Question

Consider two-dimensional turbulent flow of air over a horizontal flat plate of length 1 m. Skin friction coefficient at a length $x$ from the leading edge is given by: \[ c_f = \frac{0.06}{(Re_x)^{0.2}} \] where $Re_x$ is the local Reynolds number. Find the drag force per unit width (in N/m$^2$) on the plate if the free stream velocity is 10 m/s. Density and dynamic viscosity of air are 1.2 kg/m$^3$ and $1.83 \times 10^{-5}$ Ns/m$^2$. (Round off to three decimal places)

Hint:

Turbulent flat-plate drag uses $\tau_w = \tfrac{1}{2} c_f \rho U^2$; friction coefficient depends on $Re_x$ through a power law.

Answer 1

Correct Answer: 0.166

Reynolds number at the trailing edge: \[ Re_L = \frac{\rho U L}{\mu} \] \[ Re_L = \frac{1.2 \times 10 \times 1}{1.83 \times 10^{-5}} = 6.557 \times 10^{5} \] Skin friction coefficient: \[ c_f = \frac{0.06}{(6.557 \times 10^{5})^{0.2}} \] Compute exponent: \[ (6.557 \times 10^5)^{0.2} = 21.69 \] \[ c_f = \frac{0.06}{21.69} = 0.002766 \] Wall shear stress: \[ \tau_w = \frac{1}{2} c_f \rho U^2 \] \[ \tau_w = 0.5 (0.002766)(1.2)(100) \] \[ \tau_w = 0.16596 \approx 0.166~\text{N/m}^2 \] \[ \boxed{0.166~\text{N/m}^2} \]