For turbulent flow in circular pipes, which statement about Prandtl’s mixing length is correct with respect to its variation near the wall and across the radius?

Introduction / Context:
Prandtl’s mixing length hypothesis models turbulent momentum exchange by analogy with molecular diffusion, introducing a characteristic length scale l that represents the average distance over which eddies transport momentum before losing identity. Understanding how l varies in wall-bounded flows like pipes is essential for velocity profile modeling and shear stress estimation.

Given Data / Assumptions:

• Flow: steady, fully developed turbulent flow in a smooth circular pipe.
• We consider the qualitative behavior of mixing length l across the radius.
• Wall effects dominate near the boundary; core flow differs.

Concept / Approach:

In wall-bounded turbulence, mixing length increases with distance from the wall: l ≈ κ * y in the log-law region, where y is the wall-normal distance and κ is the von Kármán constant (~0.4). At the wall itself, fluctuations are suppressed and l → 0. Therefore, l is not a universal constant and certainly not independent of radius. While shear stress informs velocity gradients, the canonical variation of l is tied to position, not fixed by a single shear value.

Step-by-Step Solution:

Verification / Alternative check:

Empirical velocity profiles (log-law) and eddy viscosity models (νt = l^2 |du/dy|) both require l = 0 at the wall to avoid non-physical finite turbulent shear right at the boundary.

Why Other Options Are Wrong:

• Universal constant/independent of radius: Contradicts position dependence (l ∝ y in near-wall region).
• Independent of shear stress: Eddy viscosity depends on gradients; l is coupled to flow structure.
• Maximum at wall: Opposite of physical reality; turbulence production is suppressed at the wall.

Common Pitfalls:

• Assuming l is constant across the section; that leads to unrealistic profiles.
• Ignoring viscous sublayer where turbulence is damped and l remains small.

Final Answer:

zero at the pipe wall