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Chapter 8. Modeling Basic Fluid Flow

This chapter describes the basic physical models that FLUENT providesfor fluid flow and the commands for defining and using them. Models forflows in moving zones (including sliding meshes) are explained in Chap-ter 9, models for turbulence are described in Chapter 10, and modelsfor heat transfer (including radiation) are presented in Chapter 11. Anoverview of modeling species transport and reacting flows is providedin Chapter 12, details about models for species transport and reactingflows are described in Chapters 13–16, and models for pollutant forma-tion are presented in Chapter 17. An overview of multiphase modeling isprovided in Chapter 18, the discrete phase model is described in Chap-ter 19, general multiphase models are described in Chapter 20, and themelting and solidification model is described in Chapter 21. For informa-tion on modeling porous media, porous jumps, and lumped parameterfans 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 Flows

c© Fluent Inc. November 28, 2001 8-1

Modeling Basic Fluid Flow

8.1 Overview of Physical Models in FLUENT

FLUENT provides comprehensive modeling capabilities for a wide rangeof incompressible and compressible, laminar and turbulent fluid flowproblems. Steady-state or transient analyses can be performed. In FLU-ENT, a broad range of mathematical models for transport phenomena(like heat transfer and chemical reactions) is combined with the abil-ity to model complex geometries. Examples of FLUENT applicationsinclude laminar non-Newtonian flows in process equipment; conjugateheat transfer in turbomachinery and automotive engine components; pul-verized coal combustion in utility boilers; external aerodynamics; flowthrough compressors, pumps, and fans; and multiphase flows in bubblecolumns and fluidized beds.

To permit modeling of fluid flow and related transport phenomena inindustrial equipment and processes, various useful features are provided.These include porous media, lumped parameter (fan and heat exchanger),streamwise-periodic flow and heat transfer, swirl, and moving referenceframe models. The moving reference frame family of models includes theability to model single or multiple reference frames. A time-accurate slid-ing mesh method, useful for modeling multiple stages in turbomachineryapplications, for example, is also provided, along with the mixing planemodel for computing time-averaged flow fields.

Another very useful group of models in FLUENT is the set of free surfaceand multiphase flow models. These can be used for analysis of gas-liquid, gas-solid, liquid-solid, and gas-liquid-solid flows. For these typesof problems, FLUENT provides the volume-of-fluid (VOF), mixture, andEulerian models, as well as the discrete phase model (DPM). The DPMperforms Lagrangian trajectory calculations for dispersed phases (parti-cles, droplets, or bubbles), including coupling with the continuous phase.Examples of multiphase flows include channel flows, sprays, sedimenta-tion, separation, and cavitation.

Robust and accurate turbulence models are a vital component of theFLUENT suite of models. The turbulence models provided have a broadrange of applicability, and they include the effects of other physical phe-nomena, such as buoyancy and compressibility. Particular care has beendevoted to addressing issues of near-wall accuracy via the use of extended

8-2 c© Fluent Inc. November 28, 2001

8.2 Continuity and Momentum Equations

wall functions and zonal models.

Various modes of heat transfer can be modeled, including natural, forced,and mixed convection with or without conjugate heat transfer, porousmedia, etc. The set of radiation models and related submodels for mod-eling participating media are general and can take into account the com-plications of combustion. A particular strength of FLUENT is its abilityto model combustion phenomena using a variety of models, includingeddy dissipation and probability density function models. A host ofother models that are very useful for reacting flow applications are alsoavailable, including coal and droplet combustion, surface reaction, andpollutant formation models.


For all flows, FLUENT solves conservation equations for mass and mo-mentum. For flows involving heat transfer or compressibility, an addi-tional equation for energy conservation is solved. For flows involvingspecies mixing or reactions, a species conservation equation is solved or,if the non-premixed combustion model is used, conservation equationsfor the mixture fraction and its variance are solved. Additional transportequations are also solved when the flow is turbulent.

In this section, the conservation equations for laminar flow (in an in-ertial (non-accelerating) reference frame) are presented. The equationsthat are applicable to rotating reference frames are presented in Chap-ter 9. The conservation equations relevant to heat transfer, turbulencemodeling, and species transport will be discussed in the chapters wherethose models are described.

The Euler equations solved for inviscid flow are presented in Section 8.6.

The Mass Conservation Equation

The equation for conservation of mass, or continuity equation, can bewritten as follows:

∂ρ

∂t+ ∇ · (ρ~v) = Sm (8.2-1)



Equation 8.2-1 is the general form of the mass conservation equation andis valid for incompressible as well as compressible flows. The source Sm

is the mass added to the continuous phase from the dispersed secondphase (e.g., due to vaporization of liquid droplets) and any user-definedsources.

For 2D axisymmetric geometries, the continuity equation is given by

∂ρ

∂t+

∂

∂x(ρvx) +

∂

∂r(ρvr) +

ρvr

r= Sm (8.2-2)

where x is the axial coordinate, r is the radial coordinate, vx is the axialvelocity, and vr is the radial velocity.

Momentum Conservation Equations

Conservation of momentum in an inertial (non-accelerating) referenceframe is described by [10]

∂

∂t(ρ~v) + ∇ · (ρ~v~v) = −∇p+ ∇ · (τ ) + ρ~g + ~F (8.2-3)

where p is the static pressure, τ is the stress tensor (described below), andρ~g and ~F are the gravitational body force and external body forces (e.g.,that arise from interaction with the dispersed phase), respectively. ~Falso contains other model-dependent source terms such as porous-mediaand user-defined sources.

The stress tensor τ is given by

τ = µ

[(∇~v + ∇~v T) − 2

3∇ · ~vI

](8.2-4)

where µ is the molecular viscosity, I is the unit tensor, and the secondterm on the right hand side is the effect of volume dilation.

For 2D axisymmetric geometries, the axial and radial momentum con-servation equations are given by



∂

∂t(ρvx) +

1r

∂

∂x(rρvxvx) +

1r

∂

∂r(rρvrvx) = −∂p

∂x

+1r

∂

∂x

[rµ

(2∂vx

∂x− 2

3(∇ · ~v)

)]

+1r

∂

∂r

[rµ

(∂vx

∂r+∂vr

∂x

)]+ Fx (8.2-5)

and

∂

∂t(ρvr) +

1r

∂

∂x(rρvxvr) +

1r

∂

∂r(rρvrvr) = −∂p

∂r

+1r

∂

∂x

[rµ

(∂vr

∂x+∂vx

∂r

)]

+1r

∂

∂r

[rµ

(2∂vr

∂r− 2

3(∇ · ~v)

)]

− 2µvr

r2+

23µ

r(∇ · ~v) + ρ

v2z

r+ Fr (8.2-6)

where

∇ · ~v =∂vx

∂x+∂vr

∂r+vr

r(8.2-7)

and vz is the swirl velocity. (See Section 8.4 for information about mod-eling axisymmetric swirl.)



8.3 Periodic Flows

Periodic flow occurs when the physical geometry of interest and the ex-pected pattern of the flow/thermal solution have a periodically repeatingnature. Two types of periodic flow can be modeled in FLUENT. In thefirst type, no pressure drop occurs across the periodic planes. (Note toFLUENT 4 users: This type of periodic flow is modeled using a “cyclic”boundary in FLUENT 4.) In the second type, a pressure drop occursacross translationally periodic boundaries, resulting in “fully-developed”or “streamwise-periodic” flow. (In FLUENT 4, this type of periodic flowis modeled using a “periodic” boundary.)

This section discusses streamwise-periodic flow. A description of no-pressure-drop periodic flow is provided in Section 6.15, and a descriptionof streamwise-periodic heat transfer is provided in Section 11.4.

Information about streamwise-periodic flow is presented in the followingsections:

• 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 Flows

8.3.1 Overview and Limitations

Overview

FLUENT provides the ability to calculate streamwise-periodic—or “fully-developed”—fluid flow. These flows are encountered in a variety of ap-plications, including flows in compact heat exchanger channels and flowsacross tube banks. In such flow configurations, the geometry varies in


8.3 Periodic Flows

a repeating manner along the direction of the flow, leading to a peri-odic fully-developed flow regime in which the flow pattern repeats insuccessive cycles. Other examples of streamwise-periodic flows includefully-developed flow in pipes and ducts. These periodic conditions areachieved after a sufficient entrance length, which depends on the flowReynolds number and geometric configuration.

Streamwise-periodic flow conditions exist when the flow pattern repeatsover some length L, with a constant pressure drop across each repeatingmodule along the streamwise direction. Figure 8.3.1 depicts one exampleof a periodically repeating flow of this type which has been modeled byincluding a single representative module.

Velocity 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-06

Figure 8.3.1: Example of Periodic Flow in a 2D Heat Exchanger Geom-etry



Constraints on the Use of Streamwise-Periodic Flow

The following constraints apply to modeling streamwise-periodic flow:

• The flow must be incompressible.

• The geometry must be translationally periodic.

• If one of the coupled solvers is used, you can specify only the pres-sure jump; for the segregated solver, you can specify either thepressure jump or the mass flow rate.

• No net mass addition through inlets/exits or extra source terms isallowed.

• Species can be modeled only if inlets/exits (without net mass ad-dition) are included in the problem. Reacting flows are not per-mitted.

• Discrete phase and multiphase modeling are not allowed.

8.3.2 Theory

Definition of the Periodic Velocity

The assumption of periodicity implies that the velocity components re-peat themselves in space as follows:

u(~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 thedomain considered (see Figure 8.3.2).


8.3 Periodic Flows

L→

L→

A B C

uBuA uC= =

vBvA vC= = pBpA pC=pB- -

pBpA pC= =∼ ∼ ∼

Figure 8.3.2: Example of a Periodic Geometry

Definition of the Streamwise-Periodic Pressure

For viscous flows, the pressure is not periodic in the sense of Equa-tion 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 specified as a constant value.For the segregated solver, the local pressure gradient can be decomposedinto two parts: the gradient of a periodic component, ∇p(~r), and thegradient of a linearly-varying component, β ~L

|~L| :

∇p(~r) = β~L

|~L| + ∇p(~r) (8.3-3)

where p(~r) is the periodic pressure and β|~r| is the linearly-varying com-ponent of the pressure. The periodic pressure is the pressure left overafter subtracting out the linearly-varying pressure. The linearly-varyingcomponent of the pressure results in a force acting on the fluid in themomentum equations. Because the value of β is not known a priori, itmust be iterated on until the mass flow rate that you have defined isachieved in the computational model. This correction of β occurs in thepressure correction step of the SIMPLE, SIMPLEC, or PISO algorithm



where the value of β is updated based on the difference between the de-sired mass flow rate and the actual one. You have some control over thenumber of subiterations used to update β, as described in Section 8.3.3.

8.3.3 User Inputs for the Segregated Solver

If you are using the segregated solver, in order to calculate a spatiallyperiodic flow field with a specified mass flow rate or pressure derivative,you must first create a grid with translationally periodic boundaries thatare parallel to each other and equal in size. You can specify translationalperiodicity in the Periodic panel, as described in Section 6.15. (If youneed to create periodic boundaries, see Section 5.7.5.)

Once the grid has been read into FLUENT, you will complete the follow-ing inputs in the Periodicity Conditions panel (Figure 8.3.3):

Define −→Periodic Conditions...

