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Chapter 9. Zones Modeling Flows in Moving The solution of flows in moving reference frames requires the use of moving cell zones. This cell zone motion is interpreted as the motion of a 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 and solved 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 of features for solving problems in which the domain or parts of the domain are 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 turbomachinery, mixing tanks, and related devices. In each of these cases, the flow is unsteady in an inertial frame (i.e., a domain fixed in the laboratory frame) because the rotor/impeller blades sweep the domain periodically. However, in the absence of stators or baffles, it is possible to perform calculations in a domain that moves with the rotating part. In this 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 it is not possible to render the computational problem steady by choosing a calculation domain that rotates with the rotor or impeller. This situation occurs, for example, in turbomachinery applications where rotor and stator blades are in close proximity (and hence rotor-stator interaction is important). FLUENT provides three approaches to address this class of 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 is steady, with the rotor-stator or impeller-baffle effects being accounted for by approximate means. These can be acceptable models in cases where the rotor-stator interaction is weak or an approximate solution for the system is desired. The sliding mesh model, on the other hand, assumes that the flow field is unsteady, and hence models the interaction with complete fidelity. This is the model of choice if rotor-stator interaction is strong and a more accurate simulation of the system is desired. Note, however, that because the sliding mesh model requires an unsteady numerical solution, it is computationally more demanding than the MRF and mixing plane models. 9-2 c Fluent Inc. November 28, 2001 9.2 Flow in a Rotating Reference Frame 9.2 Flow in a Rotating Reference Frame 9.2.1 Overview When you create a model using FLUENT, you are typically modeling the flow in an inertial reference frame (i.e., in a non-accelerating coordinate system). However, FLUENT also has the ability to model flows in an accelerating reference frame. In this situation, the acceleration of the coordinate system is included in the equations of motion describing the flow. A common example of an accelerating reference frame in engineering applications is flow in rotating equipment. Many such flows can be modeled in a coordinate system that is moving with the rotating equipment and thus experiences a constant acceleration in the radial direction. This class of rotating flows can be treated using the rotating reference frame capability in FLUENT. Figure 9.2.1 depicts an example of a flow in a rotating reference frame, and illustrates the coordinate transformation from the stationary frame to the rotating frame. Applications Involving a Rotating Reference Frame Several examples of problems that can be modeled using a rotating reference frame are depicted in Figure 9.2.2. The applications illustrated here include: • Impellers in mixing tanks • Rotating turbomachinery blades (centrifugal impellers, axial fans, etc.) • Flows in rotating passages (e.g., cooling ducts, secondary air flow circuits, and disk cavities in rotating equipment) When such problems are defined in a rotating reference frame, the rotating boundaries become stationary relative to the rotating frame, since they are moving at the same speed as the reference frame. c Fluent Inc. November 28, 2001 9-3 Modeling Flows in Moving Zones Stationary Rotating at speed Ω y Ω x (a) Original Reference Frame Rotating at speed -Ω -Ω Stationary y1 x1 (b) Rotating Reference Frame Figure 9.2.1: Transforming Coordinates to a Rotating Reference Frame 9-4 c Fluent Inc. November 28, 2001 9.2 Flow in a Rotating Reference Frame Ω z y x (a) Rotating Impeller in a Mixing Tank Ω y x (b) Centrifugal Impeller Blades Figure 9.2.2: Applications That Can Be Modeled by FLUENT in a Rotating Reference Frame c Fluent Inc. November 28, 2001 9-5 Modeling Flows in Moving Zones Ω y z x (c) Cooling Passages in a Spinning Rotor Ω x z y (d) Axial Impeller Blades Figure 9.2.2: Applications That Can Be Modeled by FLUENT in a Rotating Reference Frame 9-6 c Fluent Inc. November 28, 2001 9.2 Flow in a Rotating Reference Frame Modeling Rotor-Stator Interaction As mentioned in Section 9.1, rotor-stator interaction problems (as illustrated in Figure 9.2.3) cannot be modeled by a simple coordinate transformation to a rotating reference frame. In FLUENT, rotor-stator interaction must be treated by applying the MRF, mixing plane, or sliding mesh 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 rotating frame of reference, the acceleration of the fluid is augmented by additional terms that appear in the momentum equations [10]. FLUENT allows you to solve rotating frame problems using either the absolute velocity, ~v , or the relative velocity, ~vr , as the dependent variable. The two velocities are related by the following equation: ~ × ~r) ~vr = ~v − (Ω (9.2-1) ~ is the angular velocity vector (that is, the angular velocity of Here, Ω the rotating frame) and ~r is the position vector in the rotating frame. The left-hand side of the momentum equations appears as follows for an inertial reference frame: ∂ (ρ~v ) + ∇ · (ρ~v~v ) ∂t (9.2-2) For a rotating reference frame, the left-hand side written in terms of absolute velocities becomes ∂ ~ × ~v ) (ρ~v ) + ∇ · (ρ~vr ~v ) + ρ(Ω ∂t (9.2-3) In terms of relative velocities the left-hand side is given by c Fluent Inc. November 28, 2001 9-7 Modeling Flows in Moving Zones Stationary Rotating Ω (a) 2D Rotor-Stator Interaction Stationary baffles Ω Rotating impeller (b) Rotating Impeller in a Baffled Tank Figure 9.2.3: Problems That Require MRF, Mixing Plane, or Sliding Meshes 9-8 c Fluent Inc. November 28, 2001 9.2 Flow in a Rotating Reference Frame ~ ∂ ~ × ~vr + Ω ~ ×Ω ~ × ~r) + ρ ∂ Ω × ~r (ρ~vr ) + ∇ · (ρ~vr ~vr ) + ρ(2Ω ∂t ∂t (9.2-4) ~ × ~vr + Ω ~ ×Ω ~ × ~r) is the Coriolis force. Note that FLUENT where ρ(2Ω ~ ∂Ω neglects the ρ ∂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, or continuity equation, can be written as follows for both the absolute and the relative velocity formulations: ∂ρ + ∇ · (ρ~vr ) = Sm ∂t 9.2.3 (9.2-5) Grid Setup for a Single Rotating Reference Frame It is important to remember the following coordinate-system constraints when you are setting up a problem involving a rotating reference frame: • For 2D problems, the axis of rotation must be parallel to the z axis. • For 2D axisymmetric problems, the axis of rotation must be the x axis. • For 3D geometries, you should generate the mesh with a specific rotational axis in mind for the rotating cell zone. Usually it is convenient to use the x, y, or z axis, but FLUENT can accommodate arbitrary rotational axes. 9.2.4 Problem Setup for a Single Rotating Reference Frame When you want to model a problem involving a single rotating reference frame, you will need to complete the following modeling inputs. (Only those steps relevant specifically to the setup of a rotating reference frame problem are listed here. You will need to set up the rest of the problem as usual.) c Fluent Inc. November 28, 2001 9-9 Modeling Flows in Moving Zones 1. Select the Velocity Formulation to be used in the Solver panel: either Relative 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 coupled solvers; 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 Origin and Rotation-Axis Direction to define the axis of rotation. (b) Also in the Fluid or Solid panel, select Moving Reference Frame in the Motion Type drop-down list and then set the Speed under Rotational Velocity in the expanded portion of the panel. Details about these inputs are presented in Section 6.17.1 for fluid zones, and in Section 6.18.1 for solid zones. 3. Define the velocity boundary conditions at walls. You can choose to define either an absolute velocity or a velocity relative to the moving reference frame (i.e., relative to the velocity of the adjacent cell zone specified in step 2). If the wall is moving at the speed of the rotating frame (and hence stationary in the rotating frame), it is convenient to specify a relative angular velocity of zero. Likewise, a wall that is stationary in the nonrotating frame of reference should be given a velocity of zero in the absolute reference frame. Specifying the wall velocities in this manner obviates the need to modify these inputs later if a change 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 direction and total pressure at any pressure inlets. For velocity inlets, you can choose to define either absolute velocities or velocities relative to the motion of the adjacent cell zone (specified in step 2). For pressure inlets, the specification of the flow direction and total 9-10 c Fluent Inc. November 28, 2001 9.2 Flow in a Rotating Reference Frame pressure will be relative or absolute, depending on the velocity formulation you selected in step 1. See Section 9.2.5 for details. (If you use one of the coupled solution algorithms, the specification is always 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 the flow 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 fluid in the domain is rotating, as in the case of a large impeller in a mixing tank. For most applications, either formulation may be used. ! When one of the coupled solution algorithms is used, the absolute formulation is always used; the relative