How to Select an Effective Mesh System in CFD Analysis
Synchronous CFD is a new type of tool in computational fluid dynamics (CFD) that helps structural engineers simulate fluid flow and heat transfer in 3D structural CAD models. For 3D simulation and analysis, the most critical steps are meshing and creating an effective mesh system.
This article discusses why rectangular adaptive meshes represent advanced technology and how to effectively select meshes for new designs, thereby significantly reducing the time required for accurate analysis and improving product design efficiency.
Before conducting any CFD analysis, it is essential to consider the required mesh system. All CFD analyses are based on differential equations governing fluid dynamics phenomena, such as the Navier-Stokes equations and energy conservation equations. As is well-known, these differential equations cannot be solved analytically (unless extensive simplifications are made). Therefore, "discretization" must be employed for solving them.
By covering the entire analysis domain with a virtual mesh system, the region under consideration is divided into numerous small volumes or cells. Assumptions are made about variables (e.g., velocity, pressure, temperature) describing the characteristics within and between these small volumes. Thus, an approximate form of these governing differential equations (known as the finite volume method) can be derived. As long as the volumes are sufficiently small, the governing equations within each volume are valid, and thus valid across the entire domain. Finally, these algebraic equations are solved iteratively to obtain the corresponding results.
Clearly, meshing is a method to achieve a reasonably accurate solution to the governing differential equations. The size and density of the selected mesh significantly affect solution accuracy. The choice of mesh system type, as well as the shape and arrangement of meshes, can be arbitrary, provided the defined mesh facilitates reliable and accurate results—a "good" mesh. However, this "provided" is a crucial qualification. Experience shows that for any practical application, the following factors must be considered when selecting a mesh system for CFD calculations:
This is why selecting a CFD mesh system is a critical task.
Two key aspects must be considered when selecting a mesh system for CFD analysis:
The main options for mesh shape include:
The main options for mesh arrangement include:
Not all combinations of mesh shape and arrangement are practically meaningful. The most commonly used meshes are:
Orthogonal meshes are further illustrated using Cartesian meshes. Strictly speaking, many annotations about Cartesian meshes also apply to "orthogonal" meshes, where mesh lines align with orthogonal coordinate axes intersecting at 90 degrees. In practice, Cartesian meshes are the most commonly used orthogonal meshes.
Orthogonal meshes based on cylindrical coordinates are also used but less frequently. Additionally, Cartesian meshes offer more advantages than other non-orthogonal meshes. This article considers various mesh shapes and arrangements but focuses on Cartesian meshes and fully unstructured (hexahedral and tetrahedral) meshes. Structured tetrahedral-body-fitted meshes, an intermediate approach, are only suitable for aerodynamic applications.
Why have Cartesian meshes become the preferred choice for many applications? Governing equations can be easily derived and explicitly expressed in the Cartesian frame of reference, and solved velocity components are almost always aligned with Cartesian coordinate directions.
Cartesian meshes exhibit higher quality than non-orthogonal meshes. Non-orthogonal meshes with greater deviations from Cartesian meshes (i.e., more severe angular distortion) show more significant "degradation" in mesh quality. Mesh quality is a key consideration when selecting a mesh system for CFD analysis. Mesh shape (especially distortion in orthogonal meshes) greatly affects the assumptions in deriving finite volume differential equations and solution methods.
Derivations for highly non-orthogonal meshes reveal a critical point: non-orthogonal meshes introduce additional "secondary terms" not present in Cartesian meshes. In fully 3D scenarios, derivations for non-orthogonal meshes generate several times more "secondary terms" than Cartesian meshes. These terms have multiple consequences:
Calculating "secondary terms" consumes more computational time. Since numerous terms are needed to describe non-orthogonal meshes similarly to Cartesian meshes, computation may take several times longer. As calculations iterate during solving, this significantly impacts overall time.
This is likely the most significant impact. Geometric parameters for each non-orthogonal mesh are typically stored (rather than computed continuously). This is why unstructured hexahedral or tetrahedral meshes require more computational storage than Cartesian meshes. In large, complex calculations, this has become a limiting factor for such methods.
