Wind and Air: Flow Through a Narrowing

Combine continuity with a carefully limited Bernoulli model.

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For steady constant-density flow with no leaks, volume flow Q = Av is constant. For steady inviscid incompressible flow along a streamline with no energy added, p + rho v^2/2 + rho gz is constant. State those conditions before linking higher speed to lower static pressure.

Worked example

Areas 2 m^2 and 1 m^2 with speeds 3 m/s and 6 m/s both give Q = 6 m^3/s. At the same height and rho = 1 kg/m^3, the ideal pressure drop from the slower to faster section is (6^2 - 3^2)/2 = 13.5 Pa.

A steady constant-density flow has area 2 square meters and speed 3 meters per second in one section, and area 1 square meter and speed 6 meters per second in the next. Both carry 6 cubic meters per second.

Would you believe it?

Invisible air presses on surfaces in many directions, not only downward. A difference in pressure on opposite sides of a surface can produce a net push.

NASA: air pressure

Question 1 What volume flow enters the 2 m^2 section at 3 m/s? 6 m^3/s 1.5 m^3/s 9 m^3/s

Question 2 At the same volume flow, speed in the 1 m^2 section is what? 3 m/s 12 m/s 6 m/s

Question 3 Using the stated ideal conditions, p_slow - p_fast equals what? 27 Pa 13.5 Pa 4.5 Pa

Question 4 Why is faster flow always means lower pressure an unsafe general rule? Energy input, height changes and losses can matter Pressure and speed cannot coexist All flows have the same density

Question 5 In a compressible steady flow, what remains constant across sections without leaks? A v regardless of density Pressure at every point Mass flow rho A v

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