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) enter image description here

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?


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?

  • $\begingroup$ 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? $\endgroup$ – Jmei May 13 '18 at 7:53
  • $\begingroup$ 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? $\endgroup$ – Chet Miller May 13 '18 at 12:12

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