Chapter 9. Modeling Flows in Moving Zones · 2001-12-03 · Chapter 9. Modeling Flows in Moving Zones The solution of ﬂows in moving reference frames requires the use of moving

Chapter 9. Modeling Flows in Moving

Zones

The solution of flows in moving reference frames requires the use ofmoving cell zones. This cell zone motion is interpreted as the motion ofa reference frame to which the cell zone is attached. With this capability,a wide variety of problems that involve moving parts can be set up andsolved using FLUENT.

The information in this chapter is divided into the following sections:

• Section 9.1: Overview of Moving Zone Approaches

• Section 9.2: Flow in a Rotating Reference Frame

• Section 9.3: The Multiple Reference Frame Model

• Section 9.4: The Mixing Plane Model

• Section 9.5: Sliding Meshes

• Section 9.6: Non-Reflecting Boundary Conditions

9.1 Overview of Moving Zone Approaches

The moving cell zone capability in FLUENT provides a powerful set offeatures for solving problems in which the domain or parts of the domainare in motion. Problems that can be addressed include the following:

• flow in a (single) rotating frame

• flow in multiple rotating and/or translating reference frames

c© Fluent Inc. November 28, 2001 9-1

Modeling Flows in Moving Zones

The single rotating frame option can be used to model flows in turbo-machinery, mixing tanks, and related devices. In each of these cases,the flow is unsteady in an inertial frame (i.e., a domain fixed in thelaboratory frame) because the rotor/impeller blades sweep the domainperiodically. However, in the absence of stators or baffles, it is possible toperform calculations in a domain that moves with the rotating part. Inthis case, the flow is steady relative to the rotating (non-inertial) frame,which simplifies the analysis.

If stators or baffles are present in addition to a rotor or impeller, then itis not possible to render the computational problem steady by choosing acalculation domain that rotates with the rotor or impeller. This situationoccurs, for example, in turbomachinery applications where rotor andstator blades are in close proximity (and hence rotor-stator interactionis important). FLUENT provides three approaches to address this classof problems:

• the multiple reference frame (MRF) model

• the mixing plane model

• the sliding mesh model

Both the MRF and mixing plane models assume that the flow field issteady, with the rotor-stator or impeller-baffle effects being accounted forby approximate means. These can be acceptable models in cases wherethe rotor-stator interaction is weak or an approximate solution for thesystem is desired. The sliding mesh model, on the other hand, assumesthat the flow field is unsteady, and hence models the interaction withcomplete fidelity. This is the model of choice if rotor-stator interactionis strong and a more accurate simulation of the system is desired. Note,however, that because the sliding mesh model requires an unsteady nu-merical solution, it is computationally more demanding than the MRFand mixing plane models.

9-2 c© Fluent Inc. November 28, 2001

9.2 Flow in a Rotating Reference Frame


9.2.1 Overview

When you create a model using FLUENT, you are typically modeling theflow in an inertial reference frame (i.e., in a non-accelerating coordinatesystem). However, FLUENT also has the ability to model flows in anaccelerating reference frame. In this situation, the acceleration of thecoordinate system is included in the equations of motion describing theflow. A common example of an accelerating reference frame in engineer-ing applications is flow in rotating equipment. Many such flows can bemodeled in a coordinate system that is moving with the rotating equip-ment and thus experiences a constant acceleration in the radial direction.This class of rotating flows can be treated using the rotating referenceframe capability in FLUENT. Figure 9.2.1 depicts an example of a flow ina rotating reference frame, and illustrates the coordinate transformationfrom the stationary frame to the rotating frame.

Applications Involving a Rotating Reference Frame

Several examples of problems that can be modeled using a rotating ref-erence frame are depicted in Figure 9.2.2. The applications illustratedhere include:

• Impellers in mixing tanks

• Rotating turbomachinery blades (centrifugal impellers, axial fans,etc.)

• Flows in rotating passages (e.g., cooling ducts, secondary air flowcircuits, and disk cavities in rotating equipment)

When such problems are defined in a rotating reference frame, the rotat-ing boundaries become stationary relative to the rotating frame, sincethey are moving at the same speed as the reference frame.



Rotating at speed Ω

Ω

Stationary

y

x

Rotating atspeed -Ω

-Ω

Stationary

y

x

1

1

(a) Original Reference Frame

(b) Rotating Reference Frame

Figure 9.2.1: Transforming Coordinates to a Rotating Reference Frame



Ω

z

y

x

y

x

Ω

(a) Rotating Impeller in a Mixing Tank

(b) Centrifugal Impeller Blades

Figure 9.2.2: Applications That Can Be Modeled by FLUENT in a Ro-tating Reference Frame



Ω

y

z

x

x

y

z

Ω

(c) Cooling Passages in a Spinning Rotor

(d) Axial Impeller Blades

Figure 9.2.2: Applications That Can Be Modeled by FLUENT in a Ro-tating Reference Frame



Modeling Rotor-Stator Interaction

As mentioned in Section 9.1, rotor-stator interaction problems (as illus-trated in Figure 9.2.3) cannot be modeled by a simple coordinate trans-formation to a rotating reference frame. In FLUENT, rotor-stator inter-action must be treated by applying the MRF, mixing plane, or slidingmesh approach. These approaches are described in detail in Sections 9.3,9.4, and 9.5.

9.2.2 Equations for a Rotating Reference Frame

When the equations of motion (see Section 8.2) are solved in a rotatingframe of reference, the acceleration of the fluid is augmented by addi-tional terms that appear in the momentum equations [10]. FLUENTallows you to solve rotating frame problems using either the absolutevelocity, ~v, or the relative velocity, ~vr, as the dependent variable. Thetwo velocities are related by the following equation:

~vr = ~v − (~Ω × ~r) (9.2-1)

Here, ~Ω is the angular velocity vector (that is, the angular velocity ofthe rotating frame) and ~r is the position vector in the rotating frame.

The left-hand side of the momentum equations appears as follows for aninertial reference frame:

∂

∂t(ρ~v) + ∇ · (ρ~v~v) (9.2-2)

For a rotating reference frame, the left-hand side written in terms ofabsolute velocities becomes

∂

∂t(ρ~v) + ∇ · (ρ~vr~v) + ρ(~Ω × ~v) (9.2-3)

In terms of relative velocities the left-hand side is given by



Stationary

Rotating

Ω

Stationarybaffles

Rotatingimpeller

Ω

(a) 2D Rotor-Stator Interaction

(b) Rotating Impeller in a Baffled Tank

Figure 9.2.3: Problems That Require MRF, Mixing Plane, or SlidingMeshes



∂

∂t(ρ~vr) + ∇ · (ρ~vr~vr) + ρ(2~Ω × ~vr + ~Ω × ~Ω × ~r) + ρ

∂~Ω∂t

× ~r (9.2-4)

where ρ(2~Ω × ~vr + ~Ω × ~Ω × ~r) is the Coriolis force. Note that FLUENT

neglects the ρ∂~Ω∂t × ~r term, so it cannot accurately model a time-varying

angular velocity using the relative velocity formulation.

For flows in rotating domains, the equation for conservation of mass, orcontinuity equation, can be written as follows for both the absolute andthe relative velocity formulations:

∂ρ

∂t+ ∇ · (ρ~vr) = Sm (9.2-5)

9.2.3 Grid Setup for a Single Rotating Reference Frame

It is important to remember the following coordinate-system constraintswhen you are setting up a problem involving a rotating reference frame:

• For 2D problems, the axis of rotation must be parallel to the zaxis.

• For 2D axisymmetric problems, the axis of rotation must be the xaxis.

• For 3D geometries, you should generate the mesh with a specificrotational axis in mind for the rotating cell zone. Usually it isconvenient to use the x, y, or z axis, but FLUENT can accommodatearbitrary rotational axes.

9.2.4 Problem Setup for a Single Rotating Reference Frame

When you want to model a problem involving a single rotating referenceframe, you will need to complete the following modeling inputs. (Onlythose steps relevant specifically to the setup of a rotating reference frameproblem are listed here. You will need to set up the rest of the problemas usual.)



1. Select the Velocity Formulation to be used in the Solver panel: eitherRelative or Absolute. (See Section 9.2.5 for details.)

Define −→ Models −→Solver...

(Note that this step is irrelevant if you are using one of the coupledsolvers; these solvers always use an absolute velocity formulation.)

2. For each cell zone in the domain, specify the angular velocity (Ω)of the reference frame and the axis about which it rotates.

Define −→Boundary Conditions...

(a) In the Fluid panel or Solid panel, specify the Rotation-Axis Ori-gin and Rotation-Axis Direction to define the axis of rotation.

(b) Also in the Fluid or Solid panel, select Moving Reference Framein the Motion Type drop-down list and then set the Speedunder Rotational Velocity in the expanded portion of the panel.

Details about these inputs are presented in Section 6.17.1 for fluidzones, and in Section 6.18.1 for solid zones.

3. Define the velocity boundary conditions at walls. You can chooseto define either an absolute velocity or a velocity relative to themoving reference frame (i.e., relative to the velocity of the adjacentcell zone specified in step 2).

If the wall is moving at the speed of the rotating frame (and hencestationary in the rotating frame), it is convenient to specify a rel-ative angular velocity of zero. Likewise, a wall that is stationaryin the nonrotating frame of reference should be given a velocity ofzero in the absolute reference frame. Specifying the wall velocitiesin this manner obviates the need to modify these inputs later if achange is made in the rotational velocity of the fluid zone.

