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Hybrid Shallow Water/3D Flow Model

This article highlights developments to be released in FLOW-3D version 10.1.

Introduction

A shallow flow model has been available in FLOW-3D for many years. The model is suitable for flows in which horizontal extents are much larger than the depth. Examples include flow in the sea, estuaries, large lakes, seasonal floods, liquid coatings, lubricating films, and water on automobile windshields.

The three-dimensional Navier-Stokes equations reduce to 2D depth-averaged ones referred to as the shallow water model [1]. This approach successfully captures free surface waves, flooding of a complex terrain, laminar and turbulent flows and even surface tension forces. Its main strength is the ability to efficiently model large-scale shallow flows.

Naturally, the shallow water approximation is not suitable to model flow having significant vertical accelerations, for example, around essentially 3D structures that often exist in a ‘shallow water’ environment, such as bridges, drilling rigs at sea and dams in reservoirs. Therefore, it is desirable to be able to combine both shallow water and 3D type flows in the same simulation.

Model Description

The hybrid shallow water/3D flow model in FLOW-3D is designed for just that purpose. The approach takes advantage of multi-block meshing, where a mesh block is designated as type shallow water or 3D. The standard shallow water or 3D equations are solved within each such block, based on the user selection, and the respective solutions are coupled at the block boundaries. Standard linked and nested mesh block arrangements are allowed (e.g., see Fig. 1), except that shallow water type nested blocks are not allowed inside a 3D mesh block.

Arrangement of two 3D mesh blocks
Figure 1. Example of an arrangement of two 3D mesh blocks
(a linked one and a nested one, shown in red) and a shallow water block.

The coupling is done differently between mixed linked blocks and mixed nested-containing blocks. The solution between linked blocks is coupled in a two-way fashion, with data passed in both directions, so that flow can take place in either direction.

For a mixed nested block case (the nested block is type 3D and the containing one is type shallow water), the solution is passed only from the containing block to the nested block; the shallow water solution in the containing block ‘does not know’ about details computed in the 3D nested block.


Example: Flooding of a Large Area

This example describes flooding of a large area, with flow originating at a weir. The weir, which can be recognized as an example simulation provided with the FLOW-3D installation, is modeled with a full 3D model, while the flooded area downstream, containing several simple structures, is described with the shallow water equations (Fig. 2). The two mesh blocks are adjacent to each other, i.e., they are linked.

Flooding of a large area with flow originating at a weir
Figure 2. Flooding of a large area with flow originating at a weir. The weir is modeled in a 3D mesh block linked to a shallow water block that contains the flood area downstream. Click the image to see a flooding animation.

In this case, a two-way coupling of the solution takes place at the boundary between the two blocks, providing the means for the flow to transition from the 3D mesh block to the shallow water one.

Example: 3D Structure in a River Bed

The second example demonstrates the use of a 3D nested block to resolve a fine structure consisting of four vertical posts placed in the middle of a dry river bed. The flow along the river is modeled with a larger containing mesh block that is a shallow water type (Fig. 3).

Mixed mesh block arrangement for the modeling of a fine 3D structure in a river bed
Figure 3. Mixed mesh block arrangement for the modeling of a fine 3D structure in a river bed. The 3D structure is described using a 3D type nested mesh block, while the large scale flow is modeled in a shallow water type containing a mesh block.

As can be seen from Fig. 4, the resolution in the containing block is insufficient to accurately describe the structure. The shallow water solution is passed from that block to the boundaries of the 3D nested block, where the mesh is sufficient to adequately describe the 3D flow (Fig. 5).

FLOW-3D's new hybrid shallow water/3D flow model
Figure 4. The direction of solution passing in a mixed, 3D/shallow water, nested block arrangement (arrow). It is clear that the containing, shallow water block (left) does not have sufficient resolution to accurately capture the 3D structure that is well described by the 3D nested block (right). Click the images above to see a shallow water animation (left) and the 3D nested block animation (right).

 

A snapshot of the combined shallow water and 3D solutions
Figure 5. A snapshot of the combined shallow water and 3D solutions. The 3D mesh block (right) is separated from the shallow water one (left), and the depth of the fluid in the shallow water solution is exaggerated for display purposes. Color denotes velocity magnitude.

Conclusion

FLOW-3D's hybrid shallow water/3D flow model greatly enhances the ability of users to model complex, mixed-scale flow problems. When post-processing the results, users can view the whole combined solution at once or select a subset of the mesh blocks that contains only the 3D or shallow water solution, thus concentrating on one aspect of the problem at a time, making analysis easier.

References

[1]  C.W. Hirt and J.E. Richardson, "The Modeling of Shallow Flows," Flow Science Technical Note # 48 (FSI-99-TN48) March 1999.

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