非圧縮性Navier-Stokes方程式
Theory and Physics
Overview
Professor, the Navier-Stokes equations are the foundation of CFD, right? Could you please explain the incompressible case in detail?
They are the most fundamental and important equations solved in CFD. The incompressible NS equations describe viscous flow of Newtonian fluids. They consist of a set of the continuity equation and the momentum equations.
Governing Equations
The incompressible NS equations are the following two equations.
Continuity Equation (Mass Conservation):
Momentum Equation (Newton's Second Law):
Could you explain the physical meaning of each term?
It's easier to understand when written in component form (for the $x$ direction).
Reynolds Number
How is the Reynolds number related to the NS equations?
When you non-dimensionalize the equations using a characteristic velocity $U$ and characteristic length $L$, the Reynolds number appears as the sole parameter.
$$ \frac{\partial \mathbf{u}^*}{\partial t^*} + (\mathbf{u}^* \cdot \nabla^*)\mathbf{u}^* = -\nabla^* p^* + \frac{1}{Re}\nabla^{*2}\mathbf{u}^* $$
$$ Re = \frac{UL}{\nu} = \frac{\text{Inertial Force}}{\text{Viscous Force}} $$
How is the Reynolds number related to the NS equations?
When you non-dimensionalize the equations using a characteristic velocity $U$ and characteristic length $L$, the Reynolds number appears as the sole parameter.
| Re Range | Flow Characteristics | Typical Examples |
|---|---|---|
| Re < 1 | Creep Flow (Stokes Flow) | Microorganism swimming, MEMS |
| 1 < Re < 2300 | Laminar Flow | Pipe flow (developed flow) |
| 2300 < Re < 4000 | Transition Region | Pipe flow (unstable) |
| Re > 4000 | Turbulent Flow | Most industrial flows |
Mathematical Properties of the Equations
The existence and uniqueness of solutions to the NS equations is an unsolved problem, right?
Exactly. The global existence and uniqueness of smooth solutions for the three-dimensional NS equations is one of the Clay Mathematics Institute's Millennium Prize Problems (with a $1 million prize). From an engineering perspective, we obtain solutions for finite time and finite domains via DNS (Direct Numerical Simulation), but a mathematical proof has not yet been established.
The nonlinearity of the equations originates from the convection term $(\mathbf{u}\cdot\nabla)\mathbf{u}$. This term is the source of complex phenomena such as turbulence, chaos, and vortex breakdown.
The $1 Million Prize Nobody Has Solved
For the incompressible Navier-Stokes equations, it hasn't even been mathematically proven that "a solution must always exist." It's one of the seven "Millennium Problems" selected by the Clay Mathematics Institute in 2000, and solving it earns you $1 million (about 150 million yen). The calculations engineers run daily in CFD are, strictly speaking, "numerically solving equations for which we don't even know if a solution truly exists"—a rather surreal situation. Moreover, of the remaining six problems, only one has been solved so far—Navier-Stokes remains uncharted territory for humanity.
Physical Meaning of Each Term
- Time Term $\partial(\rho\phi)/\partial t$: Think of the moment you turn on a faucet. At first, the water comes out unstable and splashing, but after a while, it becomes a steady flow, right? This "period of change" is described by the time term. The pulsation of blood flow from a heartbeat, or the flow fluctuation each time an engine valve opens and closes—all are unsteady phenomena. So what is steady-state analysis? It looks only at "after sufficient time has passed and the flow has settled down"—meaning setting this term to zero. Since this drastically reduces computational cost, starting with a steady-state solution 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 due to the air, the "carrier," transporting heat via convection. Here's the interesting part—this term contains "velocity × velocity," making it nonlinear. That is, as the flow becomes faster, this term rapidly strengthens, making it difficult to control. This is the root cause of turbulence. A common misconception: "Convection and conduction are similar" → 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" manner. In low Reynolds number flows (slow, viscous), diffusion is dominant. Conversely, in high Re number flows, convection overwhelms, and diffusion plays a supporting role.
- Pressure Term $-\nabla p$: When you push the plunger of a syringe, the liquid shoots out forcefully from the needle tip, right? Why? Because the piston side is high pressure, the needle tip is low pressure—this pressure difference provides the force pushing the fluid. Dam discharge works on the same principle. On a weather map, where are the isobars tightly packed? That's right, strong winds blow. "Flow is generated where there is a pressure difference"—this is the physical meaning of the pressure term in the Navier-Stokes equations. A point of confusion here: The "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 confusing absolute/gauge pressure.