Details about these inputs are presented in Section 6.13.1.

4. Define the velocity at any velocity inlets and the flow directionand total pressure at any pressure inlets. For velocity inlets, youcan choose to define either absolute velocities or velocities relativeto the motion of the adjacent cell zone (specified in step 2). Forpressure inlets, the specification of the flow direction and total



pressure will be relative or absolute, depending on the velocityformulation you selected in step 1. See Section 9.2.5 for details. (Ifyou use one of the coupled solution algorithms, the specification isalways in the absolute frame.)

Details about these inputs are presented in Sections 6.3.1 and 6.4.1.

9.2.5 Choosing the Relative or Absolute Velocity Formulation

The absolute velocity formulation is preferred in applications where theflow in most of the domain is not rotating (e.g., a fan in a large room).The relative velocity formulation is appropriate when most of the fluidin the domain is rotating, as in the case of a large impeller in a mixingtank. For most applications, either formulation may be used.

When one of the coupled solution algorithms is used, the absolute for-!mulation is always used; the relative velocity formulation is not availablein the coupled solvers.

For velocity inlets and walls, you may specify velocity in either the abso-lute or the relative frame, regardless of whether the absolute or relativevelocity is used in the computation. For pressure boundary conditions,however, FLUENT imposes several restrictions on how total pressure andflow direction are specified in rotating reference frames.

The total pressure and flow direction at a pressure inlet must be specifiedin the absolute frame if the absolute velocity formulation is used. Forcalculations using relative velocities, the total pressure and flow directionmust be specified with respect to the rotating frame.

For pressure outlets, the specified static pressure is independent of frame.When there is backflow at a pressure outlet, however, the specified staticpressure is used as the total pressure. For calculations using absolutevelocities, the specified static pressure is used as the total pressure in theabsolute frame; for the relative velocity formulation, the specified staticpressure is assumed to be the total pressure in the relative frame. As forflow direction in reverse flows, FLUENT assumes the absolute velocity tobe normal to the pressure outlet for the absolute velocity formulation;for the relative velocity formulation, it is the relative velocity that isassumed to be normal to the pressure outlet.



9.2.6 Solution Strategies for a Rotating Reference Frame

The difficulties associated with solving flows in rotating reference framesare similar to those discussed in Section 8.4.5 for axisymmetric swirlingor rotating flows. The primary issue you must confront is the high degreeof coupling between the momentum equations when the influence of therotational terms is large. A high degree of rotation introduces a largeradial pressure gradient which drives the flow in the axial and radialdirections, thereby setting up a distribution of the swirl or rotation in thefield. This coupling may lead to instabilities in the solution process, andhence require special solution techniques to obtain a converged solution.Some techniques that may be beneficial include the following:

• (Segregated solver only) Consider switching the frame in whichvelocities are solved by changing the velocity formulation settingin the Solver panel. (See Section 9.2.5 for details.)

• (Segregated solver only) Use the PRESTO! scheme (enabled in theSolution Controls panel), which is well-suited for the steep pressuregradients involved in rotating flows.

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

• (Segregated solver only) Reduce the under-relaxation factors forthe velocities, perhaps to 0.3–0.5 or lower, if necessary.

• Begin the calculations using a low rotational speed, increasing therotational speed gradually in order to reach the final desired oper-ating condition (see below).

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

Gradual Increase of the Rotational Speed to Improve SolutionStability

Because the rotation of the reference frame and the rotation defined viaboundary conditions can lead to large complex forces in the flow, your



FLUENT calculations may be less stable as the speed of rotation (andhence the magnitude of these forces) increases. One of the most effectivecontrols you can exert on the solution is to start with a low rotationalspeed and then slowly increase the rotation up to the desired level. Theprocedure you use to accomplish this is as follows:

1. Set up the problem using a low rotational speed in your inputs forboundary conditions and for the angular velocity of the referenceframe. The rotational speed in this first attempt might be selectedas 10% of the actual operating condition.

2. Solve the problem at these conditions.

3. Save this initial solution data.

4. Modify your inputs (i.e., boundary conditions and angular velocityof the reference frame). Increase the speed of rotation, perhapsdoubling it.

5. Restart or continue the calculation using the solution data savedin Step 3 as the initial guess for the new calculation. Save the newdata.

6. Continue to increment the rotational speed, following Steps 4 and5, until you reach the desired operating condition.

9.2.7 Postprocessing for a Single Rotating Reference Frame

When you solve a problem in a rotating reference frame, you can plotor report both absolute and relative velocities. For all velocity parame-ters (e.g., Velocity Magnitude and Mach Number), corresponding relativevalues will be available for postprocessing (e.g., Relative Velocity Magni-tude and Relative Mach Number). These variables are contained in theVelocity... category of the variable selection drop-down list that appearsin postprocessing panels. Relative values are also available for postpro-cessing of total pressure, total temperature, and any other parametersthat include a dynamic contribution dependent on the reference frame(e.g., Relative Total Pressure, Relative Total Temperature).



When plotting velocity vectors, you can choose to plot vectors in theabsolute frame (the default), or you can select Relative Velocity in theVectors Of drop-down list in the Vectors panel to plot vectors in therotating frame. If you plot relative velocity vectors, you might want tocolor the vectors by relative velocity magnitude (by choosing RelativeVelocity Magnitude in the Color By list); by default they will be coloredby absolute velocity magnitude. Figures 9.2.4 and 9.2.5 show absoluteand relative velocity vectors in a rotating domain with a stationary outerwall.



Velocity Vectors Colored By Velocity Magnitude (m/s)

1.29e+00

1.18e+00

1.06e+00

9.46e-01

8.31e-01

7.16e-01

6.01e-01

4.87e-01

3.72e-01

2.57e-01

1.42e-01

Figure 9.2.4: Absolute Velocity Vectors

Velocity Vectors Colored By Relative Velocity Magnitude (m/s)

1.81e+00

1.63e+00

1.45e+00

1.27e+00

1.09e+00

9.07e-01

7.27e-01

5.47e-01

3.67e-01

1.87e-01

7.09e-03

Figure 9.2.5: Relative Velocity Vectors



9.3 The Multiple Reference Frame (MRF) Model

9.3.1 Overview

As mentioned in Section 9.1, FLUENT provides three approaches formodeling problems that involve both stationary and moving zones:

• the multiple reference frame (MRF) model

• the mixing plane model

• the sliding mesh model

The MRF model [146] is the simplest of the three. It is a steady-state approximation in which individual cell zones move at differentrotational/translational speeds. This approach is appropriate when theflow at the boundary between these zones is nearly uniform (“mixedout”).

While the multiple reference frame approach is clearly an approxima-tion, it can provide a reasonable model of the time-averaged flow formany applications. For example, the MRF model can be used for a tur-bomachinery application in which rotor-stator interaction is relativelyweak. In mixing tanks, since the impeller-baffle interactions are rela-tively weak, large-scale transient effects are not present and the MRFmodel can be used. In general, any problems where transients due torotor-stator interaction are small are candidates for the MRF model.

Another potential use of the MRF model is to compute a flow fieldthat can be used as an initial condition for a transient sliding meshcalculation. This eliminates the need for a startup calculation. Themultiple reference frame model should not be used, however, if it isnecessary to actually simulate the transients that may occur in strongimpeller-baffle interactions. For such cases, the sliding mesh model (seeSection 9.5) should be used.

Examples

For a mixing tank with a single impeller, you can define a rotating ref-erence frame that encompasses the impeller and the flow surrounding it,



and use a stationary frame for the flow outside the impeller region. Anexample of this configuration is illustrated in Figure 9.3.1. (The dashesdenote the interface between the two reference frames.) Steady-stateflow conditions are assumed at the interface between the two referenceframes. That is, the velocity at the interface must be the same (in ab-solute terms) for each reference frame. The grid does not move.

Figure 9.3.1: Geometry with One Rotating Impeller

You can also model a problem that includes more than one rotating ref-erence frame. Figure 9.3.2 shows a geometry that contains two rotatingimpellers side by side. This problem would be modeled using three ref-erence frames: the stationary frame outside both impeller regions andtwo separate rotating reference frames for the two impellers. (As notedabove, the dashes denote the interfaces between reference frames.)

Restrictions

The following restrictions apply to the use of multiple reference frames:

• Use of the realizable k-ε model with multiple reference frames isnot recommended.

• The boundaries separating a moving region from adjacent regionsmust be oriented such that the component of the frame velocity



Figure 9.3.2: Geometry with Two Rotating Impellers

normal to the boundary is zero. For the example shown in Fig-ure 9.3.1, this requires the dashed boundary to be circular (notsquare or any other shape). For a translationally moving frame,the moving zone’s boundaries must be parallel to the translationalvelocity vector.

• Strictly speaking, the use of multiple reference frames is meaning-ful only for steady flow. However, FLUENT will allow you to solvean unsteady flow when multiple reference frames are being used.In this case, unsteady terms (as described in Section 22.2.8) areadded to all the governing transport equations. You should care-fully consider whether this will yield meaningful results for yourapplication, because, for unsteady flows, a sliding mesh calculationwill generally yield more meaningful results than an MRF calcula-tion.

• Particle trajectories and pathlines drawn by FLUENT use the ve-locity relative to the cell zone motion. For massless particles, theresulting pathlines follow the streamlines based on relative velocityand are meaningful. For particles with mass, however, the particletracks displayed are meaningless. Similarly, coupled discrete-phase



calculations are meaningless.

