How to Interpret this Dashboard

1. What is this Simulator?

This simulator provides a **conceptual visualization** of **2D incompressible fluid dynamics**. It numerically solves a simplified version of the Navier-Stokes equations, which are fundamental to describing how fluids like air and water move in a 2D plane. The goal is to help you build an intuition for complex fluid phenomena like turbulence, vortex dynamics, and energy dissipation.

2. The Physics & The Math

The Governing Equations

The core of the simulation is the Navier-Stokes momentum equation for an incompressible fluid in 2D:

$ \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla)\mathbf{u} = -\nabla p + \nu \nabla^2 \mathbf{u} $

• $ \mathbf{u}(x,y,t) $ is the 2D fluid velocity vector $(u,v)$.
• $ (\mathbf{u} \cdot \nabla)\mathbf{u} $ (Advection): How the fluid carries itself. This non-linear term is the source of most of the complexity, including turbulence.
• $ -\nabla p $ (Pressure Gradient): How differences in pressure ($p$) create forces that push the fluid.
• $ \nu \nabla^2 \mathbf{u} $ (Viscous Diffusion): Represents fluid friction, where $\nu$ is the kinematic viscosity.

This is coupled with the incompressibility condition, which states that the flow is divergence-free:

$ \nabla \cdot \mathbf{u} = 0 $

Key 2D Concepts

• Vorticity ($\omega_z$): In 2D, vorticity is a scalar value representing the local "spin" of a fluid parcel, defined as $ \omega_z = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} $. It is a crucial quantity in 2D flows.
• Enstrophy ($\mathcal{E}$): The integral of squared vorticity, $ \mathcal{E} = \frac{1}{2} \int \omega_z^2 dA $. In 2D turbulence, there is a cascade of enstrophy from large to small scales, where it is dissipated by viscosity.
• Inverse Energy Cascade: A unique feature of 2D turbulence where kinetic energy tends to move from small scales to larger scales, leading to the formation of large, coherent vortices. This is opposite to 3D turbulence.

3. How to Analyze the Plots & Data

The Kinetic Energy Spectrum

This log-log plot is a key tool for turbulence analysis. It shows how the fluid's kinetic energy $E$ is distributed across different length scales (wavenumber $k$; small $k$ = large scales, large $k$ = small scales). Look for:

• Inverse Energy Cascade Range: At low $k$, you might see a slope of $E(k) \propto k^{-5/3}$, indicating energy flowing to larger structures.
• Enstrophy Cascade Range: At high $k$, you might see a steeper slope of $E(k) \propto k^{-3}$, indicating enstrophy flowing to smaller, dissipative scales. The dashboard plots a $k^{-3}$ reference line.

Vector & Scalar Field Plots

This tab provides a powerful, interactive tool to inspect scalar fields like vorticity and divergence at a granular level. It features two synchronized views: a graphical canvas on the left and a numerical grid on the right.

• Interactive Hovering: Move your cursor over either the graphical canvas or the numerical grid. The corresponding cell in the other view will be highlighted, and the precise value at that point will be displayed live.
• Single-Point Detailed Analysis: Click on any cell in either the canvas or the grid. This will open a modal window showing a detailed mathematical breakdown for that specific point, including the calculated gradients used to compute the field value (e.g., $\partial v/\partial x$ and $\partial u/\partial y$ for vorticity).
• Multi-Pixel Analysis: For aggregate statistics over a custom region:
  1. Check the "Enable Multi-Pixel Analysis" box.
  2. Single-click on multiple cells in the grid to select them. Selected points will be marked with a dot.
  3. Once you have selected your points, double-click anywhere on the grid. A tooltip will appear showing the count, mean, standard deviation, min, and max of the values for your selected points.

Legal Disclaimers & Terms of Use

1. Experimental Research Tool

This 2D Fluid Dynamics Analogue Simulator (the "Tool") is provided by Quantum Q PTE. LTD. ("Quantum Q") as an experimental research and educational tool. It is not designed, certified, or intended for use in any commercial, industrial, or mission-critical applications.

2. No Warranty & Limitation of Liability

THE TOOL IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED. IN NO EVENT SHALL QUANTUM Q BE LIABLE FOR ANY CLAIM, DAMAGES, OR OTHER LIABILITY. The entire risk as to the quality, performance, and accuracy of the Tool is with you.

3. No Guarantee of Accuracy

While efforts have been made to ensure numerical integrity, Quantum Q makes no guarantee as to the accuracy, reliability, or completeness of the results. The outputs are for illustrative and conceptual purposes only and should not be used for formal applications without independent verification.

4. Intellectual Property & Governing Law

All rights are the exclusive property of Quantum Q PTE. LTD. These terms shall be governed by the laws of the Republic of Singapore.

By using this Tool, you acknowledge that you have read, understood, and agree to be bound by these terms.