構造格子
Theory and Physics
Overview
Professor, what exactly is a structured grid? I'm having trouble grasping the difference from an unstructured grid.
A structured grid (structured mesh) is a grid that fills the computational domain with regularly arranged hexahedral (3D) or quadrilateral (2D) cells. Each cell's index can be uniquely represented by $(i,j,k)$. This means the adjacency relationships between grid points are implicitly determined, so there's no need to explicitly store connectivity information.
Not having to store adjacency information means it's memory efficient, right?
Exactly. In a structured grid, neighboring cells are always at $(i\pm1, j\pm1, k\pm1)$, so data access is contiguous and cache hit rates are high. For the same number of cells, memory consumption can be about half compared to an unstructured grid, and computational speed can be 2-3 times faster, which is not uncommon.
Body-Fitted Coordinate System and Metrics
But real-world shapes are curved and twisted, right? How do you handle that with a regular grid?
That's where the body-fitted curvilinear coordinate system comes in. It maps the physical space $(x,y,z)$ to a computational space $(\xi,\eta,\zeta)$. In the computational space, the grid becomes a uniform rectangular box, allowing finite-difference schemes to be written simply.
The Jacobian matrix of the coordinate transformation is crucial and is defined as follows.
If the determinant of this Jacobian, $|J|$, becomes zero or negative, it's evidence that the grid is collapsed or inverted, causing the computation to fail.
Under what circumstances does the Jacobian become negative?
Typically around sharp corners or when grid lines intersect. The golden rule during grid generation is to verify $|J| > 0$ for all cells.
Navier-Stokes Equations on Structured Grids
Does the governing equation change form on a structured grid?
The Navier-Stokes equations in physical space are transformed into computational space. In conservative form, it becomes:
Here, $\mathbf{Q}$ is the vector of conservative variables, and $\hat{\mathbf{E}}, \hat{\mathbf{F}}, \hat{\mathbf{G}}$ are the transformed fluxes containing metric terms.
What specifically are the metric terms?
For example, $\hat{\mathbf{E}} = \frac{1}{J}(\xi_x \mathbf{E} + \xi_y \mathbf{F} + \xi_z \mathbf{G})$, where the partial differential coefficients of the coordinate transformation $\xi_x, \xi_y$, etc., multiply the fluxes. If these metric terms are not discretized accurately, freestream preservation breaks down, generating spurious numerical errors.
Geometric Progression of Grid Spacing
You want the grid to be dense near walls, right? How is that controlled?
Using a geometric progression is fundamental.
Here, $\Delta s_1$ is the thickness of the first layer, and $r$ is the growth ratio. If $r = 1.2$, each layer becomes 1.2 times thicker than the previous one. The wall-side $\Delta s_1$ is determined from $y^+$ requirements, and a growth ratio $r$ of 1.1 to 1.3 is recommended.
What happens if the growth ratio is too large?
The size ratio between adjacent cells becomes too large, increasing truncation error and worsening numerical diffusion. Especially for second-order schemes, it's desirable to keep the volume ratio of adjacent cells below 1.2.
History of Structured Grids—The 1974 Thompson-Thames-Mastin Transformation and the Birth of BFM (Body-Fitted Mesh)
The systematic method for generating "Body-Fitted Mesh" along complex object shapes was established by Joe Thompson, Frank Thames, and C. Wayne Mastin in their 1974 paper "Grid Generation by Elliptic Partial Differential Equations (TTM method)." This method maps a complex computational domain in physical space to a "computational space" described by a uniform orthogonal grid in ξ-η-ζ using "Conformal Mapping." The equations are solved in computational space and then transformed back to physical space. This first made practical structured grid generation for arbitrary shapes like airfoils, ship hulls, and engine internal flow paths possible, enabling the explosive development of CFD in the 1980s. The grid generation code GRID2D, based on the TTM method, is still used for education and research even 50 years later.
