The study of Reynolds algebras has its origin in the well-known work of O. Reynolds on fluid dynamics in 1895 and has since found broad applications. It also has close relationship with important linear operators such as algebra endomorphisms, derivations and Rota-Baxter operators. Many years ago, G. Birkhoff suggested an algebraic study of Reynolds operators, including the corresponding free algebras. We carry out such a study in this paper. We first provide examples and properties of Reynolds operators, including a multi-variant generalization of the Reynolds identity. We then construct the free Reynolds algebra on a set. For this purpose, we identify a set of bracketed words called Reynolds words which serves as the linear basis of the free Reynolds algebra. A combinatorial interpretation of Reynolds words is given in terms of rooted trees without super crowns. The closure of the Reynolds words under concatenation gives the algebra structure on the space spanned by Reynolds words. Then a linear operator is defined on this algebra such that the Reynolds identity and the desired universal property are satisfied.

雷诺代数的研究起源于O.Reynolds于1895年在流体动力学方面的著名工作，此后得到了广泛的应用。它还与重要的线性算子（如代数内同态）有着密切的关系，派生和Rota-Baxter运算符。许多年前，G。Birkhoff建议对雷诺算子进行代数研究，包括相应的自由代数。我们在本文中进行了这样的研究。我们首先提供Reynolds算子的示例和属性，包括Reynolds身份的多变量概括。然后，我们在集合上构造自由的雷诺代数。为此，我们确定了一组称为雷诺词的方括号词，它们是自由雷诺代数的线性基础。雷诺词的组合解释是根据没有超级树冠的扎根树木给出的。雷诺词在级联下的闭合给出了雷诺词跨越的空间上的代数结构。