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3. MESHING THE MODEL

When you click the Mesh command button on the Operation toolpad, GAMBIT opens the Mesh subpad. The Mesh subpad contains command buttons that allow you to perform mesh operations involving boundary layers, edges, faces, volumes, and groups.

The symbols associated with each of the Mesh subpad command sets are as follows.

Symbol Command Set
Boundary Layer
Edge
Face
Volume
Group

The following sections of this chapter describe the commands associated with each of the command buttons listed above.


3.1 Boundary Layers

3.1.1 Overview

Boundary layers define the spacing of mesh node rows in regions immediately adjacent to edges and/or faces. They are used primarily to control mesh density and, thereby, to control the amount of information available from the computational model in specific regions of interest.

As an example of a boundary layer application, consider a computational model that includes a cylinder representing a pipe through which flows a viscous fluid. Under normal circumstances, it is likely that the fluid velocity gradients are large in the region immediately adjacent to the pipe wall and small near the center of the pipe. By attaching a boundary layer to the face that represents the pipe wall, you can increase the mesh density near the wall and decrease the density near the center of the cylinder, thereby obtaining sufficient information to characterize the gradients in both regions while minimizing the total number of mesh nodes in the model.

To define a boundary layer, you must specify the following information:

You can also specify the creation of a transition boundary layer—that is, one for which the mesh node pattern changes with each succeeding layer. If you specify such a boundary layer, you must also specify the transition pattern and number of transition rows.


3.1.2 Boundary Layer Commands

The following commands are available on the Mesh/Boundary Layer subpad.

Symbol Command Description
Create Boundary Layer Creates a boundary layer attached to an edge or face
Modify Boundary Layer Modifies the definition of an existing boundary layer
View 3D Boundary Layers Meshes and displays 3-D boundary layer regions
Modify Boundary Layer Label Modifies boundary layer labels
Summarize Boundary Layers Displays existing boundary layers in the graphics window
Delete Boundary Layers Deletes boundary layers


Create Boundary Layer

The Create Boundary Layer operation (blayer create and blayer attach commands) defines the spacing of mesh nodes in the vicinity of an edge or face. The operation requires the following specifications.

The Definition specifications include the algorithm type and dimension parameters that determine the shape of the boundary layer, as well as options that govern the behavior of boundary layers in corner regions. The Transition pattern specifies the arrange­ment of mesh nodes in the boundary layer region. The Attachment parameters include the entities to which the boundary layer is attached and the entities that specify its direction.

Specifying the Definition
To define a boundary layer, you must specify the following parameters: The Algorithm specifies the method that GAMBIT uses to determine the general shape of the boundary layer. The dimension parameters specify factors such as the number of boundary layer rows and growth factor. The Internal continuity option specifies the behavior of the boundary layer in regions where it overlaps adjoining boundary layers. The Corner shape option deter­mines the shape of the mesh in regions surrounding Corner or Reversal verti­ces that connect edges to which boundary layers are attached.

Specifying the Algorithm
The Algorithm specification determines the method that GAMBIT uses to establish the heights of the elements in the first row of the boundary layer and compute the heights of all succeeding rows. GAMBIT provides the follow­ing algorithm types. Figure 3-1, Figure 3-2, and Figure 3-3 illustrate the differences between the Uniform, Aspect ratio (first), and Aspect ratio (last) algorithms for a 2-D boundary layer attached to one edge of a square planar face. In each figure, the attachment edge mesh includes five intervals and a grading ratio of 1.25, and the boundary layer includes five rows.

Figure 3-1: Uniform boundary layer algorithm (2-D)

Figure 3-2: Aspect ratio (first) boundary layer algorithm (2-D)

Figure 3-3: Aspect ratio (last) boundary layer algorithm (2-D)

For the Uniform boundary layer (Figure 3-1), the first row exhibits a uniform height across the span of the attachment edge, and the growth factor is constant; therefore, each succeeding row of elements also exhibits a uniform height. For the Aspect ratio (first) boundary layer (Figure 3-2), the first-row heights vary in proportion to the edge mesh interval lengths. Consequently, the first row of the boundary layer grows thicker from left to right across the edge, because the edge mesh interval lengths increase from left to right. For the Aspect ratio (last) boundary layer (Figure 3-3), the first row exhibits a uniform height across the span of the attachment edge, but the growth factor varies in proportion to the edge mesh interval widths. Consequently, the succeeding rows grow thicker from left to right across the edge.

