Moukalled F., Mangani L., Darwish M. The finite volume method in computational fluid dynamics. An advanced introduction with OpenFOAM and Matlab (811443), страница 12
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As thederivative of the position of a fluid parcel x0 with respect to time represents itsvelocity, the two specifications are related byvðt; xðx0 ; tÞÞ ¼@xðt; x0 Þ@tð3:1ÞBased on the above description, changes in the properties of a moving fluid canbe measured either on a fixed point in space while fluid particles are crossing it(Eulerian), or by following a fluid parcel along its path (Lagrangian).(a)z(b)zMV(t+ t)MV(t+ t)CV(t)MV(t)MV(t)yyxxFig. 3.1 a Lagrangian and b Eulerian specification of the flow field463 Mathematical Description of Physical Phenomena3.3.1 Substantial Versus Local DerivativeThe derivative (rate of change) of a field variable /ðt; xðtÞÞ, which may be a scalaror a vector quantity representing density, velocity, temperature, etc., with respect toa fixed position in space is called the Eulerian derivative ð@/=@tÞ while thederivative following a moving fluid parcel is called the Lagrangian, substantial, ormaterial derivative and is denoted by ðD/=DtÞ.
The substantial derivative of variable /, which can be derived through application of the chain rule to account forchanges induced by all independent variables along the path, is given byD/ @/ dt @/ dx@/ dy@/ dz¼þþþDt@t dt @x |{z}dt@y |{z}dt@z |{z}dtuvw@/@/@/@/þuþvþw@t@x@y@z@/¼þ v r/|fflfflffl{zfflfflffl}@t|{z}¼local rateof changeð3:2Þconvective rateof changewhere v is the velocity vector and r is the “del” or “gradient” operator definedearlier [6–8].Equation (3.2) shows that the total rate of change of the function / as a fluidparcel moves through a flow field described by its Eulerian specification v fromposition x at time t to position x þ vdt at time t þ dt (Fig.
3.2) is equal to the sum ofthe local and convective rates of change of /.Fig. 3.2 Total rate of changeof the field variable ϕ betweentime t and t + δtz( t + t,x + x )( t,x )v tyx3.3 Eulerian and Lagrangian Description of Conservation Laws47An important example of a material derivative is Dv=Dt, the rate of change ofvelocity following the flow, which is the acceleration vector given byDv @v¼þ ðv rÞvDt@tð3:3ÞIn this book, the conservation laws are described following an Eulerian formulation where the focus is on the flow within a specified region in space, called controlvolume. This choice is based on the fact that the Eulerian approach follows a field(system) rather than a particle approach, it abandons the tedious and often unnecessary task of tracking individual particles, and focuses attention on what happens ata fixed point (or volume) as different particles go by.
Moreover, a critical shortcoming of the Lagrangian approach is its inability to control the domain of interestsince fluid parcels travel to where the flow takes them, which may not be the regionof interest. This limits the usefulness of the approach as in most fluid flow applications fluid properties in a fixed region are required, e.g., the shear stress on thesurface of a moving train, and not the properties of moving material volumes.Nevertheless it should be mentioned that the Eulerian approach introduces into theconservation equations the local effect of transport by the fluid flow through theadvective rate of change term, v r/, which represents the product of an unknownvelocity field and the gradient of an unknown variable field. This nonlinearity leadsto the most interesting and most challenging phenomena of fluid flows.3.3.2 Reynolds Transport TheoremThe conservation laws mentioned above apply to moving material volumes of fluids(Fig.
3.1), and not to fixed points or control volumes. In order to express these lawsfollowing an Eulerian approach, there is a need to know the Eulerian equivalent ofan integral taken over a moving material volume of fluid. This is provided throughthe Reynolds transport theorem [9].The conversion formula differs slightly according to whether the control volumeis fixed, moving, or deformable.
To derive the formula, let B be any property of thefluid (mass, momentum, energy, etc.) and let b ¼ dB=dm be the intensive value ofB (amount of B per unit mass) in any small element of the fluid.For the arbitrary moving and deformable control volume shown in Fig. 3.1, theinstantaneous total change of B in the material volume (MV) is equal to theinstantaneous total change of B within the control volume (V) plus the net flow ofB into and out of the control volume through its control surface (S).
Let q denotesthe density of the fluid, n the outward normal to the control volume surface, vðt; xÞthe velocity of the fluid, vs ðt; xÞ the velocity of the deforming control volumesurface, and vr ðt; xÞ the relative velocity by which the fluid enters/leaves the controlvolume [i.e., vr ¼ vðt; xÞ vs ðt; xÞ], then the Reynolds transport theorem gives483 Mathematical Description of Physical Phenomena10 ZZdBdBC¼ @bqdVA þbqvr n dSdt MV dtV ðtÞð3:4ÞSðtÞFor a fixed control volume, vs ¼ 0 and the geometry is independent of timeimplying that the time derivative term on the right hand side of Eq.
