## Applied and Computational Mathematics Seminars 2009 - 2010

### Programme

These seminars take place on Tuesdays, in Room M/2.06, Senghennydd Road, Cardiff from 3pm, unless otherwise stated.

When a seminar is not scheduled there is a collaborative workshop with other groups within the College of Physical Sciences & Engineering or a SIAM Chapter Meeting. Further details can be found on the School Diary.

#### The Klebanoff modes in laminar boundary layers

3 November 2009

**Speaker:** P. Ricco (Kings College London).

We will present theoretical and numerical results on the dynamics
of the laminar streaks (or Klebanoff modes), namely the streamwise-elongated,

low-frequency disturbances appearing in pre-transitional laminar
boundary layers as a consequence of a medium-to-high level of
free-stream turbulence.

It is shown that, while the spanwise velocity component of the free-stream vortical disturbances is responsible for the streak generation in the core of the boundary layer, the penetration and confinement of the fluctuations in the outer portion of the boundary layer is due to the streamwise velocity component and the pressure disturbance. A realistic streak profile is btained, which compares favourably with experimental data.

The effect of compressibility is also discussed. Vortical disturbances are found to induce `thermal streaks', which are shown to evolve to highly-oblique Tollmien-Schlichting waves through a receptivity mechanism involving wavelength shortening of quasi three-dimensional Lam-Rott igensolution. A parametric study suggests that this receptivity mechanism could be significant when the free-stream Mach number is larger than 0.8.

#### Purely-elastic flow instabilities in extensional flows

24 November 2009

**Speaker:** Rob Poole (Liverpool).

The flow of viscoelastic fluids, e.g. polymer solutions or melts, can often give rise to spectacularly different flow phenomena compared to "simple" Newtonian fluids (water or air). One such manifestation of these differences is that in microfluidic geometries viscoelastic fluid flows can become unstable and time dependent in simple geometries at flow rates much smaller than would arise in the equivalent flow of a Newtonian fluid (typically due to inertial instabilities). Such "purely-elastic" flow instabilities arise as the inherently small scale of microfluidic flows accentuates the viscoelastic behaviour observed: the small length scale simultaneously makes the Reynolds number small and the Deborah or Weissenberg numbers, which characterize the degree of elasticity in the flow, large. In this talk I will discuss how we have been able to numerically model one such experimentally-observed instability. I will then go on to briefly examine a series of purely-elastic instabilities that we have discovered in related geometries and some recent experimental verification of one such truly a priori prediction.

#### Dynamic modelling of chorded mitral valve using immersed boundary methods

2 March 2010

**Speaker:** X. Y. Luo (Glasgow).

The dynamic behaviour of a novel chorded mitral prosthesis is studied several using immersed boundary methods. To investigate the mechanical behaviour of the mitral design under physiological flow conditions without having to model the left ventricle, we make use of in vivo magnetic resonance images of the left ventricle. The relative motion of the mitral annulus and motion of the ventricle determined from these MRI images is then used as a prescribed boundary condition for the chorded mitral valve in a dynamic cycle. This model allows us to investigate the influences of the flow vortex generated by the ventricle motion on the valve dynamics, as well as the effect of the motion of the chordae attachment points. The immersed boundary model is then improved by using an adaptive, 2nd order accurate immersed boundary model (IBAMR), which enables us to apply more realized boundary conditions and wall bending stiffness. The mitral valve IBAMR model is shown to open and close properly, with the predicted flow rate closely agrees with experimental measurements.