Navier–Stokes equations

The Navier–Stokes equations represent the fundamental principles of fluid mechanics, expressing the conservation of momentum and mass in a fluid. They provide a comprehensive description of fluid flow behavior across a wide range of physical situations. Because these equations capture the essential physics of Newtonian fluids–such as air, water, and gases under ‘standard’ conditions–they form the fundamental basis for analyzing and predicting flow dynamics.

Example use case:

• incompressible: Solving for the base flow of the cylinder wake

References:

• Barkley et al. 2006

• Towne et al. 2018

• Müller et al. 2024

• Demange et al. 2024

Nomenclature:

• \(\mathbf{u}\): velocity vector

• \(p\): pressure

• \(\rho\): density

• \(\mathbf{\tau}\): viscous stress tensor

• \(\mu\): dynamic viscosity

• \(\mathbf{I}\): identity tensor

• \(X_p\): test function for mass equation

• \(\mathbf{X}_\mathbf{u}\): test function for momentum equations

In the following, first the incompressible Navier–Stokes equations are described. Then, the details on mass and momentum equation of the compressible Navier–Stokes equations are given. For details on the energy equation, which is also needed for a compressible formulation, see the section on Energy equations.

1. Incompressible equations

Assumptions:

• primitive variables

• \(\rho = \textrm{const}\), this is reasonable if the Mach number is low, and if there are no large temperature gradients, no acoustics, no combustion

• no additional source terms

• no gravitational forces

Nonlinear incompressible Navier–Stokes equations

The mass equation is

\[ \nabla \cdot \mathbf{u} = 0\ . \]

The momentum equations are

\[ \rho \frac{\partial \mathbf{u}}{\partial t} + \rho(\mathbf{u}\cdot \nabla) \mathbf{u} + \nabla p - \nabla \cdot \mathbf{\tau} = 0\ , \]

where the viscous stress tensor \(\mathbf{\tau}\) is

\[ \mathbf{\tau} = \mu_\textrm{eff}(\nabla + \nabla ^T)\mathbf{u} \]

and where \(\mu_\textrm{eff} = \mu + \mu_t\) is the effective dynamic viscosity composed of a molecular, \(\mu\), and a turbulent, \(\mu_t\), contribution. The former is defined according to the Viscosity models selected by the user, while the latter models diffusion from unresolved turbulent fluctuations accordint to the Boussinesq hypothesis, detailed in the Turbulence models section.

Base flow equations (incompressible)

We consider the flow field to be comprised of a time-invariant base flow, which can be either a time-averaged solution or a fixed point solution of the Navier–Stokes equations, and the perturbation, such that

\[ \mathbf{q}(\mathbf{x},t) = \overline{\mathbf{q}}(\mathbf{x})+\mathbf{q}'(\mathbf{x},t), \qquad \mathbf{q}=[\mathbf u,p]^{T} \ . \]

Inserting this into the Navier–Stokes equations and taking the time-average we get the base flow equations.

The mass equation is

\[ \nabla \cdot \overline{\mathbf{u}} = 0\ . \]

The momentum equations are

\[ \rho(\overline{\mathbf{u}}\cdot \nabla) \overline{\mathbf{u}} + \nabla \overline{p} - \nabla \cdot \overline{\mathbf{\tau}} = 0\ , \]

where the mean viscous stress tensor \(\overline{\mathbf{\tau}}\) is

\[ \overline{\mathbf{\tau}} = \overline{\mu}_\textrm{eff}(\nabla + \nabla ^T)\overline{\mathbf{u}}\ . \]

Weak form of the incompressible base flow equations

Note

The weak form of the incompressible nonlinear base flow equations is only implicitly included in the generalized compressible form of the Navier–Stokes equations (see below) and is, therefore, not stated here.

Linearized incompressible Navier–Stokes equations

The mass equation is

\[ \nabla \cdot \mathbf{u}' = 0\ . \]

The momentum equations are

\[ \rho \frac{\partial \mathbf{u}'}{\partial t} + \rho [(\overline{\mathbf{u}}\cdot \nabla) \mathbf{u}' + (\mathbf{u}' \cdot \nabla) \overline{\mathbf{u}}] + \nabla p' - \nabla \cdot \mathbf{\tau}' = 0\ , \]

where the fluctuating viscous stress tensor \(\tau'\) is

\[ \mathbf{\tau}' = \mu_\textrm{eff}'[(\nabla + \nabla ^T)\overline{\mathbf{u}}]+ \overline{\mu}_\textrm{eff}[(\nabla + \nabla ^T) \mathbf{u}'] \]

and where \(\mu_\textrm{eff}'\) is the fluctuating effective dynamic viscosity (see Viscosity models and Turbulence models for details).

