Compressibility effects on drag reduction by spanwise traveling transversal surface waves in turbulent boundary layers

The effect of compressibility on drag reduction (DR) by spanwise traveling transversal surface waves (STTSW) is investigated in turbulent boundary layer (TBL) flow over a flat plate. Wall-resolved simulations are performed at Mach numbers <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:msub><a:mi>M</a:mi><a:mn>1</a:mn></a:msub><a:mo>=</a:mo><a:mn>0.2</a:mn></a:mrow></a:math> and <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:msub><b:mi>M</b:mi><b:mn>2</b:mn></b:msub><b:mo>=</b:mo><b:mn>0.7</b:mn></b:mrow></b:math> over a range of actuation parameters, i.e., wavelength, period, and amplitude. The results based on 36 cases simulated for each Mach number show that DR is more pronounced at higher Mach numbers with a maximum reduction of 19.2% at <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:msub><c:mi>M</c:mi><c:mn>1</c:mn></c:msub><c:mo>=</c:mo><c:mn>0.2</c:mn></c:mrow></c:math> compared to 27.5% at <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:msub><d:mi>M</d:mi><d:mn>2</d:mn></d:msub><d:mo>=</d:mo><d:mn>0.7</d:mn></d:mrow></d:math> for identical actuation parameters. This enhanced DR at <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mrow><e:msub><e:mi>M</e:mi><e:mn>2</e:mn></e:msub><e:mo>=</e:mo><e:mn>0.7</e:mn></e:mrow></e:math> is mainly due to the formation of spanwise shock waves at higher phase velocities. These spanwise shock waves are generated by the actuated wall deformation that redistributes the pressure field. Upstream of the shock, in the supersonic flow region, the development of streamwise vortices is weakened, resulting in reduced turbulent mixing and dissipation. Downstream of the shock, the turbulent structures are reorganized due to the compression wave leading to locally high wall-shear stress. Integrating the wall-shear stress distribution over the complete surface wave shows that the spanwise shock waves cause significant friction drag reduction. However, the pressure variations upstream and downstream of the shock wave require an increased actuation power, which results in negative net power saving at <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mrow><f:msub><f:mi>M</f:mi><f:mn>2</f:mn></f:msub><f:mo>=</f:mo><f:mn>0.7</f:mn></f:mrow></f:math>.