(3.4) can bewritten using Leibniz rule as0d@dtZ1bq dV A ¼VZV@ðbqÞdV@tð3:5ÞTherefore Eq. (3.4) simplifies to ZZdB@ðbqÞdV þ bqv n dS¼dt MV@tVð3:6ÞSApplying the divergence theorem to transform the surface integral into a volumeintegral, Eq. (3.6) becomes Z dB@ðqbÞ þ r ðqvbÞ dV¼dt MV@tð3:7ÞVAn alternative form of Eq. (3.7) can be obtained by expanding the second termin the square bracket and using the substantial derivative to get Z dBDðqbÞ þ qbr v dV¼dt MVDtð3:8ÞVEquation (3.7) or (3.8) can be used to derive the Eulerian form of the conservation laws in fixed regions.3.4 Conservation of Mass (Continuity Equation)The principle of conservation of mass [6, 10] indicates that in the absence of masssources and sinks, a region will conserve its mass on a local level.Considering the material volume of fluid shown in Fig.
3.3 of mass m, density q,and velocity v, conservation of mass in material (Lagrangian) coordinate systemcan be written as3.4 Conservation of Mass (Continuity Equation)Fig. 3.3 Conservation ofmass for a material volume ofa fluid of mass m49dSoutward directedsurface normalvolume V withenclosing surface V dm¼0dt MVð3:9ÞFor B ¼ m the corresponding intensive quantity is b ¼ 1, and based on Eq. (3.8)the equivalent expression of mass conservation in an Eulerian coordinate system isZ Dqþ qr v dV ¼ 0Dtð3:10ÞVFor the integral given in Eq.
(3.10) to be true for any control volume V, theintegrand should be equal to zero, giving the differential form of the mass conservation or continuity equation asDqþ qr v ¼ 0Dtð3:11ÞThe flux form of the continuity equation can be derived using Eq. (3.7) andleading toZ @qþ r ½qv dV ¼ 0@tð3:12ÞVAgain for the integral in Eq. (3.12) to be true for any control volume V, theintegrand should be equal to zero, giving the flux form of the mass conservation orcontinuity equation as@qþ r ½qv ¼ 0@tð3:13ÞIn the absence of any significant absolute pressure or temperature changes, it isacceptable to assume that the flow is incompressible; that is, the pressure changes503 Mathematical Description of Physical Phenomenado not have significant effects on density. This is almost invariably the case inliquids, and is a good approximation in gases at speeds much less than that ofsound.
(Note that sound waves are compressible phenomena.) The most importantconsequence in fluid dynamics is that the mass conservation (continuity) equationcan no longer be used to compute the density.The incompressibility condition indicates that q does not change with the flow,which mathematically can be expressed as Dq=Dt ¼ 0. Using the mass conservation equation given by Eq. (3.11), this is equivalent to saying that the continuityequation for incompressible flow is given byrv¼0ð3:14Þðv nÞdS ¼ 0ð3:15Þor in integral form asZSEquation (3.15) states that for incompressible flows the net flow across anycontrol volume is zero, i.e., “flow out” = “flow in”.Note also that Dq=Dt ¼ 0 does not imply that q is the same everywhere(although this happens to be the case in many hydraulic applications), but that qdoes not change along a streamline.
To be more accurate, the incompressibilityapproximation means that each fluid element keeps its original density as it moves.In practice, density differences are commonly encountered in water due to variationin salt concentration and in air due to temperature differences resulting in importantbuoyancy forces.3.5 Conservation of Linear MomentumThe principle of conservation of linear momentum [6, 10] indicates that in theabsence of any external force acting on a body, the body retains its total momentum, i.e., the product of its mass and velocity vector. Since momentum is a vectorquantity, its components in any direction will also be conserved.For the material volume of a substance, Newton’s Second Law of motion assertsthat the momentum of this specific volume can change only in the presence of a netforce acting on it, which could include both surface forces and body forces.Therefore, by considering the material volume of fluid shown in Fig.
3.4 of mass m,density q, and velocity v, Newton’s law in Lagrangian coordinates can be written asd ðmvÞdtMV0¼@ZV1f dV AMVð3:16Þ3.5 Conservation of Linear Momentum51Fig. 3.4 Conservation oflinear momentum for amaterial volume of a fluid ofmass mfndSwhere f is the external force per unit volume acting on the material volume. The termon the right hand side of Eq. (3.16) is a volume integral over material coordinatesperformed over the volume occupied instantaneously by the moving fluid, thus01ZZ@ f dV A ¼ f dVð3:17ÞVMVVThe equivalent expression of Eq. (3.16) in Eulerian coordinates can be written intwo different ways known as the conservative and non-conservative forms.3.5.1 Non-Conservative FormNoticing that in this case b ¼ v, the non-conservative form is obtained by usingEq.
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