velocity formulation is not available in the coupled solvers. For velocity inlets and walls, you may specify velocity in either the absolute or the relative frame, regardless of whether the absolute or relative velocity is used in the computation. For pressure boundary conditions, however, FLUENT imposes several restrictions on how total pressure and flow direction are specified in rotating reference frames. The total pressure and flow direction at a pressure inlet must be specified in the absolute frame if the absolute velocity formulation is used. For calculations using relative velocities, the total pressure and flow direction must 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 static pressure is used as the total pressure. For calculations using absolute velocities, the specified static pressure is used as the total pressure in the absolute frame; for the relative velocity formulation, the specified static pressure is assumed to be the total pressure in the relative frame. As for flow direction in reverse flows, FLUENT assumes the absolute velocity to be normal to the pressure outlet for the absolute velocity formulation; for the relative velocity formulation, it is the relative velocity that is assumed to be normal to the pressure outlet. c Fluent Inc. November 28, 2001 9-11 Modeling Flows in Moving Zones 9.2.6 Solution Strategies for a Rotating Reference Frame The difficulties associated with solving flows in rotating reference frames are similar to those discussed in Section 8.4.5 for axisymmetric swirling or rotating flows. The primary issue you must confront is the high degree of coupling between the momentum equations when the influence of the rotational terms is large. A high degree of rotation introduces a large radial pressure gradient which drives the flow in the axial and radial directions, thereby setting up a distribution of the swirl or rotation in the field. This coupling may lead to instabilities in the solution process, and hence 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 which velocities are solved by changing the velocity formulation setting in the Solver panel. (See Section 9.2.5 for details.) • (Segregated solver only) Use the PRESTO! scheme (enabled in the Solution Controls panel), which is well-suited for the steep pressure gradients involved in rotating flows. • Ensure that the mesh is sufficiently refined to resolve large gradients in pressure and swirl velocity. • (Segregated solver only) Reduce the under-relaxation factors for the velocities, perhaps to 0.3–0.5 or lower, if necessary. • Begin the calculations using a low rotational speed, increasing the rotational speed gradually in order to reach the final desired operating condition (see below). See Chapter 22 for details on the procedures used to make these changes to the solution parameters. Gradual Increase of the Rotational Speed to Improve Solution Stability Because the rotation of the reference frame and the rotation defined via boundary conditions can lead to large complex forces in the flow, your 9-12 c Fluent Inc. November 28, 2001 9.2 Flow in a Rotating Reference Frame FLUENT calculations may be less stable as the speed of rotation (and hence the magnitude of these forces) increases. One of the most effective controls you can exert on the solution is to start with a low rotational speed and then slowly increase the rotation up to the desired level. The procedure you use to accomplish this is as follows: 1. Set up the problem using a low rotational speed in your inputs for boundary conditions and for the angular velocity of the reference frame. The rotational speed in this first attempt might be selected as 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 velocity of the reference frame). Increase the speed of rotation, perhaps doubling it. 5. Restart or continue the calculation using the solution data saved in Step 3 as the initial guess for the new calculation. Save the new data. 6. Continue to increment the rotational speed, following Steps 4 and 5, 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 plot or report both absolute and relative velocities. For all velocity parameters (e.g., Velocity Magnitude and Mach Number), corresponding relative values will be available for postprocessing (e.g., Relative Velocity Magnitude and Relative Mach Number). These variables are contained in the Velocity... category of the variable selection drop-down list that appears in postprocessing panels. Relative values are also available for postprocessing of total pressure, total temperature, and any other parameters that include a dynamic contribution dependent on the reference frame (e.g., Relative Total Pressure, Relative Total Temperature). c Fluent Inc. November 28, 2001 9-13 Modeling Flows in Moving Zones When plotting velocity vectors, you can choose to plot vectors in the absolute frame (the default), or you can select Relative Velocity in the Vectors Of drop-down list in the Vectors panel to plot vectors in the rotating frame. If you plot relative velocity vectors, you might want to color the vectors by relative velocity magnitude (by choosing Relative Velocity Magnitude in the Color By list); by default they will be colored by absolute velocity magnitude. Figures 9.2.4 and 9.2.5 show absolute and relative velocity vectors in a rotating domain with a stationary outer wall. 9-14 c Fluent Inc. November 28, 2001 9.2 Flow in a Rotating Reference Frame 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 Velocity Vectors Colored By Velocity Magnitude (m/s) Figure 9.2.4: Absolute Velocity Vectors 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 Velocity Vectors Colored By Relative Velocity Magnitude (m/s) Figure 9.2.5: Relative Velocity Vectors c Fluent Inc. November 28, 2001 9-15 Modeling Flows in Moving Zones 9.3 The Multiple Reference Frame (MRF) Model 9.3.1 Overview As mentioned in Section 9.1, FLUENT provides three approaches for modeling 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 steadystate approximation in which individual cell zones move at different rotational/translational speeds. This approach is appropriate when the flow at the boundary between these zones is nearly uniform (“mixed out”). While the multiple reference frame approach is clearly an approximation, it can provide a reasonable model of the time-averaged flow for many applications. For example, the MRF model can be used for a turbomachinery application in which rotor-stator interaction is relatively weak. In mixing tanks, since the impeller-baffle interactions are relatively weak, large-scale transient effects are not present and the MRF model can be used. In general, any problems where transients due to rotor-stator interaction are small are candidates for the MRF model. Another potential use of the MRF model is to compute a flow field that can be used as an initial condition for a transient sliding mesh calculation. This eliminates the need for a startup calculation. The multiple reference frame model should not be used, however, if it is necessary to actually simulate the transients that may occur in strong impeller-baffle interactions. For such cases, the sliding mesh model (see Section 9.5) should be used. Examples For a mixing tank with a single impeller, you can define a rotating reference frame that encompasses the impeller and the flow surrounding it, 9-16 c Fluent Inc. November 28, 2001 9.3 The Multiple Reference Frame (MRF) Model and use a stationary frame for the flow outside the impeller region. An example of this configuration is illustrated in Figure 9.3.1. (The dashes denote the interface between the two reference frames.) Steady-state flow conditions are assumed at the interface between the two reference frames. That is, the velocity at the interface must be the same (in absolute 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 reference frame. Figure 9.3.2 shows a geometry that contains two rotating impellers side by side. This problem would be modeled using three reference frames: the stationary frame outside both impeller regions and two separate rotating reference frames for the two impellers. (As noted above, 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- model with multiple reference frames is not recommended. • The boundaries separating a moving region from adjacent regions must be oriented such that the component of the frame velocity c Fluent Inc. November 28, 2001 9-17 Modeling Flows in Moving Zones Figure 9.3.2: Geometry with Two Rotating Impellers normal to the boundary is zero. For the example shown in Figure 9.3.1, this requires the dashed boundary to be circular (not square or any other shape). For a translationally moving frame, the moving zone’s boundaries must be parallel to the translational velocity vector. • Strictly speaking, the use of multiple reference frames is meaningful only for steady flow. However, FLUENT will allow you to solve an unsteady flow when multiple reference frames are being used. In this case, unsteady terms (as described in Section 22.2.8) are added to all the governing transport equations. You should carefully consider whether this will yield meaningful results for your application, because, for unsteady flows, a sliding mesh calculation will generally yield more meaningful results than an MRF calculation. • Particle trajectories and pathlines drawn by FLUENT use the velocity relative to the cell zone motion. For massless particles, the resulting pathlines follow the streamlines based on relative velocity and are meaningful. For particles with mass, however, the particle tracks displayed are meaningless. Similarly, coupled discrete-phase 9-18 c Fluent