CFD | Rhie-Chow interpolation

Rhie-Chow Interpolation

Instead of storing and updating vector velocity \(\mathbf{U}\) directly at the cell faces (a layout known as a staggered grid), modern unstructured CFD solvers like OpenFOAM prefer storing primary variables at cell centers (a collocated grid) and using Rhie-Chow face interpolation.

This hybrid approach is used for three primary reasons:

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1. Handling Arbitrary Unstructured Meshes

• Staggered Grid Limitation: Staggered grids store scalar variables (\(p, T\)) at cell centers and component velocity vectors (\(U_x, U_y, U_z\)) on corresponding cell faces pointing normal to those faces. This layout works well on simple, structured Cartesian grids. However, on arbitrary 3D unstructured meshes with non-orthogonal, polyhedral, or skewed cells, defining distinct directional face-staggered velocity components becomes mathematically and algorithmically complex.
• Collocated Solution: Storing full 3D velocity vectors \(\mathbf{U} = (U_x, U_y, U_z)\) at cell centers allows OpenFOAM to handle complex geometries, polyhedral cells, and mesh movement seamlessly.

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2. Computational Efficiency & Memory Overhead

If full velocity vectors were stored and updated directly as state variables at every face (rather than just cell centers):

• Memory Footprint: In 3D finite volume meshes, there are typically 3 to 6 times more faces than cells. Storing all velocity vectors, matrix coefficients, and gradients at cell faces would significantly increase RAM usage.
• Matrix Solver Costs: Solving linear momentum systems at every face center would multiply computational costs and solver time significantly.

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3. Consistency with Transport Equations

Physical scalar transport equations (like enthalpy \(T\), turbulence variables \(k-\varepsilon\), or species concentration \(C\)) naturally live at cell centers.

• By updating velocity \(\mathbf{U}\) at the same cell centers, momentum is solved using the exact same matrix structures, discretization schemes, and boundary conditions as all other transported scalars.

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Summary

The Rhie-Chow method on a collocated grid provides the best of both worlds:

Approach	Primary Storage	Advantages	Drawbacks
Pure Staggered Grid	Face centers	Naturally avoids pressure-velocity decoupling without corrections.	Impractical for arbitrary unstructured or polyhedral meshes.
Pure Collocated Grid (No Rhie-Chow)	Cell centers	Simple mesh data structures and memory efficiency.	Suffers from checkerboard pressure oscillations.
OpenFOAM Approach (Collocated + Rhie-Chow)	Cell centers (with face-flux \(\phi\))	Highly efficient on unstructured grids; eliminates checkerboard oscillations.	Requires evaluating face-flux correction terms.

Rhie-Chow 插值

Rhie-Chow 插值（Rhie-Chow Interpolation）是計算流體力學（CFD）中用於同位網格（Collocated Grid）的一項關鍵技術，旨在消除非物理的棋盤式壓力震盪（Checkerboard Oscillations）。 ## 緣起與背景

• 棋盤危機：在同位網格中，速度和壓力存儲在同一個網格中心。如果對控制體介面上直接採用簡單的線性插值計算速度與壓力梯度，離散的連續性方程將無法有效感知相鄰網格間的奇偶跳躍壓力分佈，導致壓力出現如棋盤格般的鋸齒狀劇烈震盪。
• 歷史突破：1983年，工程師 C.M. Rhie 與 W.L. Chow 在《AIAA Journal》發表了關於渦輪機翼邊緣分離流動的研究論文。他們提出了一種聰明的動量插值修正技巧。

核心思想

• 引入阻尼項：Rhie-Chow 插值在計算單元面上的速度時，除了常规插值外，還人為加入了一個與「相鄰節點壓力差」相關的附加項。
• 建立強耦合：這一額外項巧妙地模仿了動量方程的一部分，將壓力梯度直接帶入介面速度的計算中。這使得連續性方程間接包含了鄰近網格的壓力差，從而充當了數值阻尼，徹底平息了棋盤效應。

後續發展

• 雖然原始的 Rhie-Chow 插值取得了巨大成功並被廣泛應用（如主流 CFD 軟體及 OpenFOAM 中），但在極小時間步長或強源項條件下，它仍會暴露出與時間步無關性、過度耗散或虛假震盪等問題。
• 後續學者（如 Perić、Majumdar、Choi 等）提出了多種改進與廣義化公式，使其在處理複雜網格、動態網格以及局部加密時更加穩健。

References

1. Rhie-Chow插值有没有对应的连续的微分方程/积分方程？
2. Collocated grids | Rhie-Chow
3. Rhie-Chow interpolation in OpenFOAM
4. Rhie-Chow 插值在OpenFOAM中的实现逻辑
   1. Rhie-Chow插值相信大家都不陌生，主要是针对同位网格（collocated Grid）中由于速度和压力梯度离散不连续（non-consecutive）导致的棋盘效应（Checkerboard Problem），即速度和压力梯度有间隔状的震荡现象
   2. 基本逻辑可以简单概括为通过不同于一般的面速度插值，来让最终的压力泊松方程离散关系不再不连续，并通过求解出来的压力去更新速度和面流量，从而消除上述的棋盘效应。