# Research overview

## Low-order models of turbulence

An example showing a packet of hairpin-like structures produced by only two modes from the model of Sharma & McKeon, (2013).

On the modelling side, we have recently published a simple model that successfully predicts turbulent structures and scaling of energy spectra in turbulent pipe and channel flow (\(10^3 < Re_\tau < 10^{10}\)). This work is in collaboration with Professor McKeon's group at Caltech. The theory has been demonstrated in experiments to successfully predict the effect of oscillating actuation at the flow surface on large-scale structures in boundary layer flow. The model is extremely computationally cheap, being essentially a linear, frequency-domain analysis of the flow resonance to the nonlinear coupling between scales.

## Turbulence control

On the control side, we are developing a methodology to synthesis low-order linear feedback controllers for turbulent flows, using distributed sensing and actuation. Alongside the widely addressed question of actuation and sensing technology, are the questions of performance specification and control logic. The work in Sharma et al (2011) presented a control theory that provided guaranteed performance characteristics in the turbulent regime. That theory was tested in low Reynolds number turbulent channel flow and used distributed body forcing. The generalisation to boundary actuation and sensing is currently being pursued in a joint project with Dr Bryn Jones at the University of Sheffield, and tests at higher Reynolds number are under way.

## Compliant metamaterial surfaces for turbulence control

One possible design for a flow surface that can provide a negative effective density in a resonant frequency band. Our most recent work on compliant surfaces is published in Luhar et al (2015)

The material specification for a flow surface with linear material properties can be expressed as bulk modulus, shear modulus, density and damping. Previously, negative values for these material properties would have been thought impossible, however such material properties are now being seriously considered to provide effects like negative refractive index, in applications such as acoustic and visual cloaking.

Such metamaterials can be surprisingly simple to implement: one material providing negative effective density consists of metal beads coated in silicone, embedded in an epoxy medium. The obvious benefit of such a passive surface is that it requires no power source and has the potential to be durable. Active metamaterials however can have their response tuned over a wide range of frequencies and wavelengths. Spatial scales of the order of the flow geometry and timescales associated with the ratio of those lengthscales to the fluid bulk velocity are most important for turbulence control (see e.g. Sharma et al. 2011); fortunately these scales are the most realistic for implementation by metamaterials in flows of practical interest.

## Model reduction for turbulence control

What remains is to generalise the control theory to more complex geometries, such as a turbulent boundary layer or flow over an aerofoil. This introduces new complexities such as spatial inhomogeneity and non-periodicity. In addition, the method requires the (off-line) solution of two Riccati equations of the same order as the number of states in the flow simulation. This is inefficient. To address this, we will explore approximate low-order solutions to these equations using Krylov subspace methods. This will yield low-order controllers with guaranteed performance characteristics in the turbulent regime.

## Other interests

Grid-based Bayesian estimation of a Lorenz system using an efficient algorithm exploiting the sparsity of the attractor. Bewley & Sharma, (2012).

State estimation of non-linear systems with non-Gaussian uncertainty, using grid-based Bayesian filtering methods.

The RZIP tokamak model. The RZIP tokamak model is freely available on GitHub.