An alternative approach for particle tracking and coupled discrete-phase calculations with multiple reference frames is to track parti-cles based on absolute velocity instead of relative velocity. To makethis change, use the define/models/dpm/tracking/track-in-absolute-frame text command. Note, however, that tracking par-ticles based on absolute velocity may result in incorrect particle-wall interaction. The particle injection velocities (specified in theSet Injection Properties panel) are defined relative to the frame ofreference in which the particles are tracked. By default, the in-jection velocities are specified relative to the local reference frame.If you enable the track-in-absolute-frame option, the injectionvelocities are specified relative to the absolute frame.

• You cannot accurately model axisymmetric swirl in the presenceof multiple reference frames using the relative velocity formula-tion. This is because the current implementation does not applythe transformation used in Equation 9.3-3 to the swirl velocityderivatives.

• Translational and rotational velocities are assumed to be constant(time varying ω, vt are not allowed).

9.3.2 The MRF Formulation

The MRF formulation will depend on the velocity formulation beingused.

Relative Velocity Formulation

In FLUENT’s implementation of the multiple reference frame feature,the calculation domain is divided into subdomains, each of which maybe rotating/translating with respect to the laboratory (inertial) frame.The governing equations in each subdomain are written with respectto that subdomain’s reference frame. Thus, the flow in stationary andtranslating subdomains is governed by the equations in Section 8.2, whilethe flow in rotating subdomains is governed by the equations presentedin Section 9.2.2.



At the boundary between two subdomains, the diffusion and other termsin the governing equations in one subdomain require values for the ve-locities in the adjacent subdomain. FLUENT enforces the continuity ofthe absolute velocity, ~v, to provide the correct neighbor values of velocityfor the subdomain under consideration. (This approach differs from themixing plane approach described in Section 9.4, where a circumferentialaveraging technique is used.)

When the relative velocity formulation is used, velocities in each subdo-main are computed relative to the motion of the subdomain. Velocitiesand velocity gradients are converted from a moving reference frame tothe absolute inertial frame as described below.

The position vector relative to the origin of the zone rotation axis isdefined as

~r = ~x− ~xo (9.3-1)

where ~x is the position in absolute Cartesian coordinates and ~xo is theorigin of the zone rotation axis, as shown in Figure 9.3.3.

The relative velocity in the moving reference frame can be converted tothe absolute (stationary) frame of reference using the following equation:

~v = ~vr + (~ω × ~r) + ~vt (9.3-2)

where ~v is the velocity in the absolute inertial reference frame, ~vr isthe velocity in the relative noninertial reference frame, and ~vt is thetranslational velocity of the noninertial reference frame.

Using this definition of absolute velocity, the gradient of the absolutevelocity vector is given by

∇~v = ∇~vr + ∇ (~ω × ~r) (9.3-3)



X

Y

Z

y

z

x

x

x

r

o

moving reference

absolute reference

frame

frame

Figure 9.3.3: Coordinate System for Relative Velocity

Absolute Velocity Formulation

When the absolute velocity formulation is used, the governing equationsin each subdomain are written with respect to that subdomain’s refer-ence frame, but the velocities are stored in the absolute frame. Thereforeno special transformation is required at the interface between two sub-domains.

9.3.3 Grid Setup for Multiple Reference Frames

Two grid setup methods are available. Choose the method that is ap-propriate for your model, noting the restrictions in Section 9.3.1.

• If the boundary between two zones that are in different referenceframes is conformal (i.e., the grid node locations are identical atthe boundary where the two zones meet), you can simply createthe grid as usual, with all cell zones contained in the same grid file.A different cell zone should exist for each portion of the domain



that is modeled in a different reference frame. Use an interior zonefor the boundary between reference frames.

• If the boundary between two zones that are in different referenceframes is non-conformal (i.e., the grid node locations are not iden-tical at the boundary where the two zones meet), follow the non-conformal grid setup procedure described in Section 5.4.3.

9.3.4 Problem Setup for Multiple Reference Frames

When you want to model a problem involving multiple reference frames,you will need to complete the following modeling inputs. (Only thosesteps relevant specifically to the setup of a multiple reference frame prob-lem are listed here. You will need to set up the rest of the problem asusual.)

The grid-setup constraints for a rotating reference frame listed in Sec-!tion 9.2.3 apply to multiple reference frames as well.

1. Select the Velocity Formulation to be used in the Solver panel: eitherAbsolute or Relative. (See Section 9.2.5 for details.)


(Note that this step is irrelevant if you are using one of the cou-pled solution algorithms; these algorithms always use an absolutevelocity formulation.)

2. For each cell zone in the domain, specify its translational velocityand/or its angular velocity (Ω) and the axis about which it rotates.


(a) If the zone is rotating, or if you plan to specify cylindricalvelocity or flow-direction components at inlets to the zone,you will need to define the axis of rotation. In the Fluid panelor Solid panel, specify the Rotation-Axis Origin and Rotation-Axis Direction.

(b) Also in the Fluid or Solid panel, select Moving Reference Framein the Motion Type drop-down list and then set the Speed



under Rotational Velocity and/or the X, Y, and Z componentsof the Translational Velocity in the expanded portion of thepanel.


3. Define the velocity boundary conditions at walls. You can chooseto define either an absolute velocity or a velocity relative to thevelocity of the adjacent cell zone specified in step 2.

If the wall is moving at the speed of the moving frame (and hencestationary relative to the moving frame), it is convenient to specifya relative angular velocity of zero. Likewise, a wall that is station-ary in the non-moving frame of reference should be given a velocityof zero in the absolute reference frame. Specifying the wall veloci-ties in this manner obviates the need to modify these inputs laterif a change is made in the rotational velocity of the fluid zone.

An example for which you would specify a relative velocity is asfollows: If an impeller is defined as wall-3 and the fluid regionwithin the impeller’s radius is defined as fluid-5, you would needto specify the angular velocity and axis of rotation for fluid-5 andthen assign wall-3 a relative velocity of 0. If you later wanted tomodel a different angular velocity for the impeller, you would needto change only the angular velocity of the fluid region; you wouldnot need to modify the wall velocity conditions.


4. Define the velocity at any velocity inlets and the flow directionand total pressure at any pressure inlets. For velocity inlets, youcan choose to define either absolute velocities or velocities relativeto the motion of the adjacent cell zone (specified in step 2). Forpressure inlets, the specification of the flow direction and totalpressure will be relative or absolute, depending on the velocityformulation you selected in step 1. See Section 9.2.5 for details. (Ifyou use one of the coupled solution algorithms, the specification isalways in the absolute frame.)

Details about these inputs are presented in Sections 6.3.1 and 6.4.1.



9.3.5 Solution Strategies for Multiple Reference Frames

No special solution strategies are necessary for translating reference frames.For multiple rotating reference frames, follow the guidelines presented inSection 9.2.6 for a single rotating reference frame.

9.3.6 Postprocessing for Multiple Reference Frames

When you solve a problem in multiple reference frames, you can plot orreport both absolute and relative velocities. For all velocity parameters(e.g., Velocity Magnitude and Mach Number), corresponding relative val-ues will be available for postprocessing (e.g., Relative Velocity Magnitudeand Relative Mach Number). These variables are contained in the Veloc-ity... category of the variable selection drop-down list that appears inpostprocessing panels. Relative values are also available for postprocess-ing of total pressure, total temperature, and any other parameters thatinclude a dynamic contribution dependent on the reference frame (e.g.,Relative Total Pressure, Relative Total Temperature).

Relative velocities are relative to the translational/rotational velocity!of the “reference zone” (specified in the Reference Values panel). Thevelocity of the reference zone is the velocity defined in the Fluid panelfor that zone.

When plotting velocity vectors, you can choose to plot vectors in theabsolute frame (the default), or you can select Relative Velocity in theVectors Of drop-down list in the Vectors panel to plot vectors relativeto the translational/rotational velocity of the “reference zone” (specifiedin the Reference Values panel). If you plot relative velocity vectors, youmight want to color the vectors by relative velocity magnitude (by choos-ing Relative Velocity Magnitude in the Color By list); by default they willbe colored by absolute velocity magnitude.

You can also generate a plot of circumferential averages in FLUENT. Thisallows you to find the average value of a quantity at several differentradial or axial positions in your model. FLUENT computes the averageof the quantity over a specified circumferential area, and then plots theaverage against the radial or axial coordinate. For more information ongenerating XY plots of circumferential averages, see Section 25.8.4.


9.4 The Mixing Plane Model


9.4.1 Overview and Limitations

The mixing plane model in FLUENT provides an alternative to the multi-ple reference frame and sliding mesh models for simulating flow throughdomains with one or more regions in relative motion. This section pro-vides a brief overview of the model and a list of its limitations.

Overview

As discussed in Section 9.3.1, the MRF model is applicable when the flowat the boundary between adjacent zones that move at different speedsis nearly uniform (“mixed out”). If the flow at this boundary is notuniform, the MRF model may not provide a physically meaningful solu-tion. The sliding mesh model (see Section 9.5) may be appropriate forsuch cases, but in many situations it is not practical to employ a slid-ing mesh. For example, in a multistage turbomachine, if the number ofblades is different for each blade row, a large number of blade passagesis required in order to maintain circumferential periodicity. Moreover,sliding mesh calculations are necessarily unsteady, and thus require sig-nificantly more computation to achieve a final, time-periodic solution.For situations where using the sliding mesh model is not feasible, themixing plane model can be a cost-effective alternative.

