Chapter 8. Modeling Basic Fluid FlowThis chapter describes the basic physical models that FLUENT provides for uid ow andthe commands for dening and using them. Models for ows in moving zones (includingsliding and dynamic meshes) are explained in Chapter 9, models for turbulence aredescribed in Chapter 10, and models for heat transfer (including radiation) are presentedin Chapter 11. An overview of modeling species transport and reacting ows is providedin Chapter 12, details about models for species transport and reacting ows are describedin Chapters 1317, and models for pollutant formation are presented in Chapter 18. Anoverview of multiphase modeling is provided in Chapter 20, the discrete phase model isdescribed in Chapter 21, general multiphase models are described in Chapter 22, and themelting and solidication model is described in Chapter 23. For information on modelingporous media, porous jumps, and lumped parameter fans and radiators, see Chapter 6.The information in this chapter is presented in the following sections: Section 8.1: Overview of Physical Models in FLUENT Section 8.2: Continuity and Momentum Equations Section 8.3: Periodic Flows Section 8.4: Swirling and Rotating Flows Section 8.5: Compressible Flows Section 8.6: Inviscid Flowsc Fluent Inc. January 28, 2003 8-1Modeling Basic Fluid Flow8.1 Overview of Physical Models in FLUENTFLUENT provides comprehensive modeling capabilities for a wide range of incompressibleand compressible, laminar and turbulent uid ow problems. Steady-state or transientanalyses can be performed. In FLUENT, a broad range of mathematical models fortransport phenomena (like heat transfer and chemical reactions) is combined with theability to model complex geometries. Examples of FLUENT applications include laminarnon-Newtonian ows in process equipment; conjugate heat transfer in turbomachineryand automotive engine components; pulverized coal combustion in utility boilers; exter-nal aerodynamics; ow through compressors, pumps, and fans; and multiphase ows inbubble columns and uidized beds.To permit modeling of uid ow and related transport phenomena in industrial equip-ment and processes, various useful features are provided. These include porous media,lumped parameter (fan and heat exchanger), streamwise-periodic ow and heat transfer,swirl, and moving reference frame models. The moving reference frame family of modelsincludes the ability to model single or multiple reference frames. A time-accurate slidingmesh method, useful for modeling multiple stages in turbomachinery applications, for ex-ample, is also provided, along with the mixing plane model for computing time-averagedow elds.Another very useful group of models in FLUENT is the set of free surface and multi-phase ow models. These can be used for analysis of gas-liquid, gas-solid, liquid-solid,and gas-liquid-solid ows. For these types of problems, FLUENT provides the volume-of-uid (VOF), mixture, and Eulerian models, as well as the discrete phase model (DPM).The DPM performs Lagrangian trajectory calculations for dispersed phases (particles,droplets, or bubbles), including coupling with the continuous phase. Examples of multi-phase ows include channel ows, sprays, sedimentation, separation, and cavitation.Robust and accurate turbulence models are a vital component of the FLUENT suite ofmodels. The turbulence models provided have a broad range of applicability, and theyinclude the eects of other physical phenomena, such as buoyancy and compressibility.Particular care has been devoted to addressing issues of near-wall accuracy via the useof extended wall functions and zonal models.Various modes of heat transfer can be modeled, including natural, forced, and mixedconvection with or without conjugate heat transfer, porous media, etc. The set of radia-tion models and related submodels for modeling participating media are general and cantake into account the complications of combustion. A particular strength of FLUENTis its ability to model combustion phenomena using a variety of models, including eddydissipation and probability density function models. A host of other models that arevery useful for reacting ow applications are also available, including coal and dropletcombustion, surface reaction, and pollutant formation models.8-2 c Fluent Inc. January 28, 20038.2 Continuity and Momentum Equations8.2 Continuity and Momentum EquationsFor all ows, FLUENT solves conservation equations for mass and momentum. For owsinvolving heat transfer or compressibility, an additional equation for energy conservationis solved. For ows involving species mixing or reactions, a species conservation equationis solved or, if the non-premixed combustion model is used, conservation equations forthe mixture fraction and its variance are solved. Additional transport equations are alsosolved when the ow is turbulent.In this section, the conservation equations for laminar ow (in an inertial (non-accelerating)reference frame) are presented. The equations that are applicable to rotating referenceframes are presented in Chapter 9. The conservation equations relevant to heat transfer,turbulence modeling, and species transport will be discussed in the chapters where thosemodels are described.The Euler equations solved for inviscid ow are presented in Section 8.6.The Mass Conservation EquationThe equation for conservation of mass, or continuity equation, can be written as follows:t + (v) = Sm (8.2-1)Equation 8.2-1 is the general form of the mass conservation equation and is valid forincompressible as well as compressible ows. The source Sm is the mass added to thecontinuous phase from the dispersed second phase (e.g., due to vaporization of liquiddroplets) and any user-dened sources.For 2D axisymmetric geometries, the continuity equation is given byt + x(vx) + r(vr) + vrr = Sm (8.2-2)where x is the axial coordinate, r is the radial coordinate, vx is the axial velocity, and vris the radial velocity.Momentum Conservation EquationsConservation of momentum in an inertial (non-accelerating) reference frame is describedby [11]t(v) + (vv) = p + () + g + F (8.2-3)c Fluent Inc. January 28, 2003 8-3Modeling Basic Fluid Flowwhere p is the static pressure, is the stress tensor (described below), and g and F arethe gravitational body force and external body forces (e.g., that arise from interactionwith the dispersed phase), respectively. F also contains other model-dependent sourceterms such as porous-media and user-dened sources.The stress tensor is given by = _(v +v T) 23 vI_ (8.2-4)where is the molecular viscosity, I is the unit