By considering the component of the Euler equations (the equations of motion for inviscid fluids) along a streamline. Integrating along a streamline for steady flow yields Bernoulli's equation.

Bernoulli's theorem

Q: Professor, I often hear about Bernoulli's theorem. Why is it so important?

It's the principle of energy conservation for steady, inviscid, incompressible flow. It's one of the most fundamental and powerful tools in fluid mechanics. Because it directly relates flow velocity and pressure, it's used everywhere—from piping design to Pitot tubes and Venturi meters.

Q: What does each term represent?

The first term, $p$, is the static pressure; the second term, $\frac{1}{2}\rho v^2$, is the dynamic pressure; and the third term, $\rho gz$, is the hydrostatic pressure. Their sum is called the total pressure, $p_0$. $$ p_0 = p + \frac{1}{2}\rho v^2 + \rho g z $$

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

CAE visualization for bernoulli equation theory - technical simulation diagram

ベルヌーイの定理

Theory and Physics

Overview

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Professor, I often hear about Bernoulli's theorem, but why is it so important?

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It's the law of energy conservation for steady, inviscid, incompressible flow. It's one of the most fundamental and powerful tools in fluid mechanics. Because it directly links flow velocity and pressure, it's used everywhere from piping design to pitot tubes and Venturi tubes.

Derivation of the Governing Equation

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How is it derived?

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Consider the component along a streamline of the Euler equations (the equations of motion for inviscid fluid). Integrating along a streamline for steady flow yields Bernoulli's equation.

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The Euler equations are as follows.

$$ \rho\frac{D\mathbf{u}}{Dt} = -\nabla p + \rho\mathbf{g} $$

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Integrating for steady flow along a streamline gives Bernoulli's equation.

$$ p + \frac{1}{2}\rho v^2 + \rho g z = \text{const (along a streamline)} $$

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What does each term represent?

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The first term $p$ is the static pressure, the second term $\frac{1}{2}\rho v^2$ is the dynamic pressure, and the third term $\rho gz$ is the hydrostatic pressure. Their sum is called the total pressure $p_0$.

$$ p_0 = p + \frac{1}{2}\rho v^2 + \rho g z $$

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It can also be written in head form, right?

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Exactly. Dividing both sides by $\rho g$ gives the head form.

$$ \frac{p}{\rho g} + \frac{v^2}{2g} + z = H = \text{const} $$

Here, $\frac{p}{\rho g}$ is the pressure head, $\frac{v^2}{2g}$ is the velocity head, $z$ is the elevation head, and $H$ is the total head.

Application Conditions and Limitations

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Can it be used for any flow?

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No, there are strict application conditions.

Condition	Meaning	If Violated
Steady Flow	$\partial/\partial t = 0$	Unsteady Bernoulli equation needed
Inviscid	$\mu = 0$ (no friction)	Add head loss term
Incompressible	$\rho = \text{const}$	Compressible Bernoulli equation needed
Same Streamline	Energy conservation is per streamline	Holds throughout space for irrotational flow

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For flows with viscous losses, use the extended form.

$$ \frac{p_1}{\rho g} + \frac{v_1^2}{2g} + z_1 = \frac{p_2}{\rho g} + \frac{v_2^2}{2g} + z_2 + h_L $$

Here, $h_L$ is the head loss (pressure loss due to friction, valves, bends, etc.).

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I see, so in practice, the key is accurately estimating the head loss.

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Exactly. In piping design, it's standard practice to combine it with the Darcy-Weisbach equation $h_L = f \frac{L}{D}\frac{v^2}{2g}$.

Coffee Break Trivia

The Textbook Explanation for "Why Wings Generate Lift" is Probably Wrong

"Because the top is longer, air flows faster and pressure drops"—this is a common explanation for wing lift in middle school science and introductory books, but it has a major flaw. The premise that "air on the top and bottom must reach the trailing edge at the same time" actually has no basis. In reality, the "equal transit time" of upper and lower air does not occur; the flow velocity on the upper surface is significantly faster than that premise suggests. The true lift is explained by circulation theory. Bernoulli's theorem is correct as a tool stating "faster flow means lower pressure," but textbooks often skip explaining why it becomes faster.

