- #1
Timtam
- 40
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I have trouble understanding why Stagnation Pressure equals Total Pressure when we apply Bernoullis theorem to a constriction.
Two different scenarios
Scenario 1.A pipe has pressure p_1 , its flow is velocity v_1 and a diameter a_1 ,it has no restriction along its length.
Total Pressure Tp_1 is therefore
Tp_1 = p_1 + \frac{1}{2}mv^2Scenario 2. A pipe pressure p_1, initial velocity of v_1 ,initially has a diameter 10a_1, it is then restricted to the same diameter a_1 as in scenario 1 . According to Bernoullis its velocity thru this restriction increases to 10v_1, and its pressure drops to p_2 = Tp - \frac{1}{2}m.10v^2
Total Pressure Tp_1 is therefore,
Before constriction
Tp_1 = p_1 + \frac{1}{2}m.v^2
within constriction
Tp_1 = p_2 + \frac{1}{2}m.10v^2
p_2 = Tp_1 - \frac{1}{2}m.10v^2In both examples Total Pressure Tp_1 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 .
Two different scenarios
Scenario 1.A pipe has pressure p_1 , its flow is velocity v_1 and a diameter a_1 ,it has no restriction along its length.
Total Pressure Tp_1 is therefore
Tp_1 = p_1 + \frac{1}{2}mv^2Scenario 2. A pipe pressure p_1, initial velocity of v_1 ,initially has a diameter 10a_1, it is then restricted to the same diameter a_1 as in scenario 1 . According to Bernoullis its velocity thru this restriction increases to 10v_1, and its pressure drops to p_2 = Tp - \frac{1}{2}m.10v^2
Total Pressure Tp_1 is therefore,
Before constriction
Tp_1 = p_1 + \frac{1}{2}m.v^2
within constriction
Tp_1 = p_2 + \frac{1}{2}m.10v^2
p_2 = Tp_1 - \frac{1}{2}m.10v^2In both examples Total Pressure Tp_1 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 .