5. Computational Meshes and Geometric Representation

Introduction and Context

A computational fluid dynamics (CFD) solution is obtained not directly on a continuous physical domain, but on a finite representation of that domain. The governing equations describe velocity, pressure, temperature, and species concentration at every point in space and time. A numerical method, however, requires a finite collection of control volumes, nodes, or elements over which these fields can be approximated. This finite representation is the computational mesh.

Mesh generation is therefore not a clerical preliminary to CFD. It is a modelling decision that influences conservation, discretisation error, solver stability, computational cost, and the credibility of engineering conclusions. A mesh may represent a simple rectangular duct with a structured array of quadrilaterals, or a complex turbomachine passage with several interacting blocks, boundary-layer layers, tetrahedral elements, polyhedral cells, and overset interfaces. In each case, the mesh must represent geometry faithfully and resolve the physical scales that control the flow.

The central difficulty is that geometric complexity and flow complexity are not identical. A geometrically simple domain may contain shocks, separated shear layers, vortices, or thin thermal boundary layers requiring substantial refinement. Conversely, a complex external shape may permit relatively coarse resolution in regions where the flow is smooth. Effective meshing consequently requires an interaction between geometry, physics, numerical method, and computational resources.

This chapter develops the principles used to construct and assess computational meshes. It examines structured, unstructured, hybrid, body-fitted, overset, and adaptive approaches; explains near-wall resolution and moving geometries; and establishes practical criteria for skewness, orthogonality, aspect ratio, and mesh convergence.

Learning Objectives

After studying this chapter, the reader should be able to:

1. distinguish between structured, unstructured, hybrid, body-fitted, overset, and adaptive meshes;

2. explain how geometry representation and mesh topology affect discretisation accuracy and solver performance;

3. select an appropriate meshing strategy for internal, external, rotating, moving, and multiphysics flows;

4. design near-wall meshes using wall functions or wall-resolved approaches and interpret the significance of the dimensionless wall distance, y+;

5. assess skewness, non-orthogonality, aspect ratio, smoothness, and cell quality;

6. formulate and interpret a mesh-convergence study, including discretisation uncertainty;

7. identify common meshing failures and improve a mesh systematically; and

8. evaluate emerging approaches involving automation, adaptive refinement, immersed boundaries, and mesh motion.

Geometry, Topology, and the Discrete Domain

A geometric model defines the physical boundaries, interfaces, volumes, and surfaces of a problem. It may originate from computer-aided design (CAD), medical imaging, experimental reconstruction, or an analytical description. Before meshing, the geometry often requires defeaturing, healing, and decomposition. Small holes, gaps, fillets, and sliver surfaces may be irrelevant to the flow but can cause severe meshing difficulties. Removing them is acceptable only when their physical influence is negligible.

Geometry and topology should be distinguished. Geometry concerns the shape and position of entities; topology concerns their connectivity and neighbourhood relationships. Two meshes can represent the same geometry while possessing different topologies. Topology determines which cells share faces, how fluxes are computed, and how data structures are assembled. In a finite-volume method, each control volume must have well-defined faces and neighbours so that conservation laws can be integrated consistently.

The mesh contains nodes, edges, faces, and cells. In two dimensions, common cells are triangles and quadrilaterals. In three dimensions, tetrahedra, hexahedra, prisms, pyramids, and polyhedra are used. A cell-centred finite-volume method stores primary variables at cell centres, whereas a vertex-centred method stores them at nodes. Finite-element methods use elements and shape functions, while finite-difference methods generally require a logically ordered grid. The same geometric mesh can therefore have different numerical interpretations.

A useful distinction is between mesh resolution and mesh quality. Resolution describes whether the cell size is sufficiently small to capture relevant physical variation. Quality describes whether the shape and connectivity of cells allow the discretisation to operate accurately and robustly. A fine but highly skewed mesh may perform worse than a moderately fine, well-shaped mesh. Likewise, a high-quality coarse mesh cannot resolve a thin boundary layer merely because its cells are geometrically regular.

Principal Mesh Families

Structured meshes

A structured mesh has an implicit, logically ordered connectivity. Each interior point or cell can be identified by indices such as i, j, and k, and its neighbours are known from those indices. Cartesian grids, curvilinear grids, and multiblock structured meshes are common examples.

