Moukalled F., Mangani L., Darwish M. The finite volume method in computational fluid dynamics. An advanced introduction with OpenFOAM and Matlab (811443), страница 17
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Show that for a steady two dimensional incompressible irrotational flowthe velocity field satisfies the following Laplace equationr2 v ¼ 0Exercise 6Consider a two-dimensional square enclosure of side L. The enclosure is filled withan incompressible fluid of viscosity l and density q. The top side of the enclosure iscovered with an infinite horizontal wall moving with a constant velocity U. Theother sides are fixed in place. Due to the motion of the top wall a flow field isestablished within the enclosure.
Using appropriate dimensionless variables, writethe simplified momentum equation for the flow field in dimensionless formshowing that the Reynolds number ðRe ¼ qUL=lÞ is the only dimensionless groupaffecting the flow.Exercise 7Consider the steady two dimensional mixed convection heat transfer in a verticalrectangular channel of width W. A cold fluid of density qin and temperature Tinenters the channel with a velocity Vin . As it flows vertically upward, the fluid isheated by the duct walls, which are maintained at the uniform hot temperature Tw .Taking buoyancy forces into consideration through the Boussinesq approximationand using the following dimensionless variablesx ¼x yu vT Tin p þ qin gy; y ¼ ; u ¼;v ¼;h ¼;p ¼WWVinVinTw Tinqin Vin2write the conservation equations of mass, momentum, and energy in dimensionlessforms. What are the dimensionless groups governing the flow and heat transfer inthe channel? What does each of them represent?Exercise 8Estimates the Reynolds number of the following flows:(a) Water flowing at a speed of 15 km/hr over a whale 10 m long.(b) Air flowing at a speed of 800 km/hr over the wing of an F16 airplane of meanchord length 3.450336 m.(c) Glycerine of dynamic viscosity 0.96 kg/ms and density 1258 kg/m3 flowing ata speed of 2.8 m/s in a pipe inclined at 25° to the horizontal and of diameter250 mm.Exercise 9 Starting from the incompressible version of the Navier-Stokes equations derive simplified equations based on the following assumptions:823 Mathematical Description of Physical Phenomena(a) Viscous effects are much more significant than any effects of fluid acceleration, i.e.,@ðvÞ þ r ½vv r ½lrv@twhich corresponds to Re ¼ qUL=l 1 (Stokes Equations).(b) Inertial effects dominate and viscous effects are considered to be negligiblethroughout the flow domain, i.e.,@ðvÞ þ r ½vv r ½lrv@twhich corresponds to Re ¼ qUL=l 1 (Euler equations).(c) Derive the Bernoulli equation from momentum conservation with the following hypothesis: one dimensional steady state conditions of a frictionlessfluid l ¼ 0.References1.
Navier CLMH (1823) Mem Acad R Sci Paris 6:389–4162. Stokes GG (1845) Trans Camb Phil Soc 8:287–3053. Hauke G (2008) An Introduction to Fluid Mechanics and transport phenomena. Springer, NewYork4. Ferziger JH, Peric M (2002) Computational methods for fluid dynamics. Springer, New York5. Patankar SV (1980) Numerical heat transfer and fluid flow. Hemisphere PublishingCorporation, USA6. Bird RB, Stewart WE, Lightfoot EN (2006) Transport phenomena, 2nd edn. Wiley, USA7.
Falkovich G (2011) Fluid mechanics (A Short Course for Physicists). Cambridge UniversityPress, Cambridge8. Emanuel G (2000) Analytical fluid dynamics. CRC Press, Boca Raton9. Reynolds O (1903) Papers on mechanical and physical subjects-the sub-mechanics of theuniverse. Collected Work, vol III, Cambridge University Press, Cambridge10. Fay JA (1994) Introduction to fluid mechanics. MIT Press, Cambridge Massachussets11.
Boussinesq J (1897) Theorie de l’ecoulement tourbillonnant et tumulueux des liquides and leslits rectilignes a grande section. Gauthier-Villars et Fils, Des Comptes Rendus des Seances deL’academie des Sciences, Paris12. Cengel YA (2003) Heat and mass transfer: a practical approach, 3rd edn. McGraw-Hill,Boston13. Incropera FP, DeWitt DP (2007) Fundamentals of heat and mass transfer, 6th edn. Wiley,Hoboken14. Bejan A (1984) Convection heat transfer.
Wiley, USA15. Moukalled F, Darwish M (2013) Double diffusive natural convection in a porous rhombicannulus. Numer Heat Transfer Part A: Appl 65(5):378–39916. Moukalled F, Darwish M (2010) Natural convection heat transfer in a porous rhombicannulus. Numer Heat Transfer, Part A: Appl 58(2):101–12417. Frohn A, Roth N (2000) Dynamics of droplets. Springer, New YorkReferences8318. Day P, Manz A, Zhang Y (2012) Microdroplet technology: principles and emergingapplications in biology and chemistry. Springer, New York19. Chanson H (2004) Hydraulics of open channel flow: an introduction, 2nd edn.