In the mixing plane approach, each fluid zone is treated as a steady-stateproblem. Flow-field data from adjacent zones are passed as boundaryconditions that are spatially averaged or “mixed” at the mixing planeinterface. This mixing removes any unsteadiness that would arise dueto circumferential variations in the passage-to-passage flow field (e.g.,wakes, shock waves, separated flow), thus yielding a steady-state re-sult. Despite the simplifications inherent in the mixing plane model, theresulting solutions can provide reasonable approximations of the time-averaged flow field.

Limitations

Note the following limitations of the mixing plane model:



• The mixing plane model requires the use of the absolute velocityformulation; you cannot use the relative velocity formulation withthe mixing plane model.

• The LES turbulence model cannot be used with the mixing planemodel.

• The models for species transport and combustion cannot be usedwith the mixing plane model.

• The general multiphase models (VOF, mixture, and Eulerian) can-not be used with the mixing plane model.

9.4.2 Mixing Plane Theory

Rotor and Stator Domains

Consider the turbomachine stages shown schematically in Figures 9.4.1and 9.4.2. Figure 9.4.1 shows a constant radial plane within a singlestage of an axial machine, while Figure 9.4.2 shows a constant θ planewithin a mixed-flow device. In each case, the stage consists of two flowdomains: the rotor domain, which is rotating at a prescribed angularvelocity, followed by the stator domain, which is stationary. The orderof the rotor and stator is arbitrary (that is, a situation where the rotoris downstream of the stator is equally valid).

In a numerical simulation, each domain will be represented by a separatemesh. The flow information between these domains will be coupled atthe mixing plane interface (as shown in Figures 9.4.1 and 9.4.2) usingthe mixing plane model. Note that you may couple any number of fluidzones in this manner; for example, four blade passages can be coupledusing three mixing planes.

Note that the stator and rotor meshes do not have to be conformal; that!is, the nodes on the stator exit boundary do not have to match the nodeson the rotor inlet boundary. In addition, the meshes can be of differenttypes (e.g., the stator can have a hexahedral mesh while the rotor has atetrahedral mesh).



rotor stator

rotor outlet: ps

mixing plane interface

stator inlet: xα yα zα εk 0 p

x

Rθ

Figure 9.4.1: Axial Rotor-Stator Interaction (Schematic Illustrating theMixing Plane Concept)

The Mixing Plane Concept

The essential idea behind the mixing plane concept is that each fluidzone is solved as a steady-state problem. At some prescribed iterationinterval, the flow data at the mixing plane interface are averaged in thecircumferential direction on both the stator outlet and the rotor inletboundaries. The FLUENT implementation uses area-weighted averages.By performing circumferential averages at specified radial or axial sta-tions, “profiles” of flow properties can be defined. These profiles—whichwill be functions of either the axial or the radial coordinate, depend-ing on the orientation of the mixing plane—are then used to updateboundary conditions along the two zones of the mixing plane interface.In the examples shown in Figures 9.4.1 and 9.4.2, profiles of averagedtotal pressure (p0), direction cosines of the local flow angles in the ra-dial, tangential, and axial directions (αr, αt, αz), total temperature (T0),turbulence kinetic energy (k), and turbulence dissipation rate (ε) arecomputed at the rotor exit and used to update boundary conditions atthe stator inlet. Likewise, a profile of static pressure (ps) is computed atthe stator inlet and used as a boundary condition on the rotor exit.

Passing profiles in the manner described above assumes specific boundary



stator

rotor

rotor outlet: ps

mixing plane

stator inlet:

xα yα zα εk0

p

interfacer

x

Ω

Figure 9.4.2: Radial Rotor-Stator Interaction (Schematic Illustrating theMixing Plane Concept)



condition types have been defined at the mixing plane interface. Thecoupling of an upstream outlet boundary zone with a downstream inletboundary zone is called a “mixing plane pair”. In order to create mixingplane pairs in FLUENT, the boundary zones must be of the followingtypes:

upstream downstreampressure outlet pressure inletpressure outlet velocity inletpressure outlet mass flow inlet

Specific instructions for setting up mixing planes are provided in Sec-tion 9.4.3.

FLUENT’s Mixing Plane Algorithm

FLUENT’s basic mixing plane algorithm can now be described:

1. Update the flow field solutions in the stator and rotor domains.

2. Average the flow properties at the stator exit and rotor inlet bound-aries, obtaining profiles for use in updating boundary conditions.

3. Pass the profiles to the boundary condition inputs required for thestator exit and rotor inlet.

4. Repeat steps 1–3 until convergence.

Note that it may be desirable to under-relax the changes in boundary!condition values in order to prevent divergence of the solution (especiallyearly in the computation). FLUENT allows you to control the under-relaxation of the mixing plane variables.

Mass Conservation

Note that the algorithm described above will not rigorously conservemass flow across the mixing plane if it is represented by a pressure in-let and pressure outlet mixing plane pair. If you use a mass flow inlet



and pressure outlet pair instead, FLUENT will force mass conservationacross the mixing plane. The basic technique consists of computing themass flow rate across the upstream zone (pressure outlet) and adjustingthe mass flux profile applied at the mass flow inlet such that the down-stream mass flow matches the upstream mass flow. This adjustmentoccurs at every iteration, thus ensuring rigorous conservation of massflow throughout the course of the calculation.

Note that, since mass flow is being fixed in this case, there will be a!jump in total pressure across the mixing plane. The magnitude of thisjump is usually small compared with total pressure variations elsewherein the flow field.

Swirl Conservation

By default, FLUENT does not conserve swirl across the mixing plane.For applications such as torque converters, where the sum of the torquesacting on the components should be zero, enforcing swirl conservationacross the mixing plane is essential, and is available in FLUENT as amodeling option. Ensuring conservation of swirl is important because,otherwise, sources or sinks of tangential momentum will be present atthe mixing plane interface.

Consider a control volume containing a stationary or rotating component(e.g., a pump impeller or turbine vane). Using the moment of momentumequation from fluid mechanics, it can be shown that for steady flow,

T =∫∫

Srvθρ~v · ndS (9.4-1)

where T is the torque of the fluid acting on the component, r is theradial distance from the axis of rotation, vθ is the absolute tangentialvelocity, ~v is the total absolute velocity, and S is the boundary surface.(The product rvθ is referred to as swirl.)

For a circumferentially periodic domain, with well-defined inlet and out-let boundaries, Equation 9.4-1 becomes



T =∫∫

outletrvθρ~v · ndS +

∫∫inlet

rvθρ~v · ndS (9.4-2)

where inlet and outlet denote the inlet and outlet boundary surfaces.

Now consider the mixing plane interface to have a finite streamwise thick-ness. Applying Equation 9.4-2 to this zone and noting that, in the limitas the thickness shrinks to zero, the torque should vanish, the equationbecomes

∫∫downstream

rvθρ~v · ndS =∫∫

upstreamrvθρ~v · ndS (9.4-3)

where upstream and downstream denote the upstream and downstreamsides of the mixing plane interface. Note that Equation 9.4-3 applies tothe full area (360 degrees) at the mixing plane interface.

Equation 9.4-3 provides a rational means of determining the tangentialvelocity component. That is, FLUENT computes a profile of tangentialvelocity and then uniformly adjusts the profile such that the swirl integralis satisfied. Note that interpolating the tangential (and radial) velocitycomponent profiles at the mixing plane does not affect mass conservationbecause these velocity components are orthogonal to the face-normalvelocity used in computing the mass flux.

9.4.3 Problem Setup for a Mixing Plane Model

The model inputs for mixing planes are presented in this section. Onlythose steps relevant specifically to the setup of a mixing plane problemare listed here. You will need to set up the rest of the problem as usual.Note that the use of wall and periodic boundaries in a mixing planemodel is consistent with their use when the model is not active.

1. Select the (default) absolute velocity formulation in the Solverpanel.




2. For each cell zone in the domain, specify its angular velocity (Ω)and the axis about which it rotates.


(a) If the zone is rotating, or if you plan to specify cylindrical-velocity or flow-direction components at inlets to the zone,you will need to define the axis of rotation. In the Fluid panelor Solid panel, specify the Rotation-Axis Origin and Rotation-Axis Direction.

(b) Also in the Fluid or Solid panel, select Moving Reference Framein the Motion Type drop-down list and then set the Speedunder Rotational Velocity and/or the X, Y, and Z componentsof the Translational Velocity in the expanded portion of thepanel.


It is important to define the axis of rotation for the cell zones on!both sides of the mixing plane interface, including the stationaryzone.

3. Define the velocity boundary conditions at walls. You can chooseto define either an absolute velocity or a velocity relative to thevelocity of the adjacent cell zone specified in step 2.

If the wall is moving at the speed of the moving frame (and hencestationary relative to the moving frame), it is convenient to specifya relative angular velocity of zero. Likewise, a wall that is station-ary in the non-moving frame of reference should be given a velocityof zero in the absolute reference frame. Specifying the wall veloci-ties in this manner obviates the need to modify these inputs laterif a change is made in the rotational velocity of the fluid zone.

An example for which you would specify a relative velocity is asfollows: If an impeller is defined as wall-3 and the fluid regionwithin the impeller’s radius is defined as fluid-5, you would needto specify the angular velocity and axis of rotation for fluid-5 andthen assign wall-3 a relative velocity of 0. If you later wanted tomodel a different angular velocity for the impeller, you would need



to change only the angular velocity of the fluid region; you wouldnot need to modify the wall velocity conditions.


