The logarithmic variance of streamwise velocity and conundrum in wall turbulence

The logarithmic dependence of streamwise turbulence intensity has been observed repeatedly in recent experimental and direct numerical simulation data. However, its spectral counterpart, a well-developed $k^{-1}$ spectrum ( $k$ is the spatial wavenumber in a wall-parallel direction), has not been convincingly observed from the same data. In the present study, we revisit the spectrum-based attached eddy model of Perry and co-workers, who proposed the emergence of a $k^{-1}$ spectrum in the inviscid limit, for small but finite $z/\delta$ and for finite Reynolds numbers ( $z$ is the wall-normal coordinate, and $\delta$ is the outer length scale). In the upper logarithmic layer (or inertial sublayer), a reexamination reveals that the intensity of the spectrum must vary with the wall-normal location at order of $z/\delta$ , consistent with the early observation argued with ‘incomplete similarity’. The streamwise turbulence intensity is subsequently calculated, demonstrating that the existence of a well-developed $k^{-1}$ spectrum is not a necessary condition for the approximate logarithmic wall-normal dependence of turbulence intensity – a more general condition is the existence of a premultiplied power-spectral intensity of $O(1)$ for $O(1/\delta ) < k < O(1/z)$ . Furthermore, it is shown that the Townsend–Perry constant must be weakly dependent on the Reynolds number. Finally, the analysis is semi-empirically extended to the lower logarithmic layer (or mesolayer), and a near-wall correction for the turbulence intensity is subsequently proposed. All the predictions of the proposed model and the related analyses/assumptions are validated with high-fidelity experimental data (Samie et al., J. Fluid Mech., vol. 851, 2018, pp. 391–415).

Resolvent-based estimation of turbulent channel flow using wall measurements

We employ a resolvent-based methodology to estimate velocity and pressure fluctuations within turbulent channel flows at friction Reynolds numbers of approximately 180, 550 and 1000 using measurements of shear stress and pressure at the walls, taken from direct numerical simulation (DNS) databases. Martini et al. (J. Fluid Mech., vol. 900, 2021, p. A2) showed that the resolvent-based estimator is optimal when the true space–time forcing statistics are utilised, thus providing an upper bound for the accuracy of any linear estimator. We use this framework to determine the flow structures that can be linearly estimated from wall measurements, and we characterise these structures and the estimation errors in both physical and wavenumber space. We also compare these results to those obtained using approximate forcing models – an eddy-viscosity model and white-noise forcing – and demonstrate the significant benefit of using true forcing statistics. All models lead to accurate results up to the buffer layer, but only using the true forcing statistics allows accurate estimation of large-scale logarithmic-layer structures, with significant correlation between the estimates and DNS results throughout the channel. The eddy-viscosity model displays an intermediate behaviour, which may be related to its ability to partially capture the forcing colour. Our results show that structures that leave a footprint on the channel walls can be accurately estimated using the linear resolvent-based methodology, and the presence of large-scale wall-attached structures enables accurate estimations through the logarithmic layer.

Velocity transformation for compressible wall-bounded turbulent flows with and without heat transfer

In this work, a transformation, which maps the mean velocity profiles of compressible wall-bounded turbulent flows to the incompressible law of the wall, is proposed. Unlike existing approaches, the proposed transformation successfully collapses, without specific tuning, numerical simulation data from fully developed channel and pipe flows, and boundary layers with or without heat transfer. In all these cases, the transformation is successful across the entire inner layer of the boundary layer (including the viscous sublayer, buffer layer, and logarithmic layer), recovers the asymptotically exact near-wall behavior in the viscous sublayer, and is consistent with the near balance of turbulence production and dissipation in the logarithmic region of the boundary layer. The performance of the transformation is verified for compressible wall-bounded flows with edge Mach numbers ranging from 0 to 15 and friction Reynolds numbers ranging from 200 to 2,000. Based on physical arguments, we show that such a general transformation exists for compressible wall-bounded turbulence regardless of the wall thermal condition.