Butterworth–Heinemann, Oxford20. Oosthuizen PH, Carscallen WE (1997) Compressible fluid flow. McGraw-Hill, Singapore21. Kreith F, Bohn MS (1993) Principles of heat transfer, 5th edn. West Publishing Company,USAChapter 4The Discretization ProcessAbstract This chapter introduces the different steps of the discretization process,which include: (i) modeling of the geometric domain and the physical phenomenaof interest; (ii) discretization of the modeled geometric domain into a grid or meshthat forms the computational domain (this process, also known as meshing ordomain discretization, results in a set of non-overlapping elements, denoted also bycells, that cover the computational domain); (iii) numerical or equation discretization that transforms the set of conservation partial differential equationsgoverning the physical processes into an equivalent system of algebraic equationsdefined over each of the elements of the computational domain; and (iv) thesolution of the resulting set of equations using an iterative solver to yield anintermediate or final solution field.
Throughout the chapter, computer implementation issues are introduced.4.1The Discretization ProcessThe numerical solution of a partial differential equation consists of finding thevalues of the dependent variable ϕ at specified points from which its distributionover the domain of interest can be constructed. These points are called grid elements, or grid nodes and result from the discretization of original geometry into aset of non overlapping discrete elements, a process known as meshing. Theresulting nodes or variables are generally positioned at cell centroids or at verticesdepending on the adopted discretization procedure.
In all methods the focus is onreplacing the continuous exact solution of the partial differential equation withdiscrete values. The distribution of ϕ is hence discretized, and it is appropriate torefer to this process of converting the governing equation into a set of algebraicequations for the discrete values of ϕ as the discretization process and the specificmethods employed to bring about this conversion as the discretization methods. Thediscrete values of ϕ are typically computed by solving a set of algebraic equationsrelating the values at neighboring grid elements to each other; these discretized or© Springer International Publishing Switzerland 2016F.
Moukalled et al., The Finite Volume Method in Computational Fluid Dynamics,Fluid Mechanics and Its Applications 113, DOI 10.1007/978-3-319-16874-6_485864 The Discretization ProcessPhysicalDomainPhysicalPhenomenaPhysical ModelingDomain ModelingSet of Governing EquationsDefined on a ComputationalDomainDomain DiscretizationStructured Grids(Cartesian, Non-Orthogonal)Block Structured gridsUnstructured GridsChimera GridsEquation DiscretizationSystem ofAlgebraicEquationsFinite DifferenceFinite VolumeFinite ElementBoundary ElementSolution MethodNumericalSolutionsCombinations ofMultigrid MethodsIterative SolversCoupled-UncoupledFig.
4.1 The discretization processalgebraic equations are derived from the conservation equation governing ϕ. Oncethe values of ϕ are computed, the data is processed to extract any needed information. The various stages of the discretization process are illustrated in Fig. 4.1.Figure 4.2 shows the process applied to the study of heat transfer from amicroprocessor connected to a heat sink with a copper base that acts as a heatspreader.
The example of Fig. 4.2 will be used to present various concepts related tothe discretization process. Since the intention is to introduce a numerical techniquefor solving the physical processes of interest and since the method has to beimplemented in a computer program, the discretization process will be explainedalong that spirit.
For example, reference will be made to how to store values in acode as related to interior elements, boundary elements, variables, etc. Throughoutthe book a Matlab® based program denoted by “uFVM” will be used as a development vehicle to present various details relating to the implementation of thesemethods. Furthermore OpenFOAM®, a very popular finite volume-based opensource code will also be presented both from a user and a developer perspective,again with the aim of moving from the numerics to the implementation details.4.1 The Discretization Process87microprocessorheat sinkheat spreader baseDomain Modelingheat sinkPhysical ModelingTsinkheat spreader baseinsulated(k T ) = qTmicroprocessormicroprocessorDomain DiscretizationPatch#2Equations DiscretizationPatch#1()ttransientterm+( v )=convectiontermaC(diffusiontermC+)+ QsourcetermaFF NB ( C )F= bCPatch#3Solution Method.........=........Fig.
4.2 An illustration of the discretization process4.1.1Step I: Geometric and Physical ModelingModeling of physical phenomena is in a way at the heart of the scientific enterprise.A physical phenomenon cannot generally be considered as understood unless it can884 The Discretization Processbe mathematically formulated, and this formulation tested and validated. For ourpurpose two levels of modeling are performed, one in relation to the geometry ofthe physical domain and a second in relation to the physical phenomena of interest.At both levels details that are neither relevant nor of interest are ignored or simplified.
For example a three dimensional domain could be turned into a twodimensional depiction, or symmetry can be taken into account to decrease the sizeof the study domain. In some cases, physical components may be removed andreplaced with appropriate mathematical representations.In the example of Fig. 4.2 a microprocessor is connected to a heat sink with acopper base that acts as a heat spreader. A first model of this system simplifies bothits physics and its geometry. The heat sink and processor are replaced by boundaryconditions that specify the estimated temperature of the heat sink and the expectedoperating temperature of the processor, respectively. The physical domain ismodeled as a two dimensional computational domain since the temperature variation through the thickness of the heat sink will be minimal.
For a steady statesolution of the heat flow and temperature distribution in the copper base, only heatconduction is considered. The result of the modeling process is a system of linear(or non-linear if k depends on T) partial differential equations, which in this caseinvolves a simplified form of the energy equation given byr ðkrT Þ ¼ q_ð4:1Þwhere k is the conductivity of the heat spreader base, and q_ is the heat source/sinkper unit volume.4.1.2Step II: Domain DiscretizationThe geometric discretization of the physical domain results in a mesh on which theconservation equations are eventually solved. This requires the subdivision of thedomain into discrete non-overlapping cells or elements that completely fill thecomputational domain to yield a grid or mesh system.
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