Structured meshes offer efficient memory use, simple neighbour addressing, and generally low numerical diffusion when aligned with the flow. They are particularly effective for channels, ducts, turbomachinery passages, and geometries that can be decomposed into regular blocks. A body-fitted structured grid maps a computational coordinate system onto a curved physical domain. Algebraic methods, elliptic grid generation, and transfinite interpolation may be used to construct such mappings.

Their principal limitation is geometric flexibility. Complex components may require many blocks, difficult block interfaces, or severe cell distortion. Block topology must be planned carefully because a poor decomposition can create abrupt changes in cell size or undesirable singularities.

Unstructured meshes

An unstructured mesh stores connectivity explicitly rather than through a simple index pattern. Triangular and tetrahedral cells can conform to highly complex boundaries and are often generated automatically. Quadrilateral, hexahedral, polygonal, and polyhedral unstructured meshes are also available.

The strength of an unstructured mesh is flexibility. Local refinement can be applied around a leading edge, a small gap, or a wake without filling the entire domain with fine cells. This is valuable for industrial geometries containing many components. The cost includes greater memory requirements, more complex data structures, and a greater risk of skewness, non-orthogonality, and irregular size transitions. Tetrahedral meshes may also require many more cells than carefully aligned hexahedral meshes for comparable resolution of directional phenomena.

Hybrid meshes

Hybrid meshes combine cell types or meshing philosophies. A common configuration uses structured or semi-structured prism layers near solid walls and tetrahedral, polyhedral, or hexahedral cells in the outer flow. Hexahedral blocks may be used in long passages, while unstructured cells fill complex junctions. Hybridisation is often the most practical industrial choice because near-wall physics and geometric flexibility impose different requirements in different regions.

The interfaces between regions must be treated carefully. Nonconformal interfaces require interpolation or conservative flux transfer, and abrupt changes in cell size or orientation can introduce error. A hybrid mesh is not automatically superior; its quality depends on the continuity and numerical treatment of the transitions.

Body-fitted meshes

A body-fitted mesh conforms to the physical boundary. Cell faces follow the shape of walls, blades, airfoils, or other immersed surfaces. This permits direct application of no-slip, heat-flux, pressure, and species boundary conditions at the boundary faces. Body fitting is especially valuable when wall shear, heat transfer, or pressure distribution is a primary quantity of interest.

The difficulty is that fitting a mesh to moving or deforming bodies may require continuous mesh deformation or remeshing. Highly curved or closely spaced surfaces can also cause small cells and poor quality. For stationary complex geometry, however, body-fitted meshes remain the dominant approach in many engineering applications.

Overset meshes

An overset, or Chimera, method uses multiple overlapping component meshes. One background mesh represents the global domain, while separate body-fitted meshes move with bodies such as blades, vehicles, or particles. Information is transferred through interpolation between donor and receiver cells. Cells covered by another mesh may be blanked, and an overset connectivity algorithm identifies valid interpolation relationships.

Overset meshes are attractive for relative motion, large body displacement, and multi-body problems because each component mesh can remain high quality as it moves. Their limitations include interpolation error, conservation challenges at overset boundaries, additional memory and connectivity operations, and possible gaps or excessive overlap. They are particularly useful when remeshing a single deforming mesh would be more damaging than using interpolation.

Adaptive meshes

Adaptive mesh refinement changes the mesh in response to estimated error or selected flow features. Refinement may be based on gradients, vorticity, density jumps, residual-based indicators, adjoint sensitivity, or user-defined geometric criteria. Coarsening may remove cells from regions that no longer require high resolution.

Adaptation can be h-refinement, which changes cell size; p-refinement, which changes the order of approximation; or r-refinement, which relocates nodes while retaining the number of cells. In practical CFD, h-adaptation is common. Adaptation is powerful for shocks, wakes, interfaces, and evolving flow structures, but it must avoid chasing numerical noise. It also requires interpolation or projection between meshes and careful conservation treatment.

Representing the Physics Through the Mesh

The correct mesh is determined by the smallest important physical scale, not simply by the smallest geometric feature. A boundary layer, shear layer, shock, recirculation zone, vortex core, flame front, or conjugate heat-transfer interface may require targeted resolution. The mesh should also reflect the directionality of the solution. Long cells aligned with a smooth boundary layer can be efficient; long cells placed across a rapidly varying direction can produce large truncation errors.