Auxiliary "cross-linkages" are introduced to compute these "secondary terms." This means heat flux calculations use temperatures from distant locations, not just adjacent ones, leading to two outcomes:
These drawbacks of non-orthogonal meshes worsen with increasing mesh distortion (non-orthogonality), making results highly application-dependent. Given these well-understood disadvantages, CFD users prefer Cartesian or other orthogonal mesh systems. Users of non-orthogonal meshes must prevent poor-quality meshes, often requiring manual "adjustments" to automatically generated meshes—one of the most time-consuming steps in CFD analysis.
If Cartesian meshes have obvious advantages, why do CFD users still use non-orthogonal meshes? This primarily stems from the need to handle complex systems, especially non-rectangular solid boundaries. Non-orthogonal meshes excel at fitting physical shapes like airfoils, ensuring mesh faces align closely with physical boundaries.
However, promising new methods have emerged in the past decade. Among them, Cartesian meshes allow non-rectangular solid shapes to intersect meshes arbitrarily, with solids within the mesh described using appropriate "cut-cell" techniques. Advantages of this method include:
Experience with complex geometries is well-documented. Four relevant examples are cited below:
Results demonstrate the accuracy of Cartesian meshes with fluid/solid descriptions. Flow around a cylinder (Re=26) was compared with experimental data and results from non-orthogonal body-fitted meshes of the same density. Both mesh types showed good agreement with experiments. Similar conclusions were drawn from other "simple" tests: Cartesian meshes with "cut-cell" technology achieve results comparable to complex non-orthogonal body-fitted meshes.
This example simulates turbulent flow around a car in a wind tunnel using nested Cartesian meshes (partially unstructured). Key findings show car surface pressure variations from Cartesian meshes match experimental measurements, comparable to results from more complex body-fitted meshes.
Partially unstructured Cartesian meshes (octree-structured) were applied to military helicopter aerodynamics (Reference 4) and extended to airfoils, full aircraft fuselages, and surrounding flows. NASA Ames also used nested Cartesian meshes ("overset structured grids") to compute flow around and behind fuselages (Reference 5). These Cartesian-based techniques simplify mesh generation and offer numerical advantages over non-orthogonal systems.
Professor Dawes reviewed turbomachinery CFD simulations, focusing on applications with complex geometries—a major simulation challenge. Early turbomachinery simulations used structured hexahedral meshes, limiting CFD adoption to "body-fitted" meshes now common in general-purpose CFD software. Professor Dawes argued this hindered usability, as mesh generation time restricted CFD in design workflows. A paradigm shift is needed: complex geometries should be processed via transformation rather than direct handling.
Professor Dawes introduced recent advances in computer graphics, where level set techniques precisely describe multi-surface geometries using 3D distance fields, storing signed distances to the nearest Cartesian mesh surface. As shown in external flow around a blade, this mesh can be directly used for flow solving. He demonstrated that modifying geometric features (e.g., holes in blades) only requires local mesh adjustments at the modified location.
In summary, these examples and other studies show:
Another key consideration in CFD mesh selection is arrangement: tightly connected structured grids (with continuous, regular lines) or fully unstructured grids with irregular node distributions.
This choice impacts computational efficiency. Unstructured meshes let users focus on specific regions, avoiding unnecessary meshes in distant areas compared to structured meshes of the same density. All else equal, results should be similar, with the main difference being computation time.
Solving fully unstructured meshes requires additional storage (for mesh connectivity) and computation time. Moreover, unstructured mesh shapes demand more storage, longer computation times, and higher mesh density due to quality issues (compared to Cartesian meshes). While unstructured meshes may have fewer cells, they often require more storage and time than structured meshes. The choice depends on the specific problem.
Structured meshes excel in problems requiring fine grids in specific regions (e.g., flow around objects), as seen in the car-in-wind-tunnel and helicopter examples. However, this is not universal: electronics enclosures or complex pumps/valves, filled with components requiring full flow field simulation, avoid "wasted" meshes, making body-fitted meshes impractical.
A compromise method combines unstructured flexibility with Cartesian advantages: nested structured meshes, where fine Cartesian grids nest within coarser ones. Efficient iteration at grid interfaces ensures high performance.
