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Which of these statements about a steady subsonic flow are correct or incorrect?

1. The static pressure increases as the streamlines diverge.
2. The velocity decreases as the streamlines diverge.

  • A
    1 is incorrect, 2 is incorrect.
  • B
    1 is correct, 2 is incorrect.
  • C
    1 is correct, 2 is correct.
  • D
    1 is incorrect, 2 is correct.

Refer to figure.
To understand the venturi effect, we must first introduce two conditions: The conservation of mass (Continuity Equation) and the Conservation of Energy (Bernoulli’s Theorem)

Conservation of Mass (Continuity Equation)

The mass of air entering the pipe, in each unit of time, equals the mass of air leaving the pipe, in the same unit of time. The mass flow through the pipe must remain constant.

The mass air flow 'm' is the product of the density, the area and the velocity:

m = ρ * A * V

Continuity equation:

m1 = m2 = m3 = ρ1 * A1 * V1 = ρ2 * A2 * V2 = ρ3 * A3 * V3 = constant

The continuity equation explains the relationship between velocity and cross-sectional area: The speed is inversely proportional to the cross-sectional area.

Conservation of Energy (Bernoulli’s Theorem)

The total energy of an airstream comes in two forms: potential energy (static pressure) and kinetic energy (dynamic pressure).
The total pressure of the airstream is the sum of the static pressure and the dynamic pressure and must remain constant, according to the law of conservation of energy.
Thus, an increase in one form of pressure, must result in a decrease in the other.

Dynamic pressure (q) is expressed by:

q = 1 / 2 * ρ * V2

Assuming the total pressure to be constant, the static pressure and the dynamic pressure vary inversely: If one increases the other decreases and vice-versa.
As an airfoil (wing) moves through the air, varying changes in velocity occur on its surfaces. Air particles move around it in specific trajectories. These trajectories are called streamlines.
Diverging streamlines mean a decrease in airspeed.
Converging streamlines imply an increase in airspeed.

In Conclusion:

With an increase in section area (from station 2 to station 3), flow speed decreases, decreasing dynamic pressure (streamlines diverge). If dynamic pressure decreases, static pressure increases. Statement 1 is therefore correct.
With an increase in section area (from station 2 to station 3), flow speed decreases, decreasing dynamic pressure (streamlines diverge). Statement 2 is therefore correct.

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