Weak form of the linearized Navier–Stokes equations

Note

The weak form of the incompressible linearized Navier–Stokes equations is only implicitly included in the generalized compressible form of the Navier–Stokes equations (see below) and is, therefore, not stated here.

2. Compressible equations

Assumptions:

• primitive variables

• no additional source terms

• no gravitational forces

In the following, details on mass and momentum equations of the compressible Navier–Stokes equations are given. For details on the energy equation, which is also needed for a compressible formulation, see the section on Energy equations.

Nonlinear compressible Navier–Stokes equations

The mass equation is

\[ \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0\ . \]

The momentum equations are

\[ \rho \frac{\partial \mathbf{u}}{\partial t} + \rho(\mathbf{u}\cdot \nabla) \mathbf{u} + \nabla p - \nabla \cdot \tau = 0\ , \]

where the viscous stress tensor \(\tau\) is

\[ \tau = \mu_\textrm{eff}[(\nabla + \nabla ^T)\mathbf{u} - \frac{2}{3} (\nabla \cdot \mathbf{u})\mathbf{I}] \]

with \(\mu_\textrm{eff}\) as the effective dynamic viscosity (see Viscosity models and Turbulence models for details) and \(\mathbf{I}\) as the identity tensor.

Base flow equations (compressible)

We consider the flow field to be comprised of a time-invariant base flow, which can be either a time-averaged flow or fixed point solution, and the perturbation, such that

\[ \mathbf{q}(\mathbf{x},t) = \overline{\mathbf{q}}(\mathbf{x})+\mathbf{q}'(\mathbf{x},t), \qquad \mathbf{q}=[\mathbf{u},p,\rho]^{T}\ . \]

Note

For the compressible equations, the overbar represents a Favre average for the velocity and a Reynolds average for pressure and density.

Inserting this into the Navier–Stokes equations and taking the average we get the base flow equations.

The mass equation is

\[ \nabla \cdot (\overline{\rho}\, \overline{\mathbf{u}}) = 0\ . \]

The momentum equations are

\[ \overline{\rho}(\overline{\mathbf{u}}\cdot \nabla) \overline{\mathbf{u}} + \nabla \overline{p} - \nabla \cdot \overline{\tau} = 0\ , \]

where the mean viscous stress tensor \(\overline{\mathbf{\tau}}\) is

\[ \overline{\tau} = \overline{\mu}_\textrm{eff}[(\nabla + \nabla ^T)\overline{\mathbf{u}} - \frac{2}{3} (\nabla \cdot \overline{\mathbf{u}})\mathbf{I}]\ . \]

Weak form of the compressible base flow equations

The weak form of the nonlinear base flow equations, as implemented in FELiCS, is

\[ \int_\Omega \mathrm{j} \nabla \cdot \left(\mathbf{X}_\mathbf{u}^* \otimes \overline{\rho}\overline{\mathbf{u}} \right) \cdot \overline{\mathbf{u}} \, \mathrm{d}\mathbf{x} - \int_{\partial\Omega} \mathrm{j} \overline{\rho}\left( \left(\overline{\mathbf{u}} \otimes \mathbf{X}_\mathbf{u}^*\right) \cdot \overline{\mathbf{u}} \right)\cdot \mathbf{n} \, \mathrm{d}\mathbf{s} + \int_\Omega \mathrm{j} \overline{p} \nabla \cdot \mathbf{X}_\mathbf{u}^* \mathrm{d}\mathbf{x} - \int_{\partial\Omega} \mathrm{j} \overline{p} \mathbf{X}_\mathbf{u}^* \cdot \mathbf{n} \, \mathrm{d}\mathbf{s} - \int_\Omega \mathrm{j} \overline{\mathbf{\tau}} : \nabla \mathbf{X}_\mathbf{u}^* \mathrm{d}\mathbf{x} + \int_{\partial\Omega} \mathrm{j} \overline{\mathbf{\tau}} \mathbf{n} \cdot \mathbf{X}_\mathbf{u}^* \mathrm{d}\mathbf{s} = 0 \]

where the mean viscous stress tensor \(\overline{\tau}\) is

\[ \mathbf{\overline{\tau}} = \overline{\mu}_\textrm{eff}[(\nabla + \nabla ^T)\overline{\mathbf{u}}] \]

and where \(\overline{\mu}_\textrm{eff}\) is the mean effective dynamic viscosity (see Viscosity models and Turbulence models for details).

