2-D Navier-Stokes Solver in a periodic box #124
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| "\n", | ||
| "## Overview\n", | ||
| "\n", | ||
| "In this notebook, we will implement a 2-D Navier-Stokes (N-S) solver in a periodic box using the vorticity-streamfunction formulation. The Poisson equation relating streamfunction to vorticity is solved spectrally using tile-based FFT in Warp.\n", |
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This intro sentence, while precise, is very technical. We can help novices by at least mentioning that we're building a solver that simulates fluid dynamics.
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You should also emphasize the Python aspect, e.g. something like "We will build our high-performance solver entirely in Python using the Warp framework..."
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"In this notebook, we will build a high-performance fluid solver entirely in Python using the Warp framework. Specifically, we will simulate 2-D turbulent flow in a box." -- How does this sound?
| "## Introduction\n", | ||
| "\n", | ||
| "This seemingly simple 2-D N-S solver example combines multiple Warp features that can be leveraged to build industrial-grade solvers:\n", | ||
| " - CUDA kernels for finite difference operators\n", |
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CUDA kernels should probably be Accelerated kernels for finite-difference operators. When I see "CUDA kernels", I think of all the nastiness associated with CUDA C++, which our users don't need to worry about.
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Updated to "Accelerated Warp kernels..."
| "### 1. Vorticity-stream function formulation\n", | ||
| "\n", | ||
| "For 2-D incompressible flow, we define:\n", | ||
| "- **Vorticity**: $\\omega = \\partial v/\\partial x - \\partial u/\\partial y$\n", | ||
| "- **Streamfunction**: $\\psi$ such that $u = \\partial\\psi/\\partial y$, $v = -\\partial\\psi/\\partial x$\n", | ||
| "\n", | ||
| "This formulation automatically satisfies continuity and eliminates pressure from the Navier-Stokes equations.\n", |
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You don't explain what the variables u and v are
| "\n", | ||
| "### 3. Spectral Poisson solver\n", | ||
| "\n", | ||
| "For periodic domains, equation (2) can be directly solved in Fourier space. The wavenumbers are defined as $k_x = \\frac{2\\pi m}{L_x}, \\quad k_y = \\frac{2\\pi n}{L_y}$, where $m$ and $n$ are the Fourier space indices. The vorticity field $\\omega$ is converted to its Fourier representation $\\hat{\\omega}_{m,n}$ using a 2-D FFT. The Poisson equation then becomes algebraic as follows\n", |
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Maybe you can add some more discussion to help novices. You start off with saying this can be solved in Fourier space, but maybe it's better to acknowledge that it might be tempting to solve the Poisson equation in physical space but we can make use of periodicity to solve the equation in spectral space, which is both efficient and simple to implement.
You can also explain what "Fourier space" is and specifically what the "FFT" means and why it deserves the "Fast" in its name
| "$\\hat{\\psi}_{m,n}$ is then converted back to physical space using inverse FFT.\n", | ||
| "\n", | ||
| "### 4. Time integration \n", | ||
| "For time stepping, we use a strong-stability preserving third-order Runge-Kutta method.\n", |
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Maybe give a layperson explanation of what SSP and third-order mean
| "\n", | ||
| "### 5. Initial condition: 2-D decaying turbulence\n", | ||
| "\n", | ||
| "In this problem, we solve the governing equations on a square box of dimensions $2\\pi \\times 2\\pi$. The initial condition is generated using the energy spectrum from San & Staples CNF (2012)\n", |
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Explain why you picked
| "\n", | ||
| "**NOTE**: Other similar time-stepping schemes can be implemented with minimal changes to the code, but we use the scheme above as a demonstration example.\n", | ||
| "\n", | ||
| "### 5. Initial condition: 2-D decaying turbulence\n", |
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I think this is too much. I would create an appendix section and put the details about the initialization there, and then in this section you just say we're gonna use this particular E(k) because of <...> and you can find more discussion behind this choice of I.C. in the appendix.
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I removed the construction of \zeta to the appendix A. I still left the definition of E(k) there.
| "$$\n", | ||
| "\\boxed{\n", | ||
| "\\begin{array}{c}\n", | ||
| "\\textbf{Solver Pipeline} \\\\[10pt]\n", |
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This "Solver Pipeline" diagram is already pretty good. I would add a "IFFT" in the first arrow that follows "Initialize
| "- Initialize $\\omega$, i.e., generate $\\hat{\\omega}$ in the Fourier space using the energy spectrum specified above and inverse FFT it to the physical space.\n", | ||
| "- Discretize the advection and diffusion terms to obtain the RHS $\\mathcal{L}(\\omega)$ of $\\frac{\\partial \\omega}{\\partial t} = \\mathcal{L}(\\omega)$ (see equation 1).\n", | ||
| "- Advancing $\\omega(t)$ to $\\omega(t+\\Delta t)$ using SSP RK3 scheme (equation 4).\n", | ||
| "- Solve Poisson equation (equation 2) in the Fourier space $\\hat{\\psi}_{m,n} = \\hat{\\omega}_{m,n}/(k_x^2 + k_y^2)$ (equation 3), and inverse FFT $\\hat{psi}$ to $\\psi$ after each RK substep. " |
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inverse FFT $\\hat{psi}$ is missing \\ in {psi}
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…ed-computing-hub into sn/ns-2d-solver
Update the notebook with the latest version from x-ac-38-v1-sn-notebook_v3 and add 15 supporting images to images/chapter-12.2/. Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>
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