tensor, and the second term on the righthand side is the eect of volume dilation.For 2D axisymmetric geometries, the axial and radial momentum conservation equationsare given byt(vx) + 1rx(rvxvx) + 1rr(rvrvx) = px + 1rx_r_2vxx 23 ( v)__+1rr_r_vxr + vrx__+ Fx(8.2-5)andt(vr) + 1rx(rvxvr) + 1rr(rvrvr) = pr + 1rx_r_vrx + vxr__+ 1rr_r_2vrr 23 ( v)__2vrr2 + 23r ( v) + v2zr + Fr (8.2-6)where v = vxx + vrr + vrr (8.2-7)and vz is the swirl velocity. (See Section 8.4 for information about modeling axisymmetricswirl.)8-4 c Fluent Inc. January 28, 20038.3 Periodic Flows8.3 Periodic FlowsPeriodic ow occurs when the physical geometry of interest and the expected pattern ofthe ow/thermal solution have a periodically repeating nature. Two types of periodic owcan be modeled in FLUENT. In the rst type, no pressure drop occurs across the periodicplanes. (Note to FLUENT 4 users: This type of periodic ow is modeled using a cyclicboundary in FLUENT 4.) In the second type, a pressure drop occurs across translationallyperiodic boundaries, resulting in fully-developed or streamwise-periodic ow. (InFLUENT 4, this type of periodic ow is modeled using a periodic boundary.)This section discusses streamwise-periodic ow. A description of no-pressure-drop pe-riodic ow is provided in Section 6.15, and a description of streamwise-periodic heattransfer is provided in Section 11.4.Information about streamwise-periodic ow is presented in the following sections: Section 8.3.1: Overview and Limitations Section 8.3.2: Theory Section 8.3.3: User Inputs for the Segregated Solver Section 8.3.4: User Inputs for the Coupled Solvers Section 8.3.5: Monitoring the Value of the Pressure Gradient Section 8.3.6: Postprocessing for Streamwise-Periodic Flows8.3.1 Overview and LimitationsOverviewFLUENT provides the ability to calculate streamwise-periodicor fully-developeduid ow. These ows are encountered in a variety of applications, including ows incompact heat exchanger channels and ows across tube banks. In such ow congura-tions, the geometry varies in a repeating manner along the direction of the ow, leadingto a periodic fully-developed ow regime in which the ow pattern repeats in succes-sive cycles. Other examples of streamwise-periodic ows include fully-developed ow inpipes and ducts. These periodic conditions are achieved after a sucient entrance length,which depends on the ow Reynolds number and geometric conguration.Streamwise-periodic ow conditions exist when the ow pattern repeats over some lengthL, with a constant pressure drop across each repeating module along the streamwisedirection. Figure 8.3.1 depicts one example of a periodically repeating ow of this typewhich has been modeled by including a single representative module.c Fluent Inc. January 28, 2003 8-5Modeling Basic Fluid FlowVelocity Vectors Colored By Velocity Magnitude (m/s) 3.57e-03 3.33e-03 3.09e-03 2.86e-03 2.62e-03 2.38e-03 2.14e-03 1.90e-03 1.67e-03 1.43e-03 1.19e-03 9.53e-04 7.15e-04 4.77e-04 2.39e-04 1.01e-06Figure 8.3.1: Example of Periodic Flow in a 2D Heat Exchanger GeometryConstraints on the Use of Streamwise-Periodic FlowThe following constraints apply to modeling streamwise-periodic ow: The ow must be incompressible. The geometry must be translationally periodic. If one of the coupled solvers is used, you can specify only the pressure jump; for thesegregated solver, you can specify either the pressure jump or the mass ow rate. No net mass addition through inlets/exits or extra source terms is allowed. Species can be modeled only if inlets/exits (without net mass addition) are includedin the problem. Reacting ows are not permitted. Discrete phase and multiphase modeling are not allowed.8.3.2 TheoryDenition of the Periodic VelocityThe assumption of periodicity implies that the velocity components repeat themselves inspace as follows:8-6 c Fluent Inc. January 28, 20038.3 Periodic Flowsu(r) = u(r + L) = u(r + 2

L) = v(r) = v(r + L) = v(r + 2

L) = (8.3-1)w(r) = w(r + L) = w(r + 2

L) = where r is the position vector and L is the periodic length vector of the domain considered(see Figure 8.3.2).LLA B CuBuA uC= =vBvA vC= = pBpA pC=pB- -pBpA pC= = Figure 8.3.2: Example of a Periodic GeometryDenition of the Streamwise-Periodic PressureFor viscous ows, the pressure is not periodic in the sense of Equation 8.3-1. Instead,the pressure drop between modules is periodic:p = p(r) p(r + L) = p(r + L) p(r + 2

L) = (8.3-2)If one of the coupled solvers is used, p is specied as a constant value. For the segregatedsolver, the local pressure gradient can be decomposed into two parts: the gradient of aperiodic component, p(r), and the gradient of a linearly-varying component, L|

L|:p(r) =

L|

L|+ p(r) (8.3-3)where p(r) is the periodic pressure and |r| is the linearly-varying component of thepressure. The periodic pressure is the pressure left over after subtracting out the linearly-varying pressure. The linearly-varying component of the pressure results in a force actingon the uid in the momentum equations. Because the value of is not known a priori,it must be iterated on until the mass ow rate that you have dened is achieved in thecomputational model. This correction of occurs in the pressure correction step of theSIMPLE, SIMPLEC, or PISO algorithm where the value of is updated based on thedierence between the desired mass ow rate and the actual one. You have some controlover the number of sub-iterations used to update , as described in Section 8.3.3.c Fluent Inc. January 28, 2003 8-7Modeling Basic Fluid Flow8.3.3 User Inputs for the Segregated SolverIf you are using the segregated solver, in order to calculate a spatially periodic ow eldwith a specied mass ow rate or pressure derivative, you must rst create a grid withtranslationally periodic boundaries that are parallel to each other and equal in size. Youcan specify translational periodicity in the Periodic panel, as described in Section 6.15.