For a short cylindrical external mouthpiece running free, the vena-contracta forms at a distance from the outlet approximately equal to:

Introduction / Context:
When fluid issues through a short external mouthpiece, the jet contracts to a minimum cross-section called the vena-contracta before expanding downstream. The distance of this section from the outlet helps estimate coefficients and practical layout clearances.

Given Data / Assumptions:

• Short external cylindrical mouthpiece on a large reservoir wall.
• Jet runs free (adequate aeration, atmospheric pressure outside).
• Negligible downstream submergence effects.

Concept / Approach:
Empirical observations and classical hydraulics indicate the vena-contracta for a short external mouthpiece occurs at about one-third of the orifice diameter downstream of the outlet plane, with a contraction coefficient close to unity when the mouthpiece runs full beyond the contraction.

Step-by-Step Solution:
Identify the regime: free jet from short external tube.Recall standard distance: approximately D/3 from the outlet face.Therefore the correct selection is one-third of the diameter.

Verification / Alternative check:
Textbook tabulations of mouthpiece characteristics list the vena-contracta location for short external tubes at roughly 0.33 D from the exit plane, confirming the rule of thumb.

Why Other Options Are Wrong:
One diameter or two-thirds diameter overestimates the distance.One-fourth diameter underestimates the location for this configuration.

Common Pitfalls:

• Confusing locations for sharp-edged orifices versus short tubes.
• Ignoring changes if the mouthpiece is running submerged or with suction effects.

Final Answer:
One-third of the orifice diameter