Aero Validation & Grid Convergence
This page presents the validation benchmarks and grid convergence studies (GCI) for Shardian Aero's AdvancedZonalModel against traditional RANS solvers and experimental wind tunnel data.
1. Grid Convergence Index (GCI) Study
To evaluate the mathematical grid sensitivity of the solver, we performed a three-grid refinement study using Roache's GCI methodology. The refinement ratio \(r = h_{coarse} / h_{fine}\) is maintained at \(r \approx 2^{1/2} \approx 1.414\) in each coordinate axis.
The grid convergence index is defined as:
Where: * \(F_s = 1.25\) is the safety factor for three-grid studies. * \(\epsilon = (\phi_{coarse} - \phi_{fine}) / \phi_{fine}\) is the relative solution difference. * \(p\) is the formal order of accuracy (empirically calculated).
GCI Analysis for Drag Coefficient (\(C_d\)) on Ahmed Body:
| Parameter | Coarse (0.9M cells) | Medium (2.5M cells) | Fine (7.2M cells) |
|---|---|---|---|
| Grid Spacing (\(h\)) | 1.41 mm | 1.00 mm | 0.71 mm |
| Calculated \(C_d\) | 0.312 | 0.298 | 0.291 |
| Relative error (\(\epsilon\)) | - | 4.69% | 2.41% |
| Calculated Order (\(p\)) | - | - | 1.94 |
| GCI (\(F_s = 1.25\)) | - | 6.11% | 3.09% |
The fine-grid GCI of 3.09% indicates that the numerical solution is within the asymptotic range of convergence, and further grid refinement will not alter the simulated drag coefficient by more than 3%.
2. Benchmark Cases
Case A: Backward-Facing Step (BFS)
The backward-facing step is a standard test case for evaluating separation under adverse pressure gradients.
* Metric: Reattachment length (\(x_r/h\)).
* Experimental Reference (Kim et al.): \(7.0 \pm 0.5\).
* Results:
* kOmegaSST: \(6.1\) (underpredicts mixing).
* AdvancedZonalModel: \(7.1\) (matches experimental bubble size).
* realizableKE: \(5.4\) (fails to resolve recirculation).
Case B: Ahmed Body (25° slant angle)
The 25° slant Ahmed Body exhibits a highly three-dimensional flow with a complex longitudinal counter-rotating vortex pair.
* Metric: Total drag coefficient (\(C_d\)).
* Experimental Reference: \(0.285 - 0.298\).
* Results:
* kOmegaSST: \(0.342\) (overpredicts pressure drag due to early slant separation).
* AdvancedZonalModel: \(0.291\) (reproduces vortex-induced downforce).
* SpalartAllmaras: \(0.365\) (underpredicts reattachment on the rear slant).
graph LR
subgraph "Ahmed Body (25° slant)"
S[Slant Flow Separation] --> V[Vortex Generation]
V --> D[Drag Forces]
end
style S fill:#bc4f4b,stroke:#0a0808,color:#fff
style V fill:#ff7a70,stroke:#0a0808,color:#fff3. Boundary Layer Velocity Profiles
The velocity profile in the logarithmic layer of a flat plate boundary layer is defined by the classical law of the wall:
Where: * \(\kappa \approx 0.41\) is the von Kármán constant. * \(B \approx 5.0\) is the log-law intercept.
Traditional models over-damp velocity profiles in highly curved walls. Shardian Aero's adaptive \(f_{IA}\) coefficient modulates the production term in the log-law region, preserving the exact log slope (\(\kappa = 0.41\)) and preventing artificial boundary layer thickening under separation.