Editing Incompressible flow (section)

== Related flow constraints ==

In fluid dynamics, a flow is considered incompressible if the divergence of the flow velocity is zero. However, related formulations can sometimes be used, depending on the flow system being modelled. Some versions are described below:

# ''Incompressible flow'': <math> {\nabla \cdot \mathbf u = 0} </math>. This can assume either constant density (strict incompressible) or varying density flow. The varying density set accepts solutions involving small perturbations in [[density]], pressure and/or temperature fields, and can allow for pressure [[atmospheric stratification|stratification]] in the domain.
# ''Anelastic flow'': <math> {\nabla \cdot \left(\rho_{o}\mathbf u\right) = 0} </math>. Principally used in the field of [[atmospheric sciences]], the anelastic constraint extends incompressible flow validity to stratified density and/or temperature as well as pressure. This allows the thermodynamic variables to relax to an 'atmospheric' base state seen in the lower atmosphere when used in the field of meteorology, for example. This condition can also be used for various astrophysical systems.<ref>{{cite journal | first= D.R. | last=Durran | title=Improving the Anelastic Approximation | journal=Journal of the Atmospheric Sciences | year=1989 | volume=46 | issue=11 | pages=1453–1461 | url=http://ams.allenpress.com/archive/1520-0469/46/11/pdf/i1520-0469-46-11-1453.pdf | doi= 10.1175/1520-0469(1989)046<1453:ITAA>2.0.CO;2 |bibcode = 1989JAtS...46.1453D | issn= 1520-0469 }} {{dead link|date=June 2010}}</ref>
# ''Low Mach-number flow'', or ''pseudo-incompressibility'': <math>\nabla \cdot \left(\alpha \mathbf u \right) = \beta</math>. The low [[Mach number|Mach-number]] constraint can be derived from the compressible Euler equations using scale analysis of non-dimensional quantities. The restraint, like the previous in this section, allows for the removal of acoustic waves, but also allows for ''large'' perturbations in density and/or temperature. The assumption is that the flow remains within a Mach number limit (normally less than 0.3) for any solution using such a constraint to be valid. Again, in accordance with all incompressible flows the pressure deviation must be small in comparison to the pressure base state.<ref>{{cite journal | first1=A.S. | last1=Almgren | first2=J.B. | last2=Bell | first3=C.A. | last3=Rendleman | first4=M. | last4=Zingale | title=Low Mach Number Modeling of Type Ia Supernovae. I. Hydrodynamics | journal=Astrophysical Journal | year=2006 | volume=637 | pages=922–936 | url=http://seesar.lbl.gov/ccse/Publications/car/LowMachSNIa.pdf | doi=10.1086/498426 | bibcode=2006ApJ...637..922A | arxiv=astro-ph/0509892 | issue=2 | access-date=2008-12-04 | archive-date=2008-10-31 | archive-url=https://web.archive.org/web/20081031174316/http://seesar.lbl.gov/CCSE/Publications/car/LowMachSNIa.pdf | url-status=dead }}</ref>

These methods make differing assumptions about the flow, but all take into account the general form of the constraint <math>\nabla \cdot \left(\alpha \mathbf u \right) = \beta</math> for general flow dependent functions <math>\alpha</math> and <math>\beta</math>.