Mesh grading describes the variation of cell size. Gradual expansion is preferable to abrupt growth because sudden changes in cell size can reduce accuracy and create solver stiffness. The appropriate growth ratio depends on the discretisation scheme and physics, but a smooth transition is generally more reliable than a nominally small number of highly expanded layers. Refinement should be concentrated where gradients, curvature, or sensitivity are high, while large nearly uniform cells can be used in smooth regions.

Geometric curvature also matters. A surface represented by planar faces may appear acceptable visually but still produce errors in normal direction, wetted area, or pressure distribution. Surface tessellation must be sufficiently fine to represent curvature, especially near leading edges, small radii, and sealing gaps. Volume cells should then be generated without losing the intended boundary shape.

In compressible flow, shocks require sufficient cells normal to the shock and suitable numerical dissipation. In incompressible separated flow, the wake and shear layers need resolution in the streamwise and cross-stream directions. In heat transfer, the thermal boundary layer may differ substantially from the velocity boundary layer, particularly when the Prandtl number is far from unity. Mesh design is therefore a physical scaling problem as much as a geometric one.

Near-Wall Resolution and the Meaning of y+

The region adjacent to a wall contains steep velocity and temperature gradients. Its treatment strongly influences skin friction, separation, drag, and heat transfer. The dimensionless wall distance is commonly written as

y+ = u_tau y / nu,

where y is the distance from the wall to the centre of the first cell, nu is kinematic viscosity, and u_tau is friction velocity, defined by u_tau = sqrt(tau_w/rho). The value of y+ is not a universal quality score; it indicates the location of the first computational point relative to the viscous sublayer and the assumptions of the turbulence-wall treatment.

For a wall-resolved approach, the first cell centre is typically placed in the viscous sublayer, often with y+ approximately one, although the suitable range depends on the turbulence model and numerical implementation. Several prism or hexahedral layers are then used to resolve the velocity profile through the boundary layer. For wall-function approaches, the first cell centre is deliberately placed in the logarithmic region, commonly with y+ of order 30 or higher, subject to the wall-function formulation. A value in the buffer layer may be undesirable because neither the viscous formulation nor the logarithmic wall function is valid there.

The first-layer height is only one part of near-wall design. Adequate points are needed across the boundary layer, layer thickness must cover the region of interest, and the growth ratio should be moderate. Layers should not collapse at sharp corners or narrow gaps. If heat transfer is modelled, the thermal wall treatment and thermal y+ must also be considered.

A common mistake is to target y+ blindly. The appropriate value depends on whether the goal is a wall-resolved large-eddy simulation, a Reynolds-averaged calculation using wall functions, or a low-Reynolds-number turbulence model. It also depends on local velocity, density, viscosity, pressure gradient, surface roughness, and the expected wall shear. Preliminary estimates should be checked after a solution because local y+ can vary considerably over the surface.

Mesh-Quality Measures

No single metric determines whether a mesh is acceptable. Quality must be assessed using several measures and interpreted relative to the solver, discretisation scheme, and flow physics.

Skewness measures how far a cell departs from an ideal shape or how far a face centre departs from the line joining adjacent cell centres. High skewness makes gradient and flux reconstruction less accurate and can cause convergence difficulties. Different software packages define skewness differently, so numerical limits should not be transferred without checking the definition.

Non-orthogonality is the angle between the line connecting neighbouring cell centres and the face normal. Orthogonal grids align these directions. Large non-orthogonality requires correction terms in the diffusive flux and may increase iteration sensitivity. A mesh with moderate non-orthogonality can be usable if the correction is bounded and the discretisation is appropriate, but concentrated extreme values are dangerous.

Aspect ratio is a measure of the ratio between the largest and smallest characteristic cell dimensions. High aspect ratio is not inherently bad. Boundary-layer cells are intentionally elongated in the wall-parallel direction. It becomes problematic when elongation is not aligned with the direction of weak variation or when it combines with skewness and abrupt rotation.

Smoothness or expansion ratio describes changes in cell size between neighbours. Rapid expansion can produce errors and impede multigrid or iterative convergence. Smooth grading is especially important near walls, shocks, and interfaces.

Jacobian and volume validity are fundamental. Cells must have positive volume, non-intersecting faces, and acceptable Jacobian determinants. Negative-volume or inverted cells invalidate the discretisation. Very small volumes may also impose severe time-step restrictions in transient simulations.

Warpage and face planarity are important for quadrilateral, hexahedral, and polyhedral cells. Non-planar faces complicate flux evaluation. At curved boundaries, some warpage is unavoidable, but it should be controlled.