Inc. November 28, 2001 9.3 The Multiple Reference Frame (MRF) Model calculations are meaningless. An alternative approach for particle tracking and coupled discretephase calculations with multiple reference frames is to track particles based on absolute velocity instead of relative velocity. To make this change, use the define/models/dpm/tracking/track-inabsolute-frame text command. Note, however, that tracking particles based on absolute velocity may result in incorrect particlewall interaction. The particle injection velocities (specified in the Set Injection Properties panel) are defined relative to the frame of reference in which the particles are tracked. By default, the injection velocities are specified relative to the local reference frame. If you enable the track-in-absolute-frame option, the injection velocities are specified relative to the absolute frame. • You cannot accurately model axisymmetric swirl in the presence of multiple reference frames using the relative velocity formulation. This is because the current implementation does not apply the transformation used in Equation 9.3-3 to the swirl velocity derivatives. • 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 being used. Relative Velocity Formulation In FLUENT’s implementation of the multiple reference frame feature, the calculation domain is divided into subdomains, each of which may be rotating/translating with respect to the laboratory (inertial) frame. The governing equations in each subdomain are written with respect to that subdomain’s reference frame. Thus, the flow in stationary and translating subdomains is governed by the equations in Section 8.2, while the flow in rotating subdomains is governed by the equations presented in Section 9.2.2. c Fluent Inc. November 28, 2001 9-19 Modeling Flows in Moving Zones At the boundary between two subdomains, the diffusion and other terms in the governing equations in one subdomain require values for the velocities in the adjacent subdomain. FLUENT enforces the continuity of the absolute velocity, ~v , to provide the correct neighbor values of velocity for the subdomain under consideration. (This approach differs from the mixing plane approach described in Section 9.4, where a circumferential averaging technique is used.) When the relative velocity formulation is used, velocities in each subdomain are computed relative to the motion of the subdomain. Velocities and velocity gradients are converted from a moving reference frame to the absolute inertial frame as described below. The position vector relative to the origin of the zone rotation axis is defined as ~r = ~x − ~xo (9.3-1) where ~x is the position in absolute Cartesian coordinates and ~xo is the origin of the zone rotation axis, as shown in Figure 9.3.3. The relative velocity in the moving reference frame can be converted to the 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 is the velocity in the relative noninertial reference frame, and ~vt is the translational velocity of the noninertial reference frame. Using this definition of absolute velocity, the gradient of the absolute velocity vector is given by ∇~v = ∇~vr + ∇ (~ω × ~r) 9-20 (9.3-3) c Fluent Inc. November 28, 2001 9.3 The Multiple Reference Frame (MRF) Model y r Y x xo Z x moving reference frame z X absolute reference frame Figure 9.3.3: Coordinate System for Relative Velocity Absolute Velocity Formulation When the absolute velocity formulation is used, the governing equations in each subdomain are written with respect to that subdomain’s reference frame, but the velocities are stored in the absolute frame. Therefore no special transformation is required at the interface between two subdomains. 9.3.3 Grid Setup for Multiple Reference Frames Two grid setup methods are available. Choose the method that is appropriate for your model, noting the restrictions in Section 9.3.1. • If the boundary between two zones that are in different reference frames is conformal (i.e., the grid node locations are identical at the boundary where the two zones meet), you can simply create the 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 c Fluent Inc. November 28, 2001 9-21 Modeling Flows in Moving Zones that is modeled in a different reference frame. Use an interior zone for the boundary between reference frames. • If the boundary between two zones that are in different reference frames is non-conformal (i.e., the grid node locations are not identical at the boundary where the two zones meet), follow the nonconformal 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 those steps relevant specifically to the setup of a multiple reference frame problem are listed here. You will need to set up the rest of the problem as usual.) ! The grid-setup constraints for a rotating reference frame listed in Section 9.2.3 apply to multiple reference frames as well. 1. Select the Velocity Formulation to be used in the Solver panel: either Absolute or Relative. (See Section 9.2.5 for details.) Define −→ Models −→Solver... (Note that this step is irrelevant if you are using one of the coupled solution algorithms; these algorithms always use an absolute velocity formulation.) 2. For each cell zone in the domain, specify its translational velocity and/or its angular velocity (Ω) and the axis about which it rotates. Define −→Boundary Conditions... (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 panel or Solid panel, specify the Rotation-Axis Origin and RotationAxis Direction. (b) Also in the Fluid or Solid panel, select Moving Reference Frame in the Motion Type drop-down list and then set the Speed 9-22 c Fluent Inc. November 28, 2001 9.3 The Multiple Reference Frame (MRF) Model under Rotational Velocity and/or the X, Y, and Z components of the Translational Velocity in the expanded portion of the panel. Details about these inputs are presented in Section 6.17.1 for fluid zones, and in Section 6.18.1 for solid zones. 3. Define the velocity boundary conditions at walls. You can choose to define either an absolute velocity or a velocity relative to the velocity of the adjacent cell zone specified in step 2. If the wall is moving at the speed of the moving frame (and hence stationary relative to the moving frame), it is convenient to specify a relative angular velocity of zero. Likewise, a wall that is stationary in the non-moving frame of reference should be given a velocity of zero in the absolute reference frame. Specifying the wall velocities in this manner obviates the need to modify these inputs later if a change is made in the rotational velocity of the fluid zone. An example for which you would specify a relative velocity is as follows: If an impeller is defined as wall-3 and the fluid region within the impeller’s radius is defined as fluid-5, you would need to specify the angular velocity and axis of rotation for fluid-5 and then assign wall-3 a relative velocity of 0. If you later wanted to model a different angular velocity for the impeller, you would need to change only the angular velocity of the fluid region; you would not need to modify the wall velocity conditions. Details about these inputs are presented in Section 6.13.1. 4. Define the velocity at any velocity inlets and the flow direction and total pressure at any pressure inlets. For velocity inlets, you can choose to define either absolute velocities or velocities relative to the motion of the adjacent cell zone (specified in step 2). For pressure inlets, the specification of the flow direction and total pressure will be relative or absolute, depending on the velocity formulation you selected in step 1. See Section 9.2.5 for details. (If you use one of the coupled solution algorithms, the specification is always in the absolute frame.) Details about these inputs are presented in Sections 6.3.1 and 6.4.1. c Fluent Inc. November 28, 2001 9-23 Modeling Flows in Moving Zones 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 in Section 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 or report both absolute and relative velocities. For all velocity parameters (e.g., Velocity Magnitude and Mach Number), corresponding relative values will be available for postprocessing (e.g., Relative Velocity Magnitude and Relative Mach Number). These variables are contained in the Velocity... category of the variable selection drop-down list that appears in postprocessing panels. Relative values are also available for postprocessing of total pressure, total temperature, and any other parameters that 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). The velocity of the reference zone is the velocity defined in the Fluid panel for that zone. When plotting velocity vectors, you can choose to plot vectors in the absolute frame (the default), or you can select Relative Velocity in the Vectors Of drop-down list in the Vectors panel to plot vectors relative to the translational/rotational velocity of the “reference zone” (specified in the Reference Values panel). If you plot relative velocity vectors, you might want to color the vectors by relative velocity magnitude (by choosing Relative Velocity Magnitude in the Color By list); by default they will be colored by absolute velocity magnitude. You can also generate a plot of circumferential averages in FLUENT. This allows you to find the average value of a quantity at several different radial or axial positions in your model. FLUENT computes the average of the quantity over a specified circumferential area, and then plots the average against the radial or axial coordinate. For more information on generating XY plots of circumferential averages, see Section 25.8.4. 