4. Define the velocity at any velocity inlets and the flow directionand total pressure at any pressure inlets or mass flow inlets. Forvelocity inlets, you can choose to define either absolute velocities orvelocities relative to the motion of the adjacent cell zone (specifiedin step 2). For pressure inlets and mass flow inlets, the specificationof the flow direction and total pressure will always be absolute,because the absolute velocity formulation is always used for mixingplane calculations. For a mass flow inlet, you do not need to specifythe mass flow rate or mass flux. FLUENT will automatically selectthe Mass Flux with Average Mass Flux specification method and setthe correct values when you create the mixing plane, as describedin Section 6.5.1.

Details about these inputs are presented in Sections 6.3.1, 6.4.1,and 6.5.1.

Note that the outlet boundary zone at the mixing plane interface!must be defined as a pressure outlet, and the inlet boundary zoneat the mixing plane interface must be defined as a velocity inlet(incompressible flow only), a pressure inlet, or a mass flow inlet.The overall inlet and exit boundary conditions can be any suitablecombination permitted by the solver (e.g., velocity inlet, pressureinlet, or mass flow inlet; pressure outlet). Keep in mind, however,that if mass conservation across the mixing plane is important, youneed to use a mass flow inlet as the downstream boundary; massconservation is not maintained across the mixing plane when youuse a velocity inlet or pressure inlet.

5. Define the mixing planes in the Mixing Planes panel (Figure 9.4.3).

Define −→Mixing Planes...

(a) Specify the two zones that comprise the mixing plane by se-lecting an upstream zone in the Upstream Zone list and adownstream zone in the Downstream Zone list. It is essentialthat the correct pairs be chosen from these lists (i.e., that the



Figure 9.4.3: The Mixing Planes Panel

boundary zones selected lie on the mixing plane interface).You can check this by displaying the grid.

Display −→Grid...

(b) (3D only) Indicate the geometry of the mixing plane interfaceby choosing one of the options under Mixing Plane Geometry.

A Radial geometry signifies that information at the mixingplane interface is to be circumferentially averaged into profilesthat vary in the radial direction, e.g., p(r), T (r). This is thecase for axial-flow machines, for example.

An Axial geometry signifies that circumferentially averagedprofiles are to be constructed that vary in the axial direction,e.g., p(x), T (x). This is the situation for a radial-flow device.

Note that the radial direction is normal to the rotation axis!for the fluid zone and the axial direction is parallel to therotation axis.

(c) (3D only) Set the number of Interpolation Points. This is the



number of radial or axial locations used in constructing theboundary profiles for circumferential averaging. You shouldchoose a number that approximately corresponds to the reso-lution of the surface mesh in the radial or axial direction. Notethat while you can use more points if you wish, the resolutionof the boundary profile will only be as fine as the resolutionof the surface mesh itself.

In 2D the flow data are averaged over the entire interface tocreate a profile consisting of a single data point. For thisreason you do not need to set the number of InterpolationPoints or select a Mixing Plane Geometry in 2D.

(d) Set the Global Parameters for the mixing plane.

i. Set the Under-Relaxation parameter. It is sometimes de-sirable to under-relax the changes in boundary values atmixing planes as these may change very rapidly duringthe early iterations of the solution and cause the calcula-tion to diverge. The changes can be relaxed by specifyingan under-relaxation less than 1. The new boundary pro-file values are then computed using

φnew = φold + α(φcalculated − φold) (9.4-4)

where α is the under-relaxation factor. Once the flowfield is established, the value of α can be increased.

ii. Click Apply to set the Global Parameters. If the Defaultbutton is visible to the right of the Apply button, clickingthe Default button will return Global Parameters back totheir default values. The Default button will then changeto be a Reset button. Clicking the Reset button willchange the Global Parameters back to the values that werelast applied.

(e) Click Create to create a new mixing plane. FLUENT will namethe mixing plane by combining the names of the zones selectedas the Upstream Zone and Downstream Zone and enter the newmixing plane in the Mixing Plane list.



If you create an incorrect mixing plane, you can select it inthe Mixing Plane list and click the Delete button to delete it.

Modeling Options

There are two options available for use with the mixing plane model: afixed pressure level for incompressible flows, and the swirl conservationdescribed in Section 9.4.2.

Fixing the Pressure Level for an Incompressible Flow

For certain turbomachinery configurations, such as a torque converter,there is no fixed-pressure boundary when the mixing plane model isused. The mixing plane model is usually used to model the three in-terfaces that connect the components of the torque converter. In thisconfiguration, the pressure is no longer fixed. As a result, the pressuremay float unbounded, making it difficult to obtain a converged solution.

To resolve this problem, FLUENT offers an option for fixing the pressurelevel. When this option is enabled, FLUENT will adjust the gauge pres-sure field after each iteration by subtracting from it the pressure valuein the cell closest to the Reference Pressure Location in the OperatingConditions panel.

This option is available only for incompressible flows calculated using!the segregated solver.

To enable the fixed pressure option, use the fix-pressure-level textcommand:

define −→ mixing-planes −→ set −→fix-pressure-level

Conserving Swirl Across the Mixing Plane

As discussed in Section 9.4.2, conservation of swirl is important for ap-plications such as torque converters. If you want to enable swirl con-servation across the mixing plane, you can use the commands in theconserve-swirl text menu:

define −→ mixing-planes −→ set −→conserve-swirl



To turn on swirl conservation, use the enable? text command. Once theoption is turned on, you can ask the solver to report information aboutthe swirl conservation during the calculation. If you turn on verbosity?,FLUENT will report for every iteration the zone ID for the zone on whichthe swirl conservation is active, the upstream and downstream swirl in-tegration per zone area, and the ratio of upstream to downstream swirlintegration before and after the correction.

To obtain a report of the swirl integration at every pressure inlet, pres-sure outlet, velocity inlet, and mass flow inlet in the domain, use thereport-swirl-integration command. You can use this informationto determine the torque acting on each component of the turbomachin-ery according to Equation 9.4-2.

9.4.4 Solution Strategies for Problems with Mixing Planes

It should be emphasized that the mixing plane model is a reasonable ap-proximation so long as there is not significant reverse flow in the vicinityof the mixing plane. If significant reverse flow occurs, the mixing planewill not be a satisfactory model of the actual flow. In a numerical simu-lation, reverse flow often occurs during the early stages of the computa-tion even though the flow at convergence is not reversed. Therefore, itis helpful in these situations to first obtain a provisional solution usingfixed conditions at the rotor-stator interface. The mixing plane modelcan then be enabled and the solution run to convergence.

Under-relaxing the changes in the mixing plane boundary values canalso help in troublesome situations. In many cases, setting the under-relaxation factor to a value less than one can be helpful. Once theflow field is established, you can gradually increase the under-relaxationfactor.

9.4.5 Postprocessing for the Mixing Plane Model

When you solve a problem using the mixing plane model, you can plotor report both absolute and relative velocities. For all velocity parame-ters (e.g., Velocity Magnitude and Mach Number), corresponding relativevalues will be available for postprocessing (e.g., Relative Velocity Magni-tude and Relative Mach Number). These variables are contained in the



Velocity... category of the variable selection drop-down list that appearsin postprocessing panels. Relative values are also available for postpro-cessing of total pressure, total temperature, and any other parametersthat include a dynamic contribution dependent on the reference frame(e.g., Relative Total Pressure, Relative Total Temperature).

Relative velocities are relative to the translational/rotational velocity!of the “reference zone” (specified in the Reference Values panel). Thevelocity of the reference zone is the velocity defined in the Fluid panelfor that zone.

When plotting velocity vectors, you can choose to plot vectors in theabsolute frame (the default), or you can select Relative Velocity in theVectors Of drop-down list in the Vectors panel to plot vectors relativeto the translational/rotational velocity of the “reference zone” (specifiedin the Reference Values panel). If you plot relative velocity vectors, youmight want to color the vectors by relative velocity magnitude (by choos-ing Relative Velocity Magnitude in the Color By list); by default they willbe colored by absolute velocity magnitude.

See also Section 25.9 for details about turbomachinery-specific postpro-cessing features.


9.5 Sliding Meshes

9.5 Sliding Meshes

9.5.1 Overview

When a time-accurate solution for rotor-stator interaction (rather than atime-averaged solution) is desired, you must use the sliding mesh modelto compute the unsteady flow field. As mentioned in Section 9.1, thesliding mesh model is the most accurate method for simulating flowsin multiple moving reference frames, but also the most computationallydemanding.

Most often, the unsteady solution that is sought in a sliding mesh sim-ulation is time-periodic. That is, the unsteady solution repeats with aperiod related to the speeds of the moving domains. However, you canmodel other types of transients, including translating sliding mesh zones(e.g., two cars or trains passing in a tunnel, as shown in Figure 9.5.1).

Figure 9.5.1: Two Passing Trains in a Tunnel

Note that for flow situations where there is no interaction between sta-tionary and moving parts (i.e., when there is only a rotor), the com-putational domain can be made stationary by using a rotating referenceframe. (See Section 9.2 for details.) When transient rotor-stator interac-tion is desired (as in the examples in Figures 9.5.2 and 9.5.3), you mustuse sliding meshes. If you are interested in a steady approximation ofthe interaction, you may use the multiple reference frame model or themixing plane model, as described in Sections 9.3 and 9.4.



flow

direction ofmotion

stationary vanes

rotating blades

Figure 9.5.2: Rotor-Stator Interaction (Stationary Guide Vanes withRotating Blades)

Figure 9.5.3: Blower


9.5 Sliding Meshes

The Sliding Mesh Technique

In the sliding mesh technique two or more cell zones are used. (If yougenerate the mesh in each zone independently, you will need to mergethe mesh files prior to starting the calculation, as described in Sec-tion 5.3.10.) Each cell zone is bounded by at least one “interface zone”where it meets the opposing cell zone. The interface zones of adjacentcell zones are associated with one another to form a “grid interface.” Thetwo cell zones will move relative to each other along the grid interface.

