In the text we derived a formula for the friction factor based on the universal velocity profile. In essence it relates \(\sqrt{2 / C_{f}}=\bar{U}^{+}\). The universal velocity profile has a kink at \(y^{+}=30\) which makes it a bit aphysical. Though we cannot determine a closed-form, analytical solution, use the van Driest formula for \(U^{+}\), equation (14.59), to determine a similar relationship. Plot \(\sqrt{2 / C_{f}}\) vs \(y^{+}\). Hint: The van Driest requires numerical integration. Expanding the denominator in a Taylor series about \(y^{+}=0\) gives: \(f\left(y^{+}\right)=2+\left(\frac{2 \kappa_{T}^{2}}{a^{2}}\right)\left(y^{+}\right)^{4}\).