Physical Meaning of Each Term
- Temporal Term $\partial(\rho\phi)/\partial t$: Imagine turning on a faucet. At first, water comes out spluttering and unstable, but after a while, the flow becomes steady, right? This "period of change" is described by the temporal term. The pulsation of blood flow from a heartbeat, or flow fluctuations each time an engine valve opens/closes—all are unsteady phenomena. So what is steady-state analysis? It looks only at "after sufficient time has passed and the flow has settled down"—meaning this term is set to zero. Since computational cost drops significantly, solving first in steady-state is a basic CFD strategy.
- Convection Term $\nabla \cdot (\rho \mathbf{u} \phi)$: What happens if you drop a leaf into a river? It gets carried downstream by the flow, right? This is "convection"—the effect where fluid motion transports things. Warm air from a heater reaching the far side of a room is also because the "carrier," air, transports heat via convection. Here's the interesting part—this term contains "velocity × velocity," making it nonlinear. That is, as flow speed increases, this term rapidly strengthens, making control difficult. This is the root cause of turbulence. A common misconception: "Convection and conduction are similar" → They're completely different! Convection is carried by flow, conduction is transmitted by molecules. There's an order of magnitude difference in efficiency.
- Diffusion Term $\nabla \cdot (\Gamma \nabla \phi)$: Have you ever put milk in coffee and left it? Even without stirring, after a while it naturally mixes, right? That's molecular diffusion. Now a question—honey and water, which flows more easily? Obviously water, right? Honey has high viscosity ($\mu$), so it flows poorly. Higher viscosity strengthens the diffusion term, making the fluid move "sluggishly." In low Reynolds number flow (slow, viscous), diffusion dominates. Conversely, in high Re number flow, convection overwhelms and diffusion plays a supporting role.
- Pressure Term $-\nabla p$: When you push a syringe plunger, liquid shoots out forcefully from the needle tip, right? Why? Because the plunger side is high pressure, the needle tip is low pressure—this pressure difference provides the force pushing the fluid. Dam discharge works on the same principle. On a weather map, where isobars are tightly packed? That's right, strong winds blow. "Flow is generated where there is a pressure difference"—this is the physical meaning of the pressure term in the Navier-Stokes equations. A point of confusion here: "Pressure" in CFD is often gauge pressure, not absolute pressure. When switching to compressible analysis, if results suddenly look wrong, mixing up absolute/gauge pressure might be the cause.
- Source Term $S_\phi$: Warmed air rises—why? Because it becomes lighter (lower density) than its surroundings, so buoyancy pushes it up. This buoyancy is added to the equation as a source term. Other examples: chemical reaction heat from a gas stove flame, Lorentz force acting on molten metal in a factory's electromagnetic pump... These are all actions that "inject energy or force into the fluid from the outside," expressed by source terms. What happens if you forget a source term? In natural convection analysis, forgetting buoyancy means the fluid doesn't move at all—a physically impossible result where warm air doesn't rise in a heated winter room.
Assumptions and Applicability Limits
- Continuum Assumption: Valid for Knudsen number Kn < 0.01 (mean free path ≪ characteristic length)
- Newtonian Fluid Assumption: Shear stress and strain rate have a linear relationship (non-Newtonian fluids require viscosity models)
- Incompressibility Assumption (for Ma < 0.3): Treat density as constant. For Mach number 0.3 and above, consider compressibility effects
- Boussinesq Approximation (natural convection): Consider density variation only in the buoyancy term, using constant density in other terms
- Non-applicable Cases: Rarefied gas (Kn > 0.1), supersonic/hypersonic flow (requires shock capturing), free surface flow (requires VOF/Level Set, etc.)
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Velocity $u$ | m/s | When converting from volumetric flow rate for inlet conditions, pay attention to cross-sectional area units |
| Pressure $p$ | Pa | Distinguish between gauge and absolute pressure. Use absolute pressure for compressible analysis |
| Density $\rho$ | kg/m³ | Air: approx. 1.225 kg/m³ @20°C, Water: approx. 998 kg/m³ @20°C |
| Viscosity Coefficient $\mu$ | Pa·s | Be careful not to confuse with kinematic viscosity coefficient $\nu = \mu/\rho$ [m²/s] |
| Reynolds Number $Re$ | Dimensionless | $Re = \rho u L / \mu$. Criterion for laminar/turbulent transition |
| CFL Number | Dimensionless | $CFL = u \Delta t / \Delta x$. Directly related to time-step stability |
Numerical Methods and Implementation
Structured Grid Generation Methods
How do you actually create a structured grid?