Note that the grid interface must be positioned so that it has fluid cells!on both sides. For example, the grid interface for the geometry shownin Figure 9.5.2 must lie in the fluid region between the rotor and stator;it cannot be on the edge of any part of the rotor or stator.

During the calculation, the cell zones slide (i.e., rotate or translate) rela-tive to one another along the grid interface in discrete steps. Figures 9.5.4and 9.5.5 show the initial position of two grids and their positions aftersome translation has occurred.

Figure 9.5.4: Initial Position of the Grids



Figure 9.5.5: Rotor Mesh Slides with Respect to the Stator

As the rotation or translation takes place, node alignment along thegrid interface is not required. Since the flow is inherently unsteady, atime-dependent solution procedure is required.

Grid Interface Shapes

The grid interface and the associated interface zones can be any shape,provided that the two interface boundaries are based on the same ge-ometry. Figure 9.5.6 shows an example with a linear grid interface andFigure 9.5.7 shows a circular-arc grid interface. (In both figures, the gridinterface is designated by a dashed line.)

If Figure 9.5.6 were extruded to 3D, the resulting sliding interface wouldbe a planar rectangle; if Figure 9.5.7 were extruded to 3D, the resultinginterface would be a cylinder. Figure 9.5.8 shows an example that woulduse a conical grid interface. (The slanted, dashed lines represent theintersection of the conical interface with a 2D plane.)


9.5 Sliding Meshes

Figure 9.5.6: 2D Linear Grid Interface

Figure 9.5.7: 2D Circular-Arc Grid Interface



Figure 9.5.8: 3D Conical Grid Interface


9.5 Sliding Meshes

For an axial rotor/stator configuration, in which the rotating and sta-tionary parts are aligned axially instead of being concentric (see Fig-ure 9.5.9), the interface will be a planar sector. This planar sector isa cross-section of the domain perpendicular to the axis of rotation at aposition along the axis between the rotor and the stator.

planar sector grid interface

portion of domain being modeled

Figure 9.5.9: 3D Planar-Sector Grid Interface



9.5.2 Sliding Mesh Theory

As discussed in Section 9.5.1, the sliding mesh model allows adjacentgrids to slide relative to one another. In doing so, the grid faces do notneed to be aligned on the grid interface. This situation requires a meansof computing the flux across the two non-conformal interface zones ofeach grid interface.

To compute the interface flux, the intersection between the interfacezones is determined at each new time step. The resulting intersectionproduces one interior zone (a zone with fluid cells on both sides) and oneor more periodic zones. If the problem is not periodic, the intersectionproduces one interior zone and a pair of wall zones (which will be emptyif the two interface zones intersect entirely), as shown in Figure 9.5.10.(You will need to change these wall zones to some other appropriateboundary type.) The resultant interior zone corresponds to where thetwo interface zones overlap; the resultant periodic zone corresponds towhere they do not. The number of faces in these intersection zones willvary as the interface zones move relative to one another. Principally,fluxes across the grid interface are computed using the faces resultingfrom the intersection of the two interface zones, rather than from theinterface zone faces themselves.

interior zone"wall" zone "wall" zone

Figure 9.5.10: Zones Created by Nonperiodic Interface Intersection

In the example shown in Figure 9.5.11, the interface zones are composedof faces A-B and B-C, and faces D-E and E-F. The intersection of thesezones produces the faces a-d, d-b, b-e, etc. Faces produced in the regionwhere the two cell zones overlap (d-b, b-e, and e-c) are grouped to form


9.5 Sliding Meshes

an interior zone, while the remaining faces (a-d and c-f) are paired up toform a periodic zone. To compute the flux across the interface into cellIV, for example, face D-E is ignored and faces d-b and b-e are used in-stead, bringing information into cell IV from cells I and III, respectively.

A B C

D E F

a b e c fd

cell zone 1

cell zone 2

interfacezone 2

interfacezone 1

I II

III

IV VI

V

Figure 9.5.11: Two-Dimensional Grid Interface



9.5.3 Setup and Solution of a Sliding Mesh Problem

Grid Requirements

Before beginning the problem setup in FLUENT, be sure that the gridyou have created meets the following requirements:

• A different cell zone exists for each portion of the domain that issliding at a different speed.

• The grid interface must be situated such that there is no motionnormal to it.

• The grid interface can be any shape (including a non-planar sur-face, in 3D), provided that the two interface boundaries are basedon the same geometry. If there are sharp features in the mesh (e.g.,90-degree angles), it is especially important that both sides of theinterface closely follow that feature.

• If you create a single grid with multiple cell zones, you must besure that each cell zone has a distinct face zone on the slidingboundary. The face zones for two adjacent cell zones will have thesame position and shape, but one will correspond to one cell zoneand one to the other. (Note that it is also possible to create aseparate grid file for each of the cell zones, and then merge themas described in Section 5.3.10.)

• If you are modeling a rotor/stator geometry using periodicity, theperiodic angle of the mesh around the rotor blade(s) must be thesame as that of the mesh around the stationary vane(s).

• All periodic zones must be correctly oriented (either rotational ortranslational) before you create the grid interface.

• For 3D cases, if the interface is periodic, only one pair of periodicboundaries can neighbor the interface.

See Section 9.5.1 for details about these restrictions and general infor-mation about how the sliding mesh model works in FLUENT.


9.5 Sliding Meshes

Setting Up the Problem

The steps for setting up a sliding mesh problem are listed below. (Notethat this procedure includes only those steps necessary for the slidingmesh model itself; you will need to set up other models, boundary con-ditions, etc. as usual.)

1. Enable the appropriate option for modeling unsteady flow in theSolver panel. (See Section 22.15 for details about the unsteadymodeling capabilities in FLUENT.)


2. Set boundary conditions for the sliding action:


(a) Change the zone type of the interface zones of adjacent cellzones to interface in the Boundary Conditions panel.

(b) In the Fluid panel or Solid panel for each moving fluid or solidzone, select Moving Mesh in the Motion Type drop-down listand set the translational and/or rotational velocity. (Notethat a solid zone cannot move at a different speed than anadjacent fluid zone.)

Note that simultaneous translation and rotation can be mod-!eled only if the rotation axis and the translation direction arethe same (i.e., the origin is fixed).

By default, the velocity of a wall is set to zero relative to the adja-cent mesh’s motion. For walls bounding a moving mesh this resultsin a “no-slip” condition in the reference frame of the mesh. There-fore, you need not modify the wall velocity boundary conditionsunless the wall is stationary in the absolute frame, and thereforemoving in the relative frame. See Section 6.13.1 for details aboutwall motion.

See Chapter 6 for details about input of boundary conditions.

3. Define the grid interfaces in the Grid Interfaces panel (Figure 9.5.12).

Define −→Grid Interfaces...



Figure 9.5.12: The Grid Interfaces Panel

(a) Enter a name for the interface in the Grid Interface field.

(b) Specify the two interface zones that comprise the grid inter-face by selecting one in the Interface Zone 1 list and one inthe Interface Zone 2 list. (The order does not matter.)

(c) Set the Interface Type, if appropriate. There are two options:

• Enable Periodic for periodic problems.

• Enable Coupled if the interface lies between a solid zoneand a fluid zone.

(d) Click on Create to create a new grid interface.

For all types of interfaces, FLUENT will create boundary zonesfor the interface (e.g., wall-9, wall-10), which will appearunder Boundary Zone 1 and Boundary Zone 2. You can use


9.5 Sliding Meshes

the Boundary Conditions panel to change them to another zonetype (e.g., pressure far-field, symmetry, pressure outlet).

If you have enabled the Coupled option, FLUENT will also cre-ate wall interface zones (e.g., wall-4, wall-4-shadow), whichwill appear under Interface Wall Zone 1 and Interface Wall Zone2.

If you create an incorrect grid interface, you can select it in theGrid Interface list and click on the Delete button to delete it. (Anyboundary zones that were created when the interface was createdwill also be deleted.)

When you have completed the problem setup, you should save an initial!case file so that you can easily return to the original grid position (i.e.,the positions before any sliding occurs). The grid position is stored inthe case file, so case files that you save at different times during theunsteady calculation will contain grids at different positions.

Solving the Problem

You will begin the sliding mesh calculation by initializing the solution(as described in Section 22.13.1) and then specifying the time step sizeand number of time steps in the Iterate panel, as for any other unsteadycalculation. (See Section 22.15 for details about time-dependent solu-tions. Note that if you wish to save the time step size in the initial casefile, you can click Apply instead of Iterate and then save a case file beforestarting to iterate.) FLUENT will iterate on the current time step so-lution until satisfactory residual reduction is achieved, or the maximumnumber of iterations per time step is reached. When it advances to thenext time step, the cell and wall zones will automatically be moved ac-cording to the specified translational or rotational velocities (set in step2b above). The new interface-zone intersections will be computed auto-matically, and resultant interior/periodic/external boundary zones willbe updated (as described in Section 9.5.2).