NOTE: If the attachment edge shown in Figure 3-1, Figure 3-2, and Figure 3-3, above, were graded uniformly (Ratio = 1), all three Algorithm options would produce boundary layers of uniform height across the span of the edge.

If you attach a boundary layer to a face (rather than an edge), GAMBIT applies the definition algorithm along the boundaries of the attachment face. For example, Figure 3-4 shows an Aspect ratio (first) boundary layer attached to one face of a cube. In this case, the boundary edges of the attach­ment face have been premeshed using five intervals and a grading ratio of 1.25.

Figure 3-4: Aspect ratio (first) boundary layer algorithm (3-D)

When attaching a boundary layer to a face, care must be taken to ensure that the boundary layer is not discontinuous at any vertices on the face boundary. In Figure 3-4, above, the boundary edges of the attachment face are graded such that the mesh interval widths on either side of any corner vertex are equal to each other. As a result, the 3-D boundary layer is continuous at all four corners of the attachment face. In Figure 3-5, the face boundary edges are graded such that edge mesh interval lengths differ on either side of three of the four corner vertices (b, c, and d). Consequently, the boundary layer exhibits discontinuities at those vertices.

Figure 3-5: Effect of discontinuous grading at face boundary vertices

Specifying the Dimensions
To specify the dimensions of any boundary layer, you must input three parameters that define its characteristics. The parameters to be specified vary according to algorithm as follows.

Uniform Algorithm

The Uniform algorithm (see Figure 3-6) definition parameters are as follows.

Figure 3-6: Boundary layer dimensions—Uniform algorithm

The First row (a) value specifies the height of the first row (a)—that is, the absolute distance between the entity to which the boundary layer is attached and the first row of mesh nodes in the boundary layer. (NOTE: For the Uniform algorithm, the first-row height, a, is uniform across the boundary layer.)

The Growth factor (b/a) value (G) represents the ratio

where b is the distance between the first and second rows and a is the height of the first row. The height of any row in the boundary layer (other than the first row) is equal to the height of the previous row times the Growth factor (b/a) value.

The Rows value specifies the total number of rows to be included in the boundary layer.

NOTE: When you specify the First row (a), Growth factor (b/a), and Rows values, GAMBIT computes the total depth (D) of the boundary layer and displays the value in the non-editable Depth (D) field on the Create Boundary Layer form.

Aspect ratio (first) Algorithm

The Aspect ratio (first) algorithm (see Figure 3-7) definition parameters are as follows.

Figure 3-7: Boundary layer dimensions—Aspect ratio (first) algorithm

The First percent (a/w) value specifies the height of any first row boundary layer node () as a percentage of mesh interval width at the associated node on the attach­ment entity. For interior nodes on the attachment entity, the general specification of first-row height can be expressed as

where is the height of the first row at node i, F is the First percent (a/w) value, and and are the widths of the attachment-entity mesh intervals on either side of node i.

For exterior nodes on the attachment entity (for example, nodes located at edge endpoints) the first-row heights can be expressed as

and

where and are the heights of the first rows at the exterior nodes.

The Growth factor (b/a) value (G) represents the ratio

where is the distance between the first and second rows at edge mesh node i and is the height of the first row at node i. (NOTE: For the Aspect ratio (first) algorithm, the Growth factor (b/a) value is constant across the bound­ary layer.) The height of any boundary layer row at a given edge node is equal to the height of the preced­ing row at that node times the growth factor, G.

The Rows value specifies the total number of rows to be included in the boundary layer.

NOTE: When you specify the First percent (a/w), Growth factor (b/a), and Rows values, GAMBIT computes the "last percent" value for the boundary layer and displays the value in the non-editable Last percent (c/w) field on the Create Boundary Layer form. The Last percent (c/w) value represents the height of the boundary layer top row at any given node relative to the corresponding mesh interval widths on the attachment entity. The Last percent (c/w) value can be computed from

where F and L are the First percent (a/w) and Last percent (c/w) values, respect­ively, G is the Growth factor (b/a) value, and R is the number of Rows.

Aspect ratio (last) Algorithm

The Aspect ratio (last) algorithm (see Figure 3-8) definition parameters are as follows.

