# Pitot tube, assumption of hydrostatic pressure distribution

We have water flowing in an open channel. A small tube is placed in the channel, and the water raises to a height "l" above the water surface. The distance from the water surface to point 1/2 (points are at same height) is d. At point 1 the fluid velocity is V1 and at point 2 it is zero (stagnation point). Calculate the water velocity V1. (Figure below for help) First I calculate the stagnation pressure Ps, by using Bernoulli from 1 to 2. This yields:

(1/2)*V1^2 + P1/rho = 0 + P2/rho

Ps = rho*(1/2)*V1^2 + P1

Then I calculate the pressure through the tube, where we have hydrostatic conditions. P0 is the atmospheric pressure.:

Ps = P0 + rhogl + rhogd.

My question is:

under which conditions can we assume that P1 = rhogd, i.e. under which conditions can we assume that the pressure at point 1 is independent of the fluid flow at that point? Is it only when the fluid flow is ONLY horizontal?

## 1 Answer

Here are the Euler (differential force balance) equations for steady, incompressible flow of an inviscid fluid:

$$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}=-\frac{1}{\rho}\frac{\partial p}{\partial x}$$

$$u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+w\frac{\partial v}{\partial z}=-\frac{1}{\rho}\frac{\partial p}{\partial y}$$

$$u\frac{\partial w}{\partial x}+v\frac{\partial w}{\partial y}+w\frac{\partial w}{\partial z}=-\frac{1}{\rho}\frac{\partial p}{\partial z}-g$$ where u is the (horizontal) velocity component in the x direction, v is the (horizontal) velocity component in the y direction, and w is the (vertical) velocity component in the z direction. What do these equations tell you about the answers to your questions?

• I guess every term on the left side in those equations turn to zero apart from the first term in the first equation, as we only have velocity in the x-direction? – Jmei May 13 '18 at 7:53
• That doesn't answer your questions. What would these equations reduce to and what would they tell you if (a) w were zero everywhere (b) w were constant everywhere, with u and v were independent of z? – Chet Miller May 13 '18 at 12:12