## Technical Document: Symbolic Regression for Periodic Hill Turbulence Modeling
### Overview
The image contains a technical specification for a symbolic regression task. The objective is to learn three scalar functions, G1(I1, I2), G2(I1, I2), and G3(I1, I2), which are used to model turbulence anisotropy in a periodic hill flow case. The document defines the flow context, the evaluation rule for the model, and the specific tensor bases (T1, T2, T3) that the scalar functions must weight to predict the anisotropy tensor.
### Components/Axes
The document is structured as a text block with the following major sections:
1. **Task Statement**: Defines the core objective.
2. **Flow Case Context**: Describes the physical setup of the turbulence problem.
3. **Evaluation Rule**: Specifies how the model's performance is measured.
4. **Tensor Bases**: Provides definitions and physical interpretations for the three tensor bases (T1, T2, T3) used in the model.
### Detailed Analysis / Content Details
**1. Task Statement**
* **Task**: Symbolic regression for periodic hill turbulence modeling.
* **Objective**: Learn three scalar functions: `G1(I1, I2)`, `G2(I1, I2)`, `G3(I1, I2)`.
**2. Flow Case Context (for description only)**
* **Bottom wall profile**:
* `y(x) = a[1 + cos(π x / L)]` for `|x| ≤ L`
* `y(x) = 0` for `|x| > L`
* Where `a` is the hill height (characteristic length `h`), and `α = L/h` controls hill steepness.
* **Training case parameter**: `α = 0.8`.
* **Boundary Conditions**:
* Streamwise (`x`) direction: periodic boundary conditions.
* Top and bottom walls: no-slip boundary conditions.
* **Reynolds Number**: `Re_h = U_b h / ν = 5600`.
**3. Evaluation Rule**
* **Input**: Tensor bases `T1`, `T2`, `T3` (each 3x3), provided by the evaluator.
* **Predicted Anisotropy (`b_hat`)**: `b_hat = G1 * T1 + G2 * T2 + G3 * T3`
* **Loss Metric**: The evaluator computes the Mean Squared Error (MSE) between the predicted `b_hat` and the target anisotropy tensor `b`.
**4. Tensor Bases: T1, T2, T3 (provided by the evaluator, do NOT compute them)**
For each sample `n`, the evaluator provides three 3x3 tensor bases (symmetric, traceless) constructed from the non-dimensionalized mean strain-rate tensor `S` and rotation tensor `R`.
* **T1 (linear strain basis)**:
* **Definition**: `T1 = S`
* **Interpretation**: Represents the Boussinesq/linear eddy-viscosity direction. It dominates in simple shear flows and acts as the baseline anisotropy response aligned with the mean strain.
* **T2 (strain-rotation coupling basis)**:
* **Definition**: `T2 = S @ R - R @ S`
* **Interpretation**: Captures the interaction between mean strain and mean rotation (curvature/turning, strong vortices). This term is crucial in separated flows, reattachment, and swirling/shear-layer regions where linear eddy-viscosity assumptions tend to fail.
* **T3 (quadratic strain nonlinearity basis)**:
* **Definition**: `T3 = S @ S - (1/3) * tr(S @ S) * I`
* **Interpretation**: Introduces nonlinear strain effects and normal-stress anisotropy beyond linear models. It helps represent differences among normal stress components and improves predictions in strongly anisotropic regions.
**Final Model Assembly**: The predicted anisotropy is assembled as:
`b_hat = G1(I1, I2) * T1 + G2(I1, I2) * T2 + G3(I1, I2) * T3`
### Key Observations
* The task is framed as a **symbolic regression** problem, implying the goal is to discover interpretable mathematical expressions for G1, G2, and G3, not just fit numerical parameters.
* The model is a **linear combination of three physically-motivated tensor bases**, where the coefficients (G1, G2, G3) are nonlinear functions of two invariants (`I1`, `I2`).
* The tensor bases progress in complexity: from linear strain (`T1`), to strain-rotation coupling (`T2`), to quadratic strain nonlinearity (`T3`). This structure suggests a hierarchy for capturing increasingly complex turbulence physics.
* The flow case (periodic hill at `Re_h=5600`) is a canonical benchmark for turbulence models, involving separation, recirculation, and reattachment, which tests the model's ability to handle non-equilibrium flows.
### Interpretation
This document defines a structured machine learning challenge for improving turbulence modeling in computational fluid dynamics (CFD). The core idea is to move beyond traditional linear eddy-viscosity models (represented by `T1`) by incorporating additional tensor terms (`T2`, `T3`) that account for complex flow physics like curvature and anisotropy.
The "symbolic regression" aspect is key. Instead of using a black-box neural network, the task seeks to find explicit, potentially simpler, mathematical formulas for the functions `G1`, `G2`, and `G3`. If successful, the resulting model would be more interpretable and physically insightful than a pure data-driven approach.
The framework explicitly links mathematical constructs to physical phenomena:
* `T1` handles basic shear.
* `T2` is critical for flows with rotation and separation (like the periodic hill).
* `T3` addresses limitations in predicting normal stress differences.
By learning how the scalar functions `G1(I1, I2)`, `G2(I1, I2)`, and `G3(I1, I2)` should vary with the flow invariants `I1` and `I2`, the task aims to create a more universal turbulence model that can adapt its behavior based on the local flow state, thereby improving predictive accuracy for complex engineering flows. The provided Reynolds number and geometric parameters (`α=0.8`) define a specific, challenging test case for this model development.
