Theory of Turbulence

Abstract

It is pointed out if there are aspects of the turbulence phenomenon which are truly universal, then they should be capable of being characterized in terms of the two parameters $\epsilon$ and $\nu$ which denote the constant rate of dissipation of energy per unit mass and the kinemetic viscosity respectively and these two parameters only without reference either to the mean square velocity ${〈{u_1}^2〉}_{\mathrm{Av}}$ or to the size of the largest energy containing eddies. This is a slight modification of Kolmogoroff's similarity principles as currently formulated. It appears that $\chi =\frac{\partial \psi (r, t)}{\partial r}, \mathrm{where} \psi (r, t)=\frac{1}{2}{〈{({u_1}^{\prime }-{u_1}^{\prime \prime })}^2〉}_{\mathrm{Av}},$ and ${u_1}^{\prime }$ and ${u_1}^{\prime \prime }$ are the velocities in the $x$-direction (say) at two points on the $x$-axis separated by a distance $r$ and at times an interval $t$ apart, can be so specified. The similarity principles require that if this is the case, $\chi$ should be of the form $\chi \equiv {\left(\frac{{\epsilon }^3}{\nu }\right)}^{\frac{1}{4}}X(r{\left(\frac{\epsilon }{{\nu }^3}\right)}^{\frac{1}{4}}, t{\left(\frac{\epsilon }{\nu }\right)}^{\frac{1}{2}}),$ where $X$ is a universal function of the arguments specified. In the limit of zero viscosity, $\chi$ must have the more special form $\chi \to r^{-\frac{1}{3}}\sigma \left(\frac{t}{r^{\frac{2}{3}}}\right) (\nu \to 0).$ The boundary conditions on $\sigma \mathrm{}\left(x\right)\mathrm{}$ are that $\sigma ={\sigma }_0\mathrm{}(>0\mathrm{})\mathrm{}$ and $\frac{d\sigma }{\mathrm{dx}}=0\mathrm{}$ at $x=0\mathrm{}$ and $\sigma \to 0$ as $x\to \infty$.