Entropy Fluctuations and Correlations in Compressible Turbulent Plane Channel Flow

The thermodynamic turbulence structure of compressible aerodynamic flows is often characterised by the correlation coefficient of entropy with pressure or temperature. We study entropy fluctuations <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>s</mi><mo>′</mo></msup></semantics></math></inline-formula> and their correlations with the fluctuations of the other thermodynamic variables in compressible turbulent plane channel flow using <span style="font-variant: small-caps;">dns</span> data. We investigate the influence of the <span style="font-variant: small-caps;">hcb</span> (Huang–Coleman–Bradshaw) friction Reynolds number (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>100</mn><mspace width="3.33333pt"></mspace><mspace width="-0.166667em"></mspace><mo>⪅</mo><mspace width="-0.166667em"></mspace><mspace width="3.33333pt"></mspace><mi>R</mi><msub><mi>e</mi><msup><mi>τ</mi><mo>★</mo></msup></msub><mspace width="3.33333pt"></mspace><mspace width="-0.166667em"></mspace><mo>⪅</mo><mspace width="-0.166667em"></mspace><mn>1000</mn><mspace width="3.33333pt"></mspace></mrow></semantics></math></inline-formula>) and of the centreline Mach number (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0.3</mn><mspace width="3.33333pt"></mspace><mspace width="-0.166667em"></mspace><mo>⪅</mo><mspace width="-0.166667em"></mspace><mspace width="3.33333pt"></mspace><msub><mover accent="true"><mi>M</mi><mo stretchy="false">¯</mo></mover><msub><mrow><mi>CL</mi></mrow><mi>x</mi></msub></msub><mspace width="3.33333pt"></mspace><mspace width="-0.166667em"></mspace><mo>⪅</mo><mspace width="-0.166667em"></mspace><mspace width="3.33333pt"></mspace><mn>2.5</mn></mrow></semantics></math></inline-formula>) on the magnitude and location of the peak of the root-mean-square <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>s</mi><mi>rms</mi><mo>′</mo></msubsup></semantics></math></inline-formula>. The complete series expansions of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>s</mi><mo>′</mo></msup></semantics></math></inline-formula> with respect to the fluctuations of the basic thermodynamic variables (pressure <i>p</i>, density <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula> and temperature <i>T</i>) are calculated for the general case of variable heat-capacity <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>c</mi><mi>p</mi></msub><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> thermodynamics. The correlation coefficients of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>s</mi><mo>′</mo></msup></semantics></math></inline-formula> with the fluctuations of the basic thermodynamic quantities (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>c</mi><mrow><msup><mi>s</mi><mo>′</mo></msup><msup><mi>p</mi><mo>′</mo></msup></mrow></msub></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>c</mi><mrow><msup><mi>s</mi><mo>′</mo></msup><msup><mi>ρ</mi><mo>′</mo></msup></mrow></msub></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>c</mi><mrow><msup><mi>s</mi><mo>′</mo></msup><msup><mi>T</mi><mo>′</mo></msup></mrow></msub></semantics></math></inline-formula>), for varying <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>R</mi><msub><mi>e</mi><msup><mi>τ</mi><mo>★</mo></msup></msub><mo>,</mo><msub><mover accent="true"><mi>M</mi><mo stretchy="false">¯</mo></mover><msub><mrow><mi>CL</mi></mrow><mi>x</mi></msub></msub><mo>)</mo></mrow></semantics></math></inline-formula>, are studied. Insight on these correlations is provided by considering the probability density function (pdf) of s′ and its joint pdfs with the other thermodynamic variables.