9-24 c Fluent Inc. November 28, 2001 9.4 The Mixing Plane Model 9.4 The Mixing Plane Model 9.4.1 Overview and Limitations The mixing plane model in FLUENT provides an alternative to the multiple reference frame and sliding mesh models for simulating flow through domains with one or more regions in relative motion. This section provides 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 flow at the boundary between adjacent zones that move at different speeds is nearly uniform (“mixed out”). If the flow at this boundary is not uniform, the MRF model may not provide a physically meaningful solution. The sliding mesh model (see Section 9.5) may be appropriate for such cases, but in many situations it is not practical to employ a sliding mesh. For example, in a multistage turbomachine, if the number of blades is different for each blade row, a large number of blade passages is required in order to maintain circumferential periodicity. Moreover, sliding mesh calculations are necessarily unsteady, and thus require significantly more computation to achieve a final, time-periodic solution. For situations where using the sliding mesh model is not feasible, the mixing plane model can be a cost-effective alternative. In the mixing plane approach, each fluid zone is treated as a steady-state problem. Flow-field data from adjacent zones are passed as boundary conditions that are spatially averaged or “mixed” at the mixing plane interface. This mixing removes any unsteadiness that would arise due to circumferential variations in the passage-to-passage flow field (e.g., wakes, shock waves, separated flow), thus yielding a steady-state result. Despite the simplifications inherent in the mixing plane model, the resulting solutions can provide reasonable approximations of the timeaveraged flow field. Limitations Note the following limitations of the mixing plane model: c Fluent Inc. November 28, 2001 9-25 Modeling Flows in Moving Zones • The mixing plane model requires the use of the absolute velocity formulation; you cannot use the relative velocity formulation with the mixing plane model. • The LES turbulence model cannot be used with the mixing plane model. • The models for species transport and combustion cannot be used with the mixing plane model. • The general multiphase models (VOF, mixture, and Eulerian) cannot 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 9.4.1 and 9.4.2. Figure 9.4.1 shows a constant radial plane within a single stage of an axial machine, while Figure 9.4.2 shows a constant θ plane within a mixed-flow device. In each case, the stage consists of two flow domains: the rotor domain, which is rotating at a prescribed angular velocity, followed by the stator domain, which is stationary. The order of the rotor and stator is arbitrary (that is, a situation where the rotor is downstream of the stator is equally valid). In a numerical simulation, each domain will be represented by a separate mesh. The flow information between these domains will be coupled at the mixing plane interface (as shown in Figures 9.4.1 and 9.4.2) using the mixing plane model. Note that you may couple any number of fluid zones in this manner; for example, four blade passages can be coupled using 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 nodes on the rotor inlet boundary. In addition, the meshes can be of different types (e.g., the stator can have a hexahedral mesh while the rotor has a tetrahedral mesh). 9-26 c Fluent Inc. November 28, 2001 9.4 The Mixing Plane Model rotor stator rotor outlet: ps stator inlet: p0α x α y α z k ε Rθ x mixing plane interface Figure 9.4.1: Axial Rotor-Stator Interaction (Schematic Illustrating the Mixing Plane Concept) The Mixing Plane Concept The essential idea behind the mixing plane concept is that each fluid zone is solved as a steady-state problem. At some prescribed iteration interval, the flow data at the mixing plane interface are averaged in the circumferential direction on both the stator outlet and the rotor inlet boundaries. The FLUENT implementation uses area-weighted averages. By performing circumferential averages at specified radial or axial stations, “profiles” of flow properties can be defined. These profiles—which will be functions of either the axial or the radial coordinate, depending on the orientation of the mixing plane—are then used to update boundary 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 averaged total pressure (p0 ), direction cosines of the local flow angles in the radial, tangential, and axial directions (αr , αt , αz ), total temperature (T0 ), turbulence kinetic energy (k), and turbulence dissipation rate () are computed at the rotor exit and used to update boundary conditions at the stator inlet. Likewise, a profile of static pressure (ps ) is computed at the stator inlet and used as a boundary condition on the rotor exit. Passing profiles in the manner described above assumes specific boundary c Fluent Inc. November 28, 2001 9-27 Modeling Flows in Moving Zones stator stator inlet: p0 αx αyα z k ε mixing plane interface r x rotor outlet: p s rotor Ω Figure 9.4.2: Radial Rotor-Stator Interaction (Schematic Illustrating the Mixing Plane Concept) 9-28 c Fluent Inc. November 28, 2001 9.4 The Mixing Plane Model condition types have been defined at the mixing plane interface. The coupling of an upstream outlet boundary zone with a downstream inlet boundary zone is called a “mixing plane pair”. In order to create mixing plane pairs in FLUENT, the boundary zones must be of the following types: upstream pressure outlet pressure outlet pressure outlet downstream pressure inlet velocity inlet mass flow inlet Specific instructions for setting up mixing planes are provided in Section 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 boundaries, obtaining profiles for use in updating boundary conditions. 3. Pass the profiles to the boundary condition inputs required for the stator 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 (especially early in the computation). FLUENT allows you to control the underrelaxation of the mixing plane variables. Mass Conservation Note that the algorithm described above will not rigorously conserve mass flow across the mixing plane if it is represented by a pressure inlet and pressure outlet mixing plane pair. If you use a mass flow inlet c Fluent Inc. November 28, 2001 9-29 Modeling Flows in Moving Zones and pressure outlet pair instead, FLUENT will force mass conservation across the mixing plane. The basic technique consists of computing the mass flow rate across the upstream zone (pressure outlet) and adjusting the mass flux profile applied at the mass flow inlet such that the downstream mass flow matches the upstream mass flow. This adjustment occurs at every iteration, thus ensuring rigorous conservation of mass flow 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 this jump is usually small compared with total pressure variations elsewhere in 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 torques acting on the components should be zero, enforcing swirl conservation across the mixing plane is essential, and is available in FLUENT as a modeling option. Ensuring conservation of swirl is important because, otherwise, sources or sinks of tangential momentum will be present at the 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 momentum equation from fluid mechanics, it can be shown that for steady flow, ZZ T = S rvθ ρ~v · n̂dS (9.4-1) where T is the torque of the fluid acting on the component, r is the radial distance from the axis of rotation, vθ is the absolute tangential velocity, ~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 outlet boundaries, Equation 9.4-1 becomes 9-30 c Fluent Inc. November 28, 2001 9.4 The Mixing Plane Model ZZ T = outlet rvθ ρ~v · n̂dS + ZZ inlet rvθ ρ~v · n̂dS (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 thickness. Applying Equation 9.4-2 to this zone and noting that, in the limit as the thickness shrinks to zero, the torque should vanish, the equation becomes ZZ downstream rvθ ρ~v · n̂dS = ZZ upstream rvθ ρ~v · n̂dS (9.4-3) where upstream and downstream denote the upstream and downstream sides of the mixing plane interface. Note that Equation 9.4-3 applies to the full area (360 degrees) at the mixing plane interface. Equation 9.4-3 provides a rational means of determining the tangential velocity component. That is, FLUENT computes a profile of tangential velocity and then uniformly adjusts the profile such that the swirl integral is satisfied. Note that interpolating the tangential (and radial) velocity component profiles at the mixing plane does not affect mass conservation because these velocity components are orthogonal to the face-normal velocity 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. Only those steps relevant specifically to the setup of a mixing plane problem are 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 plane model is consistent with their use when the model is not active. 