| Summary: | Using a direct numerical simulation (DNS), we investigate the onset of non-equilibrium effects and the subsequent emergence of a self-preserving state as a turbulent boundary layer (TBL) encounters a smooth-to-rough (STR) step change. The rough surface comprises over 2500 staggered cuboid-shaped elements where the first row is placed at <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>50</mn><mo> </mo><msub><mi>θ</mi><mn>0</mn></msub></mrow></semantics></math></inline-formula> from the inflow. A <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><msub><mi>e</mi><mi>θ</mi></msub><mo>=</mo><mn>4500</mn><mo> </mo></mrow></semantics></math></inline-formula> value is attained along with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfrac><mi>δ</mi><mi>k</mi></mfrac><mo>≈</mo><mn>35</mn></mrow></semantics></math></inline-formula> as the TBL develops. While different flow parameters adjust at dissimilar rates that further depend on the vertical distance from the surface and perhaps on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>δ</mi><mrow><mi>S</mi><mi>T</mi><mi>R</mi></mrow></msub><mo>/</mo><mi>k</mi></mrow></semantics></math></inline-formula>, an equilibrium for wall stress, mean velocity, and Reynolds stresses exists across the entire TBL by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>35</mn><mo> </mo><msub><mi>δ</mi><mrow><mi>S</mi><mi>T</mi><mi>R</mi></mrow></msub></mrow></semantics></math></inline-formula> after the step change. First-order statistics inside the inner layer adapt much earlier, i.e., at <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>10</mn></mrow></semantics></math></inline-formula>–<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>15</mn><mo> </mo><msub><mi>δ</mi><mrow><mi>S</mi><mi>T</mi><mi>R</mi></mrow></msub></mrow></semantics></math></inline-formula> after the step change. Like rough-to-smooth (RTS) scenarios, an equilibrium layer develops from the surface. Unlike RTS transitions, a nascent logarithmic layer is identifiable much earlier, at <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>4</mn><mo> </mo><msub><mi>δ</mi><mrow><mi>S</mi><mi>T</mi><mi>R</mi></mrow></msub></mrow></semantics></math></inline-formula> after the step change. The notion of equivalent sandgrain roughness does not apply upstream of this fetch because non-equilibrium advection effects permeate into the inner layer. The emergent equilibrium TBL is categorized by a fully rough state (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>k</mi><mi>s</mi><mo>+</mo></msubsup><mo>≈</mo><mn>120</mn></mrow></semantics></math></inline-formula>–<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>130</mn></mrow></semantics></math></inline-formula>; <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>k</mi><mi>s</mi></msub><mo>/</mo><mi>k</mi><mo>≈</mo><mn>2.8</mn></mrow></semantics></math></inline-formula>). Decomposition of wall stress into constituent parts reveals no streamwise dependence. Mean velocity in the outer layer is well approximated by Coles’ wake law. The wake parameter and shape factor are enhanced above their smooth-wall counterparts. Quadrant analysis shows that shear-stress-producing motions adjust promptly to the roughness, and the balance between ejections and sweeps in the outer layer remains impervious to the underlying surface.
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