Saving Case and Data Files

FLUENT’s automatic saving of case and data files (see Section 3.3.4) canbe used with the sliding mesh model. This provides a convenient wayfor you to save results at successive time steps for later postprocessing.

You must save a case file each time you save a data file because the grid!position is stored in the case file. Since the grid position changes witheach time step, reading data for a given time step will require the casefile at that time step so that the grid will be in the proper position. Youshould also save your initial case file so that you can easily return to thegrid’s original position to restart the solution if desired.

If you are planning to solve your sliding mesh model in several stages,!whereby you run the calculation for some period of time, save case anddata files, exit FLUENT, start a new FLUENT session, read the case anddata files, continue the calculation for some time, save case and data files,exit FLUENT, and so on, there may be some distortion in the mesh witheach subsequent continuation of the calculation. To avoid this problem,you can delete the grid interface before saving the case file, and thencreate it again (as described in step 3 above) after you read the case fileinto a new FLUENT session.

Time-Periodic Solutions

For some problems (e.g., rotor-stator interactions), you may be interestedin a time-periodic solution. That is, the startup transient behavior maynot be of interest to you. Once this startup phase has passed, the flow willstart to exhibit time-periodic behavior. If T is the period of unsteadiness,then for some flow property φ at a given point in the flow field:

φ(t) = φ(t+NT ) (N = 1, 2, 3, ...) (9.5-1)

For rotating problems, the period (in seconds) can be calculated by di-viding the sector angle of the domain (in radians) by the rotor speed(in radians/sec): T = θ/Ω. For 2D rotor-stator problems, T = P/vb,where P is the pitch and vb is the blade speed. The number of timesteps in a period can be determined by dividing the time period by the


9.5 Sliding Meshes

time step size. When the solution field does not change from one periodto the next (for example, if the change is less than 5%), a time-periodicsolution has been reached.

To determine how the solution changes from one period to the next, youwill need to compare the solution at some point in the flow field over twoperiods. For example, if the time period is 10 seconds, you can comparethe solution at a given point after 22 seconds with the solution after 32seconds to see if a time-periodic solution has been reached. If not, youcan continue the calculation for another period and compare the solutionsafter 32 and 42 seconds, and so on until you see little or no change fromone period to the next. You can also track global quantities, such as liftand drag coefficients and mass flow, in the same manner. Figure 9.5.13shows a lift coefficient plot for a time-periodic solution.

Z

Y

X

Cl

Time

Cl

0.110.10.090.080.070.060.050.040.030.020.010

-5.00e+00

-5.10e+00

-5.20e+00

-5.30e+00

-5.40e+00

-5.50e+00

-5.60e+00

-5.70e+00

-5.80e+00

-5.90e+00

-6.00e+00

Figure 9.5.13: Lift Coefficient Plot for a Time-Periodic Solution

The final time-periodic solution is independent of the time steps takenduring the initial stages of the solution procedure. You can thereforedefine “large” time steps in the initial stages of the calculation, sinceyou are not interested in a time-accurate solution for the startup phaseof the flow. Starting out with large time steps will allow the solution



to become time-periodic more quickly. As the solution becomes time-periodic, however, you should reduce the time step in order to achieve atime-accurate result.

If you are solving with second-order time accuracy, the temporal accu-!racy of the solution will be affected if you change the time step duringthe calculation. You may start out with larger time steps, but you shouldnot change the time step by more than 20% during the solution process.You should not change the time step at all during the last several periodsto ensure that the solution has approached a time-periodic state.

9.5.4 Postprocessing for Sliding Meshes

Postprocessing for sliding mesh problems is the same as for other un-steady problems. You will read in the case and data file for the time ofinterest and display and report results as usual. For spatially-periodicproblems, you may want to use periodic repeats (set in the Views panel,as described in Section 25.4) to display the geometry. Figure 9.5.14shows the flow field for the rotor-stator example of Figure 9.5.4 at oneinstant in time, using 1 periodic repeat.

When displaying velocity vectors, note that absolute velocities (i.e., ve-locities in the inertial, or laboratory, reference frame) are displayed bydefault. You may also choose to display relative velocities by selectingRelative Velocity in the Vectors Of drop-down list in the Vectors panel.In this case, velocities relative to the translational/rotational velocityof the “reference zone” (specified in the Reference Values panel) will bedisplayed. (The velocity of the reference zone is the velocity defined inthe Fluid panel for that zone.)

Note that you cannot create zone surfaces for the intersection boundaries(i.e., the interior/periodic/external zones created from the intersectionof the interface zones). You may instead create zone surfaces for theinterface zones. Data displayed on these surfaces will be “one-sided”.That is, nodes on the interface zones will “see” only the cells on one sideof the grid interface, and slight discontinuities may appear when youplot contour lines across the interface. Note also that, for non-planarinterface shapes in 3D, you may see small gaps in your plots of filledcontours. These discontinuities and gaps are only graphical in nature.


9.5 Sliding Meshes

Contours of Static Pressure (pascal) (Time=1.0400e-01)

1.01e+05

1.01e+05

9.98e+04

9.91e+04

9.83e+04

9.75e+04

9.68e+04

9.60e+04

9.52e+04

9.45e+04

9.37e+04

Figure 9.5.14: Contours of Static Pressure for the Rotor-Stator Example

The solution does not have these discontinuities or gaps.

You can also generate a plot of circumferential averages in FLUENT. Thisallows you to find the average value of a quantity at several differentradial or axial positions in your model. FLUENT computes the averageof the quantity over a specified circumferential area, and then plots theaverage against the radial or axial coordinate. For more information ongenerating XY plots of circumferential averages, see Section 25.8.4.



9.6 Non-Reflecting Boundary Conditions

Information about non-reflecting boundary conditions (NRBCs) is pro-vided in the following sections.

• Section 9.6.1: Overview and Limitations

• Section 9.6.2: Theory

• Section 9.6.3: Using the Non-Reflecting Boundary Conditions

9.6.1 Overview and Limitations

The standard pressure boundary conditions for compressible flow fix spe-cific flow variables at the boundary (e.g., static pressure at an outletboundary). As a result, pressure waves incident on the boundary willreflect in an unphysical manner, leading to local errors. The effects aremore pronounced for internal flow problems where boundaries are usu-ally close to geometry inside the domain, such as compressor or turbineblade rows.

The non-reflecting boundary conditions (NRBCs) permit waves to “pass”through the boundaries without spurious reflections. The method usedin FLUENT is based on the Fourier transformation of solution variablesat the non-reflecting boundary [78]. Similar implementations have beeninvestigated by other authors [155, 198].

In the method used by FLUENT, the solution is rearranged as a sum ofterms corresponding to different frequencies, and their contributions arecalculated independently. While the method was originally designed foraxial turbomachinery, it has been extended for use with radial turboma-chinery.

Limitations

Note the following limitations of NRBCs:

• NRBCs can be used only with the coupled explicit solver.



• The current implementation applies to steady compressible flows,with the density calculated using the ideal gas law.

• Inlet and outlet boundary conditions must be pressure inlets andoutlets only.

Note that the pressure inlet boundaries must be set to the cylin-!drical coordinate flow specification method when NRBCs are used.

• Quad-mapped (structured) surface meshes must be used for inflowand outflow boundaries in a 3D geometry (i.e., triangular or quad-paved surface meshes are not allowed). See Figures 9.6.1 and 9.6.2for examples.

Note that you may use unstructured meshes in 2D geometries (Fig-ure 9.6.3), and away from the inlet and outlet boundaries in 3Dgeometries.

periodicboundaries

pressure inlet, structured quad mesh

pressure outlet,structured quadmesh

Figure 9.6.1: Mesh and Prescribed Boundary Conditions in a 3D AxialFlow Problem



pressure inlet, structuredquad mesh

periodic boundaries

pressure outlet, structured quad mesh

Figure 9.6.2: Mesh and Prescribed Boundary Conditions in a 3D RadialFlow Problem



periodic boundaries

pressure inlets, structured or unstructured mesh

pressure outlets,structured or unstructured mesh

or

Figure 9.6.3: Mesh and Prescribed Boundary Conditions in a 2D Case



9.6.2 Theory

The non-reflecting boundary conditions (NRBCs) are based on Fourierdecomposition of solutions to the linearized Euler equations. The solu-tion at the inlet and outlet boundaries is circumferentially decomposedinto Fourier modes, with the 0th mode representing the average bound-ary value (which is to be imposed as a user input), and higher harmonicsthat are modified to eliminate reflections [198].

Equations in Characteristic Variable Form

In order to treat individual waves, the linearized Euler equations aretransformed to characteristic variable (Ci) form. If we first considerthe 1D form of the linearized Euler equations, it can be shown that thecharacteristic variables Ci are related to the solution variables as follows:

Q = T−1C (9.6-1)

where

Q =

ρua

ut

ur

p

, T−1 =

− 1a2 0 0 1

2a21

2a2

0 0 0 12 ρ a

12 ρ a

0 1ρ a 0 0 0

0 0 1ρ a 0 0

0 0 0 12

12

, C =

C1

C2

C3

C4

C5

where a is the average acoustic speed along a boundary zone, ρ, ua,ut, ur, and p represent perturbations from a uniform condition (e.g.,ρ = ρ− ρ, p = p− p, etc.).

Note that the analysis is performed using the cylindrical coordinate sys-tem. All overlined (averaged) flow field variables (e.g., ρ, a) are intendedto be averaged along the pitchwise direction.

