Are there cases for incompressible flow where density varies by location, such as in multiphase flow?

That's an astute observation. For example, in buoyancy-driven flows due to temperature differences (using the Boussinesq approximation), density variation is considered only in the buoyancy term, and the continuity equation $\nabla\cdot\mathbf{u}=0$ is still used. However, when dealing with water and air using the VOF (Volume of Fluid) method, density changes discontinuously, so the compressible form of the continuity equation must be used.

Continuity Equation (Mass Conservation)

Q: Professor, the continuity equation often appears alongside the Navier-Stokes equations. What does it represent?

It is the differential form of the law of mass conservation. It's the mathematical expression of the physics that 'fluid does not appear from nowhere or disappear into nowhere.' It is the foundation for all of CFD.

Q: For incompressible flow, density ρ is constant, right?

That's correct. Since Dρ/Dt = 0, the continuity equation becomes very simple. $$ \nabla \cdot \mathbf{u} = 0 $$ In component form, $$ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0 $$ This means the velocity field is 'divergence-free.' It is also called the solenoidal condition.

Category: 流体解析（CFD） | Integrated 2026-04-06

CAE visualization for continuity equation theory - technical simulation diagram

連続の式（質量保存）

Theory and Physics

Overview

🧑‍🎓

Teacher, the continuity equation often appears alongside the Navier-Stokes equations. What does it represent?

🎓

It's the differential form of the law of conservation of mass. It's the mathematical expression of the physics that "fluid does not appear out of nowhere or disappear into nowhere." It's the foundation of all CFD.

General Form (Compressible Flow)

🎓

The continuity equation for any fluid is as follows.

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

Expanding it gives,

$$ \frac{\partial \rho}{\partial t} + \frac{\partial (\rho u)}{\partial x} + \frac{\partial (\rho v)}{\partial y} + \frac{\partial (\rho w)}{\partial z} = 0 $$

🧑‍🎓

How should each term be interpreted?

🎓

$\partial\rho/\partial t$ is the rate of density change within a control volume. $\nabla\cdot(\rho\mathbf{u})$ is the net outflow rate of mass flux through the control volume surface. Their sum being zero means mass is conserved.

Material Derivative Form

🎓

Using the material derivative $D/Dt$, it can be written in another form.

$$ \frac{D\rho}{Dt} + \rho \nabla \cdot \mathbf{u} = 0 $$

This means "the rate of density change observed while riding along with a fluid particle is proportional to the divergence of the velocity field."

Case of Incompressible Flow

🧑‍🎓

For incompressible flow, $\rho$ is constant, right?

🎓

Exactly. Since $D\rho/Dt = 0$, the continuity equation becomes very simple.

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

In component form,

$$ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0 $$

This means "the velocity field is divergence-free." It is also called the solenoidal condition.

🧑‍🎓

Are there cases for incompressible flow where density differs by location (like multiphase flow)?

🎓

Sharp observation. For example, in buoyancy-driven flow due to temperature differences (Boussinesq approximation), density variation is considered only in the buoyancy term, and the continuity equation remains $\nabla\cdot\mathbf{u}=0$. However, when handling water and air with the VOF (Volume of Fluid) method, density changes discontinuously, so the compressible form of the continuity equation must be used.

Integral Form

🎓

The integral form, which is the basis of the finite volume method, is also important. For a control volume $V$,

$$ \frac{d}{dt}\int_V \rho\,dV + \oint_S \rho\mathbf{u}\cdot\mathbf{n}\,dS = 0 $$

In FVM, this integral form is discretized for each cell. The core of the SIMPLE algorithm is adjusting the pressure field so that the mass flux balance through cell faces becomes zero.

Coffee Break Trivia

When an Artery Clogs, Flow Velocity Becomes 16 Times Faster

The scariest everyday example directly linked to the continuity equation (mass conservation) is arterial stenosis. When the cross-sectional area is halved, flow velocity doubles, but if the diameter is halved, the area becomes one-fourth, so velocity quadruples. So what happens if atherosclerosis reduces the diameter by half? Velocity becomes 4 times, and since wall shear stress is proportional to the square of velocity, it becomes over 16 times. If such stress is applied to the wall of an artery with plaque buildup—well, it's going to detach, right? The continuity equation may seem like a modest formula in textbooks, but cardiac surgeons use it every day.

