Navier Stokes Equation Solver Calculator

Calculator Inputs

The form uses a responsive three-column layout on large screens, two columns on smaller screens, and one column on mobile devices.

Flow Model Choose a practical boundary-condition pattern.

Grid Points in X Higher values improve detail and increase runtime.

Grid Points in Y Use balanced grid sizes for stable comparisons.

Time Steps More steps may improve developed flow behavior.

Pressure Iterations Used in the pressure Poisson correction loop.

Time Step Size Lower values typically increase stability.

Density Fluid density used in momentum and pressure terms.

Kinematic Viscosity Higher viscosity usually smooths velocity gradients.

Domain Length X Physical size of the x-direction domain.

Domain Length Y Physical size of the y-direction domain.

Body Force X Useful for gravity-like or acceleration forcing terms.

Body Force Y Adds vertical forcing to the momentum equations.

Top Boundary Velocity Used mainly for cavity and Couette style motion.

Inlet Pressure Level Primarily used in channel-flow mode.

Pressure Relaxation Controls iterative pressure update damping.

Residual Tolerance Smaller tolerance asks for tighter convergence.

Example Data Table

These sample values show a practical setup for a lid-driven cavity style estimate.

Parameter	Example Value	Meaning
Flow Model	Lid-Driven Cavity	Moving top wall, closed box
Grid Points X	21	Horizontal discretization count
Grid Points Y	21	Vertical discretization count
Time Steps	80	Solution advancement count
Pressure Iterations	50	Poisson correction loop count
Density	1.0	Fluid density
Kinematic Viscosity	0.10	Momentum diffusion strength
Time Step	0.001	Explicit update size
Lid Velocity	1.0	Top wall speed
Residual Tolerance	0.00001	Pressure convergence target

Formula Used

1) Continuity Equation

For incompressible flow, the divergence of velocity is zero:

∂u/∂x + ∂v/∂y = 0

2) X-Momentum Equation

∂u/∂t + u∂u/∂x + v∂u/∂y = -(1/ρ)∂p/∂x + ν(∂²u/∂x² + ∂²u/∂y²) + Fx

3) Y-Momentum Equation

∂v/∂t + u∂v/∂x + v∂v/∂y = -(1/ρ)∂p/∂y + ν(∂²v/∂x² + ∂²v/∂y²) + Fy

4) Pressure Poisson Equation

∇²p = b

The source term b is built from local velocity gradients to enforce incompressibility during iteration.

This calculator applies central finite differences for spatial derivatives and forward stepping in time. It is a practical numerical solver for educational and planning use, not a certified CFD package.

How to Use This Calculator

Select the flow model that best matches your physical setup.
Enter grid sizes, time steps, and pressure-iteration count.
Provide density, viscosity, time step, and domain dimensions.
Set lid velocity, pressure level, and body forces if needed.
Choose pressure relaxation and residual tolerance.
Press Solve Equation to compute the field values.
Review the results section above the form immediately after submission.
Inspect the centerline output table and the Plotly heatmap.
Export results with the CSV or PDF download buttons.

Frequently Asked Questions

1) What does this calculator solve?

It solves a practical 2D incompressible flow approximation using iterative finite differences for velocity and pressure fields under selected boundary conditions.

2) Is this a full computational fluid dynamics package?

No. It is an educational and planning-oriented solver that demonstrates Navier Stokes behavior numerically, but it is not a replacement for industrial CFD software.

3) Why do grid points matter?

More grid points capture gradients better and usually improve field detail. They also increase runtime and may require smaller time steps for stability.

4) What is the Reynolds number estimate used for?

It gives a quick sense of flow regime behavior by relating inertia to viscous effects. Higher values often indicate stronger convective influence.

5) What does residual tolerance mean?

Residual tolerance controls how tightly the pressure correction loop must settle before the solver treats the iterative pressure update as sufficiently converged.

6) Why does time step size affect stability?

A large time step can make explicit numerical updates unstable. Smaller steps often improve stability, especially when velocity gradients are strong.

7) Can I use negative body forces?

Yes. Negative values can represent acceleration or forcing in the opposite coordinate direction, depending on your chosen physical interpretation.

8) What does the chart display?

The chart shows a heatmap of speed magnitude across the grid. Higher values indicate stronger local flow velocity within the computed domain.