Oseen's equations

The Oseen equations (after Carl Wilhelm Oseen ) are a mathematical model of the flow of incompressible liquids and gases in stationary equilibrium . In general, such fluid flows are described by the unsteady incompressible Navier-Stokes equations to which the Oseen equations are related.

In numerical mathematics , the Oseen equations are mainly used for the analysis and further development of position discretizations of the Navier-Stokes equations without having to deal with time integration and iterative resolution of the non-linearity . Especially in the field of numerical linear algebra in numerical fluid mechanics , i. H. When solving the linear system of equations , the Oseen equations are a popular benchmark .

formulation

Like the incompressible Navier-Stokes equations, the Oseen equations are a system of partial differential equations in four unknowns (velocity in three dimensions and pressure ) expressed in four equations.

The momentum equation (strictly speaking, three equations in three spatial dimensions)

${\displaystyle (\mathbf {b} \cdot \nabla )\mathbf {v} +\nabla p-\nu \Delta \mathbf {v} =\mathbf {f} }$

describes

• the flow velocity ${\displaystyle \mathbf {v} }$
• under the influence of convection with flow-independent speed , ${\displaystyle \mathbf {b} }$
• Diffusion due to the kinematic viscosity ${\displaystyle \nu }$
• under the action of an external volume force . ${\displaystyle \mathbf {f} }$

The above The momentum equation is coupled via the pressure to the continuity equation as the fourth Oseen equation, which guarantees divergence - and thus freedom from source : ${\displaystyle p}$

${\displaystyle {\begin{alignedat}{2}&\operatorname {div} \mathbf {v} +{\frac {\partial \rho }{\partial t}}&&=0\;\;{\text{mit}}\,{\frac {\partial \rho }{\partial t}}=0\\\Rightarrow \,&{\text{div}}\,\mathbf {v} &&=0\\\Leftrightarrow \,&\nabla \cdot \mathbf {v} &&=0\\\Leftrightarrow \,&{\frac {\partial v_{x}}{\partial x}}+{\frac {\partial v_{y}}{\partial y}}+{\frac {\partial v_{z}}{\partial z}}&&=0\end{alignedat}}}$

The two main differences between the Oseen and the Navier-Stokes equations are

• the stationarity of the Oseen equations, expressed as the lack of time derivative
• their flow- independent convection speed , in contrast to the Navier-Stokes equations. ${\displaystyle \mathbf {b} \neq \mathbf {v} }$ ${\displaystyle \mathbf {b} =\mathbf {v} }$

In addition, the Oseen equations can also be understood as an extension of the stationary Stokes equation to include the convection term . ${\displaystyle (\mathbf {b} \cdot \nabla )\mathbf {v} }$

Iterative approximation

The Oseen equations arise from the linearization of the stationary Navier-Stokes equations using a Picard iteration .

The above nonlinear momentum equation can be numerically approximated using an iterative process : one starts with a suitable velocity field and then solves it successively ${\displaystyle \mathbf {v} _{0}}$

${\displaystyle (\mathbf {v} _{n-1}\cdot \nabla )\mathbf {v} _{n}+\nabla p_{n}-\nu \Delta \mathbf {v} _{n}=\mathbf {f} _{n}}$

for taking into account the incompressibility until convergence occurs, d. H. the change between and is small. ${\displaystyle n=1,2,\ldots }$ ${\displaystyle \mathbf {v} _{n}}$ ${\displaystyle \mathbf {v} _{n+1}}$

The Oseen equations contain two fundamental properties that also occur in the discretization of the Navier-Stokes equations, namely the saddle point structure due to velocity-pressure coupling as well as a possibly dominant convection (compared to diffusion).

literature

• Carl Wilhelm Oseen : About the Stokes formula, and about a related task in hydrodynamics. In: Arkiv för matematik, astronomi och fysik. 6, 29, 1904, ISSN  0365-4133 , pp. 1-20.
• David Kay, Daniel Loghin, Andrew Wathen: A preconditioner for the steady-state Navier-Stokes equations. In: SIAM Journal on Scientific Computing. 24, 2002, ISSN  0196-5204 , pp. 237-256.
• Dieter Braess: Finite Elements. Theory, quick solvers, and applications in elasticity theory. 3rd corrected and supplemented edition. Springer-Verlag, Berlin et al. 2003, ISBN 3-540-00122-0 , section III.4.