SpecializedSource checked

Navier-Stokes Projection

Makes a fluid velocity field incompressible by subtracting pressure gradients.

Explore relationships Open constellationBrowse atlas

Quick Facts

Compact metadata for implementation and discovery.

Category: Simulation
Difficulty: Advanced
Family: Grid Fluid Simulation
Tier: Specialized
Status: Verified
Review: Source Checked
Sources: 1
Formulas: 1
Relationships: 4
Code example: Ready
Visualization: Ready

Overview

The projection step solves for pressure so velocity divergence is removed, a core piece of many grid-based fluid solvers.

Core Concepts

velocity divergencepressure solvePoisson equationincompressibility

Quick facts

Complexity

Runtime and memory behavior at a glance.

Best

O(cells)

Average

O(iterations * cells)

Worst

O(iterations * cells)

Space

O(cells)

When to Use

Use when a grid fluid solver needs incompressible smoke or water-like motion.
Use when smoke simulation is a central requirement.

When Not to Use

Avoid when compressible flow or high-accuracy CFD is required.
Avoid when the timestep, constraints, or stability requirements exceed what the method can handle.

What It Does

Measures velocity divergence
Solves a pressure equation
Subtracts pressure gradient from velocity

Use Cases

Smoke simulation
Water simulation
Grid-based fluid solvers

Inputs

Velocity grid
Cell size
Boundary conditions

Outputs

Divergence-free velocity field
Pressure field

Key Formulas (1)

Projection update

velocity_next = velocity − ∇(pressure)

velocity_next = velocity - gradient(pressure)

Subtracts the pressure gradient to reduce divergence.

Pseudocode

1.Compute divergence of the velocity field
2.Solve the pressure Poisson equation
3.Subtract pressure gradient from velocity
4.Apply boundary conditions
5.Use the projected velocity for the next step

Canonical Example

Input: velocity field has outward divergence in one cell
Output: projected field has near-zero divergence
Explanation: Pressure correction removes the local expansion.

Implementation Notes

Track divergence before and after projection to debug solver quality.
Keep units, timestep size, and state variables explicit in tests.
Keep a tiny example with expected output beside the implementation.

Common Pitfalls

A weak pressure solve leaves visible volume gain or loss.
Large timesteps can add energy, damp motion, or make constraints explode.
Not defining boundary, empty, or degenerate inputs before implementation.

Interactive Mechanism Walkthrough

Iterative Convergence & State Trajectory

Show a velocity grid before and after arrows are adjusted to remove divergence.

Step 1 of 5: Compute divergence of the velocity field

simulation-state

Notes

Boundary conditions strongly affect fluid behavior near solid walls.

Sources (1)

What each reference supports for this entry.

Stam 1999

Paper

Stable Fluids paper describing projection-based incompressible fluid simulation for graphics.

Open source

Explore Next

Good next jumps from the relationship graph, ranked for useful discovery.

Open discovery Open map

Often paired withSemi-Lagrangian Advection

Simulation · Extended

Stable fluid solvers commonly pair semi-Lagrangian advection with projection.

Compare with this

Often paired withVorticity Confinement

Simulation · Specialized

Vorticity confinement can restore swirling detail after projection-based fluid solves.