Figure 8.3.3: The Periodicity Conditions Panel


8.3 Periodic Flows

1. Select either the specified mass flow rate (Specify Mass Flow) optionor the specified pressure gradient (Specify Pressure Gradient) option.For most problems, the mass flow rate across the periodic boundarywill be a known quantity; for others, the mass flow rate will beunknown, but the pressure gradient (β in Equation 8.3-3) will bea known quantity.

2. Specify the mass flow rate and/or the pressure gradient (β in Equa-tion 8.3-3):

• If you selected the Specify Mass Flow option, enter the desiredvalue for the Mass Flow Rate. You can also specify an initialguess for the Pressure Gradient, but this is not required.

For axisymmetric problems, the mass flow rate is per 2π ra-!dians.

• If you selected the Specify Pressure Gradient option, enter thedesired value for Pressure Gradient.

3. Define the flow direction by setting the X,Y,Z (or X,Y in 2D) pointunder Flow Direction. The flow will move in the direction of thevector pointing from the origin to the specified point. The directionvector must be parallel to the periodic translation direction or itsopposite.

4. If you chose in step 1 to specify the mass flow rate, set the param-eters used for the calculation of β. These parameters are describedin detail below.

After completing these inputs, you can solve the periodic velocity fieldto convergence.

Setting Parameters for the Calculation of β

If you choose to specify the mass flow rate, FLUENT will need to calcu-late the appropriate value of the pressure gradient β. You can controlthis calculation by specifying the Relaxation Factor and the Number ofIterations, and by supplying an initial guess for β. All of these inputs areentered in the Periodicity Conditions panel.



The Number of Iterations sets the number of subiterations performed onthe correction of β in the pressure correction equation. Because thevalue of β is not known a priori, it must be iterated on until the MassFlow Rate that you have defined is achieved in the computational model.This correction of β occurs in the pressure correction step of the SIM-PLE, SIMPLEC, or PISO algorithm. A correction to the current valueof β is calculated based on the difference between the desired mass flowrate and the actual one. The subiterations referred to here are performedwithin the pressure correction step to improve the correction for β beforethe pressure correction equation is solved for the resulting pressure (andvelocity) correction values. The default value of 2 subiterations shouldsuffice in most problems, but can be increased to help speed conver-gence. The Relaxation Factor is an under-relaxation factor that controlsconvergence of this iteration process.

You can also speed up convergence of the periodic calculation by sup-plying an initial guess for β in the Pressure Gradient field. Note that thecurrent value of β will be displayed in this field if you have performedany calculations. To update the Pressure Gradient field with the currentvalue at any time, click on the Update button.

8.3.4 User Inputs for the Coupled Solvers

If you are using one of the coupled solvers, in order to calculate a spatiallyperiodic flow field with a specified pressure jump, you must first createa grid with translationally periodic boundaries that are parallel to eachother and equal in size. (If you need to create periodic boundaries, seeSection 5.7.5.)

Then, follow the steps below:

1. In the Periodic panel (Figure 8.3.4), which is opened from theBoundary Conditions panel, indicate that the periodicity is Transla-tional (the default).

Define −→Boundary Conditions...

2. Also in the Periodic panel, set the Periodic Pressure Jump (∆p inEquation 8.3-2).


8.3 Periodic Flows

Figure 8.3.4: The Periodic Panel

After completing these inputs, you can solve the periodic velocity fieldto convergence.

8.3.5 Monitoring the Value of the Pressure Gradient

If you have specified the mass flow rate, you can monitor the value ofthe pressure gradient β during the calculation using the Statistic Mon-itors panel. Select per/pr-grad as the variable to be monitored. SeeSection 22.16.2 for details about using this feature.

8.3.6 Postprocessing for Streamwise-Periodic Flows

For streamwise-periodic flows, the velocity field should be completelyperiodic. If a coupled solver is used to compute the periodic flow, thepressure field reported will be the actual pressure p (which is not peri-odic). If the segregated solver is used, the pressure field reported will bethe periodic pressure field p(~r) of Equation 8.3-3. Figure 8.3.5 displaysthe periodic pressure field in the geometry of Figure 8.3.1.

If you specified a mass flow rate and had FLUENT calculate the pressuregradient, you can check the pressure gradient in the streamwise direction(β) by looking at the current value for Pressure Gradient in the Periodicity



Conditions panel.

Contours 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-03

Figure 8.3.5: Periodic Pressure Field Predicted for Flow in a 2D HeatExchanger Geometry


8.4 Swirling and Rotating Flows


Many important engineering flows involve swirl or rotation and FLUENTis well-equipped to model such flows. Swirling flows are common incombustion, with swirl introduced in burners and combustors in orderto increase residence time and stabilize the flow pattern. Rotating flowsare also encountered in turbomachinery, mixing tanks, and a variety ofother applications.

Information about rotating and swirling flows is provided in the followingsubsections:

• 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 Rota-tion

When you begin the analysis of a rotating or swirling flow, it is essentialthat you classify your problem into one of the following five categoriesof flow:

• axisymmetric flows with swirl or rotation

• fully three-dimensional swirling or rotating flows

• flows requiring a rotating reference frame

• flows requiring multiple rotating reference frames or mixing planes

• flows requiring sliding meshes

Modeling and solution procedures for the first two categories are pre-sented in this section. The remaining three, which all involve “movingzones”, are discussed in Chapter 9.



8.4.1 Overview of Swirling and Rotating Flows

Axisymmetric Flows with Swirl or Rotation

Your problem may be axisymmetric with respect to geometry and flowconditions but still include swirl or rotation. In this case, you can modelthe flow in 2D (i.e., solve the axisymmetric problem) and include theprediction of the circumferential (or swirl) velocity. It is important tonote that while the assumption of axisymmetry implies that there areno circumferential gradients in the flow, there may still be non-zero swirlvelocities.