1. Select the (default) absolute velocity formulation in the Solver panel. Define −→ Models −→Solver... c Fluent Inc. November 28, 2001 9-31 Modeling Flows in Moving Zones 2. For each cell zone in the domain, specify its angular velocity (Ω) and the axis about which it rotates. Define −→Boundary Conditions... (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 panel or Solid panel, specify the Rotation-Axis Origin and RotationAxis Direction. (b) Also in the Fluid or Solid panel, select Moving Reference Frame in the Motion Type drop-down list and then set the Speed under Rotational Velocity and/or the X, Y, and Z components of the Translational Velocity in the expanded portion of the panel. Details about these inputs are presented in Section 6.17.1 for fluid zones, and in Section 6.18.1 for solid zones. ! It is important to define the axis of rotation for the cell zones on both sides of the mixing plane interface, including the stationary zone. 3. Define the velocity boundary conditions at walls. You can choose to define either an absolute velocity or a velocity relative to the velocity of the adjacent cell zone specified in step 2. If the wall is moving at the speed of the moving frame (and hence stationary relative to the moving frame), it is convenient to specify a relative angular velocity of zero. Likewise, a wall that is stationary in the non-moving frame of reference should be given a velocity of zero in the absolute reference frame. Specifying the wall velocities in this manner obviates the need to modify these inputs later if a change is made in the rotational velocity of the fluid zone. An example for which you would specify a relative velocity is as follows: If an impeller is defined as wall-3 and the fluid region within the impeller’s radius is defined as fluid-5, you would need to specify the angular velocity and axis of rotation for fluid-5 and then assign wall-3 a relative velocity of 0. If you later wanted to model a different angular velocity for the impeller, you would need 9-32 c Fluent Inc. November 28, 2001 9.4 The Mixing Plane Model to change only the angular velocity of the fluid region; you would not need to modify the wall velocity conditions. Details about these inputs are presented in Section 6.13.1. 4. Define the velocity at any velocity inlets and the flow direction and total pressure at any pressure inlets or mass flow inlets. For velocity inlets, you can choose to define either absolute velocities or velocities relative to the motion of the adjacent cell zone (specified in step 2). For pressure inlets and mass flow inlets, the specification of the flow direction and total pressure will always be absolute, because the absolute velocity formulation is always used for mixing plane calculations. For a mass flow inlet, you do not need to specify the mass flow rate or mass flux. FLUENT will automatically select the Mass Flux with Average Mass Flux specification method and set the correct values when you create the mixing plane, as described in 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 zone at 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 suitable combination permitted by the solver (e.g., velocity inlet, pressure inlet, or mass flow inlet; pressure outlet). Keep in mind, however, that if mass conservation across the mixing plane is important, you need to use a mass flow inlet as the downstream boundary; mass conservation is not maintained across the mixing plane when you use 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 selecting an upstream zone in the Upstream Zone list and a downstream zone in the Downstream Zone list. It is essential that the correct pairs be chosen from these lists (i.e., that the c Fluent Inc. November 28, 2001 9-33 Modeling Flows in Moving Zones 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 interface by choosing one of the options under Mixing Plane Geometry. A Radial geometry signifies that information at the mixing plane interface is to be circumferentially averaged into profiles that vary in the radial direction, e.g., p(r), T (r). This is the case for axial-flow machines, for example. An Axial geometry signifies that circumferentially averaged profiles 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 the rotation axis. (c) (3D only) Set the number of Interpolation Points. This is the 9-34 c Fluent Inc. November 28, 2001 9.4 The Mixing Plane Model number of radial or axial locations used in constructing the boundary profiles for circumferential averaging. You should choose a number that approximately corresponds to the resolution of the surface mesh in the radial or axial direction. Note that while you can use more points if you wish, the resolution of the boundary profile will only be as fine as the resolution of the surface mesh itself. In 2D the flow data are averaged over the entire interface to create a profile consisting of a single data point. For this reason you do not need to set the number of Interpolation Points 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 desirable to under-relax the changes in boundary values at mixing planes as these may change very rapidly during the early iterations of the solution and cause the calculation to diverge. The changes can be relaxed by specifying an under-relaxation less than 1. The new boundary profile values are then computed using φnew = φold + α(φcalculated − φold ) (9.4-4) where α is the under-relaxation factor. Once the flow field is established, the value of α can be increased. ii. Click Apply to set the Global Parameters. If the Default button is visible to the right of the Apply button, clicking the Default button will return Global Parameters back to their default values. The Default button will then change to be a Reset button. Clicking the Reset button will change the Global Parameters back to the values that were last applied. (e) Click Create to create a new mixing plane. FLUENT will name the mixing plane by combining the names of the zones selected as the Upstream Zone and Downstream Zone and enter the new mixing plane in the Mixing Plane list. c Fluent Inc. November 28, 2001 9-35 Modeling Flows in Moving Zones If you create an incorrect mixing plane, you can select it in the 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: a fixed pressure level for incompressible flows, and the swirl conservation described 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 is used. The mixing plane model is usually used to model the three interfaces that connect the components of the torque converter. In this configuration, the pressure is no longer fixed. As a result, the pressure may float unbounded, making it difficult to obtain a converged solution. To resolve this problem, FLUENT offers an option for fixing the pressure level. When this option is enabled, FLUENT will adjust the gauge pressure field after each iteration by subtracting from it the pressure value in the cell closest to the Reference Pressure Location in the Operating Conditions 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 text command: 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 applications such as torque converters. If you want to enable swirl conservation across the mixing plane, you can use the commands in the conserve-swirl text menu: define −→ mixing-planes −→ set −→conserve-swirl 9-36 c Fluent Inc. November 28, 2001 9.4 The Mixing Plane Model To turn on swirl conservation, use the enable? text command. Once the option is turned on, you can ask the solver to report information about the swirl conservation during the calculation. If you turn on verbosity?, FLUENT will report for every iteration the zone ID for the zone on which the swirl conservation is active, the upstream and downstream swirl integration per zone area, and the ratio of upstream to downstream swirl integration before and after the correction. To obtain a report of the swirl integration at every pressure inlet, pressure outlet, velocity inlet, and mass flow inlet in the domain, use the report-swirl-integration command. You can use this information to determine the torque acting on each component of the turbomachinery 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 approximation so long as there is not significant reverse flow in the vicinity of the mixing plane. If significant reverse flow occurs, the mixing plane will not be a satisfactory model of the actual flow. In a numerical simulation, reverse flow often occurs during the early stages of the computation even though the flow at convergence is not reversed. Therefore, it is helpful in these situations to first obtain a provisional solution using fixed conditions at the rotor-stator interface. The mixing plane model can then be enabled and the solution run to convergence. Under-relaxing the changes in the mixing plane boundary values can also help in troublesome situations. In many cases, setting the underrelaxation factor to a value less than one can be helpful. Once the flow field is established, you can gradually increase the under-relaxation factor. 9.4.5 Postprocessing for the Mixing Plane Model When you solve a problem using the mixing plane model, you can plot or report both absolute and relative velocities. For all velocity parameters (e.g., Velocity Magnitude and Mach Number), corresponding relative values will be available for postprocessing (e.g., Relative Velocity Magnitude and Relative Mach Number). These variables are contained in the c Fluent Inc. November 28, 2001 9-37 Modeling Flows in Moving Zones Velocity... category of the variable selection drop-down list that appears in postprocessing panels. Relative values are also available for postprocessing of total pressure, total temperature, and any other parameters that 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). The velocity of the reference zone is the velocity defined in the Fluid panel for that zone. When plotting velocity vectors, you can choose to plot vectors in the absolute frame (the default), or you can select Relative Velocity in the Vectors Of drop-down list in the Vectors panel to plot vectors relative to the translational/rotational velocity of the “reference zone” (specified in the Reference Values panel). If you plot relative velocity vectors, you might want to color the vectors by relative velocity magnitude (by choosing Relative Velocity Magnitude in the Color By list); by default they will be colored by absolute velocity magnitude. See also Section 25.9 for details about turbomachinery-specific postprocessing features. 