I heard that for incompressible flow, the continuity equation becomes div(u) = 0. What does that mean exactly?

div(u)=0 means the divergence of the velocity field is zero—volumetric flow in equals volumetric flow out for any small element. In compressible flow, the term ∂ρ/∂t (density change) appears, but in incompressible flow where ρ is constant, this term vanishes, leaving only the elegant velocity constraint div(u)=0.

In CFD for incompressible flow, I often hear about 'pressure-based solvers' and 'SIMPLE method'. Why is pressure such a big issue?

In incompressible flow, ρ is constant, so we cannot use the compressible approach p=ρRT to determine pressure. Pressure remains an independent variable that must be solved to satisfy div(u)=0. SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) is an iterative algorithm handling this pressure-velocity coupling. It is the foundation of nearly all incompressible solvers in OpenFOAM, Fluent, and similar tools.

Incompressible Flow — CAE Glossary

Q: Ma = 0.3 is about 100 m/s in air, right? So can air flows also be treated as incompressible in some cases?

Exactly. Vehicle aerodynamics at highway speeds (100 km/h ≈ Ma 0.08) or wind loads on buildings are all Ma 0.3) show significant density changes requiring compressible analysis.

Q: Is Boussinesq approximation an extension of incompressible flow theory? I've heard it's used for natural convection.

Yes. The Boussinesq approximation accounts for density variations only in the buoyancy term, treating everything else as incompressible. Specifically, ρ≈ρ₀(1−β(T−T₀)) enters only the buoyancy force. This is used for simulating warm air rising in a heated room. It breaks down if ΔT/T₀ becomes large, but for small temperature differences it is an excellent approximation.

Category: Glossary | 2026-03-28

CAE visualization for incompressible flow - technical simulation diagram

What is Incompressible Flow

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Does incompressible flow mean the fluid cannot be compressed? Since water cannot be compressed, does all water flow fall into this category?

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Strictly speaking, no fluid is truly incompressible. Even water experiences density changes at extreme depths—by about 5% at 10,000 meters. However, in engineering practice, if density changes are small enough to neglect, we treat the flow as incompressible. The criterion is Mach number: if $\mathrm{Ma} < 0.3$, it's acceptable. Most everyday water flows have $\mathrm{Ma} \ll 0.3$, so treating them as incompressible is fine.

Ma < 0.3 Criterion

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$\mathrm{Ma} = 0.3$ is about 100 m/s in air, right? So can air flows also be treated as incompressible in some cases?

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Exactly. Vehicle aerodynamics is a classic example—at highway speed (100 km/h), $\mathrm{Ma} \approx 0.08$. Building wind loads are the same way. These don't need compressible analysis. Conversely, jet engine flows or supersonic aircraft experience $\mathrm{Ma} > 0.3$ where density changes become significant, requiring compressible computation.

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Why 0.3? Why not 0.5 or 0.1?

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It comes from isentropic relations. At $\mathrm{Ma} = 0.3$, density change is about 4.5%, which aligns with the engineering rule "neglect changes below 5%." Density variation enters at $\mathrm{Ma}^2$ order, so effects drop rapidly as Ma decreases. Specifically:

$$\frac{\rho}{\rho_0} = \left(1 + \frac{\gamma-1}{2}\mathrm{Ma}^2\right)^{-\frac{1}{\gamma-1}}$$

For $\gamma = 1.4$ (air) with $\mathrm{Ma} = 0.3$, we get $\rho/\rho_0 \approx 0.956$, so a 4.4% density change. That's the basis for the "0.3 criterion."

Continuity Equation div(u) = 0

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I heard that for incompressible flow, the continuity equation becomes $\nabla \cdot \mathbf{u} = 0$. What does that mean exactly?

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The general continuity equation expresses mass conservation:

$$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0$$

Under incompressibility ($\rho = \mathrm{const.}$), we can factor out $\rho$:

$$\rho \, \nabla \cdot \mathbf{u} = 0 \quad \Rightarrow \quad \nabla \cdot \mathbf{u} = 0$$

This means "the volumetric flow entering any small element equals the flow leaving"—like water flowing into a bucket through one hole and out through another at equal rates. When you pinch a hose nozzle, the water speed increases because the reduced cross-section must satisfy $\nabla \cdot \mathbf{u} = 0$.