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 erratically and unstably, but after a while, the flow becomes steady, right? This "period of change" is described by the temporal 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? Looking only at "after sufficient time has passed and the flow has settled down"—in other words, setting this term to zero. This significantly reduces computational cost, so solving first for steady state is a basic CFD strategy.
Convection Term $\nabla \cdot (\rho \mathbf{u} \phi)$: If you drop a leaf into a river, what happens? It's 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" → 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. 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. Higher viscosity strengthens the diffusion term, making the fluid move "sluggishly." 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 a syringe plunger, 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 provides the force pushing the fluid. Dam discharge works on the same principle. On a weather map, where isobars are tightly packed? 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. When switching to compressible analysis, if results become strange, it might be due to mixing up absolute/gauge pressure.
Source Term $S_\phi$: Heated 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 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 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 (mean free path ≪ 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 above 0.3, consider compressibility effects
Boussinesq Approximation (Natural Convection): Consider density variation only in the buoyancy term, use constant density in other terms
Non-applicable Cases: Rarefied gas (Kn > 0.1), supersonic/hypersonic flow (shock capturing required), free surface flow (VOF/Level Set, etc. required)

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 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 $\nu = \mu/\rho$ [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

Numerical Application of Bernoulli's Equation

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How is Bernoulli's equation used in CFD? You don't solve it directly, right?

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Good question. In CFD, the basic approach is to solve the Navier-Stokes equations, so you don't directly discretize and solve Bernoulli's equation. However, Bernoulli's principle plays a core role in setting boundary conditions, verifying results, and 1D network analysis.

Application to Boundary Conditions

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How is it used for boundary conditions?

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For example, when setting a pressure inlet and pressure outlet, the relationship between total pressure and static pressure is needed.

$$ p_0 = p_{\text{static}} + \frac{1}{2}\rho |\mathbf{u}|^2 $$

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When setting a pressure-inlet in Ansys Fluent, you specify the Total Pressure, but internally it uses this equation to maintain consistency with the velocity field. Be careful not to confuse total and static pressure, as it can cause significant deviation from the expected flow rate.

1D Network Analysis

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Are there cases where 1D analysis suffices instead of 3D CFD?

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For preliminary studies of piping networks, 1D analysis using Bernoulli's equation plus loss terms is very effective. It solves simultaneous equations: pressure balance at nodes and flow balance in branches.

$$ p_i - p_j = \frac{1}{2}\rho v_{ij}^2 \left(f\frac{L}{D} + \sum K\right) $$

Here, $K$ is the local loss coefficient (valves, elbows, etc.), and $f$ is the Darcy friction factor. Piping analysis tools like Flownex or AFT Fathom implement this.

Use in Verification

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Can it also be used to verify CFD results?

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Of course. Comparing CFD results for a Venturi tube or contraction section with Bernoulli's theoretical values is a basic verification method.

Verification Item	Bernoulli Theoretical Value	Comparison Point with CFD Results
Velocity increase at contraction	$v_2 = v_1 \sqrt{A_1/A_2}$ (using continuity equation)	Compare centerline velocity
Pressure drop at contraction	$\Delta p = \frac{1}{2}\rho(v_2^2 - v_1^2)$	Compare cross-sectional average pressure
Total pressure from pitot tube	$p_0 = p + \frac{1}{2}\rho v^2$	Compare with stagnation pressure

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If the Reynolds number is sufficiently high (around $Re > 10^4$) and there is no flow separation, the difference from the theoretical value should be within a few percent. If it deviates significantly, there is likely a problem with the mesh or boundary conditions.

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So Bernoulli's theorem is ideal for a "sanity check" of CFD results.

Coffee Break Trivia

What Bernoulli Can and Cannot Design

The conditions for using Bernoulli's theorem are four: "inviscid, steady, incompressible, same streamline." In practice, a common pitfall is forgetting the "same streamline" condition. For example, when calculating the pressure difference across a pump using Bernoulli, you must check if the inlet and outlet streamlines are the same, or you'll get it wrong. In piping design, the standard is to use the extended form of Bernoulli with "head loss" added. By adding friction loss, bend loss, sudden expansion loss, etc., as loss coefficients, viscous effects can be treated approximately. It's originally an equation for "ideal fluid," but engineers cleverly extend its use.

Upwind Scheme

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 Peclet 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 recommended. Physical meaning: Information should not travel more than one cell per timestep.

Residual Monitoring

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

Relaxation Factor

Typical initial values: Pressure: 0.2–0.3, Velocity: 0.5–0.7. If diverging, lower the relaxation factor. After convergence, increase to accelerate.

Internal Iterations for Unsteady Calculations

Iterate within each timestep 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 tentatively calculated (predictor step), then pressure is corrected so that mass conservation is satisfied with that velocity (corrector step), and then velocity is revised using the corrected pressure—this back-and-forth is repeated to approach the correct solution. It resembles two people leveling a shelf: one adjusts the height, the other balances it, and they repeat this alternately.