9-38 c Fluent Inc. November 28, 2001 9.5 Sliding Meshes 9.5 Sliding Meshes 9.5.1 Overview When a time-accurate solution for rotor-stator interaction (rather than a time-averaged solution) is desired, you must use the sliding mesh model to compute the unsteady flow field. As mentioned in Section 9.1, the sliding mesh model is the most accurate method for simulating flows in multiple moving reference frames, but also the most computationally demanding. Most often, the unsteady solution that is sought in a sliding mesh simulation is time-periodic. That is, the unsteady solution repeats with a period related to the speeds of the moving domains. However, you can model 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 stationary and moving parts (i.e., when there is only a rotor), the computational domain can be made stationary by using a rotating reference frame. (See Section 9.2 for details.) When transient rotor-stator interaction is desired (as in the examples in Figures 9.5.2 and 9.5.3), you must use sliding meshes. If you are interested in a steady approximation of the interaction, you may use the multiple reference frame model or the mixing plane model, as described in Sections 9.3 and 9.4. c Fluent Inc. November 28, 2001 9-39 Modeling Flows in Moving Zones stationary vanes rotating blades flow direction of motion Figure 9.5.2: Rotor-Stator Interaction (Stationary Guide Vanes with Rotating Blades) Figure 9.5.3: Blower 9-40 c Fluent Inc. November 28, 2001 9.5 Sliding Meshes The Sliding Mesh Technique In the sliding mesh technique two or more cell zones are used. (If you generate the mesh in each zone independently, you will need to merge the mesh files prior to starting the calculation, as described in Section 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 adjacent cell zones are associated with one another to form a “grid interface.” The two 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 shown in 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) relative to one another along the grid interface in discrete steps. Figures 9.5.4 and 9.5.5 show the initial position of two grids and their positions after some translation has occurred. Figure 9.5.4: Initial Position of the Grids c Fluent Inc. November 28, 2001 9-41 Modeling Flows in Moving Zones Figure 9.5.5: Rotor Mesh Slides with Respect to the Stator As the rotation or translation takes place, node alignment along the grid interface is not required. Since the flow is inherently unsteady, a time-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 geometry. Figure 9.5.6 shows an example with a linear grid interface and Figure 9.5.7 shows a circular-arc grid interface. (In both figures, the grid interface is designated by a dashed line.) If Figure 9.5.6 were extruded to 3D, the resulting sliding interface would be a planar rectangle; if Figure 9.5.7 were extruded to 3D, the resulting interface would be a cylinder. Figure 9.5.8 shows an example that would use a conical grid interface. (The slanted, dashed lines represent the intersection of the conical interface with a 2D plane.) 9-42 c Fluent Inc. November 28, 2001 9.5 Sliding Meshes Figure 9.5.6: 2D Linear Grid Interface Figure 9.5.7: 2D Circular-Arc Grid Interface c Fluent Inc. November 28, 2001 9-43 Modeling Flows in Moving Zones Figure 9.5.8: 3D Conical Grid Interface 9-44 c Fluent Inc. November 28, 2001 9.5 Sliding Meshes For an axial rotor/stator configuration, in which the rotating and stationary parts are aligned axially instead of being concentric (see Figure 9.5.9), the interface will be a planar sector. This planar sector is a cross-section of the domain perpendicular to the axis of rotation at a position along the axis between the rotor and the stator. portion of domain being modeled planar sector grid interface Figure 9.5.9: 3D Planar-Sector Grid Interface c Fluent Inc. November 28, 2001 9-45 Modeling Flows in Moving Zones 9.5.2 Sliding Mesh Theory As discussed in Section 9.5.1, the sliding mesh model allows adjacent grids to slide relative to one another. In doing so, the grid faces do not need to be aligned on the grid interface. This situation requires a means of computing the flux across the two non-conformal interface zones of each grid interface. To compute the interface flux, the intersection between the interface zones is determined at each new time step. The resulting intersection produces one interior zone (a zone with fluid cells on both sides) and one or more periodic zones. If the problem is not periodic, the intersection produces one interior zone and a pair of wall zones (which will be empty if the two interface zones intersect entirely), as shown in Figure 9.5.10. (You will need to change these wall zones to some other appropriate boundary type.) The resultant interior zone corresponds to where the two interface zones overlap; the resultant periodic zone corresponds to where they do not. The number of faces in these intersection zones will vary as the interface zones move relative to one another. Principally, fluxes across the grid interface are computed using the faces resulting from the intersection of the two interface zones, rather than from the interface zone faces themselves. "wall" zone interior 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 composed of faces A-B and B-C, and faces D-E and E-F. The intersection of these zones produces the faces a-d, d-b, b-e, etc. Faces produced in the region where the two cell zones overlap (d-b, b-e, and e-c) are grouped to form 9-46 c Fluent Inc. November 28, 2001 9.5 Sliding Meshes an interior zone, while the remaining faces (a-d and c-f) are paired up to form a periodic zone. To compute the flux across the interface into cell IV, for example, face D-E is ignored and faces d-b and b-e are used instead, bringing information into cell IV from cells I and III, respectively. cell zone 1 II I III A a B b d interface zone 2 interface zone 1 C e D c f F E VI IV V cell zone 2 Figure 9.5.11: Two-Dimensional Grid Interface c Fluent Inc. November 28, 2001 9-47 Modeling Flows in Moving Zones 9.5.3 Setup and Solution of a Sliding Mesh Problem Grid Requirements Before beginning the problem setup in FLUENT, be sure that the grid you have created meets the following requirements: • A different cell zone exists for each portion of the domain that is sliding at a different speed. • The grid interface must be situated such that there is no motion normal to it. • The grid interface can be any shape (including a non-planar surface, in 3D), provided that the two interface boundaries are based on the same geometry. If there are sharp features in the mesh (e.g., 90-degree angles), it is especially important that both sides of the interface closely follow that feature. • If you create a single grid with multiple cell zones, you must be sure that each cell zone has a distinct face zone on the sliding boundary. The face zones for two adjacent cell zones will have the same position and shape, but one will correspond to one cell zone and one to the other. (Note that it is also possible to create a separate grid file for each of the cell zones, and then merge them as described in Section 5.3.10.) • If you are modeling a rotor/stator geometry using periodicity, the periodic angle of the mesh around the rotor blade(s) must be the same as that of the mesh around the stationary vane(s). • All periodic zones must be correctly oriented (either rotational or translational) before you create the grid interface. • For 3D cases, if the interface is periodic, only one pair of periodic boundaries can neighbor the interface. See Section 9.5.1 for details about these restrictions and general information about how the sliding mesh model works in FLUENT. 9-48 c Fluent Inc. November 28, 2001 9.5 Sliding Meshes Setting Up the Problem The steps for setting up a sliding mesh problem are listed below. (Note that this procedure includes only those steps necessary for the sliding mesh model itself; you will need to set up other models, boundary conditions, etc. as usual.) 1. Enable the appropriate option for modeling unsteady flow in the Solver panel. (See Section 22.15 for details about the unsteady modeling capabilities in FLUENT.) Define −→ Models −→Solver... 