Linear Response Theory for One-Point Statistics in the Inertial Sublayer of Wall-Bounded Turbulence

The idea of linear response theory well known in the statistical mechanics for thermal equilibrium systems is applied to one-point statistics in the inertial sublayer of wall-bounded turbulence (WBT). A close analogy between the energy transfer from large to small scales in isotropic turbulence and the momentum transfer in the wall normal direction in WBT plays a key role in the application. The application gives estimates of the influence of the finite Reynolds number on the statistics. The estimates are consistent with data by high-resolution direct numerical simulations of turbulent channel flow.

Figure 1

$-⟨u^{+}{v}^{+}⟩$ vs $y^{+}$: black lines, DNS by Yamamoto and Tsuji [20] for ${\mathrm{Re}}_{\tau }=1000$, 2000, 4000, and 8000 from bottom to top; symbols, DNS by Lee and Moser [19] for ${\mathrm{Re}}_{\tau }=180,550,1000$, 2000, and 5200; color lines show the value by Eq. (13) with $\kappa =0.38$ for ${\mathrm{Re}}_{\tau }=180$, 550, 1000, 2000, 4000, 5120, and 8000.

Figure 2

The same as in Fig. 1, but for (a) $-y^{+}⟨\partial (u^{+}{v}^{+})/\partial y⟩$, (b) $y^{+}⟨\partial (v^{+}{v}^{+})/\partial y⟩$, and (c) $y^{+}⟨\partial (w^{+}{w}^{+})/\partial y⟩$, and color solid lines show the value by Eq. (8) with Eq. (14).