Quality histograms and spatial plots should be examined together. A good global average can conceal a small cluster of extremely poor cells near a leading edge, gap, or interface. The location of poor cells is often more important than the domain-wide mean.

Mesh-Convergence and Discretisation Uncertainty

A mesh-convergence study asks whether the engineering quantity of interest changes acceptably as the mesh is refined. It is not sufficient to report that the solver converged residuals on one mesh. Iterative convergence concerns the solution of the algebraic equations on a fixed mesh; mesh convergence concerns the dependence of the discretised solution on cell size.

A defensible study begins with at least three systematically related meshes: coarse, medium, and fine. The refinement should occur in important regions rather than through arbitrary local changes. The same physical models, boundary conditions, solver settings, and convergence criteria should be used unless a deliberate methodological change is being examined. Quantities such as drag, lift, pressure drop, heat-transfer coefficient, mass imbalance, and peak temperature should be monitored.

If a result approaches a limiting value with refinement, the trend is informative. For a quantity phi computed on meshes with characteristic sizes h1, h2, and h3, an observed order of convergence can be estimated from the differences between solutions when the refinement ratio is known. Richardson extrapolation may then estimate the zero-grid-spacing value, and the Grid Convergence Index (GCI) can provide a structured estimate of discretisation uncertainty. These tools rely on assumptions of asymptotic behaviour and should not be applied mechanically when the meshes are not systematically refined or when the solution is not in a convergent range.

A mesh-independent result is not necessarily a physically correct result. All meshes may share an inadequate turbulence model, incorrect boundary condition, insufficient domain size, or geometry error. Verification addresses whether the equations were solved correctly; validation addresses whether the mathematical model represents reality. Mesh convergence is primarily a verification activity and does not replace experimental comparison.

Meshes for Moving and Deforming Geometries

Moving-body simulations introduce a further requirement: the mesh must remain valid as geometry changes. In an Arbitrary Lagrangian–Eulerian (ALE) formulation, the mesh moves independently or partly independently of the material. Mesh velocity enters the conservation equations, and the geometric conservation law should be satisfied so that a uniform flow remains uniform under mesh motion.

For small deformations, spring-based smoothing, Laplacian smoothing, or elasticity-based methods may reposition nodes. Smoothing is inexpensive but can concentrate distortion in narrow regions. When distortion becomes excessive, local remeshing replaces invalid cells. Dynamic layering can add or remove layers in sliding or piston-like motion. Sliding meshes allow separate zones to move relative to one another, with fluxes transferred across an interface.

Overset methods are often preferred for large rigid-body motion and rotating machinery. Deforming meshes are attractive when strict body-fitted conservation and continuous connectivity are important. The choice is governed by motion amplitude, deformation mode, topology changes, conservation requirements, and computational cost.

A Practical Mesh-Generation Workflow

A robust workflow begins with the engineering question. The analyst should identify the output quantities, dominant physics, expected flow regime, turbulence treatment, wall treatment, and transient time scales. This prevents the common error of generating a visually impressive mesh without a resolution plan.

The geometry is then cleaned and decomposed. Irrelevant details are removed, but features controlling separation, leakage, heat transfer, or structural interaction are retained. Boundary names and zones should be assigned consistently because they support reproducible setup and post-processing.

A surface mesh is generated and inspected for curvature representation, holes, intersections, and size transitions. The volume mesh is then constructed with deliberate refinement zones, boundary-layer layers, and interfaces. Automated meshing should be treated as an assistant rather than an authority: it can produce valid cells while failing to resolve the physics.

Quality checks follow. These include cell validity, skewness, orthogonality, aspect ratio, growth rate, layer continuity, local size relative to geometry, and y+ estimates. A preliminary solution often reveals additional requirements, such as refinement in a separated shear layer or a shock. Adaptation or manual modification can then be applied.

Finally, the mesh is subjected to solver and sensitivity checks. Residual convergence, global conservation, physical monitors, and mesh convergence must be considered together. The mesh, software version, generation parameters, quality statistics, and convergence evidence should be documented so that the calculation is auditable and repeatable.

Illustrative Case Study: Mesh Design for a Turbulent Heated Backward-Facing Step

Background

Consider an illustrative computational study of turbulent air flow over a backward-facing step with a heated lower wall. The purpose is to predict reattachment length, wall pressure, and local heat-transfer coefficient. The geometry is simple enough to permit systematic investigation but contains separation, recirculation, reattachment, and a developing thermal boundary layer.