Adjoint Complement to the Universal Momentum Law of the Wall

The paper is devoted to an adjoint complement to the universal Law of the Wall (LoW) for fluid dynamic momentum boundary layers. The latter typically follows from a strongly simplified, unidirectional shear flow under a constant stress assumption. We first derive the adjoint companion of the simplified momentum equation, while distinguishing between two strategies. Using mixing-length arguments, we demonstrate that the frozen turbulence strategy and a LoW-consistent (differentiated) approach provide virtually the same adjoint momentum equations, that differ only in a single scalar coefficient controlling the inclination in the logarithmic region. Moreover, it is seen that an adjoint LoW can be derived which resembles its primal counterpart in many aspects. The strategy is also compatible with wall-function assumptions for prominent RANS-type two-equation turbulence models, which ground on the mixing-length hypothesis. As a direct consequence of the frequently employed assumption that all primal flow properties algebraically scale with the friction velocity, it is demonstrated that a simple algebraic expression provides a consistent closure of the adjoint momentum equation in the logarithmic layer. This algebraic adjoint closure might also serve as an approximation for more general adjoint flow optimization studies using standard one- or two-equation Boussinesq-viscosity models for the primal flow. Results obtained from the suggested algebraic closure are verified against the primal/adjoint LoW formulations for both, low- and high-Re settings. Applications included in this paper refer to two- and three-dimensional shape optimizations of internal and external engineering flows. Related results indicate that the proposed adjoint algebraic turbulence closure accelerates the optimization process and provides improved optima at no computational surplus in comparison to the frozen turbulence approach.

Potential low bias in high-wind drag coefficient inferred from dropsonde data in hurricanes

Understanding momentum exchange at the air-sea interface is important for accurate hurricane predictions and understanding fundamental storm dynamics. One method for estimating air-sea momentum transfer in high winds is the flux-profile method, which infers surface momentum fluxes and the corresponding drag coefficient from mean velocity profiles obtained from either dropsondes or meteorological towers, under the assumption that the boundary-layer wind profile at low altitudes exhibits a logarithmic profile with height. In this study, we use dropsonde data from reconnaissance aircraft, as well as “virtual sondes” from a turbulence-resolving simulation of an intense tropical cyclone, to critically analyze the diagnosis of drag coefficient CD at hurricane-force wind speeds. In particular, the “roll-off” of the drag coefficient, where CD decreases at 10-m wind speeds ¿ 35 m s−1, is called into question based on uncertainty due to relatively low sample size and a lack of robustness of the flux-profile at high winds. In addition, multiple factors appear to favor an underestimate of CD at hurricane-force winds relative to their true values, including uncertainty in the height of recorded dropsonde data, violation of Monin-Obukhov similarity theory near the eyewall, and the short vertical extent of the logarithmic layer. Due to these and other related sources of uncertainty, it is likely that a quantitative limit has been reached in inferring the specific values of u* and CD using the flux-profile method, while at the same time the potential for underestimation may cast doubt on the CD–U10 relationship inferred from this method at high winds.

Tracking Turbulent Coherent Structures by Means of Neural Networks

The behaviours of individual flow structures have become a relevant matter of study in turbulent flows as the computational power to allow their study feasible has become available. Especially, high instantaneous Reynolds Stress events have been found to dominate the behaviour of the logarithmic layer. In this work, we present a viability study where two machine learning solutions are proposed to reduce the computational cost of tracking such structures in large domains. The first one is a Multi-Layer Perceptron. The second one uses Long Short-Term Memory (LSTM). Both of the methods are developed with the objective of taking the the structures’ geometrical features as inputs from which to predict the structures’ geometrical features in future time steps. Some of the tested Multi-Layer Perceptron architectures proved to perform better and achieve higher accuracy than the LSTM architectures tested, providing lower errors on the predictions and achieving higher accuracy in relating the structures in the consecutive time steps.