## Technical Document: Symbolic Regression for Periodic Hill Turbulence Modeling ### Overview This document outlines the task of symbolic regression for modeling periodic hill turbulence. It details the flow case context, evaluation rule, tensor bases used, and the assembly of predicted anisotropy. ### Components/Axes The document is structured into several sections: 1. **Task Definition:** Describes the objective of learning three scalar functions G1, G2, and G3, which depend on invariants I1 and I2. 2. **Flow Case Context:** Defines the bottom wall profile using equations and parameters like hill height (a), characteristic length (h), and a steepness control parameter (alpha). It also specifies boundary conditions and the Reynolds number. 3. **Evaluation Rule:** Explains how the predicted anisotropy (b_hat) is calculated using tensor bases T1, T2, and T3, along with the scalar functions G1, G2, and G3. It also mentions the use of Mean Squared Error (MSE) for evaluation. 4. **Tensor Bases:** Describes the three tensor bases (T1, T2, T3) and their interpretations: * T1 (linear strain basis): Represents the Boussinesq/linear eddy-viscosity direction. * T2 (strain-rotation coupling basis): Captures the interaction between mean strain and mean rotation. * T3 (quadratic strain nonlinearity basis): Introduces nonlinear strain effects and normal-stress anisotropy. 5. **Predicted Anisotropy Assembly:** Shows how the predicted anisotropy (b_hat) is assembled using the scalar functions and tensor bases. ### Detailed Analysis or ### Content Details **Task Definition:** * Objective: Learn three scalar functions: G1(I1, I2), G2(I1, I2), G3(I1, I2). **Flow Case Context:** * Bottom Wall Profile: * y(x) = a[1 + cos(πx / L)] for |x| ≤ L * y(x) = 0 for |x| > L * 'a' is the hill height (characteristic length h). * α = L/h controls hill steepness (training case: α = 0.8). * Streamwise (x) direction: Periodic boundary conditions. * Top and bottom walls: No-slip boundary conditions. * Reynolds number: Re_h = U_b h / v = 5600. **Evaluation Rule:** * Given tensor bases T1, T2, T3 (each 3x3), the predicted anisotropy is: * b_hat = G1 \* T1 + G2 \* T2 + G3 \* T3 * The evaluator computes MSE between b_hat and the target b. **Tensor Bases:** * 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):** * T1 = S * Interpretation: Boussinesq/linear eddy-viscosity direction. Dominates in simple shear flows and acts as the baseline anisotropy response aligned with the mean strain. * **T2 (strain-rotation coupling basis):** * T2 = S @ R - R @ S * Interpretation: Captures the interaction between mean strain and mean rotation (curvature/turning, strong vortices). Crucial in separated flows, reattachment, and swirling/shear-layer regions where linear eddy-viscosity assumptions tend to fail. * **T3 (quadratic strain nonlinearity basis):** * T3 = S @ S - (1/3) \* tr(S @ S) \* I * Interpretation: Introduces nonlinear strain effects and normal-stress anisotropy beyond linear models. Helps represent differences among normal stress components and improves predictions in strongly anisotropic regions. **Predicted Anisotropy Assembly:** * b_hat = G1(I1, I2) \* T1 + G2(I1, I2) \* T2 + G3(I1, I2) \* T3 ### Key Observations * The document provides a structured approach to modeling turbulence over a periodic hill. * It uses symbolic regression to learn scalar functions that relate to tensor bases. * The tensor bases represent different aspects of the flow, including linear strain, strain-rotation coupling, and quadratic strain nonlinearity. * The predicted anisotropy is assembled as a linear combination of these tensor bases, weighted by the learned scalar functions. ### Interpretation The document describes a methodology for modeling complex turbulent flows, specifically over a periodic hill. The approach uses symbolic regression to learn relationships between flow invariants and tensor bases that represent different physical phenomena. By combining these elements, the model aims to predict the anisotropy of the turbulence, which is crucial for understanding and simulating such flows. The use of different tensor bases allows the model to capture various aspects of the flow, including linear and nonlinear effects, as well as the interaction between strain and rotation. The ultimate goal is to improve the accuracy of turbulence models in complex flow scenarios where traditional linear eddy-viscosity assumptions may not be valid.

