Head loss due to sudden expansion — key assumption in derivation When deriving the expression for head loss in a sudden expansion of a pipe using continuity and the impulse–momentum equation, which additional assumption is made about the separated eddying fluid in the expansion zone?

Introduction / Context:
In a sudden expansion, the core jet issues into the larger pipe and separates, forming a turbulent recirculation (eddy) zone near the step. The classical derivation of head loss uses a control volume spanning just upstream and just downstream of the expansion, invoking continuity and the momentum equation. An additional pressure assumption is required to close the formulation.

Given Data / Assumptions:

• Steady, incompressible flow.
• Sections chosen sufficiently far from the step so pressures are uniform across the sections.
• A separated eddy region occupies part of the sudden expansion corner.

Concept / Approach:
The derivation assumes that the average pressure acting over the recirculating eddy region is equal to the static pressure in the fully developed downstream section. This allows the momentum balance to be written using known uniform pressures over all control surfaces. The resulting head loss expression is h_e = (V1 − V2)^2 / (2g), where V1 and V2 are mean velocities in the smaller and larger pipes.

Step-by-Step Solution:
Apply continuity: A1 V1 = A2 V2 (for incompressible steady flow).Write momentum balance between sections including pressure forces.Use the assumption p_eddy ≈ p2 to express forces over the separated region.Rearrange to obtain head loss h_e = (V1 − V2)^2 / (2g).

Verification / Alternative check:
Energy equation with a loss term between the same sections, together with the derived momentum relation, yields the same loss magnitude, validating consistency.

Why Other Options Are Wrong:
Equating eddy pressure to upstream pressure overestimates forces; neglecting eddy losses is contrary to observed dissipation; asserting frictional equality is unfounded; uniform velocity across separated sections is unrealistic at the step.

Common Pitfalls:

• Applying Bernoulli across separated regions without accounting for losses.
• Taking pressure in the separation zone as unknown, which prevents closure without the stated assumption.

Final Answer:
The mean pressure within the eddying (recirculating) region equals the downstream (expanded section) pressure