×

How to Interpret this Dashboard

1. The Big Picture: What is this Simulator?

This simulator provides a **conceptual visualization** of **3D 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. The goal is to help you build an intuition for complex fluid phenomena like turbulence, vortex dynamics, and energy dissipation.

2. Quick Start Guide

1. Set Parameters: Adjust the sliders for Grid Resolution, Viscosity, etc., on the left panel to define your fluid.
2. (or) Choose a Preset: Click a preset button like "Turbulence" or "Stable Flow" to load a pre-configured scenario.
3. Run Simulation: Click the main "Simulation" button to start the calculation.
4. Observe Live: Watch the 3D visualization evolve. You can rotate (left-click), pan (right-click), and zoom (scroll) the view. The live metrics will update on the right.
5. Stop & Analyze: Once the simulation is complete (or you click "Stop"), the detailed "Analysis" and "Metrics" sections below will become populated with data from the final state of the fluid.
6. Interpret with AI: If you have a Gemini API key, you can use the "Results", "Ask Einstein", and "Q&A" buttons for an AI-powered analysis of the simulation.

3. The Physics & The Math: A Deep Dive

The Governing Equations

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

$ \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla)\mathbf{u} = -\frac{1}{\rho}\nabla p + \nu \nabla^2 \mathbf{u} $

• $ \mathbf{u}(x,y,z,t) $ is the fluid velocity vector.
• $ \frac{\partial \mathbf{u}}{\partial t} $ (Unsteady Term): How velocity changes at a point over time.
• $ (\mathbf{u} \cdot \nabla)\mathbf{u} $ (Advection/Convection Term): How the fluid's own motion carries its momentum around. This is a non-linear term and is the source of most of the complexity, including turbulence.
• $ -\frac{1}{\rho}\nabla p $ (Pressure Gradient Term): How differences in pressure ($p$) create forces that push the fluid.
• $ \nu \nabla^2 \mathbf{u} $ (Viscous Diffusion Term): Represents fluid friction, where $\nu$ is the kinematic viscosity. This term smooths out the flow and dissipates energy.

This is coupled with the incompressibility condition, which states that mass is conserved (density is constant):

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

Key Physical Quantities & Metrics

• Vorticity ($\omega$): Defined as the curl of velocity, $ \omega = \nabla \times \mathbf{u} $, vorticity measures the local "spin" or rotation of a fluid parcel. Regions of high vorticity are where the flow is swirling intensely.
• Kinetic Energy ($E_k$): The total energy of motion, $ E_k = \frac{1}{2} \int_V |\mathbf{u}|^2 dV $. In a closed system, this energy must be dissipated by viscosity, so it typically decays over time.
• Enstrophy ($\mathcal{E}$): The integral of squared vorticity, $ \mathcal{E} = \frac{1}{2} \int_V |\omega|^2 dV $. It represents the rate of energy dissipation due to viscosity. In turbulent flows, enstrophy can increase dramatically through a process called vortex stretching.
• Helicity ($\mathcal{H}$): Defined as $ \mathcal{H} = \int_V \mathbf{u} \cdot \omega \, dV $, helicity measures the "knottedness" or alignment of vortex lines with the flow direction.
• Reynolds Number ($Re$): The crucial dimensionless ratio of inertial forces to viscous forces, $ Re = \frac{UL}{\nu} $. Low $Re$ flows are smooth and laminar; high $Re$ flows are chaotic and turbulent.
• Kolmogorov Length Scale ($\eta$): In turbulence, energy "cascades" from large eddies to small ones. The Kolmogorov scale, $ \eta = (\nu^3 / \epsilon)^{1/4} $ (where $\epsilon$ is the energy dissipation rate), is the smallest length scale in the flow, where viscosity finally dominates and dissipates the energy into heat. A good simulation requires the grid spacing to be smaller than $\eta$.

4. Analyzing the Results

Once a simulation is complete, the "Simulation Metrics" and "Detailed Analysis" sections populate with rich data from the final state of the fluid. This is where you can perform a deep dive into the physics of the flow.

Derived Physical Quantities

At the top of the "Detailed Analysis" panel, a set of key dimensionless numbers and physical scales appear. These are crucial for characterizing the nature of the flow:

• Reynolds Number ($Re$): The ratio of inertial to viscous forces. Low $Re$ signifies smooth, laminar flow, while high $Re$ suggests chaotic, turbulent flow.
• Taylor Reynolds Number ($Re_{\lambda}$): Characterizes turbulence at intermediate length scales, where vortex stretching is most active.
• Energy Dissipation ($\epsilon$): The rate at which kinetic energy is converted into heat by viscous friction. A fundamental quantity in turbulence theory.
• Kolmogorov Scale ($\eta$): The smallest length scale in a turbulent flow. Eddies smaller than this are quickly damped out by viscosity.
• Integral Scale ($L_{int}$): An estimate of the size of the largest, most energy-containing eddies in the flow.
• Resolution Ratio ($\Delta x / \eta$): Compares the grid cell size ($\Delta x$) to the Kolmogorov scale. If this ratio is greater than 1, the simulation is considered "under-resolved," meaning it may not be capturing the smallest turbulent motions accurately.

