Numerical Experiment Setup

4.4.3

Numerical Experiment Setup

Numerical experiments were conducted to establish suitable set-up parameters of the unsteady model for further numerical investigations. The main result required was the measurement of Uc for different ¯L, obtained by increasing the free-stream velocity until

an instability was induced in the fluid-structure system. The ease of measuring Uc

was therefore of paramount importance when considering initial excitation to the fluid- structure system. It was therefore important to make the initial excitation to the fluid- structure system similar in shape to the critical mode as this would reduce transient time fromt= 0 to the steady state, hence facilitating the measurement of Uc. It is proposed

that the presented results illustrate that an initial second-eigenmode deflection as the form of initial excitation would yield the lowest transient time to the perturbed state and hence the greatest ease of measuring Uc. Other forms of excitation would take a

relatively longer amount of time for oscillations to develop, as observed in figure 4.21, the initial deflection in these experiments being an approximate form of pressure impulse. To determine a suitable level of discretisation, the experiment utilising 50 flexible-surface mass points depicted in figures 4.20(a) and (b) (reshown in figures 4.22(a) and (b)) was repeated using 200 flexible-surface mass points and a smaller time step.2 _{The results for}

the more accurate experiment, see figures 4.22(c) and (d), took a substantially longer computational time to obtain. The results showed the coarser case was more stable and predicted a higher Uc than the finer case, the difference in Uc prediction between

the fine and coarse case was approximately 3%. Owing to the reduced discretisation however the computational time of the experiment was significantly reduced, the coarser case’s run time approximately one twelfth of the time that it took to run the finer case. Otherwise, the results were in good agreement, confirming that the coarser case produced no numerically generated instabilities. It was proposed that the illustrated discrepancy was an acceptable level of error to incur for the benefit of greatly reduced run time for numerical experiments initially excited using a second-eigenmode deflection.

2

The ratio of node spacing relative to the time step (an approximate form of the von Neumann number) must be kept within certain limits to allow convergence of the iterative scheme.

4.4.3 Numerical Experiment Setup 90 0 0.2 0.4 0.6 0.8 1 −1 −0.5 0 0.5 1 Non−Dimensional Length Non−Dimensional Displacement (a) 0 0.5 1 1.5 −1 −0.8 −0.6 −0.4 −0.2 0 Non−Dimensional Time

Non−Dimensional Total Energy

(b) 0 0.2 0.4 0.6 0.8 1 −1 −0.5 0 0.5 1 Non−Dimensional Length Non−Dimensional Displacement (c) 0 0.5 1 1.5 −1 −0.8 −0.6 −0.4 −0.2 0 Non−Dimensional Time

Non−Dimensional Total Energy

(d)

Figure 4.22: Fluid-structure behaviour at critical velocity where (a) and (c) depict the oscillation of the flexible surface (the thick line is the initial deflection) and (b) and (d) are the respective total energies expended: in (a) and (b), ¯U = 5.452, fifty mass points; in (c) and (d), ¯U = 5.452, two-hundred mass points; ¯L = 1, initial deflection is the second eigenmode.

4.5 Summary 91

4.5

Summary

It is proposed that the full unsteady model has been validated. There are two effects on the pressure distribution along the flexible surface: flexible-surface curvature and channel blockage. The effect of curvature is kept constant by always using the same ratio of initial deflection to flexible-surface length in experiments. A discretisation of fifty mass points or panels on the flexible surface provides suitable accuracy for unsteady experiments initiated with a low harmonic-mode initial-deflection. The channel walls are far away from the central surface if ¯H _≥ 1. Although the unsteady model has only been quantitatively validated in the absence of channel walls and rigid inlet surface, the velocity model does produce qualitatively correct results with these effects included. Therefore it is proposed that the unsteady model can be used with confidence to generate new results with rigid inlet and channel-wall surfaces included.

Chapter 5

Numerical Experiments

Herein, are presented numerical experiments and new results that give an insight into the phenomena generated by the unsteady system when the fluid and structure interact for a range of ¯L, ¯U and model modifications. The steady-state oscillation for an isolated flexible surface at ¯L = 1 was described in Chapter 4. First, fluid-structure phenomena at this value of ¯L are studied in detail. The effects on these phenomena due to mod- ifications to the unsteady model, e.g. the introduction of a rigid central surface, are then investigated. The unsteady model is then used to represent the human snoring phenomenon; realistic oscillation frequencies of the flexible surface and temporally vary- ing free-stream velocity produce unique simulations of this fluid-structure interaction. Finally, fluid-structure phenomena in the range 0.01 _≤ L¯ _≤ 1000 are investigated and further applications of the unsteady model are discussed. It is noted that when varying

¯

L, ρf is varied instead of L and therefore the effect of discretisation is kept constant in

all numerical experiments.

5.1

Fluid-Structure Phenomena at

L¯

= 1

Fluid-structure interactions observed at ¯L= 1 over a range of ¯U for an isolated flexible surface are investigated with particular attention paid toUc, the shape of the steady state