Moukalled F., Mangani L., Darwish M. The finite volume method in computational fluid dynamics. An advanced introduction with OpenFOAM and Matlab (811443), страница 16
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If the momentum fluxes are in the samedirection then the Reynolds number reveals the boundary layer characteristics of theflow. If the fluxes are defined such that the diffusion is in the cross stream direction,then as shown in Fig. 3.10 Re conveys the flow regime (i.e. laminar, transitional, orturbulent).An example showing the flow field for different values of Reynolds number isdepicted in Fig. 3.11. It represents a driven flow in a square cavity of sideL generated by the velocity U imparted to its top wall. The streamlines shown inFig.
3.11 indicates that the strength of the flow increases as Re ¼ qUL=l increases.Transition regionRe > 5x105Viscous sublayerFlowRe <5x105Fig. 3.10 Schematic of the flow over a flat plate showing the laminar, transitional, and turbulentflow regimes based on the value of Re3.9 Dimensionless NumbersRe = 1073Re = 100Re = 1000Re = 4000Fig. 3.11 Streamlines at increasing values of Reynolds number for driven flow in a square cavity3.9.2 Grashof NumberAs derived above the Grashof number [12–14] is given byGr ¼gbDTL3v2ð3:120ÞThe Grashof number represents the ratio of buoyant to viscous forces.
It plays innatural convection the same role played by the Reynolds number in forced convection. An example showing the effect of Grashof number is depicted in Fig. 3.12.The physical situation represents natural convection heat transfer in the annulusbetween a hot circular cylinder and its cold square enclosure. Isotherms displayed inthe figure are seen to become more distorted at higher values of Gr due to highernatural convection effects caused by a stronger flow field.Gr = 1.43 103Gr = 1.43 10 4Gr = 1.43 105Gr = 1.43 106Fig.
3.12 Isotherms at increasing values of Grashof number for natural convection in the annulusbetween eccentric horizontal hot circular and cold square cylinders3.9.3 Prandtl NumberThe Prandtl number [12–14] is defined as the ratio of momentum diffusivity(kinematic viscosity v) to thermal diffusivity ðaÞ, i.e.,743 Mathematical Description of Physical PhenomenaPr ¼lcpl=qv¼¼k=qcp akð3:121ÞThe Prandtl number represents the ratio of hydrodynamic boundary layer tothermal boundary layer. As displayed in Fig. 3.13, the thermal boundary layer islarger than the hydrodynamic boundary layer for Pr \ 1 (Fig. 3.13a) and theopposite is true for Pr [ 1 (Fig.
3.13b). Both layers coincide for Pr ¼ 1.(a)yVelocityHydrodynamic boundarylayerTemperatureFree Stream VelocityFree StreamTemperatureThermal boundarylayerT(x,y)u(x,y)Wall temperaturePr <1(b)VelocityTemperatureyFree Stream VelocityFree StreamTemperatureHydrodynamic boundarylayerWalltemperatureu(x,y)T(x,y)Thermal boundarylayerPr >1Fig. 3.13 The thermal and hydrodynamic boundary layer thicknesses for a Pr\1 and b Pr [ 1For the driven flow in a square cavity problem presented above, isotherms overthe domain are depicted in Fig. 3.14 for different values of Pr while holding Reconstant at 100. The increase in convection over conduction as Pr increases can beeasily inferred from the plots.Pr = 0.1Pr = 1Pr = 10Pr = 100Fig. 3.14 Isotherms at increasing values of Prandtl number for driven flow in a square cavityðRe ¼ 100Þ3.9 Dimensionless Numbers753.9.4 Péclet NumberThe Péclet number [5] is defined as the ratio of the advective transport rate of aphysical quantity to its diffusive transport rate.