In quasi-3D approaches [78, 155, 198], a procedure is developed to de-termine the changes in the characteristic variables, denoted by δCi, atthe boundaries such that waves will not reflect. These changes in char-acteristic variables are determined as follows:



δC = T δQ (9.6-2)

where

δC =

δC1

δC2

δC3

δC4

δC5

, T =

−a2 0 0 0 10 0 ρ a 0 00 0 0 ρ a 00 ρ a 0 0 10 −ρ a 0 0 1

, δQ =

δρδua

δut

δur

δp

The changes to the outgoing characteristics — one characteristic forsubsonic inflow (δC5), and four characteristics for subsonic outflow (δC1,δC2, δC3, δC4) — are determined from extrapolation of the flow fieldvariables using Equation 9.6-2.

The changes in the incoming characteristics — four characteristics forsubsonic inflow (δC1, δC2, δC3, δC4), and one characteristic for subsonicoutflow (δC5) — are split into two components: average change alongthe boundary (δCi), and local changes in the characteristic variable dueto harmonic variation along the boundary (δCiL). The incoming char-acteristics are therefore given by

δCij = δCioldj+ σ

(δCinewj

− δCioldj

)(9.6-3)

δCinewj=(δCi + δCiLj

)(9.6-4)

where i = 1, 2, 3, 4 on the inlet boundary or i = 5 on the outlet boundary,and j = 1, ...,N is the grid index in the pitchwise direction including theperiodic point once. The under-relaxation factor σ has a default value of0.75. Note that this method assumes a periodic solution in the pitchwisedirection.

The flow is decomposed into mean and circumferential components usingFourier decomposition. The 0th Fourier mode corresponds to the aver-age circumferential solution, and is treated according to the standard 1Dcharacteristic theory. The remaining parts of the solution are described



by a sum of harmonics, and treated as 2D non-reflecting boundary con-ditions [78].

Inlet Boundary

For subsonic inflow, there is one outgoing characteristic (δC5) deter-mined from Equation 9.6-2, and four incoming characteristics (δC1, δC2,δC3, δC4) calculated using Equation 9.6-3. The average changes in theincoming characteristics are computed from the requirement that the en-tropy (s), radial and tangential flow angles (αr and αt), and stagnationenthalpy (h0) are specified. Note that in FLUENT you can specify p0

and T0 at the inlet, from which sin and h0inare easily obtained. This is

equivalent to forcing the following four residuals to be zero:

R1 = p (s− sin) (9.6-5)R2 = ρ a (ut − ua tanαt) (9.6-6)R3 = ρ a (ur − ua tanαr) (9.6-7)

R4 = ρ(h0 − h0in

)(9.6-8)

where

sin = γ ln (T0in) − (γ − 1) ln (p0in

) (9.6-9)

h0in= cpT0in

(9.6-10)

The average characteristic is then obtained from residual linearizationas follows (see also Figure 9.6.4 for an illustration of the definitions forthe prescribed inlet angles):

δC1

δC2

δC3

δC4

=

−1 0 0 0tan αt

(1−γ)MM1M

−MtM tanαt

tan αtM

tan αr(γ−1)M

MtM tanαt

M2M

− tan αrM

2(γ−1)M 2Mt

M 2MrM

−2M

R1

R2

R3

R4

(9.6-11)



where

Ma =ua

a(9.6-12)

Mt =ut

a(9.6-13)

Mr =ur

a(9.6-14)

and

M = 1 + Ma − Mt tanαt + Mr tanαr (9.6-15)

M1 = −1 − Ma − Mr tanαr (9.6-16)

M2 = −1 − Ma − Mt tanαt (9.6-17)

radial

axial

theta

α

α

r

t

u

u

u

r

t

a

v

Figure 9.6.4: Prescribed Inlet Angles

where

|v| =√u2

t + u2r + u2

a (9.6-18)



et =ut

|v| (9.6-19)

er =ur

|v| (9.6-20)

ea =ua

|v| (9.6-21)

tanαt =etea

(9.6-22)

tanαr =erea

(9.6-23)

To address the local characteristic changes at each j grid point along theinflow boundary, the following relations are developed [78, 198]:

δC1Lj = p (sj − s)δC2Lj = C ′

2j− ρa

(utj − ut

)δC3Lj = −ρa

(urj − ur

)δC4Lj = −2

(1+Maj)

(1

γ−1δC1Lj + Mtj δC2Lj + MrjδC3Lj + ρ(h0j − h0

))(9.6-24)

Note that the relation for the first and fourth local characteristics forcethe local entropy and stagnation enthalpy to match their average steady-state values.

The characteristic variable C ′2j

is computed from the inverse discreteFourier transform of the second characteristic. The discrete Fouriertransform of the second characteristic in turn is related to the discreteFourier transform of the fifth characteristic. Hence, the characteristicvariable C ′

2jis computed along the pitch as follows:

C ′2j

= 2<

N2−1∑

n=1

C2n exp(i2πn

θj − θ1θN − θ1

) (9.6-25)

The Fourier coefficients C ′2n

are related to a set of equidistant distributedcharacteristic variables C∗

5jby the following [155]:



C2n =

− ut+B

N(a+ua)

∑Nj=1C

∗5j

exp(−i2π jn

N

)β > 0

−ut+Ba+ua

C5j β < 0(9.6-26)

where

B =

i√β β > 0

−sign (ut)√|β| β < 0

(9.6-27)

and

β = a2 − u2a − u2

t (9.6-28)

The set of equidistributed characteristic variables C∗5j

is computed fromarbitrary distributed C5j by using a cubic spline for interpolation, where

C5j = −ρ a(uaj − ua

)+ (pj − p) (9.6-29)

For supersonic inflow the user-prescribed static pressure (psin) along with

total pressure (p0in) and total temperature (T0in

) are sufficient for deter-mining the flow condition at the inlet.

Outlet Boundary

For subsonic outflow, there are four outgoing characteristics (δC1, δC2,δC3, and δC4) calculated using Equation 9.6-2, and one incoming char-acteristic (δC5) determined from Equation 9.6-3. The average change inthe incoming fifth characteristic is given by

δC5 = −2 (p− pout) (9.6-30)

where p is the current averaged pressure at the exit plane and pout is thedesirable average exit pressure (this value is specified by you for single-blade calculations or obtained from the assigned profile for mixing-planecalculations). The local changes (δC5Lj ) are given by



δC5Lj = C ′5j

+ ρ a(uaj − ua

)− (pj − p) (9.6-31)

The characteristic variable C′5j

is computed along the pitch as follows:

C ′5j

= 2<

N2−1∑

n=1

C5n exp(i2πn

θj − θ1θN − θ1

) (9.6-32)

The Fourier coefficients C5n are related to two sets of equidistant dis-tributed characteristic variables (C∗

2jand C∗

4j, respectively) and given

by the following [155]:

C5n =

A2N

∑Nj=1C

∗2j

exp(i2π jn

N

)− A4

N

∑Nj=1C

∗4j

exp(i2π jn

N

)β > 0

A2C2j −A4C4j β < 0(9.6-33)

where

A2 =2ua

B − ut(9.6-34)

A4 =B + ut

B − ut(9.6-35)

The two sets of equidistributed characteristic variables (C∗2j

and C∗4j

)are computed from arbitrary distributed C2j and C4j characteristics byusing a cubic spline for interpolation, where

C2j = ρ a(utj − ut

)(9.6-36)

C4j = ρ a(uaj − ua

)+ (pj − p) (9.6-37)

For supersonic outflow all flow field variables are extrapolated from theinterior.



Updated Flow Variables

Once the changes in the characteristics are determined on the inflow oroutflow boundaries, the changes in the flow variables δQ can be obtainedfrom Equation 9.6-2. Therefore, the values of the flow variables at theboundary faces are as follows:

pf = pj + δp (9.6-38)uaf

= uaj + δua (9.6-39)utf = utj + δut (9.6-40)urf

= urj + δur (9.6-41)Tf = Tj + δT (9.6-42)

9.6.3 Using the Non-Reflecting Boundary Conditions

The procedure for using the NRBCs is as follows:

1. Turn on the NRBCs using the non-reflecting text command:

define −→ boundary-conditions −→ non-reflecting −→enable?

If you are not sure whether or not NRBCs are turned on, use theshow-status text command.

2. Perform NRBC initialization using the initialize text command:

define −→ boundary-conditions −→ non-reflecting −→initialize

If the initialization is successful, a summary printout of the domainextent will be displayed. If the initialization is not successful, anerror message will be displayed indicating the source of the prob-lem.

3. If necessary, modify the parameters in the set/ submenu:

define −→ boundary-conditions −→ non-reflecting −→set



under-relaxation allows you to set the value of the under-relax-ation factor σ in Equation 9.6-3. The default value is 0.75.

discretization allows you to set the discretization scheme. Thedefault is to use higher-order reconstruction if available.

verbosity allows you to control the amount of information printedto the console during an NRBC calculation.

• 0 : silent

• 1 : basic information (default)

• 2 : detailed information (for debugging purposes only)

Using the NRBCs with the Mixing-Plane Model

If you want to use the NRBCs with the mixing-plane model you mustdefine the mixing plane interfaces as pressure-outlet and pressure-inletzone type pairs.

Using the NRBCs in Parallel FLUENT

When the NRBCs are used in conjunction with the parallel solver, allcells in each boundary zone where NRBCs will be applied must be locatedor contained within a single partition. You can ensure this by manuallypartitioning the grid (see Section 28.4.3 for more information on how todo this).


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