Influence of backward-facing steps on laminar-turbulent transition in two-dimensional boundary layers at subsonic Mach numbers

Abstract Backward-facing steps (BFS) can have a detrimental impact on laminar flow lengths because of their strong effect on boundary layer transition. BFS with normalized step heights in the range of $$h/\delta _1 \approx$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>h</mml:mi> <mml:mo>/</mml:mo> <mml:msub> <mml:mi>δ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>≈</mml:mo> </mml:mrow> </mml:math> 0.1 to 0.6 (corresponding to height-based Reynolds numbers of $$\hbox{Re}_h = (U_\infty h / \nu ) \approx$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mtext>Re</mml:mtext> <mml:mi>h</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> <mml:mi>h</mml:mi> <mml:mo>/</mml:mo> <mml:mi>ν</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>≈</mml:mo> </mml:mrow> </mml:math> 230 to 2430) were installed in a two-dimensional wind tunnel model and tested in the Cryogenic Ludwieg-Tube Göttingen, a blow-down wind tunnel with good flow quality. The influence of BFS on the location of laminar-turbulent transition was investigated over a wide range of unit Reynolds numbers from $$\hbox{Re}_1 = {17.5\times 10^{6}\,{\text {m}^{-1}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mtext>Re</mml:mtext> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:mrow> <mml:mn>17.5</mml:mn> <mml:mo>×</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mn>6</mml:mn> </mml:msup> <mml:mspace/> <mml:msup> <mml:mtext>m</mml:mtext> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:mrow> </mml:math> to $$80\times 10^{6}\,\hbox{m}^{-1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>80</mml:mn> <mml:mo>×</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mn>6</mml:mn> </mml:msup> <mml:mspace/> <mml:msup> <mml:mtext>m</mml:mtext> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> , three Mach numbers, $$M= 0.35$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0.35</mml:mn> </mml:mrow> </mml:math> , 0.50 and 0.65, and various streamwise pressure gradients. The measurement of the transition locations was accomplished non-intrusively by means of temperature-sensitive paint. Transition Reynolds numbers, calculated with the flow length up to the location of laminar-turbulent transition $$x_{T}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>T</mml:mi> </mml:msub> </mml:math> , ranged from $$\hbox{Re}_{\rm tr}\approx$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mtext>Re</mml:mtext> <mml:mi>tr</mml:mi> </mml:msub> <mml:mo>≈</mml:mo> </mml:mrow> </mml:math> 1 × 10 6 to 11 × 10 6 , and were measured as a function of step height, pressure gradient, Reynolds and Mach numbers. Incompressible linear stability analysis was used to calculate amplification rates of Tollmien–Schlichting waves; transition N -factors were determined by correlation with the measured transition locations. In parallel to earlier investigations with a similar setup, this systematic approach was used to identify functional relations between non-dimensional step parameters ( $$h/\delta _1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>h</mml:mi> <mml:mo>/</mml:mo> <mml:msub> <mml:mi>δ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:math> and $$\hbox{Re}_h$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mtext>Re</mml:mtext> <mml:mi>h</mml:mi> </mml:msub> </mml:math> ) and the relative change of the transition location. Furthermore, the change of the transition N -factor $$\Delta N$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> due to the installation of the steps was investigated. It was found that the installation of backward-facing steps with $$h/\delta _1 \lesssim 0.15$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>h</mml:mi> <mml:mo>/</mml:mo> <mml:msub> <mml:mi>δ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>≲</mml:mo> <mml:mn>0.15</mml:mn> </mml:mrow> </mml:math> and $$\hbox{Re}_h \lesssim 300$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mtext>Re</mml:mtext> <mml:mi>h</mml:mi> </mml:msub> <mml:mo>≲</mml:mo> <mml:mn>300</mml:mn> </mml:mrow> </mml:math> does not lead to a reduction of $$\hbox{Re}_{\rm tr}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mtext>Re</mml:mtext> <mml:mi>tr</mml:mi> </mml:m