(If you need to create periodic boundaries, see Section 5.7.5.)Once the grid has been read into FLUENT, you will complete the following inputs in thePeriodicity Conditions panel (Figure 8.3.3):Dene Periodic Conditions...Figure 8.3.3: The Periodicity Conditions Panel1. Select either the specied mass ow rate (Specify Mass Flow) option or the speciedpressure gradient (Specify Pressure Gradient) option. For most problems, the massow rate across the periodic boundary will be a known quantity; for others, themass ow rate will be unknown, but the pressure gradient ( in Equation 8.3-3)will be a known quantity.2. Specify the mass ow rate and/or the pressure gradient ( in Equation 8.3-3): If you selected the Specify Mass Flow option, enter the desired value for theMass Flow Rate. You can also specify an initial guess for the Pressure Gradient,but this is not required.For axisymmetric problems, the mass ow rate is per 2 radians. ! If you selected the Specify Pressure Gradient option, enter the desired value forPressure Gradient.8-8 c Fluent Inc. January 28, 20038.3 Periodic Flows3. Dene the ow direction by setting the X,Y,Z (or X,Y in 2D) point under FlowDirection. The ow will move in the direction of the vector pointing from theorigin to the specied point. The direction vector must be parallel to the periodictranslation direction or its opposite.4. If you chose in step 1 to specify the mass ow rate, set the parameters used for thecalculation of . These parameters are described in detail below.After completing these inputs, you can solve the periodic velocity eld to convergence.Setting Parameters for the Calculation of If you choose to specify the mass ow rate, FLUENT will need to calculate the appropriatevalue of the pressure gradient . You can control this calculation by specifying theRelaxation Factor and the Number of Iterations, and by supplying an initial guess for .All of these inputs are entered in the Periodicity Conditions panel.The Number of Iterations sets the number of sub-iterations performed on the correctionof in the pressure correction equation. Because the value of is not known a priori,it must be iterated on until the Mass Flow Rate that you have dened is achieved inthe computational model. This correction of occurs in the pressure correction step ofthe SIMPLE, SIMPLEC, or PISO algorithm. A correction to the current value of iscalculated based on the dierence between the desired mass ow rate and the actual one.The sub-iterations referred to here are performed within the pressure correction step toimprove the correction for before the pressure correction equation is solved for theresulting pressure (and velocity) correction values. The default value of 2 sub-iterationsshould suce in most problems, but can be increased to help speed convergence. TheRelaxation Factor is an under-relaxation factor that controls convergence of this iterationprocess.You can also speed up convergence of the periodic calculation by supplying an initial guessfor in the Pressure Gradient eld. Note that the current value of will be displayed inthis eld if you have performed any calculations. To update the Pressure Gradient eldwith the current value at any time, click on the Update button.8.3.4 User Inputs for the Coupled SolversIf you are using one of the coupled solvers, in order to calculate a spatially periodic oweld with a specied pressure jump, you must rst create a grid with translationallyperiodic boundaries that are parallel to each other and equal in size. (If you need tocreate periodic boundaries, see Section 5.7.5.)c Fluent Inc. January 28, 2003 8-9Modeling Basic Fluid FlowThen, follow the steps below:1. In the Periodic panel (Figure 8.3.4), which is opened from the Boundary Conditionspanel, indicate that the periodicity is Translational (the default).Dene Boundary Conditions...Figure 8.3.4: The Periodic Panel2. Also in the Periodic panel, set the Periodic Pressure Jump (p in Equation 8.3-2).After completing these inputs, you can solve the periodic velocity eld to convergence.8.3.5 Monitoring the Value of the Pressure GradientIf you have specied the mass ow rate, you can monitor the value of the pressuregradient during the calculation using the Statistic Monitors panel. Select per/pr-gradas the variable to be monitored. See Section 24.16.2 for details about using this feature.8.3.6 Postprocessing for Streamwise-Periodic FlowsFor streamwise-periodic ows, the velocity eld should be completely periodic. If acoupled solver is used to compute the periodic ow, the pressure eld reported will be theactual pressure p (which is not periodic). If the segregated solver is used, the pressureeld reported will be the periodic pressure eld p(r) of Equation 8.3-3. Figure 8.3.5displays the periodic pressure eld in the geometry of Figure 8.3.1.If you specied a mass ow rate and had FLUENT calculate the pressure gradient, youcan check the pressure gradient in the streamwise direction () by looking at the currentvalue for Pressure Gradient in the Periodicity Conditions panel.8-10 c Fluent Inc. January 28, 20038.4 Swirling and Rotating FlowsContours of Static Pressure (pascal) 1.68e-03 1.29e-03 8.98e-04 5.07e-04 1.16e-04-2.74e-04-6.65e-04-1.06e-03-1.45e-03-1.84e-03-2.23e-03-2.62e-03-3.01e-03-3.40e-03-3.79e-03-4.18e-03Figure 8.3.5: Periodic Pressure Field Predicted for Flow in a 2D Heat Exchanger Geom-etry8.4 Swirling and Rotating FlowsMany important engineering ows involve swirl or rotation and FLUENT is well-equippedto model such ows. Swirling ows are common in combustion, with swirl introduced inburners and combustors in order to increase residence time and stabilize the ow pattern.Rotating ows are also encountered in turbomachinery, mixing tanks, and a variety ofother applications.Information about rotating and swirling ows is provided in the following subsections: Section 8.4.1: Overview of Swirling and Rotating Flows Section 8.4.2: Physics of Swirling and Rotating Flows Section 8.4.3: Turbulence Modeling in Swirling Flows Section 8.4.4: Grid Setup for Swirling and Rotating Flows Section 8.4.5: Modeling Axisymmetric Flows with Swirl or Rotationc Fluent Inc. January 28, 2003 8-11Modeling Basic Fluid FlowWhen you begin the analysis of a rotating or swirling ow, it is essential that you classifyyour problem into one of the following ve categories of ow: axisymmetric ows with swirl or rotation fully three-dimensional swirling or rotating ows ows requiring a rotating reference frame ows requiring multiple rotating reference frames or mixing planes ows requiring sliding meshesModeling and solution procedures for the rst two categories are presented in this section.The remaining three, which all involve moving zones, are discussed in Chapter 9.8.4.1 Overview of Swirling and Rotating FlowsAxisymmetric Flows with Swirl or RotationYour problem may be axisymmetric with respect to geometry and ow conditions butstill include swirl or rotation. In this case, you can model the ow in 2D (i.e., solvethe axisymmetric problem) and include the prediction of the circumferential (or swirl)velocity. It is important to note that while the assumption of axisymmetry impliesthat there are no circumferential gradients in the ow, there may still be non-zero swirlvelocities.Momentum Conservation Equation for Swirl