- in collaboration with Prof. Martin Skote (Cranfield)*Identities for the skin-friction coefficient in incompressible turbulent boundary layers*
In this paper,
we have proved that the identity discovered by Fukagata et al. (2002) for free-stream boundary layers
simplifies to the von Karman momentum integral equation when the upper bound of integration is taken to be asymptotically large. If a finite upper bound is used, as in Fukagata et al. (2002)'s original identity, the terms of the identity depend spuriously on the bound itself. Therefore, the impact of the Reynolds stresses on the wall-shear stress cannot be quantified. In the infinite number of alternative identities obtained by performing additional integrations on the streamwise momentum equation, the relative influence of the physical terms on the wall-shear stress depends on the number of successive integrations. This analysis proves that these identities also do not reveal any physical meaning.
In the limit of large number of integrations, we have used Watson's lemma to prove that these identities degenerate (collapse) to the definition of skin-friction coefficient. We have also used new integral thicknesses to decompose the skin-friction coefficient uniquely, following the study of Renard & Deck (2016).
- in collaboration with Dr Dongdong Xu and Dr Lian Duan (Ohio State)*Identities for the skin-friction coefficient and the wall-heat flux in compressible turbulent boundary layers*
In this paper,
we have derived an integral formula for the skin-friction coefficient of compressible boundary
layers by extending the formula of Elnahhas & Johnson (2022) for incompressible boundary
layers. We have also shown that existing identities for compressible boundary layers simplify to the compressible von Karman momentum integral equation when the upper limit of integration is asymptotically large. By using asymptotic methods,
we have proved that a multiple-integration identity degenerates to the definition of the skin-friction coefficient when the number of integrations is asymptotically large. In this paper, we have studied the wall-heat flux and discussed the use of the wall-heat-flux version of our identity for the skin-friction coefficient.

- in collaboration with Dr Claudia Alvarenga*Growth of entrained disturbances in a circular pipe and in a channel*
The problem of stability and transition in pipe flows is one of the most exciting and important problems in fluid dynamics and it has been studied systematically since the pioneering work of O. Reynolds (1883). One of the central questions is: Where do the perturbations that lead the flow to turbulence come from? As recognized by Reynolds himself, disturbances at the pipe inlet are primary candidates. Motivated by this idea, we have investigated the entrainment and growth of these disturbances from the pipe mouth by asymptotic methods and numerically: see this paper. The dependence of the instability on the length scales, the frequency and the Reynolds number has been found. Our calculations show good match with the (albeit limited) experimental data. We have also studied this problem in the Cartesian geometry: the results are published in this paper.

- in collaboration with Dr Dongdong Xu, Dr Samuele Viaro, and Dr Elena Marensi*Unsteady Görtler vortices in compressible boundary layers*
Görtler vortices are ubiquitous in boundary layers when the wall is concave. They display an intense growth, leading the flow to turbulence. The main challenge when studying these flows is the strong dependence on the external flow conditions, such as free-stream disturbances, wall roughness and vibrations. We have studied these vortices in the incompressible and compressible regime, both during the initial linear growth and when they saturate nonlinearly. Neutral curves for instability have been found and our calculations show good agreement with experimental data. Here are a few papers we have written:
this paper is on the theory of small-amplitude unsteady compressible Görtler vortices, this one on neutral map in the supersonic regime, this paper on the neutral map in the incompressible regime, this paper is on nonlinear Görtler vortices and comparison with experimental data.
- in collaboration with Professor X. Wu, Professor J. Luo, and Dr Elena Marensi*Boundary-layer bypass transition induced by free-stream vortices*
Bypass transition induced by free-stream vortices in a laminar boundary layer has been studied (see this paper). For the first time, the - in collaboration with Dr M.E. Goldstein and Professor X. Wu*Receptivity of supersonic boundary layers to free-stream turbulence*
One of the key problems of fundamental fluid dynamics is to understand how external perturbing agents, such as free-stream vortical structures or fluctuations induced by wall roughness, penetrate into laminar boundary layers and trigger unstable modes thereby inducing laminar-turbulent transition. Such a process is called receptivity and it complements classical stability theory to describe the dynamics of pre-transitional boundary layers. We have focused our attention on receptivity in the compressible regime because of its relevance in technological fluid systems, such as flows over aircraft wings, over turbine blades and in jet engines. For the first time, the - in collaboration with Dr Peter D. Hicks*Attenuation of laminar streak growth by spanwise wall oscillations*
Our theoretical and numerical results (published here
and here) show that either steady or unsteady spanwise oscillating wall forcing is an effective method for suppressing the algebraic growth of laminar streaks (or Klebanoff modes) induced by free-stream vortical disturbances in a laminar boundary layer. We have shown that amplitudes as small as one tenth of the free-stream velocity can halve the streak intensity. An optimal forcing wavelength and an optimal frequency of oscillation have been found as functions of the free-stream vortical characteristics. We have also proved that the use of the classical Stokes layer as spanwise base flow leads to erroneous results.