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Does the equation become more complex in compressible flow?

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In compressible flow, $\rho$ varies in time and space, so the $\partial \rho / \partial t$ term remains. Additionally, the state equation $p = \rho R T$ couples in the energy equation. The incompressible simplification to $\nabla \cdot \mathbf{u} = 0$ drastically reduces computational cost in CFD.

Pressure-Based Solver and SIMPLE Method

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In CFD for incompressible flow, I often hear about "pressure-based solver" and "SIMPLE method." Why is pressure such a big issue?

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This is the central numerical challenge of incompressible flow. In compressible flow, the state equation $p = \rho R T$ determines pressure, but in incompressible flow where $\rho = \mathrm{const.}$, this path is unavailable. Pressure becomes an independent variable that must be solved to enforce $\nabla \cdot \mathbf{u} = 0$.

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What's the SIMPLE method algorithm roughly like?

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SIMPLE (Semi-Implicit Method for Pressure-Linked Equations), proposed by Patankar in 1972, follows this cycle:

Solve momentum equation with guessed pressure $p^*$ → obtain tentative velocity $\mathbf{u}^*$
Compute divergence of velocity field → $\nabla \cdot \mathbf{u}^* \neq 0$ (doesn't satisfy continuity yet)
Solve pressure correction equation (Poisson-type) → get correction $p'$
Update velocity and pressure → $p = p^* + p'$, $\mathbf{u} = \mathbf{u}^* + \mathbf{u}'$
Repeat until convergence

OpenFOAM's simpleFoam, Ansys Fluent's pressure-based solver, and STAR-CCM+'s SEGREGATED solver—all are built on SIMPLE-family algorithms. Variants include SIMPLEC (Consistent) and PISO (for transient).

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Is the pressure correction equation a Poisson-type like $\nabla^2 p' = \text{(source)}$?

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Exactly. When discretized, it becomes a large sparse linear system—solving this dominates SIMPLE's computational time. Tuning the solver (e.g., AMG preconditioning) makes a practical difference. In Fluent, adjusting AMG cycle counts or Courant number visibly affects wall-clock time, a lesson many CFD practitioners know firsthand.

Boussinesq Approximation and Natural Convection

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Is Boussinesq approximation an extension of incompressible flow theory? I've heard it's used for natural convection.

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Boussinesq approximation is: "account for density variation only in the buoyancy term; treat everything else as incompressible." In the momentum equation's buoyancy term, substitute:

$$\rho \approx \rho_0 \bigl[1 - \beta (T - T_0)\bigr]$$

where $\beta$ is the thermal expansion coefficient. This lets you simulate warm air accumulating at ceiling height, or natural convection cooling inside a PC case, without the computational cost of a fully compressible solver.

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When does this approximation break down?

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It needs $\beta \Delta T \ll 1$. For air, $\beta \approx 1/T_0$, so you need $\Delta T / T_0 \ll 1$. At 300 K room temperature, a 30 K difference gives $\Delta T / T_0 = 0.1$—acceptable. But fire simulation with $\Delta T = 700\mathrm{K}$ means $\Delta T / T_0 > 2$: completely invalid. Then you need low-Mach-number or fully compressible solvers.

Practical Decision Points

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In a real CFD project, how do I decide upfront whether to use incompressible or compressible solver?

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Use this checklist:

Calculate Mach number from flow speed and sound speed → if Ma < 0.3, incompressible is fine
Is there temperature variation? → small: add Boussinesq; large: skip
Does density change exceed 5%? → yes: switch to compressible
Are acoustic waves or shock propagation critical? → yes: use density-based solver

Vehicle external aerodynamics: unambiguously incompressible. F1 rear wing (Ma ≈ 0.25): borderline but doable as incompressible. Water hammer in pipes (sound waves matter): requires compressibility. These examples guide decision-making.

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