Momentum Conservation Equation for Swirl Velocity

The tangential momentum equation for 2D swirling flows may be writtenas

∂

∂t(ρw) +

1r

∂

∂x(rρuw) +

1r

∂

∂r(rρvw) =

1r

∂

∂x

[rµ∂w

∂x

]

+1r2

∂

∂r

[r3µ

∂

∂r

(w

r

)]− ρ

vw

r(8.4-1)

where x is the axial coordinate, r is the radial coordinate, u is the axialvelocity, v is the radial velocity, and w is the swirl velocity.

Three-Dimensional Swirling Flows

When there are geometric changes and/or flow gradients in the cir-cumferential direction, your swirling flow prediction requires a three-dimensional model. If you are planning a 3D FLUENT model that in-cludes swirl or rotation, you should be aware of the setup constraintslisted in Section 8.4.4. In addition, you may wish to consider simplifica-tions to the problem which might reduce it to an equivalent axisymmetricproblem, especially for your initial modeling effort. Because of the com-plexity of swirling flows, an initial 2D study, in which you can quickly



determine the effects of various modeling and design choices, can be verybeneficial.

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 want to use the cylindrical coordinate sys-tem for defining velocity-inlet boundary condition inputs, as describedin Section 6.4.1. Also, you may find the gradual increase of the rota-tional speed (set as a wall or inlet boundary condition) helpful duringthe solution process. This is described for axisymmetric swirling flowsin Section 8.4.5.

Flows Requiring a Rotating Reference Frame

If your flow involves a rotating boundary which moves through the fluid(e.g., an impeller blade or a grooved or notched surface), you will needto use a rotating reference frame to model the problem. Such applica-tions are described in detail in Section 9.2. If you have more than onerotating boundary (e.g., several impellers in a row), you can use multiplereference frames (described in Section 9.3) or mixing planes (describedin Section 9.4).

8.4.2 Physics of Swirling and Rotating Flows

In swirling flows, conservation of angular momentum (rw or r2Ω = con-stant) tends to create a free vortex flow, in which the circumferentialvelocity, w, increases sharply as the radius, r, decreases (with w finallydecaying to zero near r = 0 as viscous forces begin to dominate). Atornado is one example of a free vortex. Figure 8.4.1 depicts the radialdistribution of w in a typical free vortex.

axis

r

Figure 8.4.1: Typical Radial Distribution of w in a Free Vortex



It can be shown that for an ideal free vortex flow, the centrifugal forcescreated by the circumferential motion are in equilibrium with the radialpressure gradient:

∂p

∂r=ρw2

r(8.4-2)

As the distribution of angular momentum in a non-ideal vortex evolves,the form of this radial pressure gradient also changes, driving radialand axial flows in response to the highly non-uniform pressures thatresult. Thus, as you compute the distribution of swirl in your FLUENTmodel, you will also notice changes in the static pressure distributionand corresponding changes in the axial and radial flow velocities. It isthis high degree of coupling between the swirl and the pressure field thatmakes the modeling of swirling flows complex.

In flows that are driven by wall rotation, the motion of the wall tends toimpart a forced vortex motion to the fluid, wherein w/r or Ω is constant.An important characteristic of such flows is the tendency of fluid withhigh angular momentum (e.g., the flow near the wall) to be flung radiallyoutward (Figure 8.4.2). This is often referred to as “radial pumping”,since the rotating wall is pumping the fluid radially outward.

8.4.3 Turbulence Modeling in Swirling Flows

If you are modeling turbulent flow with a significant amount of swirl(e.g., cyclone flows, swirling jets), you should consider using one of FLU-ENT’s advanced turbulence models: the RNG k-ε model, realizable k-εmodel, or Reynolds stress model. The appropriate choice depends onthe strength of the swirl, which can be gauged by the swirl number.The swirl number is defined as the ratio of the axial flux of angularmomentum to the axial flux of axial momentum:

S =∫rw~v · d ~A

R∫u~v · d ~A (8.4-3)

where R is the hydraulic radius.



axis of rotation

Contours 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+00

Figure 8.4.2: Stream Function Contours for Rotating Flow in a Cavity(Geometry of Figure 8.4.3)

For flows with weak to moderate swirl (S < 0.5), both the RNG k-εmodel and the realizable k-ε model yield appreciable improvements overthe standard k-ε model. See Sections 10.4.2, 10.4.3, and 10.10.1 fordetails about these models.

For highly swirling flows (S > 0.5), the Reynolds stress model (RSM)is strongly recommended. The effects of strong turbulence anisotropycan be modeled rigorously only by the second-moment closure adoptedin the RSM. See Sections 10.6 and 10.10 for details about this model.

For swirling flows encountered in devices such as cyclone separators andswirl combustors, near-wall turbulence modeling is quite often a sec-ondary issue at most. The fidelity of the predictions in these cases ismainly determined by the accuracy of the turbulence model in the coreregion. However, in cases where walls actively participate in the gen-eration of swirl (i.e., where the secondary flows and vortical flows aregenerated by pressure gradients), nonequilibrium wall functions can of-ten improve the predictions since they use a law of the wall for mean



velocity sensitized to pressure gradients. See Section 10.8 for additionaldetails about near-wall treatments for turbulence.

8.4.4 Grid Setup for Swirling and Rotating Flows

Coordinate-System Restrictions

Recall that for an axisymmetric problem, the axis of rotation must bethe x axis and the grid must lie on or above the y = 0 line.

