Find the ratio of friction drag on the front half and rear half of the flat plate kept at zero incidence in a stream of uniform velocity if the boundary layer is turbulent over the whole plate.

Concept:

For the turbulent boundary layer, the average drag coefficient CD is prescribed by the relation:

\(C_D = {0.072\over {R_e^{0.2}}}\)

Drag force for flat plate(F_D):

\(={C_D} \times \frac{1}{2}\rho A{V^2}\)

\( = \frac{{0.072}}{{ {(R{e})^{0.2}} }} \times \frac{1}{2}\rho A{V^2}\)

\( = \frac{{0.072}}{{ ({{\frac{{\rho LV}}{\mu }})^{0.2}} }} \times \frac{1}{2} \times \rho \times W \times L \times {V^2}\)

i.e. \(F_D \propto{ L\over L^{0.2}}\)

Now Drag force on the front half

\(F_{Dfh} \propto{ {L\over 2}\over ({L\over 2})^{0.2}}\)

\(F_{Dfh} \propto{{1\over 2} \times 2^{0.2}}{ L\over L^{0.2}}\)

Drag force on the rear half

\(F_{DRh} = {F_D} - F_{Dfh}\)

\(F_{DRh} \propto{({1-{1\over 2} \times 2^{0.2}})}{ L\over L^{0.2}}\)

Now, the ratio of friction drag on the front half and rear half of the flat plate

\({F_{Dfh} \over F_{DRh}} ={ ({{1\over 2} \times 2^{0.2})}{ L\over L^{0.2}} \over {({1-{1\over 2} \times 2^{0.2}})}{ L\over L^{0.2}}}= {0.574 \over ({1-0.574})}\)= 1.347

Hence, the ratio of friction drag on the front half and rear half of the flat plate is 1.347.