Alexander J. Smits

Wall-pressure fluctuations beneath turbulent boundary layers (BLs) drive noise and structural fatigue through interactions between fluid and structural modes. Conventional predictive models for the spectrum – such as the widely accepted Goody model (2004 AIAA J. , vol. 42 (9), pp. 1788–1794) – fail to capture the energetic growth in the low-frequency range that occurs at high Reynolds number, while at the same time over-predicting the variance. To address these shortcomings, two semi-empirical models are proposed for the wall-pressure spectrum in canonical turbulent BLs, pipes and channels for friction Reynolds numbers $\delta ^+$ ranging from 180 to 47 000. Consistent with the approach outlined modelling the streamwise Reynolds stress in the recent work of Gustenyov et al. (2025 J. Fluid Mech., vol. 1016, A23), the models are based on consideration of two spectral components that represent the contributions to the wall-pressure fluctuations from inner-scale motions and outer-scale motions. The first model expresses the pre-multiplied spectrum as the sum of two overlapping log-normal components: an inner-scaled term that is $\delta ^+$ -invariant and an outer-scaled term whose amplitude broadens smoothly with $\delta ^+$ . Calibrated against large-eddy simulations, direct numerical simulations and recent high- $\delta ^+$ pipe data, it reproduces the inner-scaled peak and the emergence of an outer-scaled peak at large $\delta ^+$ . The second model, developed around newly available pipe data, uses theoretical arguments to prescribe the spectral shapes of the inner and outer components. Embedding the $\delta ^+$ -dependence in smooth asymptotic functions yields a formulation that varies continuously with $\delta ^+$ and generalises beyond the calibration range. Both models capture the full spectrum and recover the observed logarithmic growth of its variance, providing a compact, physics-informed empirical representation for more accurate engineering predictions of wall-pressure fluctuations.

A model is proposed for the one-dimensional spectrum and streamwise Reynolds stress in pipe flow for arbitrarily large Reynolds numbers. Constructed in wavenumber space, the model comprises four principal contributions to the spectrum: streaks, large-scale motions, very-large-scale motions and incoherent turbulence. It accounts for the broad and overlapping spectral content of these contributions from different eddy types. The model reproduces well the broad structure of the premultiplied one-dimensional spectrum of the streamwise velocity, although the bimodal shape that has been observed at certain wall-normal locations, and the $-5/3$ slope of the inertial subrange, are not captured effectively because of the simplifications made within the model. Regardless, the Reynolds stress distribution is well reproduced, even within the near-wall region, including key features of wall-bounded flows such as the Reynolds number dependence of the inner peak, the formation of a logarithmic region, and the formation of an outer peak. These findings suggest that many of these features arise from the overlap of energy content produced by both inner- and outer-scaled eddy structures combined with the viscous-scaled influence of the wall. The model is also used to compare with canonical turbulent boundary layer and channel flows, and despite some differences being apparent, we speculate that with only minor modifications to its coefficients, the model can be adapted to these flows as well.

Direct numerical simulations are performed for turbulent forced convection in a half-channel flow with a wall oscillating either as a spanwise plane oscillation or to generate a streamwise travelling wave. The friction Reynolds number is fixed at $Re_{\tau _0} = 590$ , but the Prandtl number $Pr$ is varied from 0.71 to 20. For $Pr\gt 1$ , the heat transfer is reduced by more than the drag, 40 % compared with 30 % at $Pr=7.5$ . This outcome is related to the different responses of the velocity and thermal fields to the Stokes layer. It is shown that the Stokes layer near the wall attenuates the large-scale energy of the turbulent heat flux and the turbulent shear stress, but amplifies their small-scale energy. At higher Prandtl numbers, the thinning of the conductive sublayer means that the energetic scales of the turbulent heat flux move closer to the wall, where they are exposed to a stronger Stokes layer production, increasing the contribution of the small-scale energy amplification. A predictive model is derived for the Reynolds and Prandtl number dependence of the heat-transfer reduction based on the scaling of the thermal statistics. The model agrees well with the computations for Prandtl numbers up to 20.

A new wall-wake law is proposed for the streamwise turbulence in the outer region of a turbulent boundary layer. The formulation pairs the logarithmic part of the profile (with a slope $A_1$ and additive constant $B_1$ ) to an outer linear part, and it accurately describes over 95 % of the boundary layer profile at high Reynolds numbers. Once the slope $A_1$ is fixed, $B_1$ is the only free parameter determining the fit. Most importantly, $B_1$ is shown to follow the same trend with Reynolds number as the wake factor in the wall-wake law for the mean velocity, which is tied to changes in scaling of the mean flow and the turbulence that occur at low Reynolds number.