For the case of heat transfer, thePéclet number is given byPe ¼qULcp UL¼ Re Pr¼akð3:122ÞIn this situation the Pe is equivalent to the product of the Reynolds number andthe Prandtl number. An example of the effects of Pe is shown in Fig. 3.15, whereisotherms over a flat hot plate are displayed at different values of Péclet number.Heat transfer is seen to be dominated by conduction at low values of Pe withconvection gaining increasing importance as Pe increases to become clearly thedominant heat transfer mode at Pe ¼ 1000.Pe = 1Pe = 10Pe = 100Pe = 1000Fig.
3.15 Isotherms at increasing values of Péclet number for fluid flow over a flat platemaintained at a hot uniform temperatureFor mass transport, the Péclet number is given byPe ¼UL¼ Re ScDð3:123Þwhere D is the mass diffusivity and Sc the Schmidt number. In this case Pe isequivalent to the product of the Reynolds number and the Schmidt number.A large Péclet number indicates low dependence of the flow on downstreamlocations and high dependence on upstream locations. Therefore simpler computational models can be adopted for simulating situations with high Péclet numbers.3.9.5 Schmidt NumberThe Schmidt number [14] is defined asSc ¼vDð3:124ÞThe Schmidt number in mass transfer is the counterpart of the Prandtl number inheat transfer.
It represents the ratio of the momentum diffusivity ðvÞ to mass763 Mathematical Description of Physical Phenomenadiffusivity ðDÞ. Physically, the Sc relates the thicknesses of the hydrodynamic andmass transfer boundary layers. An example showing the effect of Schmidt numberis shown in Fig. 3.16.Sc = 0.07Sc = 0.7Sc = 7Fig. 3.16 Iso-concentrations at increasing values of Schmidt number (other parameters held fixed)for natural convection mass transfer in the annulus between concentric horizontal cylinders ofrhombic cross sections with larger solute concentration on the inner wallThe figure above represents natural convection mass transfer in the annulusbetween two horizontal pipes of rhombic cross sections. The solute concentration ishigher along the inner wall of the enclosure.
The concentration non-uniformitycauses variations in density establishing a flow field. The strength of the flowincreases with increasing Sc values as manifested by the higher distortion ofiso-concentration lines that indicates an increase in convection mass transfer overdiffusion mass transfer, which dominates at low Sc values.3.9 Dimensionless Numbers773.9.6 Nusselt NumberThe Nusselt number [12–14] expressed asNu ¼hLkð3:125Þis the dimensionless form of the convection heat transfer coefficient h and providesa measure of the convection heat transfer at a solid surface.
The Nusselt numberdoes not arise as a dimensionless group when writing the conservation equations innon-dimensional forms; rather, it is widely used to report convection heat transferdata.3.9.7 Mach NumberThe Mach number ðM Þ [20] is defined as the ratio of speed of an object movingthrough a fluid and the local speed of sound [20]. Mathematically it is written asM¼j vjað3:126Þwhere jvj is the local magnitude of the fluid velocity relative to the medium inwhich it is flowing and a is the speed of sound.
The general equation for the speedof sound is given bysffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @pa¼ c@q Tð3:127ÞFor an ideal gas, it reduces toa¼pffiffiffiffiffiffiffiffifficRTð3:128Þwhere c is the ratio of specific heat at constant pressure to specific heat at constantvolume cp =cv and R is the gas constant.Flows for which the Mach number is less than 0.2 can be treated as incompressible. For M\1 the flow is called subsonic, for M ¼ 1 sonic, for 1\M\5supersonic, and for M [ 5 hypersonic. Moreover, a flow accelerating from subsonic to supersonic is called a transonic flow. The value of Mach number (less than1 or greater than 1) at the boundaries of a domain dictates the number of requiredboundary conditions there.Examples of subsonic, transonic, and supersonic flow fields are presented inFig.
3.17 via Mach contours. The physical situation represents a fluid flowing over783 Mathematical Description of Physical PhenomenaFig. 3.17 Mach contours forthe flow over a circular arcbump at a subsonic,b transonic, and c supersonicspeedsa circular arc bump with a maximum curvature of 10 % the channel height in thesubsonic and transonic cases and of 4 % in the supersonic case.