- Source Term $S_\phi$: Warmed air rises—why? Because it becomes lighter (lower density) than its surroundings, so it's pushed upward by buoyancy. This buoyancy is added to the equation as a source term. Other examples: chemical reaction heat from a gas stove flame, Lorentz force acting on molten metal in a factory's electromagnetic pump... These are all actions that "inject energy or force into the fluid from the outside," expressed by source terms. What happens if you forget a 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 heated winter room.
Assumptions and Applicability Limits
- Continuum Assumption: Valid for Knudsen number Kn < 0.01 (molecular mean free path ≪ characteristic length)
- Newtonian Fluid Assumption: Linear relationship between shear stress and strain rate (non-Newtonian fluids require viscosity models)
- Incompressibility Assumption (for Ma < 0.3): Treat density as constant. For Mach numbers above 0.3, compressibility effects must be considered.
- Boussinesq Approximation (Natural Convection): Consider density changes only in the buoyancy term, using constant density in 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, be careful with 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 time step stability. |
Numerical Methods and Implementation
Spatial Discretization
How do you solve the NS equations numerically?
The Finite Volume Method (FVM) is mainstream in commercial CFD. The computational domain is divided into cells, and the integral form of the conservation laws is discretized for each cell.
Convection Term Discretization
The choice of scheme for the convection term directly affects accuracy and stability.
| Scheme | Accuracy | Stability | Numerical Diffusion | Application |
|---|---|---|---|---|
| 1st Order Upwind | $O(h)$ | Very Stable | Large | Initial calculations, when convergence is difficult |
| 2nd Order Upwind | $O(h^2)$ | Stable | Medium | General steady-state calculations |
| QUICK | $O(h^3)$ | Somewhat Unstable | Small | High-precision calculations |
| Central Differencing | $O(h^2)$ | Unstable (high Pe) | None | LES |
| Bounded CD | $O(h^2)$ | Stable | Minimal | Standard for LES |
Why use central differencing in LES?
Because numerical diffusion would artificially dampen the SGS eddies. In LES, physical eddies need to be preserved as much as possible, with dissipation occurring only via the SGS model. However, central differencing is prone to checkerboard instability, so Bounded Central Difference with limiters is practical.
Time Integration
Let's compare time integration schemes for unsteady calculations.
| Scheme | Accuracy | Stability | CFL Constraint |
|---|---|---|---|
| 1st Order Implicit (Backward Euler) | $O(\Delta t)$ | Unconditionally Stable | None |
| 2nd Order Implicit (BDF2) | $O(\Delta t^2)$ | Unconditionally Stable | None |
| Crank-Nicolson | $O(\Delta t^2)$ | Unconditionally Stable | Possible Oscillations |
| Explicit (RK4, etc.) | $O(\Delta t^4)$ | Conditionally Stable | $CFL < 1$ |
CFL Condition
I often hear about the CFL number, but what does it mean exactly?
The Courant-Friedrichs-Lewy number is an indicator of how many cells information travels in one timestep.
$$ CFL = \frac{u\Delta t}{\Delta x} $$
I often hear about the CFL number, but what does it mean exactly?
The Courant-Friedrichs-Lewy number is an indicator of how many cells information travels in one timestep.
For explicit methods, $CFL < 1$ is a necessary condition for stability. Implicit methods have no constraint, but for time accuracy, $CFL < 5$ to 20 is recommended. In LES, it's common to maintain $CFL < 1$.
Linear Solvers
The choice of linear solver to solve the discretized system of equations is also important.
| Target Equation | Recommended Solver | Notes |
|---|---|---|
| Pressure (Poisson Equation) | AMG (Algebraic Multigrid) | Most difficult to converge, accounts for 50-80% of total computation time |
| Momentum | BiCGSTAB with ILU Preconditioning | Relatively easy to converge |
| Scalar (Temperature, etc.) | Gauss-Seidel or ILU | Linear problems |
The Secret Story of SIMPLE's Birth—The "Mind Game" Between Pressure and Velocity
The biggest hurdle for incompressible flow is that "there is no pressure equation." Trying to solve for velocity and pressure simultaneously makes the system of equations break down. In 1972, Patankar and Spalding published the algorithm "SIMPLE" that solved this difficult problem. The idea is an iterative loop: "First assume a pressure, solve for velocity, then correct the pressure from that and recalculate velocity." It seems like a simple concept, but without this algorithm, modern commercial CFD codes would hardly exist. Behind the "Iterations" screen you watch every day in Fluent, the wisdom of two people from half a century ago is at work.
Upwind Scheme (Upwind)
1st Order Upwind: Large numerical diffusion but stable. 2nd Order Upwind: Improved accuracy but risk of oscillations. Essential for high Reynolds number flows.
Central Differencing (Central Differencing)
2nd order accuracy, 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 shock waves and steep gradients.
Finite Volume Method vs Finite Element Method
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