---
title: "Euler equation, Bernoulli and Pitot tube"
subject: "Applied Computational Fluid Dynamics"
chapter: 2
key: "semester_7/applied_computational_fluid_dynamics:euler-equation-pitot-numerical"
source: https://www.bctengineer.com/q/semester_7/applied_computational_fluid_dynamics/euler-equation-pitot-numerical
site: BCT Engineering
---

# Euler equation, Bernoulli and Pitot tube

Applied Computational Fluid Dynamics · Chapter 2: Governing Equations and Numerical Foundations

- Asked in: year not given, 7 marks
- Video: [Understanding Bernoulli's Equation](https://www.youtube.com/watch?v=DW4rItB20h4) — The Efficient Engineer
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/applied_computational_fluid_dynamics/euler-equation-pitot-numerical
- Whole chapter: https://www.bctengineer.com/md/semester_7/applied_computational_fluid_dynamics/chapter-2

## Question

(a) Write the Euler equations of motion for inviscid flow and show that they give Bernoulli's equation along a streamline for steady, incompressible flow. (b) A Pitot-static tube in an air duct ($\rho = 1.2$ kg/m³) shows a pressure difference between the stagnation and static ports of 450 Pa. Find the air velocity.

## Short answer

Euler equations: ρ(∂u/∂t + u·∇u) = −∇p + ρg. For steady incompressible irrotational flow, integration gives p + ½ρV² + ρgz = const. Pitot tube velocity: V = √(2Δp/ρ) = √(2×450/1.2) = 27.4 m/s.

## Answer

#### (a) Euler equations
For an inviscid fluid ($\mu=0$) the Navier-Stokes equation loses the viscous term:
$$
\rho\left(\frac{\partial\vec V}{\partial t}+(\vec V\cdot\nabla)\vec V\right)=-\nabla p+\rho\vec g
$$
In components (x-direction): $\dfrac{\partial u}{\partial t}+u\dfrac{\partial u}{\partial x}+v\dfrac{\partial u}{\partial y}+w\dfrac{\partial u}{\partial z}=-\dfrac1\rho\dfrac{\partial p}{\partial x}+g_x$.

**Bernoulli from Euler:** take steady flow along a streamline of length $ds$. The momentum equation along $s$ becomes
$$
V\frac{dV}{ds}=-\frac1\rho\frac{dp}{ds}-g\frac{dz}{ds}
$$
Multiply by $ds$ and integrate with constant $\rho$:
$$
\int\frac{dp}{\rho}+\int V\,dV+\int g\,dz=\text{const}\ \Rightarrow\ \frac p\rho+\frac{V^2}{2}+gz=\text{constant}
$$
Equivalently $p+\tfrac12\rho V^2+\rho gz=$ const along a streamline. It is valid for steady, incompressible, inviscid flow with no heat or work addition. If the flow is also irrotational, the constant is the same for the whole field.

In CFD, Euler solvers are used for high-speed external flows where viscous effects outside the boundary layer are small, as they are cheaper than full N-S solvers.

#### (b) Pitot tube
At the stagnation port the velocity is zero, so
$$
p_0-p=\tfrac12\rho V^2
$$
$$
V=\sqrt{\frac{2(p_0-p)}{\rho}}=\sqrt{\frac{2\times450}{1.2}}=27.4\ \text{m/s}
$$
(The Mach number is about 0.08, so incompressible treatment is valid.)

**Answer:** $V=27.4$ m/s.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The Euler equations are obtained from Navier-Stokes by:
   A. Removing the pressure term
   B. Removing the viscous term (mu = 0)
   C. Removing the convective term
   D. Removing the body force only
   Answer: B. For inviscid fluid mu = 0, so only inertia, pressure and body force remain. (Key point: Euler equation)

2. Bernoulli's equation along a streamline requires:
   A. Unsteady, compressible, viscous flow
   B. Steady, incompressible, inviscid flow
   C. Rotational flow only
   D. Flow with heat addition
   Answer: B. Integrating Euler along a streamline gives p + 1/2 rho V^2 + rho g z = const for steady, incompressible, inviscid flow. (Key point: Bernoulli conditions)

3. At the stagnation port of a Pitot tube:
   A. Velocity is maximum
   B. Pressure equals static pressure
   C. Velocity is zero and p0 - p = 1/2 rho V^2
   D. Pressure is zero
   Answer: C. Fluid is brought to rest, converting dynamic pressure to stagnation pressure. (Key point: Pitot principle)

4. A pressure difference of 450 Pa in air (1.2 kg/m^3) gives V of:
   A. 19.4 m/s
   B. 375 m/s
   C. 13.7 m/s
   D. 27.4 m/s
   Answer: D. V = sqrt(2 x 450/1.2) = 27.4 m/s. (Key point: Pitot velocity)

## Related in Applied Computational Fluid Dynamics

- [Streamline and acceleration of a flow field](https://www.bctengineer.com/question.md?key=semester_7%2Fapplied_computational_fluid_dynamics%3Astreamline-acceleration-numerical) — Applied Computational Fluid Dynamics, chapter 1
- [Integral continuity for a compressing duct](https://www.bctengineer.com/question.md?key=semester_7%2Fapplied_computational_fluid_dynamics%3Aintegral-continuity-duct-numerical) — Applied Computational Fluid Dynamics, chapter 2
- [Plane Poiseuille flow from Navier-Stokes](https://www.bctengineer.com/question.md?key=semester_7%2Fapplied_computational_fluid_dynamics%3Aplane-poiseuille-flow-numerical) — Applied Computational Fluid Dynamics, chapter 2
- [Streamline, pathline and streakline differentiated](https://www.bctengineer.com/question.md?key=semester_7%2Fapplied_computational_fluid_dynamics%3Astreamline-pathline-streakline) — Applied Computational Fluid Dynamics, chapter 1
- [Derive differential continuity equation](https://www.bctengineer.com/question.md?key=semester_7%2Fapplied_computational_fluid_dynamics%3Acontinuity-equation-differential-form) — Applied Computational Fluid Dynamics, chapter 2
- [RANS equations and the closure problem](https://www.bctengineer.com/question.md?key=semester_7%2Fapplied_computational_fluid_dynamics%3Areynolds-averaging-closure-problem) — Applied Computational Fluid Dynamics, chapter 5

## Related in other subjects

- [Air velocity in a duct by Pitot-static tube](https://www.bctengineer.com/question.md?key=semester_4%2Ffluid_mechanics_mechanical%3Apitot-static-air-duct-numerical) — Fluid Mechanics, chapter 7
- [Deriving Bernoulli's equation from Euler's equation](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Aeuler-bernoulli-derivation) — Hydro Power, chapter 1
- [Bernoulli from Euler with head losses](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Aeuler-to-bernoulli-with-losses) — Fluid Mechanics, chapter 5
- [Air jet on curved vane with pitot](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Apitot-air-jet-curved-vane-force) — Fluid Mechanics, chapter 7