Grid Sensitivity in Swirling and Rotating Flows

In addition to the setup constraint described above, you should be awareof the need for sufficient resolution in your grid when solving flows thatinclude swirl or rotation. Typically, rotating boundary layers may bevery thin, and your FLUENT model will require a very fine grid near arotating wall. In addition, swirling flows will often involve steep gradientsin the circumferential velocity (e.g., near the centerline of a free-vortextype flow), and thus require a fine grid for accurate resolution.

8.4.5 Modeling Axisymmetric Flows with Swirl or Rotation

As discussed in Section 8.4.1, you can solve a 2D axisymmetric problemthat includes the prediction of the circumferential or swirl velocity. Theassumption of axisymmetry implies that there are no circumferentialgradients in the flow, but that there may be non-zero circumferentialvelocities. Examples of axisymmetric flows involving swirl or rotationare depicted in Figures 8.4.3 and 8.4.4.

Problem Setup for Axisymmetric Swirling Flows

For axisymmetric problems, you will need to perform the following stepsduring the problem setup procedure. (Only those steps relevant specif-ically to the setup of axisymmetric swirl/rotation are listed here. Youwill need to set up the rest of the problem as usual.)

1. Activate solution of the momentum equation in the circumferentialdirection by turning on the Axisymmetric Swirl option for Space inthe Solver panel.



Region tobe modeled

Rotating Cover

Ω

x

y

Figure 8.4.3: Rotating Flow in a Cavity

Region to be modeled

Ω

Figure 8.4.4: Swirling Flow in a Gas Burner



Define −→ Models −→Solver...

2. Define the rotational or swirling component of velocity, rΩ, at inletsor walls.


Remember to use the axis boundary type for the axis of rotation.!

The procedures for input of rotational velocities at inlets and at wallsare described in detail in Sections 6.4.1 and 6.13.1.

Solution Strategies for Axisymmetric Swirling Flows

The difficulties associated with solving swirling and rotating flows area result of the high degree of coupling between the momentum equa-tions, which is introduced when the influence of the rotational terms islarge. A high level of rotation introduces a large radial pressure gradientwhich drives the flow in the axial and radial directions. This, in turn,determines the distribution of the swirl or rotation in the field. Thiscoupling may lead to instabilities in the solution process, and you mayrequire special solution techniques in order to obtain a converged solu-tion. Solution techniques that may be beneficial in swirling or rotatingflow calculations include the following:

• (Segregated solver only) Use the PRESTO! scheme (enabled in thePressure list for Discretization in the Solution Controls panel), whichis well-suited for the steep pressure gradients involved in swirlingflows.

• Ensure that the mesh is sufficiently refined to resolve large gradi-ents in pressure and swirl velocity.

• (Segregated solver only) Change the under-relaxation parameterson the velocities, perhaps to 0.3–0.5 for the radial and axial veloc-ities and 0.8–1.0 for swirl.

• (Segregated solver only) Use a sequential or step-by-step solutionprocedure, in which some equations are temporarily left inactive(see below).



• If necessary, start the calculations using a low rotational speed orinlet swirl velocity, increasing the rotation or swirl gradually inorder to reach the final desired operating condition (see below).

See Chapter 22 for details on the procedures used to make these changesto the solution parameters. More details on the step-by-step procedureand on the gradual increase of the rotational speed are provided below.

Step-By-Step Solution Procedures for Axisymmetric Swirling Flows

Often, flows with a high degree of swirl or rotation will be easier to solveif you use the following step-by-step solution procedure, in which onlyselected equations are left active in each step. This approach allows youto establish the field of angular momentum, then leave it fixed while youupdate the velocity field, and then finally to couple the two fields bysolving all equations simultaneously.

Since the coupled solvers solve all the flow equations simultaneously, the!following procedure applies only to the segregated solver.

In this procedure, you will use the Equations list in the Solution Con-trols panel to turn individual transport equations on and off betweencalculations.

1. If your problem involves inflow/outflow, begin by solving the flowwithout rotation or swirl effects. That is, enable the Axisymmetricoption instead of the Axisymmetric Swirl option in the Solver panel,and do not set any rotating boundary conditions. The resultingflow-field data can be used as a starting guess for the full problem.

2. Enable the Axisymmetric Swirl option and set all rotating/swirlingboundary conditions.

3. Begin the prediction of the rotating/swirling flow by solving onlythe momentum equation describing the circumferential velocity.This is the Swirl Velocity listed in the Equations list in the SolutionControls panel. Let the rotation “diffuse” throughout the flow field,based on your boundary condition inputs. In a turbulent flow



simulation, you may also want to leave the turbulence equationsactive during this step. This step will establish the field of rotationthroughout the domain.

4. Turn off the momentum equations describing the circumferentialmotion (Swirl Velocity). Leaving the velocity in the circumferen-tial direction fixed, solve the momentum and continuity (pressure)equations (Flow in the Equations list in the Solution Controls panel)in the other coordinate directions. This step will establish the ax-ial and radial flows that are a result of the rotation in the field.Again, if your problem involves turbulent flow, you should leavethe turbulence equations active during this calculation.

5. Turn on all of the equations simultaneously to obtain a fully cou-pled solution. Note the under-relaxation controls suggested above.

In addition to the steps above, you may want to simplify your calcula-tion by solving isothermal flow before adding heat transfer or by solvinglaminar flow before adding a turbulence model. These two methods canbe used for any of the solvers (i.e., segregated or coupled).

Gradual Increase of the Rotational or Swirl Speed to Improve SolutionStability

Because the rotation or swirl defined by the boundary conditions canlead to large complex forces in the flow, your FLUENT calculations willbe less stable as the speed of rotation or degree of swirl increases. Hence,one of the most effective controls you can apply to the solution is to solveyour rotating flow problem starting with a low rotational speed or swirlvelocity 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 velocityin your inputs for boundary conditions. The rotation or swirl inthis first attempt might be selected as 10% of the actual operatingconditions.