VelocityThe tangential momentum equation for 2D swirling ows may be written ast(w)+1rx(ruw)+1rr(rvw) = 1rx_rwx_+ 1r2r_r3 r_wr__vwr (8.4-1)where x is the axial coordinate, r is the radial coordinate, u is the axial velocity, v is theradial velocity, and w is the swirl velocity.Three-Dimensional Swirling FlowsWhen there are geometric changes and/or ow gradients in the circumferential direction,your swirling ow prediction requires a three-dimensional model. If you are planninga 3D FLUENT model that includes swirl or rotation, you should be aware of the setupconstraints listed in Section 8.4.4. In addition, you may wish to consider simplications8-12 c Fluent Inc. January 28, 20038.4 Swirling and Rotating Flowsto the problem which might reduce it to an equivalent axisymmetric problem, especiallyfor your initial modeling eort. Because of the complexity of swirling ows, an initial2D study, in which you can quickly determine the eects of various modeling and designchoices, can be very benecial.For 3D problems involving swirl or rotation, there are no special inputs required during !the problem setup and no special solution procedures. Note, however, that you may wantto use the cylindrical coordinate system for dening velocity-inlet boundary conditioninputs, as described in Section 6.4.1. Also, you may nd the gradual increase of therotational speed (set as a wall or inlet boundary condition) helpful during the solutionprocess. This is described for axisymmetric swirling ows in Section 8.4.5.Flows Requiring a Rotating Reference FrameIf your ow involves a rotating boundary which moves through the uid (e.g., an impellerblade or a grooved or notched surface), you will need to use a rotating reference frame tomodel the problem. Such applications are described in detail in Section 9.2. If you havemore than one rotating boundary (e.g., several impellers in a row), you can use multiplereference frames (described in Section 9.3) or mixing planes (described in Section 9.4).8.4.2 Physics of Swirling and Rotating FlowsIn swirling ows, conservation of angular momentum (rw or r2 = constant) tends tocreate a free vortex ow, in which the circumferential velocity, w, increases sharply as theradius, r, decreases (with w nally decaying to zero near r = 0 as viscous forces beginto dominate). A tornado is one example of a free vortex. Figure 8.4.1 depicts the radialdistribution of w in a typical free vortex.axisrFigure 8.4.1: Typical Radial Distribution of w in a Free VortexIt can be shown that for an ideal free vortex ow, the centrifugal forces created by thecircumferential motion are in equilibrium with the radial pressure gradient:pr = w2r (8.4-2)As the distribution of angular momentum in a non-ideal vortex evolves, the form of thisradial pressure gradient also changes, driving radial and axial ows in response to thehighly non-uniform pressures that result. Thus, as you compute the distribution of swirlin your FLUENT model, you will also notice changes in the static pressure distributionc Fluent Inc. January 28, 2003 8-13Modeling Basic Fluid Flowand corresponding changes in the axial and radial ow velocities. It is this high degreeof coupling between the swirl and the pressure eld that makes the modeling of swirlingows complex.In ows that are driven by wall rotation, the motion of the wall tends to impart a forcedvortex motion to the uid, wherein w/r or is constant. An important characteristicof such ows is the tendency of uid with high angular momentum (e.g., the ow nearthe wall) to be ung radially outward (Figure 8.4.2). This is often referred to as radialpumping, since the rotating wall is pumping the uid radially outward. axis of rotationContours of Stream Function (kg/s) 7.69e-03 6.92e-03 6.15e-03 5.38e-03 4.62e-03 3.85e-03 3.08e-03 2.31e-03 1.54e-03 7.69e-04 0.00e+00Figure 8.4.2: Stream Function Contours for Rotating Flow in a Cavity (Geometry ofFigure 8.4.3)8.4.3 Turbulence Modeling in Swirling FlowsIf you are modeling turbulent ow with a signicant amount of swirl (e.g., cyclone ows,swirling jets), you should consider using one of FLUENTs advanced turbulence models:the RNG k- model, realizable k- model, or Reynolds stress model. The appropriatechoice depends on the strength of the swirl, which can be gauged by the swirl number.The swirl number is dened as the ratio of the axial ux of angular momentum to theaxial ux of axial momentum:S =_ rwv d

AR_ uv d

A(8.4-3)8-14 c Fluent Inc. January 28, 20038.4 Swirling and Rotating Flowswhere R is the hydraulic radius.For ows with weak to moderate swirl (S < 0.5), both the RNG k- model and therealizable k- model yield appreciable improvements over the standard k- model. SeeSections 10.4.2, 10.4.3, and 10.10.1 for details about these models.For highly swirling ows (S > 0.5), the Reynolds stress model (RSM) is strongly recom-mended. The eects of strong turbulence anisotropy can be modeled rigorously only bythe second-moment closure adopted in the RSM. See Sections 10.6 and 10.10 for detailsabout this model.For swirling ows encountered in devices such as cyclone separators and swirl combustors,near-wall turbulence modeling is quite often a secondary issue at most. The delity of thepredictions in these cases is mainly determined by the accuracy of the turbulence modelin the core region. However, in cases where walls actively participate in the generationof swirl (i.e., where the secondary ows and vortical ows are generated by pressuregradients), non-equilibrium wall functions can often improve the predictions since theyuse a law of the wall for mean velocity sensitized to pressure gradients. See Section 10.8for additional details about near-wall treatments for turbulence.8.4.4 Grid Setup for Swirling and Rotating FlowsCoordinate-System RestrictionsRecall that for an axisymmetric problem, the axis of rotation must be the x axis and thegrid must lie on or above the y = 0 line.Grid Sensitivity in Swirling and Rotating FlowsIn addition to the setup constraint described above, you should be aware of the needfor sucient resolution in your grid when solving ows that include swirl or rotation.Typically, rotating boundary layers may be very thin, and your FLUENT model willrequire a very ne grid near a rotating wall. In addition, swirling ows will often involvesteep gradients in the circumferential velocity (e.g., near the centerline of a free-vortextype ow), and thus require a ne grid for accurate resolution.8.4.5 Modeling Axisymmetric Flows with Swirl or RotationAs discussed in Section 8.4.1, you can solve a 2D axisymmetric problem that includesthe prediction of the