Figure 3-8: Boundary layer dimensions—Aspect ratio (last) algorithm

The First row (a) value specifies the height of the first row (a)—that is, the absolute distance between the entity to which the boundary layer is attached and the first row of mesh nodes in the boundary layer. (NOTE: For the Aspect ratio (last) algorithm, the first-row height, a, is uniform across the boundary layer.)

The Rows value specifies the total number of rows to be included in the boundary layer.

The Last percent (c/w) value specifies the height of the boundary layer top row at any node relative to the corresponding mesh interval widths on the attachment entity. At any interior mesh node on the attachment entity (for example, the endpoints of an attachment edge), the relationship between the top row height (), the Last percent (c/w) value (L), and the mesh interval widths (w) can be expressed as

where and are the widths of the attachment-entity mesh intervals on either side of node i.

NOTE: For the Aspect ratio (last) algorithm, the growth factor varies across the boundary layer and is computed at each mesh node on the attachment entity. For mesh nodes that are interior to the entity, the growth factor at any node i can be expressed as

where is the node-specific growth factor and R is the number of Rows.

Because the growth factor is not constant across the boundary layer, GAMBIT does not display the Growth factor (b/a) field on the Create Boundary Layer form.

Specifying Internal Continuity

When you attach a boundary layer to a face that constitutes part of a volume, GAMBIT imprints the boundary layer on all adjoining faces that are also part of the volume (see Figure 3-9(a)). If you attach boundary layers to two or more adjoining faces of a volume, the boundary-layer imprints overlap on any faces that are common neighbors to the faces to (see Figure 3-9(b)).

Figure 3-9: Boundary-layer imprints (with shaded attachment faces)

The Internal continuity option on the Create Boundary Layer form determines the manner in which GAMBIT imprints boundary layers on adjoining faces as well as the mesh pattern in regions of imprint overlap.

Figure 3-10: Effect of the Internal continuity option

The effect of the Internal continuity option depends, in part, on the values of two GAMBIT default variables:

The ANGLE_SMOOTH_FACTOR default variable specifies whether or not the boundary-layer angling in the corner region is smoothed out across the adjacent edges. The ADJUST_EDGE_BL_HEIGHT default variable specifies whether or not GAMBIT adjusts the boundary layer heights along the adjacent edges to maintain constant heights with respect to the edges. Each default variable can take the values 0 (off) and 1 (on).

Figure 3-11 shows the effect of these default variables on the boundary layer created using the Internal continuity option. In Figure 3-11(a), both variables are set to zero; therefore, the angling of the boundary layer is confined to the corner region. In Figure 3-11(b), ANGLE_SMOOTH_FACTOR is set to 1; therefore, GAMBIT spreads the boundary-layer angling across the entire edge. In Figure 3-11(c), ADJUST_EDGE_BL_HEIGHT is also set to 1; therefore, GAMBIT adjusts the boundary-layer heights to maintain constant heights with respect to the edges adjacent to the corner.

Figure 3-11: Effect of default variables on Internal continuity option

In addition to affecting the mesh pattern in the imprint overlap regions, the Internal continuity option directly affects which types of meshing schemes are appropriate for volumes to which boundary layers have been applied. For example, the volume shown in Figure 3-10(b) can be meshed using a Map meshing scheme-resulting in the mesh shown in Figure 3-12(a). By contrast, the volume shown in Figure 3-10(a) cannot be meshed using a Map scheme, because the vertex located at the lower right corner of the front face (and imprint overlap region) is necessarily treated as a Side vertex. To mesh the volume shown in Figure 3-10(a), it is most reasonable to apply a Pave meshing scheme to the front face, then apply a Cooper meshing scheme to the volume, using the front and back faces as source faces (see Figure 3-12(b)).

Figure 3-12: Effect of Internal continuity option on allowable meshing schemes

Specifying the Wedge Corner Shape

GAMBIT allows you to control the shape of the mesh in the region surrounding a Corner or Reversal vertex that connects two edges to which boundary layers are attached. To do so, you must select or unselect (default) the Wedge corner shape option on the Create Boundary Layer form. The Wedge corner shape option produces the following effects (see Figure 3-13):

Figure 3-13: Effect of Wedge corner shape option

If two edges meet at a Corner or Reversal vertex, and each edge possesses a separate boundary layer, then to create a wedge-shaped boundary layer at the corner, you must select the Wedge corner shape option when creating each separate boundary layer.