Situation and problem

An initial coarse tetrahedral mesh produces a converged pressure field, but the predicted reattachment point changes substantially when the solver order is modified. The wall heat-transfer coefficient is also oscillatory near the step. Inspection shows that the mesh has only a few poorly aligned cells in the separated region, abrupt growth downstream, and an uncontrolled first-layer height.

Analysis

The important physical regions are the step corner, separated shear layer, recirculation bubble, reattachment zone, and downstream thermal boundary layer. The original mesh resolves the geometry but not the directional gradients. Its near-wall cells have y+ values that fall partly in the buffer region, inconsistent with the selected wall function. Because the tetrahedral cells are nearly isotropic, many cells are spent in the outer flow while the shear layer remains under-resolved.

Approach and solution

The geometry is retained, but the volume mesh is redesigned as a hybrid mesh. Several prism layers are placed on the heated wall and the upper wall, with a first-layer height selected for the intended wall treatment and a smooth growth ratio. The step corner receives local refinement, while a stretched but aligned mesh is introduced through the expected shear layer and recirculation region. The downstream domain is refined gradually rather than abruptly.

Three systematically refined meshes are generated. Reattachment length, pressure coefficient, wall heat flux, and integrated heat-transfer rate are monitored. The medium and fine meshes give similar reattachment lengths, while the local heat-transfer peak continues to change near the sharp corner. The corner peak is therefore reported with an uncertainty qualification, whereas the integrated heat-transfer rate is judged sufficiently insensitive for the study objective.

Outcome

The redesigned mesh improves stability, reduces oscillation in the wall heat-transfer distribution, and produces a more credible convergence trend. The exercise demonstrates that a converged residual history on the original mesh did not establish discretisation adequacy. It also shows that a single global quality number would not have identified the principal problem: poor physical resolution and inappropriate near-wall placement.

Learning points

The mesh must be designed around the outputs of interest. Separation and heat transfer require different but interacting resolution strategies. Near-wall requirements should be selected from the turbulence-wall model, and mesh convergence should be assessed using engineering quantities rather than residuals alone.

Challenges, Risks, and Common Mistakes

One frequent mistake is refining every region uniformly. This increases computational cost without necessarily improving the result. Another is refining only where the geometry is complicated while neglecting smooth-looking regions containing wakes or boundary layers. A third is relying on default quality limits without understanding the metric definitions or the capabilities of the discretisation scheme.

Poor CAD preparation can create duplicate faces, non-manifold edges, internal leaks, and unintended fluid regions. Excessive defeaturing can remove physically important gaps or radii. Excessively thin cells can cause time-step restrictions and ill-conditioned linear systems. Abrupt changes in cell size can create numerical reflections or convergence problems.

Near-wall errors are often hidden by attractive contour plots. A wall-function calculation with y+ in the buffer layer may produce plausible fields but unreliable wall quantities. Similarly, a mesh may have acceptable average skewness while containing a few cells that dominate the error. Local inspection, physical monitors, and sensitivity tests are essential.

Adaptive refinement can also be misused. Refining on raw gradients may focus on numerical discontinuities or turbulence-model artefacts. A meaningful indicator should be linked to truncation error, adjoint sensitivity, or a clearly relevant physical feature. Interpolation during adaptation must preserve conservation as far as the method permits.

Professional and Governance Considerations

Engineering decisions based on CFD should be traceable. A professional report should identify the geometry source, simplifications, mesh type and count, quality criteria, wall treatment, y+ range, boundary conditions, solver tolerances, convergence monitors, and mesh-sensitivity evidence. Omitting this information makes independent assessment difficult.

For safety-critical applications, mesh procedures should be subject to verification plans, independent review, configuration control, and documented acceptance criteria. Automated meshing and adaptive algorithms should not be treated as exempt from engineering judgement. If a mesh limitation affects a design decision, that limitation should be disclosed rather than concealed by excessive numerical precision.

Digital Tools and Emerging Directions

Modern meshing systems increasingly combine CAD healing, geometry-aware sizing, boundary-layer extrusion, parallel generation, and automated quality improvement. Machine-learning tools are being explored for predicting refinement regions, selecting meshing parameters, and constructing surrogate error indicators. Such tools can accelerate workflows, but their recommendations require physical and numerical verification.