2. Solve the problem at these conditions, perhaps using the step-by-step solution strategy outlined above.


8.5 Compressible Flows

3. Save this initial solution data.

4. Modify your inputs (boundary conditions). Increase the speed ofrotation, perhaps doubling it.

5. Restart the calculation using the solution data saved in step 3 asthe initial solution for the new calculation. Save the new data.

6. Continue to increment the speed of rotation, following steps 4 and5, until you reach the desired operating condition.

Postprocessing for Axisymmetric Swirling Flows

Reporting of results for axisymmetric swirling flows is the same as forother flows. The following additional variables are available for postpro-cessing when axisymmetric swirl is active:

• Swirl Velocity (in the Velocity... category)

• Swirl-Wall Shear Stress (in the Wall Fluxes... category)


Compressibility effects are encountered in gas flows at high velocityand/or in which there are large pressure variations. When the flow ve-locity approaches or exceeds the speed of sound of the gas or when thepressure change in the system (∆p/p) is large, the variation of the gasdensity with pressure has a significant impact on the flow velocity, pres-sure, and temperature. Compressible flows create a unique set of flowphysics for which you must be aware of the special input requirementsand solution techniques described in this section. Figures 8.5.1 and 8.5.2show examples of compressible flows computed using FLUENT.

Information about compressible flows is provided in the following sub-sections:

• Section 8.5.1: When to Use the Compressible Flow Model

• Section 8.5.2: Physics of Compressible Flows



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-01

Figure 8.5.1: Transonic Flow in a Converging-Diverging Nozzle

Contours 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+04

Figure 8.5.2: Mach 0.675 Flow Over a Bump in a 2D Channel



• 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 Flows

8.5.1 When to Use the Compressible Flow Model

Compressible flows can be characterized by the value of the Mach num-ber:

M ≡ u/c (8.5-1)

Here, c is the speed of sound in the gas:

c =√γRT (8.5-2)

and γ is the ratio of specific heats (cp/cv).

When the Mach number is less than 1.0, the flow is termed subsonic. AtMach numbers much less than 1.0 (M < 0.1 or so), compressibility effectsare negligible and the variation of the gas density with pressure can safelybe ignored in your flow modeling. As the Mach number approaches1.0 (which is referred to as the transonic flow regime), compressibilityeffects become important. When the Mach number exceeds 1.0, the flowis termed supersonic, and may contain shocks and expansion fans whichcan impact the flow pattern significantly. FLUENT provides a wide rangeof compressible flow modeling capabilities for subsonic, transonic, andsupersonic flows.

8.5.2 Physics of Compressible Flows

Compressible flows are typically characterized by the total pressure p0

and total temperature T0 of the flow. For an ideal gas, these quantitiescan be related to the static pressure and temperature by the following:



p0

p=(

1 +γ − 1

2M2)γ/(γ−1)

(8.5-3)

T0

T= 1 +

γ − 12

M2 (8.5-4)

These relationships describe the variation of the static pressure and tem-perature in the flow as the velocity (Mach number) changes under isen-tropic conditions. For example, given a pressure ratio from inlet to exit(total to static), Equation 8.5-3 can be used to estimate the exit Machnumber which would exist in a one-dimensional isentropic flow. For air,Equation 8.5-3 predicts a choked flow (Mach number of 1.0) at an isen-tropic pressure ratio, p/p0, of 0.5283. This choked flow condition will beestablished at the point of minimum flow area (e.g., in the throat of anozzle). In the subsequent area expansion the flow may either acceler-ate to a supersonic flow in which the pressure will continue to drop, orreturn to subsonic flow conditions, decelerating with a pressure rise. If asupersonic flow is exposed to an imposed pressure increase, a shock willoccur, with a sudden pressure rise and deceleration accomplished acrossthe shock.

Basic Equations for Compressible Flows

Compressible flows are described by the standard continuity and mo-mentum equations solved by FLUENT, and you do not need to activateany special physical models (other than the compressible treatment ofdensity as detailed below). The energy equation solved by FLUENT cor-rectly incorporates the coupling between the flow velocity and the statictemperature, and should be activated whenever you are solving a com-pressible flow. In addition, if you are using the segregated solver, youshould activate the viscous dissipation terms in Equation 11.2-1, whichbecome important in high-Mach-number flows.

The Compressible Form of the Gas Law

For compressible flows, the ideal gas law is written in the following form:



ρ =pop + p

RMw

T(8.5-5)

where pop is the operating pressure defined in the Operating Conditionspanel, p is the local static pressure relative to the operating pressure,R is the universal gas constant, and Mw is the molecular weight. Thetemperature, T , will be computed from the energy equation.

8.5.3 Modeling Inputs for Compressible Flows

In order to set up a compressible flow in FLUENT, you will need to followthe steps listed below. (Only those steps relevant specifically to the setupof compressible flows are listed here. You will need to set up the rest ofthe problem as usual.)

1. Set the Operating Pressure in the Operating Conditions panel.

Define −→Operating Conditions...

(You can think of pop as the absolute static pressure at a pointin the flow where you will define the gauge pressure p to be zero.See Section 7.12 for guidelines on setting the operating pressure.For time-dependent compressible flows, you may want to specifya floating operating pressure instead of a constant operating pres-sure. See Section 8.5.4 for details.)

2. Activate solution of the energy equation in the Energy panel.

Define −→ Models −→Energy...

3. (Segregated solver only) If you are modeling turbulent flow, acti-vate the optional viscous dissipation terms in the energy equationby turning on Viscous Heating in the Viscous Model panel. Notethat these terms can be important in high-speed flows.

