Fluid Dynamics and Acoustics

Hydrodynamic stability theory (HST) is concerned with predicting how a flow of fluid (e.g. air or water) becomes unstable and leads to turbulence. Turbulence is one of the outstanding problems in classical mechanics. HST is fundamental to many areas of fluid mechanics because the appearance of turbulence dramatically changes properties of the flow, e.g. the drag on an aeroplane wing or wind turbine blade, or the mixing and transport of heat in the atmosphere and ocean, which is needed in climate models. Clouds can sometimes reveal the instability that amplifies waves in the atmosphere, see Figure 1. The HST group at Keele is interested in all areas of application and has studied flows inspired by aeronautics, internal combustion engines and geophysical flows.

Figure 1: These clouds look just like waves on the sea. This is because they are both produced by the so-called `Kelvin-Helmholtz' instability mechanism (google this scientific term for lots of amazing images). 

Neutral curves separate stable from unstable regions of parameter planes for fluid flows. Instability can enhance mixing, which is important in many applications. The linear stability equation investigated in [1] was first published in 1931, and has been studied in connection with a great many geophysical and astrophysical flows ever since. The paper reveals that, completely unexpectedly, this equation can generate neutral curves with an underlying fractal structure (i.e. with self-similar properties of infinite complexity, obtained by advanced asymptotic analysis). This behaviour seems not to have been found previously in hydrodynamic stability theory (a major field of fluid mechanics), nor in the study of linear ordinary differential equations. See Figure 2.

Figure 2: This figure presents the stability of the simplest flow between two parallel plates moving in their own planes relative to one-another (called plane Couette flow), but for fluid whose density varies between the plates. Along each axis is measured a parameter that describes the density profile. Shaded regions represent instability of the flow. A theory giving the values of each red dot has been obtained, and at every red dot, three stability boundaries intersect. The red dots follow a fractal pattern. See [1] for details.

1. Healey, J. J. 2019 Fractal sets of neutral curves for stably stratified plane Couette flow. Journal of Fluid Mechanics. Vol. 872, pp. 697-728.

Our results in [1] challenge the universally adopted paradigm for the description of random nonlinear waves.  By direct numerical simulation of the Zakharov equation it is shown that the evolution of the wave spectra considered in the absence of forcing essentially differs from the predictions of the universally used kinetic equation, although the integral quantities behave in the similar way. Currently all wave modelling and forecasting is based on the kinetic equation.

The work in [2] suggests a new paradigm for the description of water waves trapped by a jet current. The untractable 2d boundary value problem for waves trapped on currents has been asymptotically reduced to a solvable 1d Sturm-Liouville one. This unlocks the possibility of developing asymptotically consistent theory for weakly non- linear waves on a current. 

The work in [3] challenges the foundations of the hundred year old Ekman paradigm for describing wind induced currents in the ocean. A broad novel class of exact solutions of the Navier-Stokes equations describing transient Ekman currents in the model with time dependent eddy viscosity has been derived. It has been shown that the Ekman currents caused by an arbitrarily varying wind quickly become unstable to finite length perturbations, which undermines the Ekman paradigm fundamental assumption of horizontal uniformity.

The work in [4] presents a new class of robust solitary wave solutions for nonlinear water waves trapped on a jet currents found by solving numerically the full Euler equations. The long-lived envelope solitary waves provide a model of rogue waves frequently occurring on jet currents.

1. S. Annenkov, V. Shrira (2018) Spectral evolution of weakly nonlinear random wave fields: kinetic equations vs direct numerical simulations, J. Fluid Mech., 844, 766-795.
2. V. Shrira, A. Slunyaev (2014) Trapped waves on jet currents: asymptotic modal approach, J. Fluid Mech., 738, 65-104.
3. V. Shrira, R. Almelah (2020) Upper-ocean Ekman current dynamics: a new perspective, J. Fluid Mech., 887, A24-32.
4. V. Shrira, A. Slunyaev (2014) Nonlinear dynamics of trapped waves on jet currents and rogue waves, Phys. Review E, 89, 041002(R)

We study sound generation by an aircraft engine. The paper [1] determined fully three-dimensional sharp-edged sound fields produced by gusts striking a supersonic edge of an aircraft fan blade.  No gust fields of this complexity have previously been calculated.

The paper [2] tackled a similar problem to the one above, but in the slightly easier case of a single-frequency sound field.  Nevertheless, many completely original three-dimensional sound fields were determined here for the first time. 

John Chapman lecturing at the Isaac Newton Institute, Cambridge

1. Chapman, C.J. & Powles, C. J., 2019 Basic singular fields in the theory of impulsive supersonic leading-edge noise, Wave Motion 89, pp. 79-82
2. Powles, C.J. & Chapman, C. J. , 2019 Canonical sound fields in the frequency-domain theory of supersonic leading-edge noise, Wave Motion 86, pp. 125-136

We have investigated stability of a spreading surfactant-laden drop down an inclined plane to small-amplitude disturbances with transverse variation [1]. A dispersion relationship is described using long-wavelength asymptotics and numerical simulations, which reveals, for the first time, the physical mechanisms and new scaling properties of a novel fingering instability at the drop’s leading edge. This information will be useful to practitioners who are interested in suppressing such instabilities that are undesirable in many applications, such as in coating flows and surfactant-replacement therapy in neonatal lungs.

We are the first to implement r-adaptive numerical schemes to fourth-order parabolic PDEs [2, 3]. This method accurately resolves the solution and reduces the computational effort compared to that using a fixed uniform mesh. We also introduce a new mesh density function that resolves multiple solution structures observed in many thin-film flows.

1. Goddard J. and Naire S. 2015. The spreading and stability of a surfactant-laden drop on an inclined prewetted substrate. J. Fluid Mech., vol. 772, 535-568.
2. Alharbi A. and Naire S. 2017. An adaptive moving mesh method for thin film flow equations with surface tension. J. Computational and Applied Mathematics, vol. 319, 365-384.
3. Alharbi A. and Naire S. 2019. An adaptive moving mesh method for two-dimensional thin film flow equations with surface tension, Journal of Computational and Applied Mathematics, vol. 356, 2019, 219-230.

Figure 1 An r-adaptive moving mesh method for simulating the time and spatial evolution of a spreading droplet down an inclined plane.

Figure 2 Comparison of r-adaptive moving mesh method (bottom panel) with a fixed uniform mesh (top and middle panel) scheme.