Incompressible flow with variations in density

Abstract

ENGLISH ABSTRACT : This study involves the investigation of incompressible flow with variable density. The fact that variable density does not necessarily imply that the flow is compressible, may require some clarification. An attempt is made in this thesis to clarify this ambiguity by investigating examples of incompressible flow with density that varies with pressure, temperature and salinity. In order to investigate incompressible flow with variations in density, the conditions of incompressibility that will simplify the continuity equation are determined by using scaling analysis. The Boussinesq approximation as well as the hydrostatic approximation is then applied to simplify the momentum equations of incompressible fluid flow with variations in density. Depth-averaging is also used to re-derive the shallow water equations, also with variable density. A numerical method for solving the one-dimensional shallow water equations (suggested by Benkaldoun and Saiëd) is then reviewed. It is also implemented and applied to solve some typical examples in order to illustrate the behaviour of the flow under the assumptions of incompressible flow with density that varies with temperature and salinity. The main results of this study can be summarized as follows: The scaling analysis serves to explain in a systematic way some conditions of incompressible flow, such as that the speed of sound must be large compared to the flow velocity, and that the diffusion of heat and salt should be negligible. Next, the solution of the one-dimensional shallow water equations, using the stated numerical method, yields qualitatively expected results.

AFRIKAANSE OPSOMMING : Hierdie studie behels ’n ondersoek na onsamedrukbare vloei met veranderlike digtheid. Die feit dat veranderlike digtheid nie noodwendig beteken dat die vloei samedrukbaar is nie, mag ’n verduideliking verg. ’n Poging om hierdie oënskynlike dubbelsinnigheid uit te klaar word in hierdie tesis aangewend deur voorbeelde van onsamedrukbare vloei wat met druk, temperatuur en soutgehalte verander, te ondersoek. Ten einde onsamedrukbare vloei met veranderlike digtheid te ondersoek, is die voorwaardes van onsamedrukbaarheid wat tot vereenvoudiging in die kontinuïteitsvergelyking lei, deur skaal-analise vasgestel. Die Boussinesq benadering sowel as die hidrostatiese benadering word dan toegepas om die momentumvergelykings vir onsamedrukbare vloei met veranderlike digtheid, te vereenvoudig. Diepte-gemiddeldes word ook gebruik om die vlak-water-vergelykings weer te herlei, hier ook met veranderlike digtheid. ’n Numeriese metode om die vlak-water-vergelykings op te los (voorgestel deur Benkaldoun en Saiëd) word hersien. Dit word ook geïmplementeer en aangewend omtipiese voorbeelde op te los waar die gedrag van vloei onder die aannames van onsamedrukbaarheid met digtheid wat verander met temperatuur en soutgehalte, geïllustreer word. Die hoof resultate van die studie kan as volg opgesom word: Die skaalanalise dien goed om die voorwaardes van onsamedrukbare vloei in ’n sistematiese manier te verduidelik, byvoorbeeld dat die spoed van klank groot moet wees in vergelyking met die vloeisnelheid, en dat die diffusie van hitte en sout weglaatbaar moet wees. Verder toon die oplossing van die gemelde numeriese metode kwalitatief verwagte resultate.