| This thesis investigates the nonlinear development and secondary instability of streaks or G(?)rtler vortices induced by elevated free-stream vortical disturbances(FSVD)in the boundary layers which form over flat or concave walls and may be subject to a pressure gradient.A study is also made of the impact of nonlinear streaks or G(?)rtler vortices excited by elevated FSVD on boundary-layer separation,which would occur in the absence of FSVD.We first consider the nonlinear evolution and secondary instability of G(?)rtler vortices induced by FSVD in a zero-pressure-gradient boundary layer.The focus is on lowfrequency(long-wavelength)components of FSVD,to which the boundary layer is most receptive.For simplification,FSVD are modelled by a pair of oblique modes with opposite spanwise wavenumbers±k3,and their intensity is strong enough(but still of low level)that the excitation and evolution of G(?)rtler vortices are nonlinear.For the general case that the G(?)rtler number GΛ(based on the spanwise wavelengthΛ of the disturbances)is O(1),the formation and evolution of G¨ortler vortices are governed by the nonlinear unsteady boundary-region equations(NUBRE),supplemented by appropriate upstream and far-field boundary conditions,which characterise the impact of FSVD on the boundary layer.This initial-boundary-value problem is solved numerically.FSVD excite steady and unsteady G(?)rtler vortices,which undergo nonmodal growth,modal growth and nonlinear saturation for FSVD of moderate intensity.However,for sufficiently strong FSVD the modal stage is bypassed.Nonlinear interactions cause G(?)rtler vortices to saturate,with the saturated amplitude being independent of FSVD intensity when GΛ 0.The predicted modified mean-flow profiles and structure of G(?)rtler vortices are in excellent agreement with several steady experimental measurements.As the frequency increases,the nonlinearly generated harmonic component(0,2)(which has zero frequency and wavenumber 2k3)becomes larger,and as a result the G(?)rtler vortices appear almost steady.The secondary instability analysis indicates that G(?)rtler vortices become inviscidly unstable in the presence of FSVD with a high enough intensity.Three types of inviscid unstable modes,referred to as sinuous(odd)modes I,II and varicose(even)modes I,are identified,and their relevance is delineated.The characteristics of dominant unstable modes,including their frequency ranges and eigenfunctions,are in good agreement with experiments.The secondary instability is intermittent when FSVD are unsteady and of low frequency.However,the intermittence diminishes as the frequency increases.We then investigate streaks and G(?)rtler vortices in a boundary layer over a flat or concave wall in a contracting or expanding stream,which provides a favourable or adverse pressure gradient,respectively.An important effect of a pressure gradient is that the oncoming FSVD are distorted by the non-uniform inviscid flow outside the boundary layer through convection and stretching.This process is accounted for by using the rapid distortion theory.The impact of the distorting FSVD is analysed to provide the appropriate initial and boundary conditions,which form,along with the NUBRE,the appropriate initial-boundary-value problem describing the excitation and nonlinear evolution of the vortices.Numerical results show that in the presence of an adverse/favourable pressure gradient G(?)rtler vortices saturate earlier/later,but the intensity of the saturated vortices is lower/higher than that in the case of zero pressure gradient.With the increase of the favourable pressure gradient,the nonlinearly generated harmonic component(0,2)dominates eventually,causing two mushrooms to appear within one spanwise period.On the other hand,for the same pressure gradient and at low levels of FSVD,the vortices saturate earlier and at a higher amplitude as the G(?)rtler number increases.Raising FSVD intensity reduces the effects of the pressure gradient and curvature on the growth and saturation of the vortices.At a high turbulence level of T u = 14%,the curvature does not impact the evolution of vortices,while the pressure gradient only influences the saturation intensity.The unsteadiness of FSVD is found to reduce the boundary-layer response significantly at low turbulence levels,but that effect weakens as the turbulence level increases.A secondary instability analysis of the vortices is performed for two moderate pressure gradients,one adverse and the other favourable.Three families of unstable modes have been identified,which may become dominant depending on the frequency and streamwise location.In the presence of an adverse pressure gradient,the secondary instability occurs earlier,but the unstable modes appear in a smaller band of streamwise wavenumber or frequency,and their growth rates are smaller.The opposite is true in the case of a favourable pressure gradient.The present theoretical framework,which accounts for the influence of the curvature,the turbulence level and the pressure gradient,allows for a detailed and integrated description of the key transition processes,and represents a useful step towards predicting,on the basis of first principles,the pre-transitional flow and transition itself of the boundary layer over a blade in turbo-machinery.We further study the elevated FSVD preventing the separation in a boundary layer over a plate or concave wall subject to streamwise adverse pressure gradients.It is assumed as before that the FSVD are strong enough,i.e.the turbulent Reynolds number is O(1),so that the streaks or G(?)rtler vortices generated in the boundary layer are fully nonlinear and can alter the mean-flow profile by order one amount.The excitation and evolution of streaks and G(?)rtler vortices are governed by the NUBRE supplemented by appropriate initial and boundary conditions.The flow variables are decomposed into two parts: the steady spanwise-averaged,and the unsteady or spanwise-varying components.These two parts are coupled and are computed simultaneously.Numerical results show that the separation is removed when FSVD exceeds a critical level.It is inferred that nonlinear streaks or G(?)rtler vortices prevent the separation.The critical turbulence level to eliminate the separation depends on the streamwise curvature,the pressure gradient and the frequency of FSVD.The critical turbulence level decreases significantly with the G(?)rtler number,indicating that the G(?)rtler vortices inhibit separation.A higher critical turbulence level is required to prevent the separation in the case of stronger adverse pressure gradient.The unsteady streaks excited by unsteady FSVD with lowfrequencies are found to be more effective in suppressing the separation,consistent with experimental findings.The present theoretical framework accounts for both the receptivity to physically realisable FSVD and the nonlinear evolution of the resulting streaks or G(?)rtler vortices.It may be exploited to develop an efficient and physics-based approach for controlling separation. |
