Boussinesq approximation

The Boussinesq approximation or Boussinesq approximation refers to various approximations in hydrodynamics , all of which go back to Joseph Boussinesq .

1. On the one hand, Boussinesq looked at water waves in shallow water, and the approximations he made led to Boussinesq equations .

2. In the theory of turbulence , the Boussinesq approximation is used in eddy viscosity models .

3. A Boussinesq approximation to the incompressible Navier-Stokes equations is also used to describe flows in liquids (especially convection ) that are caused by density variations due to temperature fluctuations (in the following, as in the notation in the article, Navier-Stokes -Equation vectors highlighted in font). In addition, the not too large temperature fluctuations (in ) are only taken into account in the density and pressure variation with the thermal expansion coefficient . The following applies to the fluctuation of the pressure : ${\displaystyle T_{s}}$ ${\displaystyle T=T_{0}+T_{s}}$ ${\displaystyle \rho =\rho _{0}(1-\beta T_{s})}$ ${\displaystyle \beta }$ ${\displaystyle p_{s}}$ ${\displaystyle p_{0}=\rho _{0}\mathbf {g} \cdot \mathbf {x} }$

${\displaystyle {\frac {\nabla p}{\rho }}=\mathbf {g} +{\frac {\nabla p_{s}}{\rho _{0}}}+\beta T_{s}\mathbf {g} }$

The incompressible Navier-Stokes equation in the gravitational field with gravitational acceleration becomes: ${\displaystyle \mathbf {g} }$

${\displaystyle {\frac {\partial \mathbf {v} }{\partial t}}+(\mathbf {v} \cdot \nabla )\mathbf {v} =-{\frac {\nabla p}{\rho }}+\nu \Delta \mathbf {v} +\mathbf {g} =-{\frac {\nabla p_{s}}{\rho _{0}}}+\nu \Delta \mathbf {v} -\beta T_{s}\mathbf {g} }$

For the description of the convection in the Boussinesq approximation, the equation of the freedom from divergence of the velocity field is added (derived from the continuity equation, neglecting the density fluctuations):

${\displaystyle \nabla \cdot \mathbf {v} =0}$

and the equation for the variation in temperature due to heat flow:

${\displaystyle {\frac {\partial T_{s}}{\partial t}}+\mathbf {v} \cdot \nabla T_{s}=a\nabla ^{2}T_{s}}$

where is the thermal diffusivity (for ) (internal heat sources in the liquid are not assumed here). ${\displaystyle a}$ ${\displaystyle \rho _{0},T_{0}}$

Individual evidence

1. ↑ For example, Wolfgang Polifke, Jan Kopitz, heat transfer, 2nd edition, Pearson 2009, p. 469f