The change in theflow type from elliptic to hyperbolic (with discontinuities in the form of shockwaves) as the Mach number increases from subsonic ðM\1Þ to supersonicðM [ 1Þ is apparent.3.9.8 Eckert NumberThe Eckert number ðEcÞ [21] is a dimensionless number relating the kinetic energyof the flow to its enthalpy and is computed asEc ¼vvcp DTð3:129Þwhere DT is a characteristic temperature difference.
This dimensionless numberappears as a factor multiplying the viscous dissipation term U, when nondimensionalizing the compressible energy equation. A large value of Ec indicateshigh viscous dissipation occurring at high speed of the flow (high kinetic energy). Forsmall Eckert number ðEc 1Þ several terms in the energy equation become negligible (e.g., viscous dissipation, body forces, etc.). This reduces the energy equation toits incompressible form (i.e., a balance between conduction and convection).3.9 Dimensionless Numbers793.9.9 Froude NumberThe Froude number ðFr Þ [19] is a dimensionless number defined as the ratio of apffiffiffiffiffifficharacteristic velocity ðU Þ to a gravitational wave velocity ð gLÞ asUFr ¼ pffiffiffiffiffiffigLð3:130ÞIt is a measure of the resistance of partially immersed objects moving throughfluids, with higher Fr values indicating higher fluid resistance.For the free-surface flow shown in Fig.
3.18, the nature of the flow is dictated bythe value of Froude number. For Fr [ 1 the flow is supercritical and for Fr\1 it issubcritical. The flow at the interface between the two regions, known as the“hydraulic jump”, is just critical and is characterized by a Froude number value of 1.Fig. 3.18 A free surface flowshowing the supercritical,critical, and subcriticalregions3.9.10 Weber NumberThe dimensionless Weber number ðWeÞ [17, 18] is defined asWe ¼qU 2 Lrð3:131Þwhere U (m/s) and L (m) are the characteristic velocity and length, respectively, andr the surface tension (N/m).
The Weber number, which represents the ratio ofinertia to surface tension forces, is helpful in analyzing multiphase flows involvinginterfaces between two different fluids, with curved surfaces such as droplets andbubbles.803 Mathematical Description of Physical Phenomena3.10 ClosureThis chapter has shown that many physical phenomena can be modeled throughconservation equations. These equations are derived from first principles by writingbalances over a finite volume.
It was also shown that the conservation equationsgoverning the transport of mass, momentum, energy, and other specific quantitieshave a common form embodied in the general scalar transport equation. Thisequation has transient, convection, diffusion and source terms. Each term brings acharacteristic contribution to the equation that needs to be reproduced by the discretization procedure.3.11 ExercisesExercise 1By comparing the continuity, momentum, and energy equations with the generalscalar transport equation, derive expressions for /, C/ and S/ .Exercise 2Show that for an incompressible flow of constant viscosity the following holds:n hior l rv þ ðrvÞT ¼ lr2 vExercise 3A steady incompressible flow field is defined by the following velocity vector:v ¼ ðx þ yÞi þ ðy þ zÞj þ 2ðx zÞk(a) Verify that it satisfies the continuity equation.(b) Assuming constant viscosity l, calculate the viscous stress tensor s.(c) Denoting the fluid density by q and neglecting body forces, develop anequation for the pressure gradient.Exercise 4The vorticity x of a flow field is defined as the curl of the velocity vector, i.e.,x¼rvUsing the above definition of vorticity, show that for an incompressible fluid, thefollowing relation between the velocity and vorticity vectors holds:r ½ðv rÞv ¼ 0:5r2 ðv vÞ v r2 v x x3.11Exercises81Exercise 5A flow is said to be irrotational if its vorticity (defined in Exercise 4 above) is zero,i.e., x ¼ 0.
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