2. Set boundary conditions for the sliding action: Define −→Boundary Conditions... (a) Change the zone type of the interface zones of adjacent cell zones to interface in the Boundary Conditions panel. (b) In the Fluid panel or Solid panel for each moving fluid or solid zone, select Moving Mesh in the Motion Type drop-down list and set the translational and/or rotational velocity. (Note that a solid zone cannot move at a different speed than an adjacent fluid zone.) ! Note that simultaneous translation and rotation can be modeled only if the rotation axis and the translation direction are the same (i.e., the origin is fixed). By default, the velocity of a wall is set to zero relative to the adjacent mesh’s motion. For walls bounding a moving mesh this results in a “no-slip” condition in the reference frame of the mesh. Therefore, you need not modify the wall velocity boundary conditions unless the wall is stationary in the absolute frame, and therefore moving in the relative frame. See Section 6.13.1 for details about wall 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... c Fluent Inc. November 28, 2001 9-49 Modeling Flows in Moving Zones 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 interface by selecting one in the Interface Zone 1 list and one in the 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 zone and a fluid zone. (d) Click on Create to create a new grid interface. For all types of interfaces, FLUENT will create boundary zones for the interface (e.g., wall-9, wall-10), which will appear under Boundary Zone 1 and Boundary Zone 2. You can use 9-50 c Fluent Inc. November 28, 2001 9.5 Sliding Meshes the Boundary Conditions panel to change them to another zone type (e.g., pressure far-field, symmetry, pressure outlet). If you have enabled the Coupled option, FLUENT will also create wall interface zones (e.g., wall-4, wall-4-shadow), which will appear under Interface Wall Zone 1 and Interface Wall Zone 2. If you create an incorrect grid interface, you can select it in the Grid Interface list and click on the Delete button to delete it. (Any boundary zones that were created when the interface was created will 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 in the case file, so case files that you save at different times during the unsteady 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 size and number of time steps in the Iterate panel, as for any other unsteady calculation. (See Section 22.15 for details about time-dependent solutions. Note that if you wish to save the time step size in the initial case file, you can click Apply instead of Iterate and then save a case file before starting to iterate.) FLUENT will iterate on the current time step solution until satisfactory residual reduction is achieved, or the maximum number of iterations per time step is reached. When it advances to the next time step, the cell and wall zones will automatically be moved according to the specified translational or rotational velocities (set in step 2b above). The new interface-zone intersections will be computed automatically, and resultant interior/periodic/external boundary zones will be updated (as described in Section 9.5.2). c Fluent Inc. November 28, 2001 9-51 Modeling Flows in Moving Zones Saving Case and Data Files FLUENT’s automatic saving of case and data files (see Section 3.3.4) can be used with the sliding mesh model. This provides a convenient way for 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 with each time step, reading data for a given time step will require the case file at that time step so that the grid will be in the proper position. You should also save your initial case file so that you can easily return to the grid’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 and data files, exit FLUENT, start a new FLUENT session, read the case and data 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 with each subsequent continuation of the calculation. To avoid this problem, you can delete the grid interface before saving the case file, and then create it again (as described in step 3 above) after you read the case file into a new FLUENT session. Time-Periodic Solutions For some problems (e.g., rotor-stator interactions), you may be interested in a time-periodic solution. That is, the startup transient behavior may not be of interest to you. Once this startup phase has passed, the flow will start 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 + N T ) (N = 1, 2, 3, ...) (9.5-1) For rotating problems, the period (in seconds) can be calculated by dividing 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 time steps in a period can be determined by dividing the time period by the 9-52 c Fluent Inc. November 28, 2001 9.5 Sliding Meshes time step size. When the solution field does not change from one period to the next (for example, if the change is less than 5%), a time-periodic solution has been reached. To determine how the solution changes from one period to the next, you will need to compare the solution at some point in the flow field over two periods. For example, if the time period is 10 seconds, you can compare the solution at a given point after 22 seconds with the solution after 32 seconds to see if a time-periodic solution has been reached. If not, you can continue the calculation for another period and compare the solutions after 32 and 42 seconds, and so on until you see little or no change from one period to the next. You can also track global quantities, such as lift and drag coefficients and mass flow, in the same manner. Figure 9.5.13 shows a lift coefficient plot for a time-periodic solution. -5.00e+00 -5.10e+00 -5.20e+00 -5.30e+00 -5.40e+00 Cl -5.50e+00 -5.60e+00 -5.70e+00 -5.80e+00 -5.90e+00 -6.00e+00 0 Y X 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 Time Z Cl Figure 9.5.13: Lift Coefficient Plot for a Time-Periodic Solution The final time-periodic solution is independent of the time steps taken during the initial stages of the solution procedure. You can therefore define “large” time steps in the initial stages of the calculation, since you are not interested in a time-accurate solution for the startup phase of the flow. Starting out with large time steps will allow the solution c Fluent Inc. November 28, 2001 9-53 Modeling Flows in Moving Zones to become time-periodic more quickly. As the solution becomes timeperiodic, however, you should reduce the time step in order to achieve a time-accurate result. ! If you are solving with second-order time accuracy, the temporal accuracy of the solution will be affected if you change the time step during the calculation. You may start out with larger time steps, but you should not 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 periods to 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 unsteady problems. You will read in the case and data file for the time of interest and display and report results as usual. For spatially-periodic problems, you may want to use periodic repeats (set in the Views panel, as described in Section 25.4) to display the geometry. Figure 9.5.14 shows the flow field for the rotor-stator example of Figure 9.5.4 at one instant in time, using 1 periodic repeat. When displaying velocity vectors, note that absolute velocities (i.e., velocities in the inertial, or laboratory, reference frame) are displayed by default. You may also choose to display relative velocities by selecting Relative Velocity in the Vectors Of drop-down list in the Vectors panel. In this case, velocities relative to the translational/rotational velocity of the “reference zone” (specified in the Reference Values panel) will be displayed. (The velocity of the reference zone is the velocity defined in the 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 intersection of the interface zones). You may instead create zone surfaces for the interface zones. Data displayed on these surfaces will be “one-sided”. That is, nodes on the interface zones will “see” only the cells on one side of the grid interface, and slight discontinuities may appear when you plot contour lines across the interface. Note also that, for non-planar interface shapes in 3D, you may see small gaps in your plots of filled contours. These discontinuities and gaps are only graphical in nature. 9-54 c Fluent Inc. November 28, 2001 9.5 Sliding Meshes 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 Contours of Static Pressure (pascal) (Time=1.0400e-01) 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. This allows you to find the average value of a quantity at several different radial or axial positions in your model. FLUENT computes the average of the quantity over a specified circumferential area, and then plots the average against the radial or axial coordinate. For more information on generating XY plots of circumferential averages, see Section 25.8.4. c Fluent Inc. November 28, 2001 9-55 Modeling Flows in Moving Zones 9.6 Non-Reflecting Boundary Conditions Information about non-reflecting boundary conditions (NRBCs) is provided 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 specific flow variables at the boundary (e.g., static pressure at an outlet boundary). As a result, pressure waves incident on the boundary will reflect in an unphysical manner, leading to local errors. The effects are more pronounced for internal flow problems where boundaries are usually close to geometry inside the domain, such as compressor or turbine blade rows. The non-reflecting boundary conditions (NRBCs) permit waves to “pass” through the boundaries without spurious reflections. The method used in FLUENT is based on the Fourier transformation of solution variables at the non-reflecting boundary [78]. Similar implementations have been investigated by other authors [155, 198]. In the method used by FLUENT, the solution is rearranged as a sum of terms corresponding to different frequencies, and their contributions are calculated independently. While the method was originally designed for axial turbomachinery, it has been extended for use with radial turbomachinery. Limitations Note the following limitations of NRBCs: • NRBCs can be used only with the coupled explicit solver. 