Adjoint-based adaptation is especially valuable when a specific output, such as drag or pressure loss, is more important than the global field. It refines regions that influence the chosen functional, which may differ from regions of largest local gradient. Immersed-boundary and cut-cell approaches avoid body-fitted remeshing for some complex or moving geometries, although accurate wall representation, small cut cells, and conservation remain significant issues.

High-order methods increase the importance of geometric representation because curved boundaries and element quality directly affect the accuracy of polynomial approximations. Isogeometric and spline-based approaches seek closer integration between CAD geometry and numerical representation. These developments do not eliminate meshing challenges; they shift attention toward parametrisation, curvature continuity, and robust error control.

Best-Practice Principles

A reliable mesh strategy can be summarised as follows:

• begin with the physical quantities and scales that must be resolved;
• simplify geometry only after assessing its physical significance;
• select topology and cell types to suit both geometry and flow direction;
• use boundary-layer meshes deliberately and verify y+ after solution;
• maintain smooth size transitions and avoid isolated poor cells;
• inspect quality spatially, not only through global averages;
• distinguish solver convergence from mesh convergence;
• use at least three systematically related meshes for important studies;
• document mesh generation, quality, assumptions, and uncertainty; and
• use adaptive, overset, immersed, or automated methods only with appropriate conservation and verification checks.

Chapter Summary

A computational mesh is the finite geometric and topological representation on which the CFD equations are discretised. Its design affects accuracy, stability, computational cost, and the credibility of engineering results. Structured meshes provide efficient connectivity and alignment, while unstructured meshes offer geometric flexibility. Hybrid meshes combine these advantages; body-fitted meshes represent walls directly; overset meshes support moving components; and adaptive meshes allocate resolution according to estimated need.

Mesh resolution must follow physical scales such as boundary layers, wakes, shocks, vortices, and thermal interfaces. Near-wall design requires compatibility between first-cell placement, y+, prism-layer structure, and the selected wall treatment. Skewness, non-orthogonality, aspect ratio, growth rate, warpage, and cell validity should be assessed together and interpreted locally.

A converged residual history does not prove mesh adequacy. Mesh-convergence studies compare engineering outputs on systematically refined meshes and may employ Richardson extrapolation or the Grid Convergence Index when their assumptions are satisfied. Moving geometries require appropriate smoothing, remeshing, sliding, ALE, or overset strategies. Professional CFD practice also requires documentation, uncertainty awareness, and independent verification of automated or adaptive procedures.

Key Terms

Mesh: A finite spatial representation of a computational domain.

Structured mesh: A mesh with logically ordered, implicit connectivity.

Unstructured mesh: A mesh whose cell connectivity is explicitly stored and irregularly organised.

Hybrid mesh: A mesh combining different cell types or meshing strategies.

Body-fitted mesh: A mesh whose boundaries conform to the physical geometry.

Overset mesh: Overlapping component meshes connected through interpolation.

Adaptive mesh refinement: Automatic or guided alteration of mesh resolution using an error or feature indicator.

Topology: The connectivity relationships among mesh entities.

Skewness: A measure of departure from an ideal cell or face-centre arrangement.

Non-orthogonality: The angular mismatch between a face normal and the line joining adjacent cell centres.

Aspect ratio: The ratio of a cell’s largest to smallest characteristic dimensions.

Mesh grading: The spatial variation of cell size.

y+: Dimensionless wall distance based on friction velocity, wall distance, and kinematic viscosity.

Wall function: A model that represents near-wall flow without resolving the viscous sublayer directly.

Boundary layer: A thin region near a wall in which velocity or thermal variables vary rapidly.

Mesh convergence: Reduction of solution dependence on cell size through systematic refinement.

Discretisation uncertainty: Uncertainty arising from approximating continuous equations on a finite mesh.

Overset interpolation: Transfer of solution information between overlapping meshes.

Geometric conservation law: A condition ensuring consistent conservation under mesh motion.

Immersed-boundary method: A method representing solid boundaries without requiring a body-fitted volume mesh.

Review Questions

Short Answer Questions

1. What is the difference between mesh resolution and mesh quality?

2. Define skewness and explain why excessive skewness is undesirable.

3. What does y+ represent in a near-wall mesh?

4. Distinguish between a body-fitted mesh and an overset mesh.

5. Why is residual convergence insufficient to establish mesh independence?

Long Answer Questions

1. Compare structured, unstructured, and hybrid meshes in terms of topology, computational cost, geometric flexibility, and numerical accuracy.