circumferential or swirl velocity. The assumption of axisymmetryimplies that there are no circumferential gradients in the ow, but that there may benon-zero circumferential velocities. Examples of axisymmetric ows involving swirl orrotation are depicted in Figures 8.4.3 and 8.4.4.c Fluent Inc. January 28, 2003 8-15Modeling Basic Fluid FlowRegion tobe modeledRotating CoverxyFigure 8.4.3: Rotating Flow in a CavityRegion to be modeledFigure 8.4.4: Swirling Flow in a Gas Burner8-16 c Fluent Inc. January 28, 20038.4 Swirling and Rotating FlowsProblem Setup for Axisymmetric Swirling FlowsFor axisymmetric problems, you will need to perform the following steps during the prob-lem setup procedure. (Only those steps relevant specically to the setup of axisymmetricswirl/rotation are listed here. You will need to set up the rest of the problem as usual.)1. Activate solution of the momentum equation in the circumferential direction byturning on the Axisymmetric Swirl option for Space in the Solver panel.Dene Models Solver...2. Dene the rotational or swirling component of velocity, r, at inlets or walls.Dene Boundary Conditions...Remember to use the axis boundary type for the axis of rotation. !The procedures for input of rotational velocities at inlets and at walls are described indetail in Sections 6.4.1 and 6.13.1.Solution Strategies for Axisymmetric Swirling FlowsThe diculties associated with solving swirling and rotating ows are a result of thehigh degree of coupling between the momentum equations, which is introduced when theinuence of the rotational terms is large. A high level of rotation introduces a large radialpressure gradient which drives the ow in the axial and radial directions. This, in turn,determines the distribution of the swirl or rotation in the eld. This coupling may leadto instabilities in the solution process, and you may require special solution techniquesin order to obtain a converged solution. Solution techniques that may be benecial inswirling or rotating ow calculations include the following: (Segregated solver only) Use the PRESTO! scheme (enabled in the Pressure listfor Discretization in the Solution Controls panel), which is well-suited for the steeppressure gradients involved in swirling ows. Ensure that the mesh is suciently rened to resolve large gradients in pressureand swirl velocity. (Segregated solver only) Change the under-relaxation parameters on the velocities,perhaps to 0.30.5 for the radial and axial velocities and 0.81.0 for swirl. (Segregated solver only) Use a sequential or step-by-step solution procedure, inwhich some equations are temporarily left inactive (see below). If necessary, start the calculations using a low rotational speed or inlet swirl velocity,increasing the rotation or swirl gradually in order to reach the nal desired operatingcondition (see below).c Fluent Inc. January 28, 2003 8-17Modeling Basic Fluid FlowSee Chapter 24 for details on the procedures used to make these changes to the solutionparameters. More details on the step-by-step procedure and on the gradual increase ofthe rotational speed are provided below.Step-By-Step Solution Procedures for Axisymmetric Swirling FlowsOften, ows with a high degree of swirl or rotation will be easier to solve if you use thefollowing step-by-step solution procedure, in which only selected equations are left activein each step. This approach allows you to establish the eld of angular momentum, thenleave it xed while you update the velocity eld, and then nally to couple the two eldsby solving all equations simultaneously.Since the coupled solvers solve all the ow equations simultaneously, the following pro- !cedure applies only to the segregated solver.In this procedure, you will use the Equations list in the Solution Controls panel to turnindividual transport equations on and o between calculations.1. If your problem involves inow/outow, begin by solving the ow without rotationor swirl eects. That is, enable the Axisymmetric option instead of the AxisymmetricSwirl option in the Solver panel, and do not set any rotating boundary conditions.The resulting ow-eld data can be used as a starting guess for the full problem.2. Enable the Axisymmetric Swirl option and set all rotating/swirling boundary condi-tions.3. Begin the prediction of the rotating/swirling ow by solving only the momentumequation describing the circumferential velocity. This is the Swirl Velocity listedin the Equations list in the Solution Controls panel. Let the rotation diusethroughout the ow eld, based on your boundary condition inputs. In a turbulentow simulation, you may also want to leave the turbulence equations active duringthis step. This step will establish the eld of rotation throughout the domain.4. Turn o the momentum equations describing the circumferential motion (SwirlVelocity). Leaving the velocity in the circumferential direction xed, solve themomentum and continuity (pressure) equations (Flow in the Equations list in theSolution Controls panel) in the other coordinate directions. This step will establishthe axial and radial ows that are a result of the rotation in the eld. Again, ifyour problem involves turbulent ow, you should leave the turbulence equationsactive during this calculation.5. Turn on all of the equations simultaneously to obtain a fully coupled solution. Notethe under-relaxation controls suggested above.In addition to the steps above, you may want to simplify your calculation by solvingisothermal ow before adding heat transfer or by solving laminar ow before adding a8-18 c Fluent Inc. January 28, 20038.4 Swirling and Rotating Flowsturbulence model. These two methods can be used for any of the solvers (i.e., segregatedor coupled).Gradual Increase of the Rotational or Swirl Speed to Improve Solution StabilityBecause the rotation or swirl dened by the boundary conditions can lead to large com-plex forces in the ow, your FLUENT calculations will be less stable as the speed ofrotation or degree of swirl increases. Hence, one of the most eective controls you canapply to the solution is to solve your rotating ow problem starting with a low rotationalspeed or swirl velocity and then slowly increase the magnitude up to the desired level.The procedure for accomplishing this is as follows:1. Set up the problem using a low rotational speed or swirl velocity in your inputs forboundary conditions. The rotation or swirl in this rst attempt might be selectedas 10% of the actual operating conditions.2. Solve the problem at these conditions, perhaps using the step-by-step solutionstrategy outlined above.3. Save this initial