9-56 c Fluent Inc. November 28, 2001 9.6 Non-Reflecting Boundary Conditions • 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 and outlets only. ! Note that the pressure inlet boundaries must be set to the cylindrical coordinate flow specification method when NRBCs are used. • Quad-mapped (structured) surface meshes must be used for inflow and outflow boundaries in a 3D geometry (i.e., triangular or quadpaved surface meshes are not allowed). See Figures 9.6.1 and 9.6.2 for examples. Note that you may use unstructured meshes in 2D geometries (Figure 9.6.3), and away from the inlet and outlet boundaries in 3D geometries. pressure inlet, structured quad mesh periodic boundaries pressure outlet, structured quad mesh Figure 9.6.1: Mesh and Prescribed Boundary Conditions in a 3D Axial Flow Problem c Fluent Inc. November 28, 2001 9-57 Modeling Flows in Moving Zones pressure inlet, structured quad mesh periodic boundaries pressure outlet, structured quad mesh Figure 9.6.2: Mesh and Prescribed Boundary Conditions in a 3D Radial Flow Problem 9-58 c Fluent Inc. November 28, 2001 9.6 Non-Reflecting Boundary Conditions or periodic boundaries pressure outlets, structured or unstructured mesh pressure inlets, structured or unstructured mesh Figure 9.6.3: Mesh and Prescribed Boundary Conditions in a 2D Case c Fluent Inc. November 28, 2001 9-59 Modeling Flows in Moving Zones 9.6.2 Theory The non-reflecting boundary conditions (NRBCs) are based on Fourier decomposition of solutions to the linearized Euler equations. The solution at the inlet and outlet boundaries is circumferentially decomposed into Fourier modes, with the 0th mode representing the average boundary value (which is to be imposed as a user input), and higher harmonics that are modified to eliminate reflections [198]. Equations in Characteristic Variable Form In order to treat individual waves, the linearized Euler equations are transformed to characteristic variable (Ci ) form. If we first consider the 1D form of the linearized Euler equations, it can be shown that the characteristic variables Ci are related to the solution variables as follows: Q̃ = T −1 C (9.6-1) where ρ̃ ũa Q̃ = ũ t ũ r p̃ , T −1 = − a12 0 0 0 0 0 0 1 ρ a 0 0 0 0 0 1 ρ a 0 1 2a2 1 2 ρ a 1 2a2 1 2 ρ a 0 0 0 0 1 2 1 2 , C = C 1 C2 C 3 C 4 C5 where a is the average acoustic speed along a boundary zone, ρ̃, ũa , ũt , ũr , and p̃ represent perturbations from a uniform condition (e.g., ρ̃ = ρ − ρ, p̃ = p − p, etc.). Note that the analysis is performed using the cylindrical coordinate system. All overlined (averaged) flow field variables (e.g., ρ, a) are intended to be averaged along the pitchwise direction. In quasi-3D approaches [78, 155, 198], a procedure is developed to determine the changes in the characteristic variables, denoted by δCi , at the boundaries such that waves will not reflect. These changes in characteristic variables are determined as follows: 9-60 c Fluent Inc. November 28, 2001 9.6 Non-Reflecting Boundary Conditions δC = T δQ (9.6-2) where δC1 δC2 δC = δC3 , T = δC4 δC5 −a2 0 0 0 0 0 ρa 0 0 0 0 ρa 0 ρa 0 0 0 −ρ a 0 0 1 0 0 1 1 δρ δua , δQ = δut δur δp The changes to the outgoing characteristics — one characteristic for subsonic inflow (δC5 ), and four characteristics for subsonic outflow (δC1 , δC2 , δC3 , δC4 ) — are determined from extrapolation of the flow field variables using Equation 9.6-2. The changes in the incoming characteristics — four characteristics for subsonic inflow (δC1 , δC2 , δC3 , δC4 ), and one characteristic for subsonic outflow (δC5 ) — are split into two components: average change along the boundary (δC i ), and local changes in the characteristic variable due to harmonic variation along the boundary (δCiL ). The incoming characteristics are therefore given by δCij = δCioldj + σ δCinewj − δCioldj δCinewj = δC i + δCiLj (9.6-3) (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 the periodic point once. The under-relaxation factor σ has a default value of 0.75. Note that this method assumes a periodic solution in the pitchwise direction. The flow is decomposed into mean and circumferential components using Fourier decomposition. The 0th Fourier mode corresponds to the average circumferential solution, and is treated according to the standard 1D characteristic theory. The remaining parts of the solution are described c Fluent Inc. November 28, 2001 9-61 Modeling Flows in Moving Zones by a sum of harmonics, and treated as 2D non-reflecting boundary conditions [78]. Inlet Boundary For subsonic inflow, there is one outgoing characteristic (δC5 ) determined from Equation 9.6-2, and four incoming characteristics (δC1 , δC2 , δC3 , δC4 ) calculated using Equation 9.6-3. The average changes in the incoming characteristics are computed from the requirement that the entropy (s), radial and tangential flow angles (αr and αt ), and stagnation enthalpy (h0 ) are specified. Note that in FLUENT you can specify p0 and T0 at the inlet, from which sin and h0in are 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 tan αt ) (9.6-6) R3 = ρ a (ur − ua tan αr ) (9.6-7) R4 = ρ h0 − h0in (9.6-8) where sin = γ ln (T0in ) − (γ − 1) ln (p0in ) (9.6-9) h0in = cp T0in (9.6-10) The average characteristic is then obtained from residual linearization as follows (see also Figure 9.6.4 for an illustration of the definitions for the prescribed inlet angles): δC 1 δC = δC 3 2 δC 4 −1 0 tan αt (1−γ)M tan αr (γ−1)M 2 (γ−1)M M1 M Mt M tan αt t 2M M −Mt M 0 tan αt M2 M r 2M M 0 tan αt M − tan αr M −2 M R1 R2 R 3 R4 (9.6-11) 9-62 c Fluent Inc. November 28, 2001 9.6 Non-Reflecting Boundary Conditions where Ma = Mt = Mr = ua a ut a ur a (9.6-12) (9.6-13) (9.6-14) and M = 1 + Ma − Mt tan αt + Mr tan αr (9.6-15) M1 = −1 − Ma − Mr tan αr (9.6-16) M2 = −1 − Ma − Mt tan αt (9.6-17) radial ur v αr ut ua axial αt theta Figure 9.6.4: Prescribed Inlet Angles where |v| = c Fluent Inc. November 28, 2001 q u2t + u2r + u2a (9.6-18) 9-63 Modeling Flows in Moving Zones et = er = ea = ut |v| ur |v| ua |v| (9.6-19) (9.6-20) (9.6-21) et ea er tan αr = ea tan αt = (9.6-22) (9.6-23) To address the local characteristic changes at each j grid point along the inflow boundary, the following relations are developed [78, 198]: δC1Lj = p (sj − s) δC2Lj = C20 j − ρa utj − ut δC3Lj = −ρa urj − ur δC4Lj = −2 (1+Maj ) 1 γ−1 δC1Lj + Mtj δC2Lj + Mrj δC3Lj + ρ h0j − h0 (9.6-24) Note that the relation for the first and fourth local characteristics force the local entropy and stagnation enthalpy to match their average steadystate values. The characteristic variable C20 j is computed from the inverse discrete Fourier transform of the second characteristic. The discrete Fourier transform of the second characteristic in turn is related to the discrete Fourier transform of the fifth characteristic. Hence, the characteristic variable C20 j is computed along the pitch as follows: θj − θ1 X Ĉ2n exp i2πn = 2< C20 j N 2 −1 n=1 θN − θ1 (9.6-25) The Fourier coefficients C20 n are related to a set of equidistant distributed characteristic variables C5∗j by the following [155]: 9-64 c Fluent Inc. November 28, 2001 9.6 Non-Reflecting Boundary Conditions ( Ĉ2n = PN t +B − Nu(a+u a) t +B − ua+u C 5j a ∗ j=1 C5j exp −i2π jn N β>0 β<0 (9.6-26) where ( √ B= i β β>0 p −sign (ut ) |β| β < 0 (9.6-27) β = a2 − u2a − u2t (9.6-28) and The set of equidistributed characteristic variables C5∗j is computed from arbitrary 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 determining 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 characteristic (δC5 ) determined from Equation 9.6-3. The average change in the incoming fifth characteristic is given by δC 5 = −2 (p − pout ) (9.6-30) where p is the current averaged pressure at the exit plane and pout is the desirable average exit pressure (this value is specified by you for singleblade calculations or obtained from the assigned profile for mixing-plane calculations). The local changes (δC5Lj ) are given by c Fluent Inc. November 28, 2001 9-65 Modeling Flows in Moving Zones δC5Lj = C50 j + ρ a uaj − ua − (pj − p) (9.6-31) 0 The characteristic variable C5j is computed along the pitch as follows: −1 X θj − θ1 Ĉ5n exp i2πn = 2< C50 j N 2 (9.6-32) θN − θ1 n=1 The Fourier coefficients Ĉ5n are related to two sets of equidistant distributed characteristic variables (C2∗j and C4∗j , respectively) and given by the following [155]: ( Ĉ5n = PN jn ∗ j=1 C2j exp i2π N − A2 C2j − A4 C4j A2 N A4 N PN jn ∗ j=1 C4j exp i2π N β>0 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 (C2∗j and C4∗j ) are computed from arbitrary distributed C2j and C4j characteristics by using 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 the interior. 9-66 c Fluent Inc. November 28, 2001 9.6 Non-Reflecting Boundary Conditions Updated Flow Variables Once the changes in the characteristics are determined on the inflow or outflow boundaries, the changes in the flow variables δQ can be obtained from Equation 9.6-2. Therefore, the values of the flow variables at the boundary 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) = Tj + δT (9.6-42) Tf 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 the show-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 domain extent will be displayed. If the initialization is not successful, an error message will be displayed indicating the source of the problem. 3. If necessary, modify the parameters in the set/ submenu: define −→ boundary-conditions −→ non-reflecting −→set c Fluent Inc. November 28, 2001 9-67 Modeling Flows in Moving Zones under-relaxation allows you to set the value of the under-relaxation factor σ in Equation 9.6-3. The default value is 0.75. discretization allows you to set the discretization scheme. The default is to use higher-order reconstruction if available. verbosity allows you to control the amount of information printed to 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 must define the mixing plane interfaces as pressure-outlet and pressure-inlet zone type pairs. Using the NRBCs in Parallel FLUENT When the NRBCs are used in conjunction with the parallel solver, all cells in each boundary zone where NRBCs will be applied must be located or contained within a single partition. You can ensure this by manually partitioning the grid (see Section 28.4.3 for more information on how to do this). 9-68 c Fluent Inc. November 28, 2001