\n ## Text Document: Symbolic Regression for Periodic Hill Turbulence Modeling ### Overview This document outlines the task of symbolic regression for modeling periodic hill turbulence. It details the problem setup, evaluation rules, and the tensor bases used in the regression process. The document primarily consists of text with mathematical formulations and descriptions of the underlying physical phenomena. ### Components/Axes The document is structured as a series of paragraphs describing the task, the flow context, evaluation rules, and tensor bases. There are no explicit axes or charts. The document uses mathematical notation to define the hill profile and the anisotropy tensor. ### Detailed Analysis or Content Details **Task:** Symbolic regression for periodic hill turbulence modeling. Learn three scalar functions G1(I1, I2), G2(I1, I2), G3(I1, I2). **Flow case context (for description only):** * Bottom wall profile: * y(x) = a(1 + cos(πx / 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: α = 0.8). * Streamwise (x) direction uses periodic boundary conditions; top and bottom walls use no-slip boundary conditions. * Reynolds number: Re\_h = U\_b h / ν = 5600. **Evaluation rule:** Given tensor bases T1, T2, T3 (each 3x3), the predicted anisotropy is: b\_hat = G1 \* T1 + G2 \* T2 + G3 \* T3 The evaluator computes MSE between b\_hat and the target b. **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):** * T1 = S * Interpretation: 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):** * 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):** * 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. The predicted anisotropy is assembled as: b\_hat = G1(I1, I2) \* T1 + G2(I1, I2) \* T2 + G3(I1, I2) \* T3 ### Key Observations The document focuses on defining the mathematical framework for modeling turbulence over periodic hills. The key components are the hill profile, the Reynolds number, the tensor bases (T1, T2, T3), and the scalar functions G1, G2, and G3 that need to be learned through symbolic regression. The tensor bases represent different aspects of the anisotropy, ranging from linear strain effects to nonlinear strain and strain-rotation coupling. ### Interpretation This document describes a problem in computational fluid dynamics (CFD) where the goal is to model the anisotropy of Reynolds stresses in turbulent flow over periodic hills. The use of symbolic regression suggests that the researchers are attempting to discover analytical expressions for the scalar functions G1, G2, and G3 that best fit the observed data. The tensor bases T1, T2, and T3 provide a structured way to represent the anisotropy, and their interpretations highlight the physical mechanisms that contribute to it. The document emphasizes the limitations of linear eddy-viscosity models and the need for more sophisticated approaches that can capture nonlinear effects and strain-rotation coupling. The Reynolds number of 5600 indicates a relatively high Reynolds number flow, where turbulence is fully developed. The hill steepness parameter α = 0.8 suggests a moderately steep hill, which is likely to induce significant flow separation and turbulence. The overall goal is to develop a more accurate and physically meaningful model for turbulence over periodic hills, which can be used for predicting the flow behavior and optimizing the design of engineering systems.

## 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.

## Screenshot: Technical Document on Symbolic Regression for Periodic Hill Turbulence Modeling ### Overview The image displays a technical document outlining a symbolic regression framework for modeling periodic hill turbulence. It includes flow case context, evaluation rules, tensor basis definitions, and a final equation for predicted anisotropy. ### Components/Axes - **Sections**: 1. **Task**: Symbolic regression for periodic hill turbulence modeling. 2. **Flow case context**: Describes bottom wall profile equations and boundary conditions. 3. **Evaluation rule**: Defines predicted anisotropy using tensor bases. 4. **Tensor bases**: Describes T1 (linear strain), T2 (strain-rotation coupling), and T3 (quadratic strain nonlinearity). 5. **Final equation**: Combines tensor bases with coefficients G1, G2, G3. ### Detailed Analysis - **Flow case context**: - Bottom wall profile: - \( y(x) = a[1 + \cos(\pi x / L)] \) for \( |x| \leq L \) - \( y(x) = 0 \) for \( |x| > L \) - \( a \) = hill height (characteristic length \( h \)), \( \alpha = L/h \) controls steepness (training case: \( \alpha = 0.8 \)). - Reynolds number: \( Re_h = 5600 \). - Boundary conditions: No-slip at bottom wall, periodic at top wall. - **Tensor bases**: - **T1 (linear strain basis)**: - \( T1 = S \) (Boussinesq/linear eddy-viscosity direction). - Dominates in simple shear flows; baseline anisotropy aligned with mean strain. - **T2 (strain-rotation coupling basis)**: - \( T2 = S @ R - R @ S \) (interaction between mean strain and rotation). - Critical in separated flows, reattachment, and swirling regions. - **T3 (quadratic strain nonlinearity basis)**: - \( T3 = S @ S - \frac{1}{3} \cdot \text{tr}(S @ S) \cdot I \). - Captures nonlinear strain effects and normal-stress anisotropy. - **Final equation**: - Predicted anisotropy: \[ b_{\text{hat}} = G1(II, I2) \cdot T1 + G2(II1, I2) \cdot T2 + G3(II1, I2) \cdot T3 \] - Coefficients \( G1, G2, G3 \) are learned via symbolic regression. ### Key Observations - The model uses symbolic regression to learn scalar functions \( G1, G2, G3 \) that weight tensor bases. - Tensor bases are constructed from the non-dimensionalized mean strain-rate tensor \( S \) and rotation tensor \( R \). - T2 and T3 address limitations of linear eddy-viscosity models in complex flow regions. ### Interpretation This framework extends linear turbulence modeling by incorporating nonlinear and rotational effects through tensor bases. The symbolic regression approach allows data-driven optimization of coefficients \( G1, G2, G3 \), enabling the model to adapt to specific flow conditions (e.g., hill steepness \( \alpha \)). The inclusion of T2 and T3 suggests an emphasis on capturing anisotropic turbulence in regions with strong vortices or separation, where linear models fail. The Reynolds number \( Re_h = 5600 \) indicates a moderately turbulent flow regime.