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Title Incompressible flow with variations in density
Publication Date
Date Available
Degree MSc
Discipline/Department Mathematical Sciences
Degree Level masters
University/Publisher Stellenbosch University

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…

Subjects/Keywords Density; Fluids  – Migration; Navier-Stokes equations; Fluid mechanics; Boussinesq approximation; UCTD
Contributors Diedericks, G. P. J.; Maritz, M. F.; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Division Applied Mathematics.
Language en
Rights Stellenbosch University
Country of Publication za
Format viii, 109 pages : illustrations (some colour)
Record ID handle:10019.1/104909
Repository stellenbosch
Date Indexed 2020-07-20

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Stellenbosch University https://scholar.sun.ac.za List of Figures 2.1 Different types of fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Lagrangian description of the flow . . . . . . . . . . . . . . . . . . . . . . . 11…

…94 5.10 Flow with non-flat bottom . . . . . . . . . . . . . . . . . . . . . . . . . . 95 vii Stellenbosch University https://scholar.sun.ac.za viii LIST OF FIGURES 5.11 The free-surface profile at time t = 1 s to t = 150 s…

…101 5.18 Salinity profile at times t = 10 s to t = 3000 s . . . . . . . . . . . . . . . 102 5.19 Density profile at times t = 10 s to t = 3000 s . . . . . . . . . . . . . . . 102 Stellenbosch University https://scholar.sun.ac.za Nomenclature Symbol…

…Temperature S Salinity Cl Chlorinity p Pressure m Mass ρ Density V Volume Fnormal Force e Internal energy h The specific enthalpy 1 Stellenbosch University https://scholar.sun.ac.za η The specific entropy cp The specific heat at…

…Numerical flux Uin Approximate value of the average of U at time tn 2 2 Stellenbosch University https://scholar.sun.ac.za Qni Approximate value of the average of Q at time tn τ Parameter X Parametric representation of the characteristic curves 3…

Stellenbosch University https://scholar.sun.ac.za Chapter 1 Introduction 1.1 Introduction and problem statement For different studies in fluid mechanics and various applications in Computational Fluid Dynamics, it is a common assumption to take the fluid…

…existing numerical method will be reviewed in order to solve the set of governing equations 4 Stellenbosch University https://scholar.sun.ac.za corresponding to the flow which satisfies the conditions of incompressibility with varying density. 1.2…

…Taking the first objective, many authors have discussed incompressible fluids and flow in literature. As stated before, incompressibility is a well-used assumption for 5 Stellenbosch University https://scholar.sun.ac.za studying different types of flow…