Effect of Favourable Pressure Gradient on Turbulence in Boundary Layers

Abstract

This thesis explores the effects of favourable pressure gradient on the structure of turbulent boundary layers (TBL). In this context, the structure of three types of boundary layers namely a zero-pressure-gradient boundary layer, equilibrium boundary layers under favourable pressure gradient and relaminarising boundary layers is investigated mostly from the point of view of large-scale dynamics. This covers a whole range of flows on the so-called Reynolds number - pressure gradient diagram - from turbulent zero pressure gradient flows to relaminarising flows at relatively low Reynolds numbers. The study of turbulent and relaminarising boundary layers is carried out primarily using direct numerical analyses and some limited experiments in this thesis. The direct numerical simulations (DNS) of a zero-pressure-gradient turbulent boundary layer (ZPG TBL) is validated against the experimental and DNS data available in the literature. Furthermore, the important question of time-averaged signature of a large scale vortex structure and its relation with the two-point correlations in the context of ZPG TBL is addressed. In this context, a synthetic flow consisting of hairpin vortex structures is generated. The two-point correlations in the synthetic TBL and a real TBL are found to be qualitatively similar. This shows that the vortex structure leaves a time-averaged footprint in the form of correlations of velocity and vorticity. A study of two-point correlations in a real TBL shows that the structure angle deduced from two-point correlations varies with wall-normal location. The structure angle is small near the wall and increases away from the wall in agreement with the previous studies. The small angle close to the wall signifies the presence of streamwise structure. Away from the wall, this streamwise coherence is lost and the correlation contours become more isotropic. The presence of the wall and the mean shear affects smaller scales making them anisotropic close to the wall. Towards the edge of the boundary layer, smaller scales tend to become isotropic leading to -5/3 law in the energy spectrum. Further, a relation between a passive scalar in a flow and vorticity is explored. It is found that the scalar product of vorticity and scalar gradient is conserved in a non-diffusive situation. This assertion is demonstrated under various flow conditions. Despite the differences in Schmidt numbers, the structures observed in the outer layer are similar in both numerical and experimental flow visualisations. Further, the equilibrium turbulent boundary layers under favourable pressure gradient are studied. The numerical simulations of equilibrium sink flow TBL are validated against the experimental results of Dixit (2010). A study of two-point correlations reveals that the near-wall structure angle decreases with a favourable pressure gradient in sink flow TBLs. In the outer region, the loss of streamwise coherence occurs at a wall-normal location closer to the wall than in an ZPG TBL. Edge intermittency study reveals that the flow is non-turbulent beyond y/δ = 0.8 inside the mean boundary layer edge. The variation of the ratio of pressure gradient to Reynolds shear stress gradient shows that this ratio is very large (> 50) beyond y/δ = 0.8. The dominance of pressure gradient makes this part of sink flow TBL to behave like a Euler-region. Small scales in sink flow TBL tend to be isotropic near the edge of the boundary layer and spectra shows -5/3 law akin to ZPG TBL, albeit at lower Reynolds numbers. The concept of equilibrium is extended to flows with wall transpiration. The sink flow TBL is a special case of more generalised equilibrium TBLs with wall transpiration. Conditions required for the flow with wall transpiration are derived. It is observed that there is a systematic variation of various statistical properties with wall velocity. Further, it is observed that the motion in these equilibrium flows is purely active like in sink flow TBL. In equilibrium TBL, the Reynolds shear stress is directly related to mean velocity. So we have at our disposal an exact relation between the Reynolds shear stress and the mean velocity gradient without the need to do any ad-hoc modelling for the sink flow. This is an interesting observation from the point of view of modelling TBLs using eddy-viscosity. Eddy-viscosity model derived from sink flow TBL data is found to predict the mean velocity profiles in flows with wall transpiration with a sufficient accuracy. Similarly, it is plausible that any general non-equilibrium flow may be treated as a departure from equilibrium. With suitable modifications, eddy viscosity obtained from equilibrium TBL may be used to model them without invoking ad-hoc assumptions. Finally, the effect of initial Reynolds number on the process of relaminarisation is studied numerically and experimentally. ZPG TBLs with two different initial Reynolds number are subjected to different degrees of acceleration. However, the pressure gradient history is same in both the cases. It is observed that the flow with a higher initial Reynolds number relaminarises at a lower pressure gradient value than the flow with a lower initial Reynolds number. Assessment of different parameter criteria reveals that the criterion proposed by Narasimha & Sreenivasan (1973) is appropriate for the prediction of the onset of relaminarisation. Further, the structures in relaminarising flows are studied. The near-wall structure angle is found to decrease with the increasing FPG and the streamwise length of the structure also increases. The low and high speed streaks in the near-wall region are found to become longer and less undulating with an increase in the spanwise spacing. A stabilisation mechanism of near-wall streaks is also presented which suggests that the kinematic effect of mean vertical velocity directed towards the wall is responsible for the stabilisation of streaks.