Specifying the Transition Characteristics

The boundary-layer transition characteristics consist of two components:

Specifying the Transition Pattern

The transition pattern determines the arrangement of mesh nodes in the region near the outermost row of the boundary layer. Boundary layer transition patterns are defined by the ratio

A:B

where B is the number of mesh intervals in a given row and A is the number of mesh intervals in the immediately preceding full row. GAMBIT allows you to specify any of four transition patterns—1:1, 4:2, 3:1, or 5:1.

Figure 3-14 shows four different two-row boundary layers representing each of the four transition patterns listed above.

Figure 3-14: Boundary layer transition patterns

NOTE: Edges can host any of the four transition patterns, but faces can host only the 1:1 transition pattern.

Specifying the Number of Transition Rows

When you specify any transition pattern other than 1:1, you must also specify the number of transition rows—that is, the number of outermost rows to which the transition pattern is applied. GAMBIT applies the 1:1 pattern to all rows other than the transition rows. Figure-3-15 shows the effect of the number of transition rows on a boundary layer consisting of three rows with the transition pattern 4:2.

Figure 3-15: Effect of number of transition rows

Specifying the Attachment Entity

To define the location of a boundary layer, you must specify the edge or face to which the boundary layer is attached. If the edge or face is shared by two or more faces or volumes, respectively, you must also specify the face or volume that defines the direction of the boundary layer. For example, each edge of a rectangular brick volume is shared by two rectangular faces. If you attach a boundary layer to one of the edges of the volume, you must specify which of the corresponding faces defines the direction of the boundary layer.

Specifying the Boundary Layer Direction

When you specify an edge or face to which to attach a boundary layer, GAMBIT highlights the edge or face in the graphics window and displays the following items:

You can change the direction of the boundary layer either by means of the Attachment (Edge or Face) list box on the Create Boundary Layer form or by means of the mouse.

NOTE: If the boundary-layer attachment entity serves as an attachment entity for a size function or is part of a higher-topology entity to which a size function is attached, GAMBIT might or might not reflect the size-function definition in the temporary display of the boundary layer. Specifically, the boundary-layer display reflects the definition of the size function only if the background grid for the size function has already been generated—for example, by meshing an edge that is also part of the size-function attachment entity.

Changing Direction by Means of the List Box

When you specify an edge or face in the Attachment list box on the Create Boundary Layer form, the list box displays both the specified entity itself and the face or volume that defines the direction of the boundary layer. To change the direction of the boundary layer by means of the list box, you can perform either of the following operations.

  1. Specify the edge or face again in the Attachment list box
  2. Use the Edge List or Face List paired pick-list form to specify the entity and direction of the boundary layer (see "Using the Edge List or Face List Form," below).
Changing Direction by Means of the Mouse

To change the direction of the boundary layer by means of the mouse, Shift-middle-click the entity to which the boundary layer is attached.

Specifying Multiple Boundary Layers

GAMBIT allows you to apply a given boundary layer definition to more than one edge or face at a time. To do so, you must include in the Attachment entity pick list all of the entities to which the currently defined boundary layer is to be attached.

You can add an edge or face to the Attachment entity pick list on one of the following ways:

Smoothing the Mesh at Boundary Layer Transition Points

If you attach 2-D boundary layers to adjacent edges that share a common face or attach 3-D boundary layers to adjacent faces that share a common volume, GAMBIT automatically smoothes the resulting mesh at the transition points between the boundary layers. You can control the range of elements over which the boundary layers are smoothed by means of the HEIGHT_TRANSIT_RATIO default variable. As an example of mesh smoothing at boundary layer transition points, con­sider the 2-D boundary layers shown in Figure 3-16. In this case, the boundary layers are attached to adjacent edges that constitute one side of a square face. They differ from each other only with respect to their growth factors.

Figure 3-16: Example 2-D boundary layers on adjacent edges

If you retain the default value for the HEIGHT_TRANSIT_RATIO default variable and mesh the face shown in Figure 3-16, GAMBIT creates the mesh shown in Figure 3-17. In this case, the discontinuity between the boundary layers is smoothed over three intervals on either side of the transition point.

