ANS:A - V²

In turbulent flow in a pipe, the head loss hfh_fhf​ is generally proportional to the square of the velocity VVV of the fluid. Therefore, the correct answer is: V2V^2V2

Explanation:

1. Head Loss in Turbulent Flow:
   • In turbulent flow through a pipe, head loss hfh_fhf​ is caused by frictional effects between the flowing fluid and the pipe wall.
   • The Darcy-Weisbach equation is commonly used to calculate head loss in turbulent flow: hf=f⋅LD⋅V22gh_f = f \cdot \frac{L}{D} \cdot \frac{V^2}{2g}hf​=f⋅DL​⋅2gV2​ where:
     • fff is the Darcy-Weisbach friction factor,
     • LLL is the length of the pipe,
     • DDD is the diameter of the pipe,
     • VVV is the velocity of the fluid,
     • ggg is the acceleration due to gravity.
2. Proportionality to Velocity:
   • From the Darcy-Weisbach equation, it can be seen that hfh_fhf​ is directly proportional to V2V^2V2.
   • This means that doubling the velocity will result in a fourfold increase in head loss due to friction.
3. Other Options:
   • 1V2\frac{1}{V^2}V21​ and 1V\frac{1}{V}V1​ are inversely proportional to velocity and do not represent the correct relationship for head loss in turbulent flow.
   • VVV alone is not the correct proportional term for head loss; it is V2V^2V2 that represents the frictional losses associated with turbulent flow in a pipe.

Conclusion: