D channel, or the line integral of the tangential component of the vector potential around the channel in 3D, the flow being given by Stokes' theorem. Discussion will be restricted to 2D in the following.

We further restrict discussion to continuous Hermite finite elements which have at least first-derivative degrees-of-freedom. With this, one can draw a large number of candidate triangular and rectangular elements from the plate-bending literature. These elements have derivatives as components of the gradient. In 2D, the gradient and curl of a scalar are clearly orthogonal, given by the expressions, \[\begin{aligned} \nabla\varphi &= \left(\frac{\partial \varphi}{\partial x},\,\frac{\partial \varphi}{\partial y}\right)^\mathsf{T}, \\[5pt] \nabla\times\varphi &= \left(\frac{\partial \varphi}{\partial y},\,-\frac{\partial \varphi}{\partial x}\right)^\mathsf{T}. \end{aligned}\]

Adopting continuous plate-bending elements, interchanging the derivative degrees-of-freedom and changing the sign of the appropriate one gives many families of stream function elements.

Taking the curl of the scalar stream function elements gives divergence-free velocity elements. The requirement that the stream function elements be continuous assures that the normal component of the velocity is continuous across element interfaces, all that is necessary for vanishing divergence on these interfaces.

Boundary conditions are simple to apply. The stream function is constant on no-flow surfaces, with no-slip velocity conditions on surfaces. Stream function differences across open channels determine the flow. No boundary conditions are necessary on open boundaries, though consistent values may be used with some problems. These are all Dirichlet conditions.

The algebraic equations to be solved are simple to set up, but of course are non-linear, requiring iteration of the linearized equations.

Condensed: the full section is in Wikipedia.

Recovering pressure from the velocity field is easy. The discrete weak equation for the pressure gradient is, \[(\mathbf{g}_i, \nabla p) = -\bigl(\mathbf{g}_i, \left(\mathbf{u}\cdot\nabla\right)\mathbf{u}_j\bigr) - \nu\left(\nabla\mathbf{g}_i: \nabla\mathbf{u}_j\right) + \left(\mathbf{g}_i, \mathbf{f}^I\right)\]

where the test/weight functions are irrotational. Any conforming scalar finite element may be used. However, the pressure gradient field may also be of interest. In this case, one can use scalar Hermite elements for the pressure. For the test/weight functions \(\mathbf{g}_i\) one would choose the irrotational vector elements obtained from the gradient of the pressure element.

The rotating frame of reference introduces some interesting pseudo-forces into the equations through the material derivative term. Consider a stationary inertial frame of reference \(K\) , and a non-inertial frame of reference \(K'\), which is translating with velocity \(\mathbf{U}(t)\) and rotating with angular velocity \(\Omega(t)\) with respect to the stationary frame. The Navier-Stokes equation observed from the non-inertial frame then becomes

\(\begin{aligned} \rho \left( \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} \right) = & - \nabla p + \nabla \cdot \left( \mu \left[\nabla\mathbf{u} + ( \nabla\mathbf{u} )^\mathsf{T} - \tfrac23 (\nabla\cdot\mathbf{u})\mathbf I\right] \right) + \nabla[\zeta (\nabla\cdot\mathbf{u})] + \rho\mathbf{f} \\ & - \rho \left[2\mathbf\Omega\times\mathbf u + \mathbf\Omega\times(\mathbf\Omega\times\mathbf x)+ \frac{\mathrm{d} \mathbf U}{\mathrm{d} t} + \frac{\mathrm{d} \mathbf \Omega}{\mathrm{d} t}\times\mathbf x\right] \end{aligned}\)

Here \(\mathbf{x}\) and \(\mathbf{u}\) are measured in the non-inertial frame. The first term in the parenthesis represents Coriolis acceleration, the second term is due to centrifugal acceleration, the third is due to the linear acceleration of \(K'\) with respect to \(K\) and the fourth term is due to the angular acceleration of \(K'\) with respect to \(K\).

