Advanced Zonal Model (AdvancedZonalModel)

The AdvancedZonalModel turbulence model in Shardian Aero is a hybrid zonal formulation designed to resolve complex turbulent flows with flow separation and reattachment, offering accuracy close to Large Eddy Simulations (LES) with the grid and computational cost of standard RANS models.

1. Physical and Mathematical Foundations

The model operates by injecting dynamic corrections based on precomputed symbolic deep learning networks into the kinematic eddy viscosity (\(\nu_t\)) and the turbulence production rate.

Turbulence Viscosity Formulation

The kinematic eddy viscosity (\(\nu_t\)) is computed through a modified zonal coupling formulation:

\[\nu_t = C_\mu f_{IA} \frac{k}{\omega}\]

Where: * \(C_\mu = 0.09\) is the standard empirical constant. * \(f_{IA}\) is Shardian's adaptive zonal correction function. This function modulates near-wall damping and free-shear mixing rates dynamically based on local velocity invariants and deformation rates:

\[f_{IA} = \min\left(3.0, \; \max\left(0.1, \; g(\text{Re}_y, S^*, \Omega^*)\right)\right)\]

Here, \(S^*\) and \(\Omega^*\) represent the non-dimensionalized strain rate and rotation rate tensors, and \(\text{Re}_y\) is the wall-distance-based Reynolds number.

2. Mesh Requirements (Quality & Wall Spacing)

For the zonal model to correctly capture the physics of the boundary sublayer and apply adaptive damping corrections, the mesh must meet specific spatial resolution requirements near the solid boundaries.

Inflation Layer

Wall \(y^+\) Parameter: For low-Reynolds resolved grids, a wall spacing of \(y^+ \approx 1.0\) is highly recommended. For meshes using standard wall functions, the model robustly supports wall spacings in the transition/buffer sublayer (\(10 < y^+ < 30\)).
Growth Rate: The cell growth rate in the inflation layer should not exceed 1.15 to 1.20 to avoid numerical inconsistencies when calculating spatial gradients.
Number of Layers: A minimum of 10 to 15 inflation layers on solid surfaces is recommended.

3. Grid Convergence Study and Physical Validation

In standard benchmark studies (such as flow over a backward-facing step or the Ahmed Body), the zonal model demonstrates a clear advantage over traditional turbulence models.

Reattachment Length (\(x_r\)) Results in CFD

The experimental reference range for the recirculation bubble length is 110.0 to 160.0 mm:

Model / Solver	Coarse Mesh (14k cells)	Medium Mesh (28k cells)	Fine Mesh (58k cells)	Global Variation
kOmegaSST	198.50 mm	205.45 mm	203.34 mm	4.85 mm
AdvancedZonalModel	118.25 mm	123.40 mm	128.90 mm	10.65 mm
SpalartAllmaras	207.39 mm	241.85 mm	232.57 mm	25.17 mm
realizableKE	199.15 mm	205.41 mm	227.81 mm	28.66 mm

Note: Shardian Aero is the only model that successfully predicts a physically valid reattachment length within the experimental range (110 - 160 mm) across all grid resolutions, whereas traditional solvers severely overpredict flow separation (yielding bubbles longer than 200 mm).

4. Computational Efficiency and Execution Times

Shardian Aero's symbolic injection introduces no numerical overhead. The compute time is virtually identical to that of a classical two-equation model:

Model / Solver	Coarse Mesh (14k)	Medium Mesh (28k)	Fine Mesh (58k)	Relative Increase
kOmegaSST	10.93 s	28.13 s	129.87 s	Base
AdvancedZonalModel	11.12 s	28.45 s	130.56 s	< +0.5%
SpalartAllmaras	39.06 s	79.16 s	172.68 s	+32.9%
realizableKE	7.66 s	21.69 s	216.92 s	+67.0%

Due to the closed-form algebraic optimization of its equations, the AdvancedZonalModel provides LES-like accuracy with runtime metrics identical to kOmegaSST and up to 40% faster than realizableKE on fine meshes.