Stoichiometric mixing length L_s of reacting coaxial jet flames is a critical scaling parameter for liquid rocket engine combustors. Previous studies have shown that L_s for shear coaxial flames can be scaled like their nonreacting counterparts using a nondimensional momentum flux ratio J. In addition, stoichiometric mixing lengths of reacting and nonreacting coaxial jets collapse upon a single line by altering J using an effective outer flow gas density. This effective density is calculated from a modified version of the equivalence principle, originally developed by Tacina and Dahm [1, 2] and accounts for the effects of heat release on mixing. However, previous studies also required a second nonphysical scaling constant S_c for the reacting jets, which is not predicted by the equivalence principle [3]. It was originally hypothesized that S_c is attributed to the limitation of hydroxyl (OH) planar laser-induced fluorescence, which only infers L_s. Direct quantitative measurement of conserved scalar fields using conventional optical diagnostics is difficult due to the lack of a tracer that easily fluoresces, survives high temperature oxygen flames, and is not dominated by quenching effects. To measure a conserved scalar field, this work implements x-ray fluorescence of Kr and Ar tracers to obtain quantitative mixture fraction fields. From these mixture fraction fields, stoichiometric mixing lengths for two CH₄/O₂ flames are calculated and scaled against nonreacting coaxial mixing lengths using the equivalence principle. By directly measuring the stoichiometric mixing length, it is established that the additional constant is a byproduct of the OH measurement technique and the equivalence principle fully captures the scaling. Comparison with high-fidelity simulation of the flame further supports this conclusion. In addition to further strengthening this scaling method, this work represents the first use of x-ray fluorescence to make quantitative conserved scalar measurements in turbulent flames.

反应同轴喷射火焰的化学计量混合长度L _s是液体火箭发动机燃烧器的一个关键比例参数。先前的研究表明，剪切同轴火焰的L _s可以像使用无量纲动量通量比J的非反应对应物一样缩放。此外，反应和非反应同轴射流的化学计量混合长度通过改变J使用有效的外流气体密度。该有效密度是根据等效原理的修改版本计算得出的，该原理最初由 Tacina 和 Dahm [1, 2] 开发，并考虑了热释放对混合的影响。然而，以前的研究还需要反应射流的第二个非物理比例常数S _c，这不是由等效原理 [3] 预测的。最初假设S _c归因于羟基（OH）平面激光诱导荧光的限制，这仅推断L _s. 使用传统光学诊断法直接定量测量守恒标量场是困难的，因为缺乏易于发出荧光、在高温氧火焰中存活且不受淬火效应支配的示踪剂。为了测量守恒的标量场，这项工作实施了 Kr 和 Ar 示踪剂的 X 射线荧光，以获得定量的混合分数场。从这些混合分数场，两个 CH ₄ /O _{2的化学计量混合长度}使用等效原理计算火焰并根据非反应同轴混合长度进行缩放。通过直接测量化学计量混合长度，确定附加常数是 OH 测量技术的副产品，并且等效原理完全捕获了比例。与火焰的高保真模拟的比较进一步支持了这一结论。除了进一步加强这种定标方法外，这项工作还首次使用 X 射线荧光在湍流火焰中进行定量守恒标量测量。