The response of a turbulent boundary layer to different shaped transverse grooves

The response of a turbulent boundary layer to three different shaped transverse grooves was investigated at two values of momentum thickness Reynolds numbers ( R =1000 and 3000). A 20-mm wide square, semicircular and triangular groove with depth to width ( d / w) ratio of unity was used. In general, the effects of the grooves are more significant at the higher R , with the most pronounced effects caused by the square groove. An increase in wall shear stress _w was observed just downstream of the groove for all three shapes. The increase in _w is followed by a small decrease in _w below the smooth-wall value before it relaxes back to the corresponding smooth-wall value at x / ₀3. At the higher R , the maximum increase in _w for the square groove is about 50% higher than for the semicircular groove and almost twice that for the triangular groove. The effect of the square groove on U / U ₀, u / U ₀ and v / U ₀ is much more significant than the effect of the semicircular and triangular grooves. There is an increase in the bursting frequency ( f _B⁺) on the grooved-wall compared to the smooth-wall case. The distribution of f _B⁺ downstream of the different shaped grooves is similar to the _w distribution.Symbols C _f skin friction coefficient, C _f2 _w/( ( U ₀)²) - C _f,0 skin friction coefficient on the smooth wall - d groove depth - D _h diameter of the idealized primary eddy inside the groove - D _h,s diameter of the idealized secondary eddies inside the groove - d _i internal layer thickness - E turbulent energy spectrum - f _B bursting frequency - f _B⁺ normalized bursting frequency, f _B⁺ f _B/( u )² - k wave number, k =2f/ U - q _i ⁺ contributing quadrant to the total Reynolds stress – uv , q _i ⁺ – uv _i /( u )², i =1, 2, 3, 4 - R Reynolds number based on , R U ₀ / - R Reynolds number based on , R U ₀ / - U mean velocity in the streamwise direction - U ₀ free stream velocity - U ⁺ normalized U by inner variable, U ⁺ U / u - u root-mean-square of velocity fluctuation in the streamwise direction - u ⁺ normalized u by inner variable, u ⁺ u / u - u friction velocity, u ( _w/ )^0.5 - – uv Reynolds stress - v root-mean-square of velocity fluctuation in the wall-normal direction - w groove width - x streamwise coordinate measured from the groove trailing edge - y wall-normal coordinate - y ⁺ normalized y by inner variables, y ⁺ yu / Greek symbols boundary layer thickness - ₀ boundary layer thickness just upstream of the groove, unless otherwise stated - fluid kinematic viscosity - momentum thickness - fluid density - _w wall shear stress