Figure 3-17: Mesh with boundary layer smoothing at transition point

As noted above, you can use the HEIGHT_TRANSIT_RATIO default variable to control the number of intervals over which the mesh is smoothed. The effect of the default variable depends on whether its value is greater or less than one (1) and can be summarized as follows:

By default, the HEIGHT_TRANSIT_RATIO value is equal to 0.5; therefore, the boundary layer is smoothed over half of the intervals on each side of the transition point (see Figure 3-17, above). If you specify a HEIGHT_TRANSIT_RATIO value greater than one (1), GAMBIT rounds the value up or down to the closest integer and uses the rounded value as the number of intervals on either side of the transition point over which the mesh is smoothed. For example, if you specify a value of 2 for the default variable, GAMBIT smoothes the mesh as shown in Figure 3-18.

Figure 3-18: Boundary layer smoothing—HEIGHT_TRANSIT_RATIO = 2

In this case, the mesh is smoothed over a distance of two intervals on either side of the transition point.

Using the Create Boundary Layer Form

To open the Create Boundary Layer form (see below), click the Create Boundary Layer command button on the Mesh/Boundary Layer subpad.

The Create Boundary Layer form contains the following specifications.

Show displays the boundary layer(s) in the graphics window as they are created and defined.

Definition Specifications
Definition:
Algorithm: contains radio buttons that specify the boundary layer definition algorithm. GAMBIT provides the following algorithm options.
  • Uniform
  • Aspect ratio (first)
  • Aspect ratio (last)
For a description of the algorithm options, see "Specifying the Algorithm," above.

The definition specifications differ according to Algorithm option as follows.

Uniform Algorithm Specifications

When you specify the Algorithm:Uniform option, GAMBIT displays the Definition fields as shown on the Create Boundary Layer form, above.

First row (a) specifies the height of the boundary layer first row.
Growth Factor (b/a) specifies the growth factor—that is, the ratio of the height of each row relative to that of the preceding row.
Rows specifies the total number of rows in the boundary layer.
Depth (D) displays (non-editable field) the total depth of the boundary layer.

Aspect ratio (first) Algorithm Specifications

When you specify the Algorithm:Aspect ratio (first) option, GAMBIT displays the following Definition fields on the Create Boundary Layer form.

First percent (a/w) specifies the height of the boundary layer first row as a percentage of the edge element width on the attachment entity.
Growth Factor (b/a) specifies the growth factor—that is, the ratio of the height of each row relative to that of the preceding row.
Rows specifies the total number of rows in the boundary layer.
Last percent (c/w) displays (non-editable field) the height of the top row as a percentage of the average interval width.

Aspect ratio (last) Algorithm Specifications

When you specify the Algorithm:Aspect ratio (last) option, GAMBIT displays the following Definition fields on the Create Boundary Layer form.

First row (a) specifies the height of the boundary layer first row.
Rows specifies the total number of rows in the boundary layer.
Last percent (c/w) specifies the height of the top row as a percentage of the average interval width.

General Specifications

The following Definition specifications are common to all of the Algorithm options.

Internal continuity

specifies that boundary-layer imprints are dovetailed in overlapping regions (see "Specifying Internal Continuity," above).

Wedge corner shape

specifies that the boundary-layer forms a wedge shape in the region surrounding a Corner or Reversal vertex (see "Specifying the Corner Shape," above).

Transition Specifications

Transition Pattern:

contains four radio buttons that specify the transition pattern. The pattern options are 1:1, 4:2, 3:1, and 5:1. (See "Specifying the Transition Pattern," above.)

Transition Rows

specifies the number of transition rows for transition patterns 4:2, 3:1, and 5:1. (NOTE: You must use the slide bar, rather than the associated text box, to set the number of transition rows.)

Attachment Specifications

Attachment:


Edges
Faces
specifies whether the boundary layer is attached to an edge or a face.
Edges
Faces
specifies the edge or face to which the boundary layer is attached.

NOTE: When you click the pick list button on the Attachment entity list box, GAMBIT opens a paired pick list form titled Edge List or Face List. For instructions in using the paired pick list form, see "Using the Edge List or Face List Form," below.

Label

specifies a label for the boundary layer.