Define −→ Models −→Viscous...

This step is not necessary if you are using one of the coupled solvers,because the coupled solvers always include the viscous dissipationterms in the energy equation.



4. Set the following items in the Materials panel:

Define −→Materials...

(a) Select ideal-gas in the drop-down list next to Density.

(b) Define all relevant properties (specific heat, molecular weight,thermal conductivity, etc.).

5. Set boundary conditions (using the Boundary Conditions panel),being sure to choose a well-posed boundary condition combinationthat is appropriate for the flow regime. See below for details. Recallthat all inputs for pressure (either total pressure or static pressure)must be relative to the operating pressure, and the temperature in-puts at inlets should be total (stagnation) temperatures, not statictemperatures.


These inputs should ensure a well-posed compressible flow problem. Youwill also want to consider special solution parameter settings, as notedin Section 8.5.5, before beginning the flow calculation.

Boundary Conditions for Compressible Flows

Well-posed inlet and exit boundary conditions for compressible flow arelisted below:

• For flow inlets:

– Pressure inlet: Inlet total temperature and total pressure and,for supersonic inlets, static pressure

– Mass flow inlet: Inlet mass flow and total temperature

• For flow exits:

– Pressure outlet: Exit static pressure (ignored if flow is super-sonic at the exit)

It is important to note that your boundary condition inputs for pres-sure (either total pressure or static pressure) must be in terms of gauge



pressure—i.e., pressure relative to the operating pressure defined in theOperating Conditions panel, as described above.

All temperature inputs at inlets should be total (stagnation) tempera-tures, not static temperatures.

8.5.4 Floating Operating Pressure

FLUENT provides a “floating operating pressure” option to handle time-dependent compressible flows with a gradual increase in the absolutepressure in the domain. This option is desirable for slow subsonic flowswith static pressure build-up, since it efficiently accounts for the slowchanging of absolute pressure without using acoustic waves as the trans-port 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 domain

Limitations

The floating operating pressure option should not be used for transonicor incompressible flows. In addition, it cannot be used if your modelincludes any pressure inlet, pressure outlet, exhaust fan, inlet vent, intakefan, outlet vent, or pressure far field boundaries.

Theory

The floating operating pressure option allows FLUENT to calculate thepressure rise (or drop) from the integral mass balance, separately fromthe solution of the pressure correction equation. When this option isactivated, the absolute pressure at each iteration can be expressed as

pabs = pop,float + p (8.5-6)



where p is the pressure relative to the reference location, which in thiscase is in the cell with the minimum pressure value. Thus the referencelocation itself is floating.

pop,float is referred to as the floating operating pressure, and is defined as

pop,float = 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 floating operating pressure pop,float,rather than in the pressure p, helps to prevent roundoff error. If the pres-sure rise were included in p, the calculation of the pressure gradient forthe momentum equation would give an inexact balance due to precisionlimits for 32-bit real numbers.

Enabling Floating Operating Pressure

When time dependence is active, you can turn on the Floating OperatingPressure option in the Operating Conditions panel.

Define −→Operating Conditions...

(Note that the inputs for Reference Pressure Location will disappear whenyou enable Floating Operating Pressure, since these inputs are no longerrelevant.)

The floating operating pressure option should not be used for transonic!flows or for incompressible flows. It is meaningful only for slow subsonicflows of ideal gases, when the characteristic time scale is much largerthan the sonic time scale.

Setting the Initial Value for the Floating Operating Pressure

When the floating operating pressure option is enabled, you will need tospecify a value for the Initial Operating Pressure in the Solution Initializa-tion panel.

Solve −→ Initialize −→Initialize...



This initial value is stored in the case file with all your other initialvalues.

Storage and Reporting of the Floating Operating Pressure

The current value of the floating operating pressure is stored in the datafile. If you visit the Operating Conditions panel after a number of timesteps have been performed, the current value of the Operating Pressurewill be displayed.

Note that the floating operating pressure will automatically be reset tothe initial operating pressure if you reset the data (i.e., start over at thefirst iteration of the first time step).

Monitoring Absolute Pressure

You can monitor the absolute pressure during the calculation using theSurface Monitors panel (see Section 22.16.4 for details). You can also gen-erate graphical plots or alphanumeric reports of absolute pressure whenyour solution is complete. The Absolute Pressure variable is containedin the Pressure... category of the variable selection drop-down list thatappears in postprocessing panels. See Chapter 27 for its definition.

8.5.5 Solution Strategies for Compressible Flows

The difficulties associated with solving compressible flows are a resultof the high degree of coupling between the flow velocity, density, pres-sure, and energy. This coupling may lead to instabilities in the solutionprocess and, therefore, may require special solution techniques in or-der to obtain a converged solution. In addition, the presence of shocks(discontinuities) in the flow introduces an additional stability problemduring the calculation. Solution techniques that may be beneficial incompressible flow calculations include the following:

• (segregated solver only) Use conservative under-relaxation param-eters on the velocities, perhaps values of 0.2 or 0.3.

• (segregated solver only) Set the under-relaxation on pressure to avalue of 0.1 or so and use the SIMPLE algorithm.



• Set reasonable limits for the temperature and pressure (in the So-lution Limits panel) to avoid solution divergence, especially at thestart of the calculation. If FLUENT prints messages about temper-ature or pressure being limited as the solution nears convergence,the high or low computed values may be physical, and you willneed to change the limits to allow these values.

• If required, begin the calculations using a reduced pressure ratioat the boundaries, increasing the pressure ratio gradually in or-der to reach the final desired operating condition. You can alsoconsider starting the compressible flow calculation from an incom-pressible flow solution (although the incompressible flow solutioncan in some cases be a rather poor initial guess for the compressiblecalculation).

