[mod] readme

README.md

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+# Simple TC Solver
+
+[![License](https://img.shields.io/github/license/NaokiHori/SimpleTCSolver)](https://opensource.org/license/MIT)
+[![Last Commit](https://img.shields.io/github/last-commit/NaokiHori/SimpleTCSolver/main)](https://github.com/NaokiHori/SimpleTCSolver/commits/main)
+[![CI](https://github.com/NaokiHori/SimpleTCSolver/actions/workflows/ci.yml/badge.svg)](https://github.com/NaokiHori/SimpleTCSolver/actions/workflows/ci.yml)
+[![Documentation](https://github.com/NaokiHori/SimpleTCSolver/actions/workflows/doc.yml/badge.svg)](https://github.com/NaokiHori/SimpleTCSolver/actions/workflows/doc.yml)
+
+![Thumbnail](https://github.com/NaokiHori/SimpleTCSolver/blob/main/docs/source/thumbnail.png)
+
+## Overview
+
+This library numerically solves the incompressible Navier-Stokes equations in both two-dimensional and three-dimensional planar and curved channels using the finite-difference method. While the primary objective is to simulate Taylor-Couette flows, the solver can also be applied to planar flow simulations. 
+
+For details on the governing equations and numerical methods ensuring conservative and consistent schemes, refer to the [documentation](https://naokihori.github.io/SimpleTCSolver).
+
+## Features
+
+- Energy-consistent treatment of advective, pressure-gradient, and diffusive terms, ensuring proper conservation properties.
+- [MPI parallelization](https://github.com/NaokiHori/SimpleDecomp).
+- Efficient FFT-based direct Poisson solver.
+- Explicit and implicit treatments of diffusive terms in all spatial directions.
+- Scalar transport simulation.
+
+## Dependencies
+
+- [C compiler](https://gcc.gnu.org)
+- [MPI](https://www.open-mpi.org)
+- [FFTW3](https://www.fftw.org)
+- [Python3](https://www.python.org) (only required for initializing flow fields in `NPY` format)
+
+## Quick Start
+
+1. **Set up your workspace**
+
+   ```console
+   mkdir -p /path/to/your/directory
+   cd /path/to/your/directory
+   ```
+
+2. **Clone the repository**
+
+   ```console
+   git clone --recurse-submodules https://github.com/NaokiHori/SimpleTCSolver
+   cd SimpleTCSolver
+   ```
+
+3. **Set initial conditions**
+
+   `Python3` is used to generate the initial flow fields as `NPY` files.
+
+   ```console
+   cd initial_condition
+   make output
+   bash main.sh
+   cd ..
+   ```
+
+4. **Build the solver**
+
+   ```console
+   make output
+   make all
+   ```
+
+## Example Results
+
+### Instantaneous Velocity Field
+
+![Velocity Field](https://raw.githubusercontent.com/NaokiHori/SimpleTCSolver/artifacts/artifacts/typical/snapshot.png)
+
+### Maximum Divergence
+
+![Divergence](https://raw.githubusercontent.com/NaokiHori/SimpleTCSolver/artifacts/artifacts/typical/divergence.png)
+
+### Normalized Energy Injection and Dissipation
+
+![Energy Balance](https://raw.githubusercontent.com/NaokiHori/SimpleTCSolver/artifacts/artifacts/typical/balance_main.png)
+
+The black-dashed line represents reference results from Ostilla et al., *J. Fluid Mech. (719), 2013*.
+
+The numerical scheme is designed to ensure energy injection and dissipation remain perfectly balanced (up to rounding error) in steady states:
+
+![Energy Dissipation](https://raw.githubusercontent.com/NaokiHori/SimpleTCSolver/artifacts/artifacts/typical/balance_dif.png)
+
+## 2D Version
+
+Since Taylor-Couette flows are inherently three-dimensional, the default solver is designed for 3D simulations. However, a 2D version that extracts radial-azimuthal motions is available in the `2d` branch.
+
+## Planar Flows
+
+This solver can also be used for planar flow simulations (e.g., normal channel flows). To enable this mode, set the `is_curved` flag in the flow initializer to `false`. For more details, refer to the [documentation](https://naokihori.github.io/SimpleTCSolver).
+
+## Acknowledgement
+
+I would like to thank [Dr. Kazuyasu Sugiyama](https://researchmap.jp/50466786) for insightful discussions at *Flow for Future - PoF25* and the *37th CFD Symposium*.
+