solution data.4. Modify your inputs (boundary conditions). Increase the speed of rotation, perhapsdoubling it.5. Restart the calculation using the solution data saved in step 3 as the initial solutionfor the new calculation. Save the new data.6. Continue to increment the speed of rotation, following steps 4 and 5, until youreach the desired operating condition.Postprocessing for Axisymmetric Swirling FlowsReporting of results for axisymmetric swirling ows is the same as for other ows. Thefollowing additional variables are available for postprocessing when axisymmetric swirlis active: Swirl Velocity (in the Velocity... category) Swirl-Wall Shear Stress (in the Wall Fluxes... category)c Fluent Inc. January 28, 2003 8-19Modeling Basic Fluid Flow8.5 Compressible FlowsCompressibility eects are encountered in gas ows at high velocity and/or in which thereare large pressure variations. When the ow velocity approaches or exceeds the speed ofsound of the gas or when the pressure change in the system (p/p) is large, the variationof the gas density with pressure has a signicant impact on the ow velocity, pressure,and temperature. Compressible ows create a unique set of ow physics for which youmust be aware of the special input requirements and solution techniques described inthis section. Figures 8.5.1 and 8.5.2 show examples of compressible ows computedusing FLUENT.Contours of Mach Number 1.57e+00 1.43e+00 1.29e+00 1.16e+00 1.02e+00 8.82e-01 7.45e-01 6.07e-01 4.70e-01 3.32e-01 1.95e-01Figure 8.5.1: Transonic Flow in a Converging-Diverging NozzleInformation about compressible ows is provided in the following subsections: Section 8.5.1: When to Use the Compressible Flow Model Section 8.5.2: Physics of Compressible Flows Section 8.5.3: Modeling Inputs for Compressible Flows Section 8.5.4: Floating Operating Pressure Section 8.5.5: Solution Strategies for Compressible Flows Section 8.5.6: Reporting of Results for Compressible Flows8-20 c Fluent Inc. January 28, 20038.5 Compressible FlowsContours of Static Pressure (pascal) 2.02e+04 1.24e+04 4.68e+03-3.07e+03-1.08e+04-1.86e+04-2.63e+04-3.41e+04-4.18e+04-4.95e+04-5.73e+04Figure 8.5.2: Mach 0.675 Flow Over a Bump in a 2D Channel8.5.1 When to Use the Compressible Flow ModelCompressible ows can be characterized by the value of the Mach number:M u/c (8.5-1)Here, c is the speed of sound in the gas:c =_RT (8.5-2)and is the ratio of specic heats (cp/cv).When the Mach number is less than 1.0, the ow is termed subsonic. At Mach numbersmuch less than 1.0 (M < 0.1 or so), compressibility eects are negligible and the variationof the gas density with pressure can safely be ignored in your ow modeling. As the Machnumber approaches 1.0 (which is referred to as the transonic ow regime), compressibilityeects become important. When the Mach number exceeds 1.0, the ow is termedsupersonic, and may contain shocks and expansion fans which can impact the ow patternsignicantly. FLUENT provides a wide range of compressible ow modeling capabilitiesfor subsonic, transonic, and supersonic ows.c Fluent Inc. January 28, 2003 8-21Modeling Basic Fluid Flow8.5.2 Physics of Compressible FlowsCompressible ows are typically characterized by the total pressure p0 and total tem-perature T0 of the ow. For an ideal gas, these quantities can be related to the staticpressure and temperature by the following:p0p =_1 + 12 M2_/(1)(8.5-3)T0T = 1 + 12 M2(8.5-4)These relationships describe the variation of the static pressure and temperature in theow as the velocity (Mach number) changes under isentropic conditions. For example,given a pressure ratio from inlet to exit (total to static), Equation 8.5-3 can be used toestimate the exit Mach number which would exist in a one-dimensional isentropic ow.For air, Equation 8.5-3 predicts a choked ow (Mach number of 1.0) at an isentropicpressure ratio, p/p0, of 0.5283. This choked ow condition will be established at thepoint of minimum ow area (e.g., in the throat of a nozzle). In the subsequent areaexpansion the ow may either accelerate to a supersonic ow in which the pressure willcontinue to drop, or return to subsonic ow conditions, decelerating with a pressure rise.If a supersonic ow is exposed to an imposed pressure increase, a shock will occur, witha sudden pressure rise and deceleration accomplished across the shock.Basic Equations for Compressible FlowsCompressible ows are described by the standard continuity and momentum equationssolved by FLUENT, and you do not need to activate any special physical models (otherthan the compressible treatment of density as detailed below). The energy equationsolved by FLUENT correctly incorporates the coupling between the ow velocity and thestatic temperature, and should be activated whenever you are solving a compressibleow. In addition, if you are using the segregated solver, you should activate the viscousdissipation terms in Equation 11.2-1, which become important in high-Mach-numberows.The Compressible Form of the Gas LawFor compressible ows, the ideal gas law is written in the following form: = pop + pRMwT (8.5-5)where pop is the operating pressure dened in the Operating Conditions panel, p is thelocal static pressure relative to the operating pressure, R is the universal gas constant,8-22 c Fluent Inc. January 28, 20038.5 Compressible Flowsand Mw is the molecular weight. The temperature, T, will be computed from the energyequation.8.5.3 Modeling Inputs for Compressible FlowsTo set up a compressible ow in FLUENT, you will need to follow the steps listed below.(Only those steps relevant specically to the setup of compressible ows are listed here.You will need to set up the rest of the problem as usual.)1. Set the Operating Pressure in the Operating Conditions panel.Dene Operating Conditions...(You can think of pop as the absolute static pressure at a point in the ow where youwill dene the gauge pressure p to be zero. See Section 7.12 for guidelines on settingthe operating pressure. For time-dependent compressible ows, you may want tospecify a oating operating pressure instead of a constant operating pressure. SeeSection 8.5.4 for details.)2. Activate solution of the energy equation in the Energy panel.Dene Models Energy...3. (Segregated solver only) If you are modeling turbulent ow, activate the optionalviscous dissipation terms in the energy equation by turning on Viscous Heating inthe Viscous Model panel. Note that these terms can be important in high-speedows.Dene Models Viscous...This step is not necessary if you are using one of the coupled solvers, because thecoupled solvers always include the viscous dissipation terms in the energy equation.4. Set the following items in the Materials panel:Dene Materials...