The Navier-Stokes equations are strictly a statement of the balance of momentum. To fully describe fluid flow, more information is needed, how much depending on the assumptions made. This additional information may include boundary data (no-slip, capillary surface, etc.), conservation of mass, balance of energy, and/or an equation of state.

Regardless of the flow assumptions, a statement of the conservation of mass is generally necessary. This is achieved through the mass continuity equation, as discussed above in the "General continuum equations" within this article, as follows: \[\begin{aligned} \frac{\mathbf{D}m}{{\mathbf{Dt}}}&={\iiint\limits_V}\left({\frac{\mathbf{D}\rho}{{\mathbf{Dt}}} + \rho (\nabla \cdot \mathbf{u})}\right)dV \\ \frac{\mathbf{D}\rho}{{\mathbf{Dt}}} + \rho (\nabla \cdot{\mathbf{u}})&=\frac{\partial\rho}{\partial t} + ({\nabla \rho}) \cdot{\mathbf{u}} + {\rho}(\nabla \cdot \mathbf{u})= \frac{\partial\rho}{\partial t} + \nabla\cdot({\rho \mathbf{u}})= 0 \end{aligned}\] A fluid media for which the density \(\rho\) is constant is called incompressible. Therefore, the rate of change of \(\rho\) with respect to time \(\frac{\partial\rho}{\partial t}\) and the gradient of density \(\nabla \rho\) are equal to zero. In this case the general equation of continuity, \(\frac{\partial\rho}{\partial t} + \nabla\cdot({\rho \mathbf{u}})= 0\), reduces to: \[\rho(\nabla{\cdot}{\mathbf{u}}) = 0\] Furthermore, assuming that \(\rho \neq 0\) means that the right-hand side of the equation (zero) is divisible by density \(\rho\). Therefore, the continuity equation for an incompressible fluid reduces further to:\[(\nabla{\cdot{\mathbf{u}}}) = 0\] This relationship, \((\nabla{\cdot{\mathbf{u}}}) = 0\), identifies that the divergence of the flow velocity vector \(\mathbf{u}\) is equal to zero, which means that for an incompressible fluid the flow velocity field is a solenoidal vector field or a divergence-free vector field. Note that this relationship can be expanded upon due to its uniqueness with the vector Laplace operator \(\nabla ^{2} \mathbf{u} =\nabla (\nabla \cdot \mathbf{u} )-\nabla \times (\nabla \times \mathbf{u} )\), and vorticity \(\boldsymbol \omega = \nabla \times \mathbf{u}\) which is now expressed like so, for an incompressible fluid:\[\nabla ^{2}\mathbf {u} = - \bigl(\nabla \times (\nabla \times \mathbf {u} )\bigr) = - (\nabla \times \boldsymbol \omega)\]

Taking the curl of the incompressible Navier-Stokes equation results in the elimination of pressure. This is especially easy to see if 2D Cartesian flow is assumed (like in the degenerate 3D case with \(u_z = 0\) and no dependence of anything on \(z\)), where the equations reduce to: \[\begin{aligned} \rho \left(\frac{\partial u_x}{\partial t} + u_x \frac{\partial u_x}{\partial x} + u_y \frac{\partial u_x}{\partial y}\right) &= -\frac{\partial p}{\partial x} + \mu \left(\frac{\partial^2 u_x}{\partial x^2} + \frac{\partial^2 u_x}{\partial y^2}\right) + \rho g_x \\ \rho \left(\frac{\partial u_y}{\partial t} + u_x \frac{\partial u_y}{\partial x} + u_y \frac{\partial u_y}{\partial y}\right) &= -\frac{\partial p}{\partial y} + \mu \left(\frac{\partial^2 u_y}{\partial x^2} + \frac{\partial^2 u_y}{\partial y^2}\right) + \rho g_y. \end{aligned}\]

Differentiating the first with respect to \(y\), the second with respect to \(x\) and subtracting the resulting equations will eliminate pressure and any conservative force. For incompressible flow, defining the stream function \(\psi\) through \[u_x = \frac{\partial \psi}{\partial y}; \quad u_y = -\frac{\partial \psi}{\partial x}\] results in mass continuity being unconditionally satisfied (given the stream function is continuous), and then incompressible Newtonian 2D momentum and mass conservation condense into one equation: \[\frac{\partial}{\partial t}\left(\nabla^2 \psi\right) + \frac{\partial \psi}{\partial y} \frac{\partial}{\partial x}\left(\nabla^2 \psi\right) - \frac{\partial \psi}{\partial x} \frac{\partial}{\partial y}\left(\nabla^2 \psi\right) = \nu \nabla^4 \psi\]

where \(\nabla^4\) is the 2D biharmonic operator and \(\nu\) is the kinematic viscosity, \(\nu = \frac{\mu}{\rho}\). We can also express this compactly using the Jacobian determinant: \[\frac{\partial}{\partial t}\left(\nabla^2 \psi\right) + \frac{\partial\left(\psi, \nabla^2\psi \right)}{\partial(y,x)} = \nu \nabla^4 \psi.\]

This single equation together with appropriate boundary conditions describes 2D fluid flow, taking only kinematic viscosity as a parameter. Note that the equation for creeping flow results when the left side is assumed zero.

In axisymmetric flow another stream function formulation, called the Stokes stream function, can be used to describe the velocity components of an incompressible flow with one scalar function.

The incompressible Navier-Stokes equation is a differential algebraic equation, having the inconvenient feature that there is no explicit mechanism for advancing the pressure in time. Consequently, much effort has been expended to eliminate the pressure from all or part of the computational process. The stream function formulation eliminates the pressure but only in two dimensions and at the expense of introducing higher derivatives and elimination of the velocity, which is the primary variable of interest.

The Navier-Stokes equations are nonlinear partial differential equations in the general case and so remain in almost every real situation. In some cases, such as one-dimensional flow and Stokes flow (or creeping flow), the equations can be simplified to linear equations. The nonlinearity makes most problems difficult or impossible to solve and is the main contributor to the turbulence that the equations model.

The nonlinearity is due to convective acceleration, which is an acceleration associated with the change in velocity over position. Hence, any convective flow, whether turbulent or not, will involve nonlinearity. An example of convective but laminar (nonturbulent) flow would be the passage of a viscous fluid (for example, oil) through a small converging nozzle. Such flows, whether exactly solvable or not, can often be thoroughly studied and understood.

Turbulence is the time-dependent chaotic behaviour seen in many fluid flows. It is generally believed that it is due to the inertia of the fluid as a whole: the culmination of time-dependent and convective acceleration; hence flows where inertial effects are small tend to be laminar (the Reynolds number quantifies how much the flow is affected by inertia). It is believed, though not known with certainty, that the Navier-Stokes equations describe turbulence properly.

The numerical solution of the Navier-Stokes equations for turbulent flow is extremely difficult, and due to the significantly different mixing-length scales that are involved in turbulent flow, the stable solution of this requires such a fine mesh resolution that the computational time becomes significantly infeasible for calculation or direct numerical simulation. Attempts to solve turbulent flow using a laminar solver typically result in a time-unsteady solution, which fails to converge appropriately. To counter this, time-averaged equations such as the Reynolds-averaged Navier-Stokes equations (RANS), supplemented with turbulence models, are used in practical computational fluid dynamics (CFD) applications when modeling turbulent flows. Some models include the Spalart-Allmaras, k, ω, k, ε, and SST models, which add a variety of additional equations to bring closure to the RANS equations. Large eddy simulation (LES) can also be used to solve these equations numerically. This approach is computationally more expensive, in time and in computer memory, than RANS, but produces better results because it explicitly resolves the larger turbulent scales.