• In some cases, computing an inviscid solution as a starting pointmay be helpful.

See Chapter 22 for details on the procedures used to make these changesto the solution parameters.

8.5.6 Reporting of Results for Compressible Flows

You can display the results of your compressible flow calculations in thesame manner that you would use for an incompressible flow. The vari-ables listed below are of particular interest when you model compressibleflow:

• Total Temperature

• Total Pressure

• Mach Number

These variables are contained in the variable selection drop-down listthat appears in postprocessing panels. Total Temperature is in the Tem-perature... category, Total Pressure is in the Pressure... category, andMach Number is in the Velocity... category. See Chapter 27 for theirdefinitions.


8.6 Inviscid Flows

8.6 Inviscid Flows

Inviscid flow analyses neglect the effect of viscosity on the flow and areappropriate for high-Reynolds-number applications where inertial forcestend to dominate viscous forces. One example for which an inviscidflow calculation is appropriate is an aerodynamic analysis of some high-speed projectile. In a case like this, the pressure forces on the body willdominate the viscous forces. Hence, an inviscid analysis will give you aquick estimate of the primary forces acting on the body. After the bodyshape has been modified to maximize the lift forces and minimize thedrag forces, you can perform a viscous analysis to include the effects ofthe fluid viscosity and turbulent viscosity on the lift and drag forces.

Another area where inviscid flow analyses are routinely used is to providea good initial solution for problems involving complicated flow physicsand/or complicated flow geometry. In a case like this, the viscous forcesare important, but in the early stages of the calculation the viscous termsin the momentum equations will be ignored. Once the calculation hasbeen started and the residuals are decreasing, you can turn on the viscousterms (by enabling laminar or turbulent flow) and continue the solutionto convergence. For some very complicated flows, this is the only way toget the calculation started.

Information about inviscid flows 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 Flows

8.6.1 Euler Equations

For inviscid flows, FLUENT solves the Euler equations. The mass con-servation equation is the same as for a laminar flow, but the momentumand energy conservation equations are reduced due to the absence ofmolecular diffusion.



In this section, the conservation equations for inviscid flow in an inertial(non-rotating) reference frame are presented. The equations that are ap-plicable to non-inertial reference frames are described in Chapter 9. Theconservation equations relevant for species transport and other modelswill be discussed in the chapters where those models are described.

The Mass Conservation Equation

The equation for conservation of mass, or continuity equation, can bewritten as follows:

∂ρ

∂t+ ∇ · (ρ~v) = Sm (8.6-1)

Equation 8.6-1 is the general form of the mass conservation equation andis valid for incompressible as well as compressible flows. The source Sm

is the mass added to the continuous phase from the dispersed secondphase (e.g., due to vaporization of liquid droplets) and any user-definedsources.

For 2D axisymmetric geometries, the continuity equation is given by

∂ρ

∂t+

∂

∂x(ρvx) +

∂

∂r(ρvr) +

ρvr

r= Sm (8.6-2)

where x is the axial coordinate, r is the radial coordinate, vx is the axialvelocity, and vr is the radial velocity.

Momentum Conservation Equations

Conservation of momentum is described by

∂

∂t(ρ~v) + ∇ · (ρ~v~v) = −∇p+ ρ~g + ~F (8.6-3)

where p is the static pressure and ρ~g and ~F are the gravitational bodyforce and external body forces (e.g., forces that arise from interactionwith the dispersed phase), respectively. ~F also contains other model-dependent source terms such as porous-media and user-defined sources.


8.6 Inviscid Flows

For 2D axisymmetric geometries, the axial and radial momentum con-servation equations are given by

∂

∂t(ρvx) +

1r

∂

∂x(rρvxvx) +

1r

∂

∂r(rρvrvx) = −∂p

∂x+ Fx (8.6-4)

and

∂

∂t(ρvr) +

1r

∂

∂x(rρvxvr) +

1r

∂

∂r(rρvrvr) = −∂p

∂r+ Fr (8.6-5)

where

∇ · ~v =∂vx

∂x+∂vr

∂r+vr

r(8.6-6)

Energy Conservation Equation

Conservation of energy is described by

∂

∂t(ρE) + ∇ · (~v(ρE + p)) = −∇ ·

∑

j

hjJj

+ Sh (8.6-7)

8.6.2 Setting Up an Inviscid Flow Model

For inviscid flow problems, you will need to perform the following stepsduring the problem setup procedure. (Only those steps relevant specif-ically to the setup of inviscid flow are listed here. You will need to setup the rest of the problem as usual.)

1. Activate the calculation of inviscid flow by selecting Inviscid in theViscous Model panel.

Define −→ Models −→Viscous...

2. Set boundary conditions and flow properties.


Define −→Materials...



3. Solve the problem and examine the results.

8.6.3 Solution Strategies for Inviscid Flows

Since inviscid flow problems will usually involve high-speed flow, youmay have to reduce the under-relaxation factors for momentum (if youare using the segregated solver) or reduce the Courant number (if you areusing the coupled solver), in order to get the solution started. Once theflow is started and the residuals are monotonically decreasing, you canstart increasing the under-relaxation factors or Courant number back upto the default values.

Modifications to the under-relaxation factors and the Courant numbercan be made in the Solution Controls panel.

Solve −→ Controls −→Solution...

The solution strategies for compressible flows apply also to inviscid flows.See Section 8.5.5 for details.

8.6.4 Postprocessing for Inviscid Flows

If you are interested in the lift and drag forces acting on your model, youcan use the Force Reports panel to compute them.

Report −→Forces...

See Section 26.3 for details.


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