(a) Select ideal-gas in the drop-down list next to Density.(b) Dene all relevant properties (specic heat, molecular weight, thermal con-ductivity, etc.).5. Set boundary conditions (using the Boundary Conditions panel), being sure to choosea well-posed boundary condition combination that is appropriate for the ow regime.See below for details. Recall that all inputs for pressure (either total pressure orstatic pressure) must be relative to the operating pressure, and the temperatureinputs at inlets should be total (stagnation) temperatures, not static temperatures.Dene Boundary Conditions...c Fluent Inc. January 28, 2003 8-23Modeling Basic Fluid FlowThese inputs should ensure a well-posed compressible ow problem. You will also want toconsider special solution parameter settings, as noted in Section 8.5.5, before beginningthe ow calculation.Boundary Conditions for Compressible FlowsWell-posed inlet and exit boundary conditions for compressible ow are listed below: For ow inlets: Pressure inlet: Inlet total temperature and total pressure and, for supersonicinlets, static pressure Mass ow inlet: Inlet mass ow and total temperature For ow exits: Pressure outlet: Exit static pressure (ignored if ow is supersonic at the exit)It is important to note that your boundary condition inputs for pressure (either totalpressure or static pressure) must be in terms of gauge pressurei.e., pressure relative tothe operating pressure dened in the Operating Conditions panel, as described above.All temperature inputs at inlets should be total (stagnation) temperatures, not statictemperatures.8.5.4 Floating Operating PressureFLUENT provides a oating operating pressure option to handle time-dependent com-pressible ows with a gradual increase in the absolute pressure in the domain. This optionis desirable for slow subsonic ows with static pressure build-up, since it eciently ac-counts for the slow changing of absolute pressure without using acoustic waves as thetransport mechanism for the pressure build-up.Examples of typical applications include the following: combustion or heating of a gas in a closed domain pumping of a gas into a closed domainLimitationsThe oating operating pressure option should not be used for transonic or incompressibleows. In addition, it cannot be used if your model includes any pressure inlet, pressureoutlet, exhaust fan, inlet vent, intake fan, outlet vent, or pressure far eld boundaries.8-24 c Fluent Inc. January 28, 20038.5 Compressible FlowsTheoryThe oating operating pressure option allows FLUENT to calculate the pressure rise(or drop) from the integral mass balance, separately from the solution of the pressurecorrection equation. When this option is activated, the absolute pressure at each iterationcan be expressed aspabs = pop,oat + p (8.5-6)where p is the pressure relative to the reference location, which in this case is in the cellwith the minimum pressure value. Thus the reference location itself is oating.pop,oat is referred to as the oating operating pressure, and is dened aspop,oat = p0op + pop (8.5-7)where p0op is the initial operating pressure and pop is the pressure rise.Including the pressure rise pop in the oating operating pressure pop,oat, rather than inthe pressure p, helps to prevent roundo error. If the pressure rise were included in p, thecalculation of the pressure gradient for the momentum equation would give an inexactbalance due to precision limits for 32-bit real numbers.Enabling Floating Operating PressureWhen time dependence is active, you can turn on the Floating Operating Pressure optionin the Operating Conditions panel.Dene Operating Conditions...(Note that the inputs for Reference Pressure Location will disappear when you enableFloating Operating Pressure, since these inputs are no longer relevant.)The oating operating pressure option should not be used for transonic ows or for !incompressible ows. It is meaningful only for slow subsonic ows of ideal gases, whenthe characteristic time scale is much larger than the sonic time scale.Setting the Initial Value for the Floating Operating PressureWhen the oating operating pressure option is enabled, you will need to specify a valuefor the Initial Operating Pressure in the Solution Initialization panel.Solve Initialize Initialize...This initial value is stored in the case le with all your other initial values.c Fluent Inc. January 28, 2003 8-25Modeling Basic Fluid FlowStorage and Reporting of the Floating Operating PressureThe current value of the oating operating pressure is stored in the data le. If you visitthe Operating Conditions panel after a number of time steps have been performed, thecurrent value of the Operating Pressure will be displayed.Note that the oating operating pressure will automatically be reset to the initial oper-ating pressure if you reset the data (i.e., start over at the rst iteration of the rst timestep).Monitoring Absolute PressureYou can monitor the absolute pressure during the calculation using the Surface Monitorspanel (see Section 24.16.4 for details). You can also generate graphical plots or alphanu-meric reports of absolute pressure when your solution is complete. The Absolute Pressurevariable is contained in the Pressure... category of the variable selection drop-down listthat appears in postprocessing panels. See Chapter 29 for its denition.8.5.5 Solution Strategies for Compressible FlowsThe diculties associated with solving compressible ows are a result of the high degreeof coupling between the ow velocity, density, pressure, and energy. This coupling maylead to instabilities in the solution process and, therefore, may require special solutiontechniques in order to obtain a converged solution. In addition, the presence of shocks(discontinuities) in the ow introduces an additional stability problem during the cal-culation. Solution techniques that may be benecial in compressible ow calculationsinclude the following: (segregated solver only) Use conservative under-relaxation parameters on the ve-locities, perhaps values of 0.2 or 0.3. (segregated solver only) Set the under-relaxation on pressure to a value of 0.1 orso and use the SIMPLE algorithm. Set reasonable limits for the temperature and pressure (in the Solution Limits panel)to avoid solution divergence, especially at the start of the calculation. If FLUENTprints messages about temperature or pressure being limited as the solution nearsconvergence, the high or low computed values may be physical, and you