Physical Meaning of Each Term

Temporal Term $\partial(\rho\phi)/\partial t$: Think of the moment you turn on a faucet. At first, water comes out in an unstable, spluttering manner, but after a while, the flow stabilizes, right? This "period of change" is described by the temporal term. The pulsation of blood flow from a heartbeat, the fluctuation of flow each time an engine valve opens and closes—all are unsteady phenomena. So what is steady-state analysis? Looking only at "after sufficient time has passed and the flow has settled down"—meaning setting this term to zero. Since computational cost drops significantly, solving first with a steady-state assumption is a basic CFD strategy.
Convection Term $\nabla \cdot (\rho \mathbf{u} \phi)$: What happens if you drop a leaf into a river? It gets carried downstream by the flow, right? This is "convection"—the effect where fluid motion transports things. Warm air from a heater reaching the far corner of a room is also because the "carrier," air, transports heat via convection. Here's the interesting part—this term contains "velocity × velocity," making it nonlinear. That is, as flow becomes faster, this term rapidly strengthens, making control difficult. This is the root cause of turbulence. A common misconception: "Convection and conduction are similar things" → They're completely different! Convection is carried by flow, conduction is transmitted by molecules. There's an order of magnitude difference in efficiency.
Diffusion Term $\nabla \cdot (\Gamma \nabla \phi)$: Have you ever put milk in coffee and left it? Even without stirring, after a while it naturally mixes, right? That's molecular diffusion. Now a question—honey and water, which flows more easily? Obviously water, right? Honey has high viscosity ($\mu$), so it flows poorly. When viscosity is large, the diffusion term becomes strong, and the fluid moves in a "thick, sluggish" manner. In low Reynolds number flow (slow, viscous), diffusion dominates. Conversely, in high Re number flow, convection overwhelms, and diffusion plays a supporting role.
Pressure Term $-\nabla p$: When you push the plunger of a syringe, liquid shoots out forcefully from the needle tip, right? Why? Because the plunger side is high pressure, the needle tip is low pressure—this pressure difference becomes the force pushing the fluid. Dam discharge works on the same principle. On a weather map, where isobars are tightly packed? That's right, strong winds blow. "Where there is a pressure difference, flow is generated"—this is the physical meaning of the pressure term in the Navier-Stokes equations. A point of confusion here: "Pressure" in CFD is often gauge pressure, not absolute pressure. If results go wrong immediately after switching to compressible analysis, it might be due to mixing up absolute/gauge pressure.
Source Term $S_\phi$: Warmed air rises—why? Because it becomes lighter (lower density) than its surroundings, so it's pushed up by buoyancy. This buoyancy is added to the equation as a source term. Other examples: chemical reaction heat generated by a gas stove flame, Lorentz force applied to molten metal by an electromagnetic pump in a factory... These are all actions that "inject energy or force into the fluid from the outside," expressed by the source term. What happens if you forget the source term? In natural convection analysis, forgetting to include buoyancy means the fluid doesn't move at all—a physically impossible result where warm air doesn't rise in a room with the heater on in winter.

Assumptions and Applicability Limits

Continuum Assumption: Valid for Knudsen number Kn < 0.01 (mean free path of molecules ≪ characteristic length)
Newtonian Fluid Assumption: Shear stress and strain rate have a linear relationship (non-Newtonian fluids require viscosity models)
Incompressibility Assumption (for Ma < 0.3): Treat density as constant. For Mach number ≥ 0.3, compressibility effects must be considered.
Boussinesq Approximation (Natural Convection): Consider density variation only in the buoyancy term, using constant density for other terms.
Non-applicable Cases: Rarefied gas (Kn > 0.1), supersonic/hypersonic flow (requires shock capturing), free surface flow (requires VOF/Level Set, etc.)

Dimensional Analysis and Unit Systems

Variable	SI Unit	Notes / Conversion Memo
Velocity $u$	m/s	When converting from volumetric flow rate for inlet conditions, pay attention to cross-sectional area units.
Pressure $p$	Pa	Distinguish between gauge pressure and absolute pressure. Use absolute pressure for compressible analysis.
Density $\rho$	kg/m³	Air: approx. 1.225 kg/m³ @20°C, Water: approx. 998 kg/m³ @20°C
Viscosity Coefficient $\mu$	Pa·s	Be careful not to confuse with kinematic viscosity coefficient $\nu = \mu/\rho$ [m²/s]
Reynolds Number $Re$	Dimensionless	$Re = \rho u L / \mu$. Indicator for laminar/turbulent transition.
CFL Number	Dimensionless	$CFL = u \Delta t / \Delta x$. Directly related to timestep stability.

Numerical Methods and Implementation

Role of Pressure in Incompressible Flow

🧑‍🎓

For incompressible flow, pressure isn't in the continuity equation, right? How is pressure determined?

🎓

A question that gets to the core. In the incompressible NS equations, since density is known, the continuity equation $\nabla\cdot\mathbf{u}=0$ acts as a constraint to determine pressure, replacing the equation of state. Pressure can be interpreted as a "Lagrange multiplier to keep the velocity field divergence-free."

