Methods for Measuring and Computing the Reference Temperature in Newton’s Law of Cooling for External Flows

Newton’s law of cooling requires a reference temperature (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>r</mi><mi>e</mi><mi>f</mi></mrow></msub></mrow></semantics></math></inline-formula>) to define the heat-transfer coefficient (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>h</mi></mrow></semantics></math></inline-formula>). For external flows with multiple temperatures in the freestream, obtaining <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>r</mi><mi>e</mi><mi>f</mi></mrow></msub></mrow></semantics></math></inline-formula> is a challenge. One widely used method, referred to as the adiabatic-wall (AW) method, obtains <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>r</mi><mi>e</mi><mi>f</mi></mrow></msub></mrow></semantics></math></inline-formula> by requiring the surface of the solid exposed to convective heat transfer to be adiabatic. Another widely used method, referred to as the linear-extrapolation (LE) method, obtains <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>r</mi><mi>e</mi><mi>f</mi></mrow></msub></mrow></semantics></math></inline-formula> by measuring/computing the heat flux (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mrow><mi>q</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>′</mo><mo>′</mo></mrow></msubsup></mrow></semantics></math></inline-formula>) on the solid surface at two different surface temperatures (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>s</mi></mrow></msub></mrow></semantics></math></inline-formula>) and then linearly extrapolating to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mrow><mi>q</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>′</mo><mo>′</mo></mrow></msubsup><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>. A third recently developed method, referred to as the state-space (SS) method, obtains <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>r</mi><mi>e</mi><mi>f</mi></mrow></msub></mrow></semantics></math></inline-formula> by probing the temperature space between the highest and lowest in the flow to account for the effects of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>s</mi></mrow></msub></mrow></semantics></math></inline-formula> or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mrow><mi>q</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>′</mo><mo>′</mo></mrow></msubsup></mrow></semantics></math></inline-formula> on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>r</mi><mi>e</mi><mi>f</mi></mrow></msub></mrow></semantics></math></inline-formula>. This study examines the foundation and accuracy of these methods via a test problem involving film cooling of a flat plate where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mrow><mi>q</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>′</mo><mo>′</mo></mrow></msubsup></mrow></semantics></math></inline-formula> switches signs on the plate’s surface. Results obtained show that only the SS method could guarantee a unique and physically meaningful <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>r</mi><mi>e</mi><mi>f</mi></mrow></msub></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>=</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>r</mi><mi>e</mi><mi>f</mi></mrow></msub></mrow></semantics></math></inline-formula> on a nonadiabatic surface <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mrow><mi>q</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>′</mo><mo>′</mo></mrow></msubsup><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>. The AW and LE methods both assume <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>r</mi><mi>e</mi><mi>f</mi></mrow></msub></mrow></semantics></math></inline-formula> to be independent of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>s</mi></mrow></msub></mrow></semantics></math></inline-formula>, which the SS method shows to be incorrect. Though this study also showed the adiabatic-wall temperature, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>A</mi><mi>W</mi></mrow></msub></mrow></semantics></math></inline-formula>, to be a good approximation of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>r</mi><mi>e</mi><mi>f</mi></mrow></msub></mrow></semantics></math></inline-formula> (<10% relative error), huge errors can occur in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>h</mi></mrow></semantics></math></inline-formula> about the solid surface where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>|</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>−</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>A</mi><mi>W</mi></mrow></msub><mo>|</mo></mrow></semantics></math></inline-formula> is near zero because where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>=</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>A</mi><mi>W</mi></mrow></msub></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mrow><mi>q</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>′</mo><mo>′</mo></mrow></msubsup><mo>≠</mo><mn>0</mn></mrow></semantics></math></inline-formula>.