will needto change the limits to allow these values. If required, begin the calculations using a reduced pressure ratio at the boundaries,increasing the pressure ratio gradually in order to reach the nal desired operatingcondition. You can also consider starting the compressible ow calculation froman incompressible ow solution (although the incompressible ow solution can insome cases be a rather poor initial guess for the compressible calculation).8-26 c Fluent Inc. January 28, 20038.5 Compressible Flows In some cases, computing an inviscid solution as a starting point may be helpful.See Chapter 24 for details on the procedures used to make these changes to the solutionparameters.8.5.6 Reporting of Results for Compressible FlowsYou can display the results of your compressible ow calculations in the same mannerthat you would use for an incompressible ow. The variables listed below are of particularinterest when you model compressible ow: Total Temperature Total Pressure Mach NumberThese variables are contained in the variable selection drop-down list that appears inpostprocessing panels. Total Temperature is in the Temperature... category, Total Pressureis in the Pressure... category, and Mach Number is in the Velocity... category. SeeChapter 29 for their denitions.c Fluent Inc. January 28, 2003 8-27Modeling Basic Fluid Flow8.6 Inviscid FlowsInviscid ow analyses neglect the eect of viscosity on the ow and are appropriate forhigh-Reynolds-number applications where inertial forces tend to dominate viscous forces.One example for which an inviscid ow calculation is appropriate is an aerodynamicanalysis of some high-speed projectile. In a case like this, the pressure forces on the bodywill dominate the viscous forces. Hence, an inviscid analysis will give you a quick estimateof the primary forces acting on the body. After the body shape has been modied tomaximize the lift forces and minimize the drag forces, you can perform a viscous analysisto include the eects of the uid viscosity and turbulent viscosity on the lift and dragforces.Another area where inviscid ow analyses are routinely used is to provide a good ini-tial solution for problems involving complicated ow physics and/or complicated owgeometry. In a case like this, the viscous forces are important, but in the early stages ofthe calculation the viscous terms in the momentum equations will be ignored. Once thecalculation has been started and the residuals are decreasing, you can turn on the viscousterms (by enabling laminar or turbulent ow) and continue the solution to convergence.For some very complicated ows, this is the only way to get the calculation started.Information about inviscid ows is provided in the following subsections: Section 8.6.1: Euler Equations Section 8.6.2: Setting Up an Inviscid Flow Model Section 8.6.3: Solution Strategies for Inviscid Flows Section 8.6.4: Postprocessing for Inviscid Flows8.6.1 Euler EquationsFor inviscid ows, FLUENT solves the Euler equations. The mass conservation equationis the same as for a laminar ow, but the momentum and energy conservation equationsare reduced due to the absence of molecular diusion.In this section, the conservation equations for inviscid ow in an inertial (non-rotating)reference frame are presented. The equations that are applicable to non-inertial refer-ence frames are described in Chapter 9. The conservation equations relevant for speciestransport and other models will be discussed in the chapters where those models aredescribed.8-28 c Fluent Inc. January 28, 20038.6 Inviscid FlowsThe Mass Conservation EquationThe equation for conservation of mass, or continuity equation, can be written as follows:t + (v) = Sm (8.6-1)Equation 8.6-1 is the general form of the mass conservation equation and is valid forincompressible as well as compressible ows. The source Sm is the mass added to thecontinuous phase from the dispersed second phase (e.g., due to vaporization of liquiddroplets) and any user-dened sources.For 2D axisymmetric geometries, the continuity equation is given byt + x(vx) + r(vr) + vrr = Sm (8.6-2)where x is the axial coordinate, r is the radial coordinate, vx is the axial velocity, and vris the radial velocity.Momentum Conservation EquationsConservation of momentum is described byt(v) + (vv) = p + g + F (8.6-3)where p is the static pressure and g and F are the gravitational body force and externalbody forces (e.g., forces that arise from interaction with the dispersed phase), respectively.

F also contains other model-dependent source terms such as porous-media and user-dened sources.For 2D axisymmetric geometries, the axial and radial momentum conservation equationsare given byt(vx) + 1rx(rvxvx) + 1rr(rvrvx) = px + Fx (8.6-4)andt(vr) + 1rx(rvxvr) + 1rr(rvrvr) = pr + Fr (8.6-5)wherec Fluent Inc. January 28, 2003 8-29Modeling Basic Fluid Flow v = vxx + vrr + vrr (8.6-6)Energy Conservation EquationConservation of energy is described byt(E) + (v(E + p)) = __

jhjJj__+ Sh (8.6-7)8.6.2 Setting Up an Inviscid Flow ModelFor inviscid ow problems, you will need to perform the following steps during the prob-lem setup procedure. (Only those steps relevant specically to the setup of inviscid oware listed here. You will need to set up the rest of the problem as usual.)1. Activate the calculation of inviscid ow by selecting Inviscid in the Viscous Modelpanel.Dene Models Viscous...2. Set boundary conditions and ow properties.Dene Boundary Conditions...Dene Materials...3. Solve the problem and examine the results.8.6.3 Solution Strategies for Inviscid FlowsSince inviscid ow problems will usually involve high-speed ow, you may have to reducethe under-relaxation factors for momentum (if you are using the segregated solver) orreduce the Courant number (if you are using the coupled solver), in order to get thesolution started. Once the ow is started and the residuals are monotonically decreasing,you can start increasing the under-relaxation factors or Courant number back up to thedefault values.Modications to the under-relaxation factors and the Courant number can be made inthe Solution Controls panel.Solve Controls Solution...The solution strategies for compressible ows apply also to inviscid ows. See Sec-tion 8.5.5 for details.8-30 c Fluent Inc. January 28, 20038.6 Inviscid Flows8.6.4 Postprocessing for Inviscid FlowsIf you are interested in the lift and drag forces acting on your model, you can use theForce Reports panel to compute them.Report Forces...See Section 28.3 for details.c Fluent Inc. January 28, 2003 8-31Modeling Basic Fluid Flow8-32 c Fluent Inc. January 28, 2003