*Hydrophobic surfaces*
In this paper we have studied laminar and turbulent flows over hydrophobic surfaces featuring shear-dependent slip length. The laminar channel-flow and pipe-flow solutions are derived analytically and a nonlinear Lyapunov stability analysis is performed on the laminar channel flow to establish the stability conditions. The feedback law extracted through the stability analysis is recognized for the first time to coincide with the slip-length model used to represent the hydrophobic surfaces, thereby providing a precise physical interpretation for the feedback law advanced by Balogh et al. (2001). The theoretical framework by K. Fukagata, N. Kasagi, and P. Koumoutsakos (2006) is used to model the drag-reduction effect and the theoretical drag-reduction values are in very good agreement with our direct numerical simulation data. The turbulent drag reduction is measured as a function of the hydrophobic-surface parameters and is found to be a function of the time- and space-averaged slip length, irrespectively of the local and instantaneous slip behaviour at the wall. The power spent by the turbulent flow on the hydrophobic walls is computed for the first time and is found to be a non-negligible portion of the power saved through drag reduction, thereby recognizing the hydrophobic surfaces as a passive-absorbing drag-reduction method. The turbulent flow is further investigated through flow visualizations and statistics of the relevant quantities, such as vorticity and strain rates.
- in collaboration with Daniel Wise*Oscillating discs*
Our new turbulent drag-reduction technique based on flush-mounted moving discs has been taken a step further and the effect of sinusoidally oscillating discs on wall turbulence has been investigated (see this paper). The new parameter, the period of oscillation, plays now a new crucial role to fix the maximum drag reduction to about 20%. The laminar Rosenblat viscous pump flow is used to predict accurately the power spent for disc motion in the fully-developed turbulent channel flow case and to estimate localized and transient regions over the disc surface subjected to the turbulent regenerative braking effect, for which the wall turbulence exerts work on the discs.
The Fukagata-Iwamoto-Kasagi identity is also used effectively to show that the drag reduction is due to two combined effects, one linked to the direct shearing action of the oscillating disc boundary layer on the wall turbulence, and the other one is due the additional disc-flow Reynolds stresses produced by the streamwise-elongated structures which form between discs and modulate slowly in time. We have also found scalings for these two contributions.
- in collaboration with Stanislav Hahn, Dr Daniel Wise, and Dr Claudia Alvarenga*Spinning discs and rings*
In this paper, the active open-loop turbulent drag reduction technique proposed by L. Keefe in 1998 has been studied by direct numerical simulations for the first time. A turbulent channel flow is modified by the steady rotation of rigid flush-mounted discs, located next to one another on the walls. For a fixed
maximum disc tip velocity, drag reduction can be achieved when the disc diameter is larger than a threshold, while below this threshold the drag increases. A maximum drag reduction of 23% is computed. The net power saved, obtained by taking into account the power spent to enforce the rotational motion against the fluid viscous resistance, is found to be positive and reach 10%. We have also discussed the disc-flow parameters required in flows of technological interest, such as over wings of commercial aircraft in flight conditions and over high-speed trains and ship hulls, and presented some ideas for future implementations based on existing micro-electromagnetic motor and micro-air turbine technologies.
More recently, in this paper, we have discovered unexpected properties of the disc flow technique. For example, it has been shown that, for certain conditions, the drag reduction can increase by halving the number of discs. This means that the drag reduction does not scale linearly with the actuated area. The radial flow induced by the discs is found to be responsible for the additional reduction of wall friction. Rotating half-discs and spinning rings have also been studied.
- in collaboration with Professor M. Quadrio*Wall traveling waves*
Sinusoidal streamwise traveling waves of spanwise wall velocity alter the near-wall turbulence significantly. When the wave phase speed matches the wall turbulent convection velocity, drag increase occurs, while drag reduction is found for backward traveling waves and for forward traveling waves with a wave speed sufficiently different from the convection velocity (see this paper). This other paper of ours shows that the laminar solution for the generalized Stokes layer agrees with the turbulent spanwise profile and the amount of drag reduction relates linearly with the boundary layer thickness of the spanwise layer under specified conditions. An optimal thickness for drag reduction exists and the net energetic balance is positive, a favourable result for future practical applications.
- in collaboration with Professor M. Quadrio*Spanwise wall oscillations*
Our numerical calculations have clarified previously conflicting results on the drag reduction effects of wall sinusoidal oscillations, such as the dependence of drag reduction on the wall forcing parameters. The optimal periods for drag reduction at constant maximum wall velocity and displacement have been found and their values have been related to the survival time of the turbulent coherent structures near the wall (see this paper and this one). Our flow visualizations have clearly shown how the wall motion affects the near-wall turbulence dynamics involving low-speed streaks and quasi-streamwise vortices (see this paper), while this paper presents experimental results on the turbulence statistics in a free-stream boundary layer with wall motion.
This paper of ours proves why the turbulent kinetic energy decreases during the wall motion, which in turn leads to the increased mass flow rate for fixed-pressure-gradient conditions in a turbulent channel flow.

- in collaboration with Prof. A. Baron and Dr P. Molteni*Pressure waves generated by a high-speed train running through a tunnel*
This paper presents an experimental and numerical study on the pressure waves generated by a train entering and running through a tunnel aims at a detailed characterization of the flow in the standard tunnel geometry and in the configuration with airshafts along the tunnel surface. Laboratory experiments were conducted in a scaled facility where train models travelled at a maximum velocity of about 150 km/h through a 6-meter-long tunnel. The flow was simulated by a one-dimensional numerical code modified to include the effect of the separation bubble forming near the train head. The numerical simulations reproduced well the experimental results. We tested the influence of the train cross-sectional shape and length on the compression wave produced by the vehicle entering the confined area. The cross-section shape was not found to be influential as long as the blockage ratio, namely the ratio between the train and tunnel cross-sectional areas, is constant. The pressure waves are one-dimensional sufficiently downwind of the tunnel mouth, thus validating the
comparison between the experimental and computational results. It is further shown that the numerical code can satisfactorily reproduce the pressure variations for the case with airshaft apertures along the tunnel surface.

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