- #1

- 42

- 0

Two different scenarios

Scenario 1.A pipe has pressure [itex]p_1[/itex] , its flow is velocity [itex]v_1[/itex] and a diameter [itex]a_1[/itex] ,it has no restriction along its length.

Total Pressure [itex]Tp_1 [/itex] is therefore

[itex]Tp_1 = p_1 + \frac{1}{2}mv^2[/itex]

Scenario 2. A pipe pressure [itex]p_1[/itex], initial velocity of [itex]v_1[/itex] ,initially has a diameter [itex]10a_1[/itex], it is then restricted to the same diameter [itex]a_1[/itex] as in scenario 1 . According to Bernoullis its velocity thru this restriction increases to [itex]10v_1[/itex], and its pressure drops to [itex]p_2 = Tp - \frac{1}{2}m.10v^2[/itex]

Total Pressure [itex]Tp_1 [/itex] is therefore,

Before constriction

[itex]Tp_1 = p_1 + \frac{1}{2}m.v^2[/itex]

within constriction

[itex]Tp_1 = p_2 + \frac{1}{2}m.10v^2[/itex]

[itex]p_2 = Tp_1 - \frac{1}{2}m.10v^2[/itex]

In both examples Total Pressure [itex]Tp_1 [/itex] or Stagnation Pressure are equivalent .

I agree that, over time, the pressure will normalise to the same value but how can this be correct instantaneously ? Wouldn't scenario 2 exert an initially higher stagnation pressure (Than in Scenario 1 and Total pressure) against the obstruction and take fractionally longer to normalise ?

Reasoning:

Energy Mass have been conserved by transferring some scalar random particle velocity (hydrostatic pressure) to vector kinetic energy (Dynamic Pressure)

In the second scenario the kinetic energy of the first particles initially hitting the obstruction is 10x times greater due to their velocity in one degree of freedom along the streamline, whereas the hydrostatic pressure reduction must be shared amongst all degrees of freedom at this point and thus wouldn't fully compensate

I liken it to Surge Pressure/Water Hammer in a tube but the explanations I have seen explain this by way of the mass/energy of all of the particles 'piling up' behind the obstruction explaining the shockwave but I am thinking the increase in velocity will also contribute a initial pressure spike above static pressure .