Question

For a fully developed turbulent flow of an incompressible Newtonian fluid through a pipe of constant diameter, which of the following statements is/are CORRECT?

• A. Reynolds stress, averaged over a sufficiently long time, is zero everywhere inside the pipe.
• B. Reynolds stress at the pipe wall is zero.
• C. Average velocity of the fluid is half of its center-line velocity.
• D. Average pressure gradient in the flow direction is constant.

Hint:

- At the wall: \(u'=v'=0 \Rightarrow \overline{u'v'}=0\).
- Fully developed flow: velocity profile does not change with \(x\) \(\Rightarrow\) constant \(dp/dx\).
- Mean vs. centerline velocity ratio is \(< 2\) for turbulent pipe flow.

Answer 1

The Correct Option is B, D

Step 1: Reynolds stress distribution. In turbulent pipe flow the time-averaged Reynolds shear stress \(-\rho \overline{u'v'}\) is generally nonzero across the cross-section, vanishing at the pipe centerline and at the wall (due to no-slip and no-penetration implying \(u'=v'=0\) at the wall). Hence (A) is false, while (B) is true. Step 2: Mean vs. centerline velocity. For fully developed turbulent flow, the centerline velocity is typically about \(1.2{-}1.3\) times the mean velocity (not \(2\) times). Therefore statement (C) is false. Step 3: Streamwise pressure gradient. In fully developed pipe flow (laminar or turbulent), the wall shear stress and thus the axial pressure gradient are uniform along \(x\); hence the average pressure gradient is constant. So (D) is correct.