Linearized compressible Navier–Stokes equations

The mass equation is

\[ \frac{\partial \rho'}{\partial t} + \nabla \cdot (\overline{\rho} \mathbf{u}' + \rho' \overline{\mathbf{u}}) = 0\ . \]

The momentum equations are

\[ \overline{\rho} \frac{\partial \mathbf{u}'}{\partial t} + \overline{\rho} [(\overline{\mathbf{u}}\cdot \nabla) \mathbf{u}' + (\mathbf{u}' \cdot \nabla) \overline{\mathbf{u}}] + \rho' (\overline{\mathbf{u}}\cdot \nabla)\overline{\mathbf{u}} + \nabla p' - \nabla \cdot \tau ' = 0\ , \]

where the fluctuating viscous stress tensor \(\tau'\) is

\[ \tau' = \mu_\textrm{eff}'[(\nabla + \nabla ^T)\overline{\mathbf{u}} - \frac{2}{3} \nabla \cdot \overline{\mathbf{u}} \mathbf{I}]+ \overline{\mu}_\textrm{eff}[(\nabla + \nabla ^T) \mathbf{u}' - \frac{2}{3}( \nabla \cdot \mathbf{u}' )\mathbf{I}] \]

and where \(\mu_\textrm{eff}'\) is the fluctuating effective dynamic viscosity (see Viscosity models and Turbulence models for details).

Weak form of the linearized Navier–Stokes equations

The weak form of the linearized Navier–Stokes equations with normal mode ansatz, as implemented in FELiCS, is

\[ \int_\Omega \omega \hat{\rho}X_p^* \mathrm{d}\mathbf{x} = \int_\Omega \mathrm{j} \widehat{\rho\mathbf{u}} \cdot \nabla X_p^* \, \mathrm{d}\mathbf{x} - \int_{\partial\Omega} \mathrm{j} \widehat{\rho\mathbf{u}} \cdot \mathbf{n} X_p^* \, \mathrm{d}\mathbf{s} \]

and

\[ \int_\Omega \omega \overline{\rho} \hat{\mathbf{u}} \cdot \mathbf{X}_\mathbf{u}^* \mathrm{d}\mathbf{x} = - \int_\Omega \mathrm{j} \left(\overline{\rho}\overline{\mathbf{u}} \cdot \nabla \right) \hat{\mathbf{u}} \cdot \mathbf{X}_\mathbf{u}^* \, \mathrm{d}\mathbf{x} -\int_\Omega \mathrm{j} \left(\overline{\rho}\hat{\mathbf{u}} \cdot \nabla \right) \overline{\mathbf{u}} \cdot \mathbf{X}_\mathbf{u}^* \, \mathrm{d}\mathbf{x} - \int_\Omega \mathrm{j} \left(\hat{\rho}\overline{\mathbf{u}} \cdot \nabla \right) \overline{\mathbf{u}} \cdot \mathbf{X}_\mathbf{u}^* \, \mathrm{d}\mathbf{x} + \int_\Omega \mathrm{j} \hat{p} \nabla \cdot \mathbf{X}_\mathbf{u}^* \mathrm{d}\mathbf{x} - \int_{\partial\Omega} \mathrm{j} \hat{p} \mathbf{X}_\mathbf{u}^* \cdot \mathbf{n} \, \mathrm{d}\mathbf{s} - \int_\Omega \mathrm{j} \hat{\mathbf{\tau}} : \nabla \mathbf{X}_\mathbf{u}^* \mathrm{d}\mathbf{x} + \int_{\partial\Omega} \mathrm{j} \hat{\mathbf{\tau}} \mathbf{n} \cdot \mathbf{X}_\mathbf{u}^* \mathrm{d}\mathbf{s} \]

with

\[ \widehat{\rho\mathbf{u}} = \overline{\rho}\mathbf{\hat{u}} + \hat{\rho}\mathbf{\overline{u}}\ . \]

Note

In the linearized equations, the convective term is not integrated by parts.