On the inertial energy spectrum of turbulence in a reacting gas (Q1111416)

\textit{A. Q. Eschenroeder} [AJAA J. 3, 1839 ff (1965)] investigated the energy spectrum of turbulence in a reacting gas. In this approach in order to determine the spectral distribution of turbulent energy within the framework of similarity theory, a source term is appended to the quasi stationary dynamic equation for isotropic turbulence. This equation is written as \[ (*)\quad \epsilon (t)+\int^{k}_{0}G(k',t)dk'+\int^{k}_{0}F(k',t)dk'-2\nu \int^{k}_{0}k^{'2}E(k',t)dk'=0. \] Here \(\epsilon\) represents the power, as derived from large eddy decay or shear strain by the mean motion which is supplied near the origin of wave number space. The second term on the left hand side of (*) describes the stored energy feeding turbulent motion in the wave number range 0 to k, as contributed by the homogeneous source term. The third term on the left hand side of (*) represents the inertial transfer of energy through the hierarchy of eddies. The last term describes the dissipation of turbulent energy into heat under the influence of viscosity. We show that a variety of solutions fo the energy spectrum are obtainable depending on the assumptions of more general expressions for the transfer function and the source term. Critical examinations of such solutions may be of some help in obtaining more insight into the present problem. We use Von Kármán's form for the transfer function and a general expression, based on dimensional reasonings for the source term. Solving the dynamical equation (*), after neglecting the last term, the inertial energy spectrum is determined in terms of a defined source effective parameter \(\eta\). Eschenroeder's solution for the inertial energy spectrum follows as a particular case.