Let me introduce three representative methods.
Algebraic Method (Transfinite Interpolation: TFI)
Given the distribution of grid points on the boundary, interior grid points are determined by interpolation. Computationally fast, but controlling grid quality for complex shapes is difficult.
Elliptic Grid Generation (Elliptic Grid Generation)
Grid points are placed by solving Poisson equations.
Grid concentration near walls and orthogonality can be controlled by adjusting the control functions $P, Q, R$. Produces the highest grid quality but requires iterative computation and takes time to generate.
Hyperbolic Grid Generation
A method that "grows" the grid from the wall in the normal direction. Particularly effective for boundary layer mesh generation, naturally ensuring wall orthogonality.
Which method should I choose?
In practice, a hybrid approach is common: first create an initial grid with TFI, then smooth it with an elliptic solver. This is the standard workflow in tools like ICEM or Pointwise.
Multi-Block Structured Grid
For complex shapes, a single block seems impossible...
In that case, use a multi-block structured grid. The computational domain is divided into multiple blocks, each containing a structured grid, and information is exchanged between blocks via interfaces.
There are two main types of connections between blocks.
- Conformal: Grid points match at the interface. No interpolation error.
- Non-conformal: Grid points do not match. Interpolation is required.
Conformal is obviously better, right?
In terms of accuracy, yes, but for complex shapes, maintaining conformity makes topology design extremely difficult. In practice, block topology editors like ICEM CFD Hexa are invaluable. However, this topology design is the biggest bottleneck for structured grids; even an expert can take days to weeks to mesh a single car.
Compatibility with Finite-Difference Schemes
I've heard finite-difference methods are easier to use with structured grids. What does that mean?
Because grid points are arranged regularly in structured grids, high-order accuracy finite-difference schemes can be naturally constructed. For example, 5th-order WENO schemes or compact finite-difference methods (Pade-type) truly shine on structured grids.
Achieving equivalent high-order accuracy on unstructured grids requires gradient reconstruction via least squares and handling of non-uniform spacing, which skyrockets computational cost.
So that's why structured grids are preferred for DNS (Direct Numerical Simulation) and LES.
Exactly. Structured grids are still mainstream for high-precision academic computations. Especially for DNS of relatively simple shapes like channel flow or pipe flow, structured grids are essentially the only choice.
Choosing Between O-Type, C-Type, and H-Type Grids—Decision Criteria for Structured Grid Design Around Airfoils
For structured grids around airfoils, the choice of grid topology directly impacts accuracy. O-type grids (concentric elliptical shape surrounding the airfoil) can uniformly and with high quality resolve the entire airfoil perimeter, naturally allowing continuous mesh from the leading edge to the trailing edge. However, grid lines bend at the airfoil's sharp trailing edge, potentially reducing shape accuracy for thin trailing edges. C-type grids (wrapping around in a C-shape from the front of the airfoil) make it easier to capture the separation region near the trailing edge in detail, suitable for stall analysis. H-type grids (deformed rectangular orthogonal grids) are simplest but cannot match the high quality of O-type grids around the airfoil. In practice, a hybrid OC-H type composite grid, where "an O-type or C-type grid is generated based on topology, and the far field is covered by an H-type structured grid," is the standard choice for high-precision RANS/LES of airfoils.
Upwind Differencing (Upwind)
First-order upwind: Large numerical diffusion but stable. Second-order upwind: Improved accuracy but risk of oscillations. Essential for high Reynolds number flows.
Central Differencing (Central Differencing)
Second-order accurate, but numerical oscillations occur for Pe number > 2. Suitable for low Reynolds number diffusion-dominated flows.
TVD Schemes (MUSCL, QUICK, etc.)
Suppress numerical oscillations while maintaining high accuracy via limiter functions. Effective for capturing shock waves or steep gradients.
Finite Volume Method vs Finite Element Method
FVM: Naturally satisfies conservation laws. Mainstream in CFD. Related Topics
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