Using the Edge List or Face List Form

When you specify an edge or face to which a boundary layer is attached, GAMBIT adds the edge or face to a paired pick list. The paired pick list includes both the attachment entity itself (edge or face) and the entity that defines the direction of the boundary layer (face or volume). You can modify the edge or face paired pick list by means of either the Edge List or Face List pick-list form, respectively. Both forms operate according to the following general principles described for the Edge List form.

To open the Edge List form (see below), select Edge in the Attachment field on the Create Boundary Layer form and click the associated pick list button.

The Edge List paired pick-list form operates in a manner similar to that of conventional pick-list forms (see GAMBIT User's Guide, Chapter 3). It differs from the conventional forms only in that the Picked scroll list includes two columns.

When you add an edge to the Picked scroll list by means of the right-arrow command button, GAMBIT adds the edge to the Edge column and automatically includes one of its associated faces in the Face column. (The face defines the direction of the boundary layer.) If you add the same edge again to the Picked scroll list, GAMBIT creates a second entry for the edge in the Edge column and includes another of its associated faces in the Face column. When the Face column includes all faces associated with a given edge, GAMBIT removes that edge from the Available column.


Modify Boundary Layer

The Modify Boundary Layer operation (blayer modify and blayer attach commands) modifies the specifications for any existing boundary layer.

Using the Modify Boundary Layer Form

To open the Modify Boundary Layer form (see below), click the Modify Boundary Layer command button on the Mesh/Boundary Layer subpad.

(For a description of the options and specifications available on the Modify Boundary Layer form, see "Create Boundary Layer," above.)


View 3D Boundary Layers

The View 3D Boundary Layers operation (blayer mesh command) allows you to examine volume meshes in regions affected by 3-D boundary layers. When you execute the View 3D Boundary Layers command for any 3-D boundary layer, GAMBIT meshes the volume associated with the boundary layer, renders the mesh invisible outside the boundary layer region, and automatic­ally opens the Examine Mesh form.

Figure 3-19 illustrates the effect of the View 3D Boundary Layers operation for a cube with a uniform boundary layer attached to two adjoining faces. In this case, the boundary layer was created using the Internal continuity option; therefore, the boundary layer dovetails in its overlapping regions.

Figure 3-19: View 3D Boundary Layers operation

If you execute the View 3D Boundary Layers operation for the boundary layer shown in Figure 3-19(a), GAMBIT meshes the cube, renders the mesh invisible outside the boundary layer region, and automatically opens the Examine Mesh form to display the mesh (Figure 3-19(b)). By default, GAMBIT selects the Range option on the Examine Mesh form and displays all volume elements in the boundary layer region; however, you can use any of the Examine Mesh options (for example, Plane or Sphere) to customize the mesh display.

NOTE: It is advisable to close the Examine Mesh form before executing subsequent GAMBIT operations. When you close the Examine Mesh form, GAMBIT automatically executes an undo command to undo the blayer mesh command that generated the boundary layer mesh(es).

Using the View 3D Boundary Layers Form

To open the View 3D Boundary Layers form (see below), click the View command button on the Mesh/Boundary Layers subpad.

The View 3D Boundary Layers form includes the following specification.

B.L.s specifies the boundary layer region(s) to be displayed.


Modify Boundary Layer Label

The Modify Boundary Layer Label operation (blayer modify command) changes the label associated with any boundary layer.

Using the Modify Boundary Layer Label Form

To open the Modify Boundary Layer Label form (see below), click the Modify Label command button on the Mesh/Boundary Layer subpad.

The Modify Boundary Layer Label form includes the following specifications.

B.L. specifies the boundary layer to be modified.
Label specifies a new label for the boundary layer.


Summarize Boundary Layers

The Summarize Boundary Layers operation (blayer summarize command) displays one or more existing boundary layers in the graphics window.

Using the Summarize Boundary Layers Form

To open the Summarize Boundary Layers form (see below), click the Summarize command button on the Mesh/Boundary Layer subpad.

The Summarize Boundary Layers form contains the following specification.

B.L.s specifies the boundary layer(s) for which summary information is to be displayed.


Delete Boundary Layers

The Delete Boundary Layers operation (blayer delete command) deletes one or more existing boundary layers.

Using the Delete Boundary Layers Form

To open the Delete Boundary Layers form (see below), click the Delete command button on the Mesh/Boundary Layer subpad.

The Delete Boundary Layers form includes the following specification.

B.L.s specifies the boundary layer(s) to be deleted.


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