Pressure-Velocity Coupling Methods

🎓

Let's organize the representative algorithms.

Method	Overview	Application
SIMPLE	Pressure Correction Method. Provisional velocity → pressure correction → velocity correction	Steady / Unsteady
SIMPLEC	Convergence-accelerated version of SIMPLE. No need for correction under-relaxation.	Steady problems
PISO	Two pressure corrections within one step. Suitable for unsteady problems.	Unsteady problems
Fractional Step Method	Velocity prediction → pressure Poisson → velocity projection	Unsteady / Academic codes
Coupled	Solves velocity and pressure simultaneously	When fast convergence is needed

🧑‍🎓

I feel like SIMPLE is the one I see most often.

🎓

The pressure correction equation in SIMPLE is a Poisson-type equation derived from the continuity equation.

$$ \nabla \cdot \left(\frac{1}{a_P}\nabla p'\right) = \nabla \cdot \mathbf{u}^* $$

Here $p'$ is the pressure correction value, $\mathbf{u}^$ is the provisional velocity (velocity obtained tentatively from the momentum equation), $a_P$ is the diagonal coefficient of the momentum equation. It corrects the residual $\nabla\cdot\mathbf{u}^ \neq 0$ using $p'$.

Verification of Mass Conservation

🧑‍🎓

How can I check if mass conservation is satisfied in the calculation results?

🎓

Check the balance of mass flow rates across each face.

$$ \sum_{\text{inlets}} \dot{m}_{\text{in}} - \sum_{\text{outlets}} \dot{m}_{\text{out}} \approx 0 $$

In Fluent, you can check the flow rate at each boundary face via Report > Fluxes > Mass Flow Rate. An imbalance below 0.1% of the total flow rate is considered sufficiently converged.

Rhie-Chow Interpolation

🎓

On collocated grids (where velocity and pressure are placed at the same location), checkerboard instability occurs. Rhie-Chow interpolation is a technique that adds a pressure gradient correction to the cell face velocity to resolve this issue, and is standardly adopted in modern finite volume method solvers.

🧑‍🎓

The continuity equation seems simple at first glance, but implementing it numerically requires considerable ingenuity, doesn't it?

🎓

Yes. Most of the difficulty in incompressible CFD boils down to the problem of "how to satisfy the continuity equation," i.e., the pressure-velocity coupling problem.

Coffee Break Trivia

All Flow Meters Operate on the Continuity Equation

The mechanism of differential pressure flow meters (orifice flow meters) used in factory piping is the continuity equation itself. By reducing the cross-sectional area at the constriction to increase flow velocity, pressure drops according to Bernoulli's theorem. Measuring the pressure difference before and after allows calculating the flow rate inversely. Aircraft airspeed indicators (pitot tubes) work on the same principle. In numerical analysis too, it's fundamental to check after calculation whether "inlet flow rate = outlet flow rate" holds. If the continuity equation is not satisfied, there is either a leak somewhere in the mesh or a contradiction in the boundary conditions.

Upwind Differencing (Upwind)

First-order upwind: Large numerical diffusion but stable. Second-order upwind: Improved accuracy but risk of oscillations. Essential for high Reynolds number flows.

Central Differencing

Second-order accurate, but numerical oscillations occur for Pe number > 2. Suitable for low Reynolds number diffusion-dominated flows.

TVD Schemes (MUSCL, QUICK, etc.)

Maintain high accuracy while suppressing numerical oscillations via limiter functions. Effective for capturing shocks and steep gradients.

Finite Volume Method vs Finite Element Method

FVM: Naturally satisfies conservation laws. Mainstream in CFD. FEM: Advantageous for complex shapes and multiphysics. Mesh-free methods like SPH are also developing.

CFL Condition (Courant Number)

Explicit methods: CFL ≤ 1 is the stability condition. Implicit methods: Stable even for CFL > 1, but affects accuracy and iteration count. LES: CFL ≈ 1 is recommended. Physical meaning: Information should not travel more than one cell per timestep.

Residual Monitoring

Convergence is judged when residuals for continuity equation, momentum, and energy drop by 3-4 orders of magnitude. The mass conservation residual is particularly important.

Relaxation Factors

Pressure: 0.2~0.3, Velocity: 0.5~0.7 are typical initial values. If diverging, lower the relaxation factors. After convergence, increase to accelerate.

Internal Iterations for Unsteady Calculations

Within each timestep, iterate until a steady solution converges. Internal iteration count: 5~20 iterations is a guideline. If residuals fluctuate between timesteps, review the timestep size.

Analogy for the SIMPLE Method

The SIMPLE method is an "alternating adjustment" technique. First, velocity is provisionally determined (prediction