Question

The velocity distribution in the boundary layer for the turbulent flow over the plate generally follows the law

• A. parabolic
• B. hyperbolic
• C. linear
• D. logarithmic

Hint:

Distinguish between laminar and turbulent boundary layer velocity profiles. Laminar boundary layers often follow a parabolic (or cubic for Blasius) profile, while turbulent boundary layers are characterized by a logarithmic velocity profile (the "log law of the wall"). This difference is due to the dominant effect of turbulent mixing in the latter.

Answer 1

The Correct Option is D

Step 1: Understand boundary layers and different flow regimes.
Boundary Layer: A thin layer of fluid near a solid surface where the fluid velocity changes from zero at the surface (due to the no-slip condition) to the free-stream velocity away from the surface.
Flow Regimes:
Laminar Flow: Characterized by smooth, orderly fluid motion in layers. In a laminar boundary layer over a flat plate, the velocity profile is typically parabolic (e.g., Blasius solution).
Turbulent Flow: Characterized by chaotic, irregular, and unsteady fluid motion with significant mixing. Turbulent boundary layers are much thicker than laminar ones for the same Reynolds number and exhibit a different velocity profile.
Step 2: Recall the velocity distribution law for turbulent boundary layers.
For turbulent flow over a flat plate, the velocity distribution within the boundary layer is generally described by a logarithmic law, often referred to as the "log law of the wall" or the "logarithmic velocity profile".
This law states that the mean velocity \(u\) at a distance \(y\) from the wall is proportional to the logarithm of \(y\). The log law is typically expressed as:
\[ \frac{u}{u_} = \frac{1}{\kappa} \ln\left(\frac{y u_}{\nu}\right) + B \] where:
\(u\) is the mean velocity at distance \(y\) from the wall.
\(u_\) is the friction velocity (\(u_ = \sqrt{\tau_w/\rho}\), where \(\tau_w\) is wall shear stress and \(\rho\) is fluid density).
\( \kappa \) is the von Kármán constant (approximately 0.41).
\( \nu \) is the kinematic viscosity.
\(B\) is a constant (approximately 5.0 for a smooth wall).
This logarithmic profile accurately describes the velocity distribution in the inner region of the turbulent boundary layer (the "overlap layer") and is characteristic of turbulent flows. While there's also a viscous sublayer near the wall (linear profile) and an outer defect law, the "logarithmic law" is the most general and defining characteristic of turbulent boundary layer velocity distribution.
Step 3: Evaluate the given options.
(1) parabolic: This describes the velocity profile for laminar flow in pipes or ducts, or approximately for laminar boundary layers.
(2) hyperbolic: This is not a standard description for velocity profiles in boundary layers.
(3) linear: This describes the velocity profile very close to the wall within the viscous sublayer of both laminar and turbulent flows, but not the overall turbulent boundary layer. (4) logarithmic: This correctly describes the general velocity distribution in the turbulent boundary layer. Therefore, the velocity distribution in the boundary layer for turbulent flow over a plate generally follows the logarithmic law. The final answer is \( \boxed{\text{logarithmic}} \).