Question

The velocity distribution in the turbulent boundary layer follows

• A. Straight line law
• B. Parabolic law
• C. Hyperbolic law
• D. Logarithmic law

Hint:

Remember this clear distinction for boundary layer velocity profiles: - Laminar flow regime \(\implies\) Parabolic velocity profile. - Turbulent flow regime \(\implies\) Logarithmic velocity profile (often approximated by the \(1/7\)-th power law).

Answer 1

The Correct Option is D

Concept: When a fluid flows past a solid boundary wall, a boundary layer develops due to viscous shear stresses. Depending on the local Reynolds number, this boundary layer can be structurally classified as either laminar or turbulent. In a laminar boundary layer, fluid particles move in smooth parallel lines, and viscous forces dominate, yielding a classic smooth quadratic profile. In contrast, a turbulent boundary layer is characterized by random fluid motion, intense eddy mixing, and rapid momentum transfer across fluid layers. This changes the distribution of mean velocity across the boundary profile.

Step 1: Analyzing velocity behavior in different flow regimes.
Let us analyze the distinct profile shapes across different regimes to see why they differ:
• Laminar Boundary Layer Laminar Pipe Flow: The velocity distribution follows a quadratic or parabolic profile derived directly from Newton's law of viscosity, expressed generally as \( u(y) \propto y^2 \).
• Turbulent Boundary Layer: Due to continuous eddy fluctuations and intense momentum mixing, the velocity profile flattens significantly across the central core region, while exhibiting a sharp, steep gradient immediately adjacent to the solid surface boundary.

Step 2: Examining the mathematical law for turbulent layers.
Extensive experimental work by Prandtl and von Kármán showed that the mean velocity distribution within the inner region of a fully developed turbulent boundary layer closely obeys the Logarithmic Law of the Wall. This relationship is expressed mathematically as: \[ u^+ = \frac{1}{\kappa} \ln(y^2) + B \] Where the non-dimensionalized parameters are defined as:
• \( u^+ = \frac{u}{u_{\tau}} \) (where \(u\) is mean velocity and \(u_{\tau}\) is the friction velocity).
• \( y^+ = \frac{y \cdot u_{\tau}}{\nu} \) (dimensionless distance from the solid wall boundary).
• \(\kappa\) is the empirical von Kármán constant (\(\approx 0.41\)).
• \(B\) is a constant dependent on surface roughness characteristics.

Step 3: Concluding the velocity profile type.
Because the fluid velocity varies logarithmically with the distance \(y\) away from the bounding wall across a major portion of the turbulent layer, this distribution profile is governed by a logarithmic law.