The Detailed Analysis Tabs

The panel below the derived quantities is tabbed, allowing you to explore the data from different perspectives:

Distribution Analysis Tab:

These histograms show the statistical distribution of key quantities (Velocity, Vorticity, Pressure, etc.) throughout the entire 3D domain at the final simulation time. Long "tails" in the vorticity distribution, for example, are a hallmark of turbulence, indicating the presence of rare but extremely intense vortex structures.

Energy Spectrum Tab:

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

• Energy-Containing Range (low $k$): The largest scales where energy is introduced into the flow.
• Inertial Range (mid $k$): Energy cascades from large eddies to smaller ones without much dissipation. Here, developed turbulence follows the famous Kolmogorov -5/3 power law, $ E(k) \propto k^{-5/3} $. The dashboard plots a reference line with this slope. If your spectrum matches this slope, you are observing a key feature of turbulence.
• Dissipation Range (high $k$): At the smallest scales (near $\eta$), viscosity becomes dominant and dissipates the energy. The spectrum falls off very steeply here.

2D & 3D Slice Viewer Tabs:

These powerful interactive tools allow you to "slice" open the 3D data cube to inspect its internal structure. You can select the physical variable (Vorticity, Velocity, Pressure), the axis to slice along (X, Y, Z), and the specific slice index using the shared controls.

• 2D Slice Viewer: Shows a traditional 2D heatmap and numerical grid of the selected slice. Click any point to open a detailed analysis modal for that specific location.
• 3D Slice Viewer: Provides two new perspectives. The left view is a full 3D volumetric rendering of the selected variable, allowing you to see the entire structure. The right view is a 3D bar chart of the selected 2D slice, where bar height and color represent the value. Both are fully rotatable.
• Multi-Pixel Analysis (in 2D View): Check the "Enable Multi-Pixel Analysis" box. In this mode, you can single-click multiple cells on the numerical grid to add them to a selection. When you are finished, double-click anywhere on the grid. A tooltip will appear showing aggregate statistics for all the points you selected.

×

Legal Disclaimers & Terms of Use

1. Experimental Research Tool

This 3D Fluid Dynamics Simulator (the "Tool") is provided by Quantum Q PTE. LTD. ("Quantum Q") as an experimental research and educational tool, currently in a beta development phase. It is intended solely for use by the scientific, mathematical, and engineering communities for conceptual exploration and educational purposes. The Tool is not designed, certified, or intended for use in any commercial, industrial, clinical, or mission-critical applications.

2. No Warranty

THE TOOL IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE, AND NONINFRINGEMENT. IN NO EVENT SHALL QUANTUM Q, ITS DIRECTORS, EMPLOYEES, OR AFFILIATES BE LIABLE FOR ANY CLAIM, DAMAGES, OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT, OR OTHERWISE, ARISING FROM, OUT OF, OR IN CONNECTION WITH THE TOOL OR THE USE OR OTHER DEALINGS IN THE TOOL.

3. Limitation of Liability & Risk Assumption

You expressly understand and agree that Quantum Q, its directors, and its employees shall not be liable for any direct, indirect, incidental, special, consequential, or exemplary damages, including but not limited to, damages for loss of profits, goodwill, use, data, or other intangible losses (even if Quantum Q has been advised of the possibility of such damages), resulting from the use or the inability to use the Tool. The entire risk as to the quality, performance, and accuracy of the Tool is with you. This includes, but is not limited to, any damages or injury caused by any failure of performance, error, omission, interruption, defect, or bug within the software, particularly given its acknowledged experimental and beta status. Should the Tool prove defective, you assume the entire cost of all necessary servicing, repair, or correction, and you agree to indemnify and hold harmless Quantum Q and its employees from any and all claims arising from your use of the Tool.

4. No Guarantee of Accuracy

While efforts have been made to ensure the mathematical and numerical integrity of the simulation, Quantum Q makes no guarantee as to the accuracy, reliability, or completeness of the results generated by the Tool. The outputs are for illustrative and conceptual purposes only and should not be used for peer-reviewed publications, engineering design, or any other formal application without independent verification and validation. As stated prominently within the Tool, this simulator is an *analogue* and is not a direct simulation of the 3D Navier-Stokes equations.

5. Intellectual Property

All rights, title, and interest in and to the Tool, including its source code, design, and branding, are the exclusive property of Quantum Q PTE. LTD. You may not copy, modify, distribute, sell, or lease any part of our Tool or its included software, nor may you reverse engineer or attempt to extract the source code of that software, unless laws prohibit those restrictions or you have our written permission.

6. Governing Law & Jurisdiction

These terms shall be governed by and construed in accordance with the laws of the Republic of Singapore, without regard to its conflict of law provisions. Any disputes arising out of or in connection with these terms shall be subject to the exclusive jurisdiction of the courts of the Republic of Singapore.

7. International Applicability and English Law

While the primary governing law is that of the Republic of Singapore, these terms are intended to be broadly applicable to users globally. For users in jurisdictions where English law is applicable or preferred, these terms shall be interpreted in a manner consistent with the general principles of English contract law, to the extent that such interpretation does not conflict with the mandatory laws of Singapore. The parties acknowledge and agree that the principles of common law and equity, as applied in English law, may be considered in interpreting these terms where Singapore law is silent or where such principles provide additional clarity, provided always that Singapore law remains the ultimate governing law.

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