# Experimental study on drag-reduction effect due to sinusoidal riblets in turbulent channel flow

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## Abstract

The drag-reduction effect of a three-dimensional sinusoidal riblet surface is experimentally evaluated in a fully developed turbulent channel flow. The lateral spacing of the adjacent walls of the riblet is varied sinusoidally in the streamwise direction. The obtained maximum total drag-reduction rate is approximately 12 % at a bulk Reynolds number of 3,400. The flow structure over the sinusoidal riblet surface is also analyzed in the velocity field by using two-dimensional particle image velocimetry. The velocity field is compared with the corresponding flow over a flat surface. It is found through pathlines and Reynolds shear stress analyses that the drag-reduction mechanism is similar to those of two-dimensional riblets. A different point is that the present riblet respectively induces a downward and upward flows in the expanded and contracted regions, which prevent vortices from hitting the bottom wall with wider lateral spacing of the riblet. In consequence, the wetted area of the present sinusoidal riblet is smaller than those of two-dimensional riblets, resulting in the high drag-reduction effect.

### List of symbols

- \(x\)
Position in the streamwise direction (m)

- \(y\)
Position in the wall-normal direction (m)

- \(z\)
Position in the spanwise direction (m)

- \(u\)
Streamwise velocity (m/s)

- \(v\)
Wall-normal velocity (m/s)

- \(w\)
Spanwise velocity (m/s)

- \(\delta\)
Channel half-width (m)

- \({\hbox{d}}p\)
Differential pressure (Pa)

- \(\nu\)
Kinematic viscosity (\(\hbox {m}^2/\hbox {s}\))

- \(\rho\)
Density (\(\hbox {kg/m}^3\))

- \(Q\)
Mass flow rate (\(\hbox {m}^3/\hbox {s}\))

- \(L_z\)
Channel spanwise width (m)

- \(C_{\mathrm{T}}\)
Total drag coefficient

- \(D_{\mathrm{P}}\)
Pressure drag on the riblet (Pa)

- \(R_{\mathrm{D}}\)
Drag-reduction rate (%)

- \(l_t\)
Riblet thickness (m)

- \(h\)
Height of riblet wall (m)

- \(l\)
Distance between pressure taps (m)

- \(L_p\)
Distance of the center position between pressure taps from the beginning of the test section (m)

- \(l_{x}\)
Streamwise length of one cycle of the riblet (m)

- \(l_{z}\)
Lateral spacing of the riblet (m)

- \(S\)
Wetted area (\(\hbox {m}^2\))

- \(u_{b}\)
bulk velocity (m/s) (=\(\frac{1}{2\delta }\int ^{2\delta }_0 \overline{u(y)}{\hbox{d}}y\))

- \(\mu\)
Viscosity (\(\hbox {Pa}\,\hbox {s}\))

- \(\tau _{w}\)
Wall shear stress (\(\hbox {N/m}^2\)) (=\(\mu {\hbox{d}}\overline{u}/{\hbox{d}}y_{\mathrm{wall}}\))

- \(u_{\tau }\)
Friction velocity (\(\hbox {m/s}^2\)) (\(= \sqrt{\tau _{w, \,{\mathrm{flat}}}/\rho }\))

- \(Re_{\tau }\)
Friction Reynolds number (–) (=\(u_{\tau , \,{\mathrm{flat}}} \delta / \nu\))

- \(Re_{b}\)
Bulk Reynolds number (–) (=\(u_b 2\delta / \nu\))

- \(B_{i}\)
Event probability of quadrant

- \(\omega _{z}\)
Spanwise vorticity (1/s)

- \(t\)
Time (s)

- \(T\)
Measurement time (s)

- \(()^{+}\)
Non-dimensionalization by \(u_{\tau , \,{\mathrm{flat}}}\) and \(\nu\) (wall-unit)

- \(()^\prime\)
Fluctuation from the spatiotemporal average

- \(()^{\prime \prime }\)
Fluctuation from time average (random component)

- \(( ),_{\, {\mathrm{flat}}}\)
Experimental result for flat-flat case (both side walls are flat surface)

- \(( ),_{\, {\mathrm{rib}}}\)
Experimental result for flat-riblet case (riblet boards are installed on a lower wall)

- \(( ),_{\, {\text {2-D}}}\)
Result for a 2-D riblet surface

- \(( ),_{\, {\text {3-D}}}\)
Result for a 3-D riblet surface

- \(()_{i}\)
Direction or quadrant

- \(\overline{( )}\)
Average over time

- \(\langle \rangle\)
Average over space

- \(\widetilde{( )}\)
Periodic fluctuation

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