Computing the scaling exponents in fluid turbulence from first principles: Demonstration of multiscaling.

*(English)*Zbl 0953.76036Summary: We develop a consistent closure procedure for the calculation of the scaling exponents \(\zeta_n\) of the \(n\)th-order correlation functions in fully developed hydrodynamic turbulence, starting from first principles. The closure procedure is constructed to respect the fundamental rescaling symmetry of the Euler equation. The starting point of the procedure is an infinite hierarchy of coupled equations that obey identically the scaling for any set of scaling exponents \(\zeta_n\).

We present in detail the lowest non-trivial closure of this infinite set of equations, and prove that this closure leads to the determination of the scaling exponents from solvability conditions. The equations under consideration after this closure are nonlinear integrodifferential equations, reflecting the nonlinearity of the original Navier-Stokes equations. Nevertheless, they have a very special structure such that the determination of the scaling exponents requires a procedure that is very similar to the solution of linear homogeneous equations, in which amplitudes are determined by fitting to the boundary conditions in the space of scales. The renormalization scale that is necessary for any anomalous scaling appears at this point. The Hölder inequalities on the scaling exponents select the renormalization scaled as the outer scale of turbulence \(L\). We demonstrate that the solvability condition for fur equations leads to non-Kolmogorov values of the scaling exponents \(\zeta_n\). Finally, we show that this solutions is a first approximation in a systematic series of improving approximations for the calculation of anomalous exponents in turbulence.

We present in detail the lowest non-trivial closure of this infinite set of equations, and prove that this closure leads to the determination of the scaling exponents from solvability conditions. The equations under consideration after this closure are nonlinear integrodifferential equations, reflecting the nonlinearity of the original Navier-Stokes equations. Nevertheless, they have a very special structure such that the determination of the scaling exponents requires a procedure that is very similar to the solution of linear homogeneous equations, in which amplitudes are determined by fitting to the boundary conditions in the space of scales. The renormalization scale that is necessary for any anomalous scaling appears at this point. The Hölder inequalities on the scaling exponents select the renormalization scaled as the outer scale of turbulence \(L\). We demonstrate that the solvability condition for fur equations leads to non-Kolmogorov values of the scaling exponents \(\zeta_n\). Finally, we show that this solutions is a first approximation in a systematic series of improving approximations for the calculation of anomalous exponents in turbulence.