2. Explain the design of a near-wall mesh for a wall-resolved turbulence calculation and for a wall-function calculation.

3. Describe the principal steps in a professional CFD mesh-generation workflow.

4. Explain how skewness, non-orthogonality, aspect ratio, and mesh grading influence a finite-volume solution.

5. Discuss the methods available for CFD simulations involving moving or deforming geometries.

Analytical / Application Questions

1. A simulation of flow over an airfoil gives acceptable lift but strongly mesh-dependent drag. Develop a diagnostic plan using surface representation, near-wall resolution, quality metrics, and mesh convergence.

2. You must simulate a rotating impeller inside a stationary casing with large relative motion. Compare a sliding-mesh, deforming body-fitted, and overset strategy, and justify a selection.

3. Three meshes produce reattachment lengths of 6.8, 6.4, and 6.3 step heights, while local wall heat flux near a sharp corner continues to change significantly. Explain how you would report the results and what additional study would be appropriate.

22. Multiple Choice Questions with Answers

Question 1

Which feature most directly distinguishes a structured mesh from an unstructured mesh?

A. The structured mesh always contains quadrilateral cells

B. The structured mesh has logically ordered, implicit connectivity

C. The unstructured mesh cannot represent curved boundaries

D. The unstructured mesh always contains tetrahedral cells

Correct Answer: B. The structured mesh has logically ordered, implicit connectivity

Question 2

A CFD model uses wall functions, but most first-cell centres lie in the buffer layer. What is the principal concern?

A. The cells have excessively low aspect ratio

B. The mesh is necessarily body-fitted

C. The wall treatment may be applied outside its appropriate range

D. The flow must be laminar

Correct Answer: C. The wall treatment may be applied outside its appropriate range

Question 3

Why can a high aspect ratio be acceptable in a boundary-layer mesh?

A. Elongation is useful when aligned with the direction of weak variation

B. Aspect ratio has no influence on discretisation

C. High aspect ratio eliminates the need for y+ control

D. Boundary layers contain no velocity gradients

Correct Answer: A. Elongation is useful when aligned with the direction of weak variation

Question 4

Which meshing approach is especially suited to large rigid-body motion while allowing each body mesh to remain high quality?

A. Uniform Cartesian refinement only

B. Overset meshing

C. Single undeformed body-fitted meshing

D. Random node deletion

Correct Answer: B. Overset meshing

Question 5

What does large non-orthogonality indicate?

A. The face normal and line joining neighbouring cell centres are poorly aligned

B. The cell has zero volume

C. The mesh has perfect flow alignment

D. The first-cell y+ is exactly one

Correct Answer: A. The face normal and line joining neighbouring cell centres are poorly aligned

Question 6

Which statement best describes mesh convergence?

A. Residuals decrease on one fixed mesh

B. The solution becomes less sensitive to systematic mesh refinement

C. The CAD model contains no small faces

D. All cells have identical volume

Correct Answer: B. The solution becomes less sensitive to systematic mesh refinement

Question 7

A mesh has excellent average quality, but a small cluster of highly skewed cells lies at a separation point. What is the most appropriate interpretation?

A. The cluster can be ignored because the global average is good

B. The local defect may dominate the error and should be investigated

C. The solution is automatically mesh independent

D. The mesh must be changed to a Cartesian grid everywhere

Correct Answer: B. The local defect may dominate the error and should be investigated

Question 8

Which quantity is most appropriate for judging whether a mesh is adequate for a specific engineering objective?

A. Only the total number of cells

B. Only the largest aspect ratio in the domain

C. The sensitivity of the relevant output, such as drag or pressure drop, to refinement

D. The visual colour of the mesh display

Correct Answer: C. The sensitivity of the relevant output, such as drag or pressure drop, to refinement

Question 9

What is a major limitation of overset meshes?

A. They cannot represent moving bodies

B. They require interpolation between overlapping meshes, which can introduce error

C. They always require a single conformal grid

D. They cannot be used with body-fitted component meshes

Correct Answer: B. They require interpolation between overlapping meshes, which can introduce error

Question 10

Why should a mesh-convergence study normally use at least three systematically related meshes?

A. To establish a refinement trend rather than relying on a single comparison

B. To guarantee that the turbulence model is physically correct

C. To remove the need for iterative convergence

D. To ensure every cell has unit volume

Correct Answer: A. To establish a refinement trend rather than relying on a single comparison