Taylor's frozen hypothesis of the pressure fluctuations in turbulent channel flow at high Reynolds numbers

Abstract

We study the application of Taylor's frozen hypothesis to the pressure fluctuations in turbulent channels by means of spatio-temporal data from direct numerical simulations with large computational domains up to the friction Reynolds number $R{e_\tau } = 2000$ . The applicability of the hypothesis is quantitatively verified by comparing the wavenumber and Taylor (frequency) premultiplied spectra of the pressure fluctuations at each distance y from the wall. Using the local mean velocity $U(y)$ for the hypothesis leads to a large difference between both spectra with a value of $O(50\,{\%})$ for its maximum from the wall to $y/h \approx 0.2$ , where h is the channel half-depth. Alternatively, the convection velocity of the pressure fluctuations ${C_p}(y)$ , originally defined by Del Álamo & Jiménez (J. Fluid Mech., vol. 640, 2009, pp. 5–26) as a function of y, is investigated and adopted for the hypothesis. It is nearly equal to $U(y)$ from ${y^ + } = 20$ to the channel centre, where ${y^ + } = y{u_\tau }/\nu $ , in which ${u_\tau }$ and $\nu $ represent the friction velocity and kinematic viscosity, respectively. For ${y^ + } \le 20$ , ${C_p}(y)$ is almost constant with a value of around $12{u_\tau }$ . Applying ${C_p}(y)$ for the hypothesis decreases the difference between both spectra down to a value of $O(10\,{\%})$ for its maximum from the wall to $y/h \approx 0.2$ . Finally, the difference between the pressure wavenumber and frequency premultiplied spectra near the wall is reduced further via applying a wavenumber-dependent convection velocity. This wavenumber-dependent convection velocity is the linear combination of the convection velocities of the turbulent structures associated with the pressure field weighted by their relative contributions to the pressure variance.