In conclusion, structured vs. unstructured meshes is an efficiency issue. Judging by total cell count alone is misleading: Cartesian meshes’ efficient solving may offset "wasted" cells. Properly used, nested structured and octree grids in Cartesian coordinates match the flexibility of tetrahedral/hexahedral meshes (with cut-cell technology enabling accurate description of arbitrary geometries).
When setting up a CFD problem, users must consider mesh generation, which involves:
For Cartesian meshes, defining data is straightforward: only X, Y, Z coordinates are needed. For 100,000 cells, this might involve 138 values (e.g., 46×46×46). Users can set parameters to control generation (e.g., FloEFD), simplifying adjustments for less experienced users to achieve good quality.
In contrast, unstructured mesh generation is poles apart. Due to irregular node ordering, each cell requires X, Y, Z coordinates—300,000 values for 100,000 cells, impossible to set manually. Methods like Delaunay Triangulation or Advancing Front Method automate this, but limited user control often necessitates post-generation adjustments to ensure quality and convergence. Even a single poor-quality cell can compromise results, requiring manual fixes.
Meshes must be dense enough to capture flow details, but interpolation cannot predict required density. Users rely on experience to adjust meshes, with "automatic" generation guided by mathematical criteria. Complex generation processes and time investments make post-adjustments difficult for users.
Geometry modifications compound challenges: changing a model requires re-generating and re-adjusting meshes. Professor Dawes (Reference 6) noted that CAD models are often "dirty" even after simplification, risking mesh generation failures. Attention to surface meshes is critical for resolving near-wall boundary layers, distinguishing between analysis (understanding performance) and design (iterating geometry).
From a design perspective, rapid geometry modification and mesh regeneration are vital. Body-fitted meshes, requiring long generation and manual adjustments, suit analysis but hinder design workflows.
FloEFD uses an octree mesh with further refinement capabilities, employing cut-cell technology at fluid-solid interfaces. Defining the initial mesh starts with a base grid, fully automated via a dialog box (with manual adjustment available by disabling "Automatic settings").
The initial mesh builds on a nearly uniform Cartesian base grid. The "Level of Initial Mesh" slider controls base grid density, and "Show basic mesh" visualizes it. This base grid can be refined to capture model features, with options like "Minimum gap size" and "Minimum wall thickness" driving local refinement. The slider auto-sets refinement levels for small solid features, curved surfaces, and narrow channels, enabling automatic generation.
After automatic generation, users can disable "Automatic settings" for manual adjustments, controlling mesh refinement globally or locally (e.g., for specific components, surfaces, edges, points, or fluid regions).
Adaptive meshing adjusts grids during solving based on results, critical for capturing unknown flow features (e.g., shocks in high-Mach flows) or refining regions with steep velocity, temperature, or pressure gradients.
Octree meshes simplify adaptation: refining by splitting cells into 8 sub-cells or coarsening by merging 8 sub-cells. A FloEFD example (Reference 7) illustrates this for 2D supersonic flow in a sudden contraction-expansion pipe.
A uniform supersonic air flow (Mach 3, 293.2K, 1atm) enters parallel walls, with flow weakening in the contraction due to oblique shocks. Initial wall-refined meshes failed to capture shocks, but adaptive refinement during solving focused cells on shock regions, reducing total count while improving accuracy. Mach number contours showed precise shock capture, with centerline results matching theoretical solutions.
[1] S. V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing, 1980.
[2] S. V. Patankar, Unpublished Presentation at 6th International FLOTHERM User Conference, October 1997.
[3] D. B. Spalding, “CAD to SFT, with Aeronautical Applications”, Plenary Lecture at 38th Israel Annual Conference on Aerospace Sciences, February 1998.
[4] M. J. Aftomis, M. J. Berger, and J. E. Melton, “Robust and Efficient Cartesian Mesh Generation for Component-Based Geometry”, Paper no. AIAA 97-0196, Presented at 35th AIAA Aerospace Sciences Meeting and Exhibit, January 1997.
[5] R. L. Meakin, “On Adaptive Refinement and Overset Structured Grids”, Paper No. AIAA-97-1858, 1997.
[6] W. N. Dawes, “Turbomachinery computational fluid dynamics: asymptotes and paradigm shifts”, Phil. Trans. R. Soc. A, Vol. 365, No. 1859, pp. 2553-2585, May 2007.
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