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Title:  Instabilities In Supersonic Couette Flow 
Authors:  Malik, M 
Advisors:  Dey, Jyotirmoy 
Keywords:  Supersonic Flow (Aeronautics) Compressible Couette Flow Uniform Shear Flow Nonmodal Transient Growth Linear Stability Spatial Transient Growth Shear Flow  Stability Shear Flow Transition Compressible Flow (Aerodynamics) Supersonic Couette Flow Secondary Instability 
Submitted Date:  Jun2008 
Series/Report no.:  G22945 
Abstract:  Compressible plane Couette flow is studied with superposed small perturbations. The steady mean flow is characterized by a nonuniform shearrate and a varying temperature across the wallnormal direction for an appropriate perfect gas model. The studies are broadly into four main categories as said briefly below.
Nonmodal transient growth studies and estimation of optimal perturbations have been made. The maximum amplification of perturbation energy over time, G max, is found to increase with Reynolds number Re, but decreases with Mach number M. More specifically, the optimal energy amplification Gopt (the supremum of G max over both the streamwise and spanwise wavenumbers) is maximum in the incompressible limit and decreases monotonically as M increases. The corresponding optimal streamwise wavenumber, αopt, is nonzero at M = 0, increases with increasing M, reaching a maximum for some value of M and then decreases, eventually becoming zero at high Mach numbers. While the pure streamwise vortices are the optimal patterns at high Mach numbers (in contrast to incompressible Couette flow), the modulated streamwise vortices are the optimal patterns for lowtomoderate values of the Mach number. Unlike in incompressible shear flows, the streamwiseindependent modes in the present flow do not follow the scaling law G(t/Re) ~ Re2, the reasons for which are shown to be tied to the dominance of some terms (related to density and temperature fluctuations) in the linear stability operator. Based on a detailed nonmodal energy anlaysis, we show that the transient energy growth occurs due to the transfer of energy from the mean flow to perturbations via an inviscid algebraic instability. The decrease of transient growth with increasing Mach number is also shown to be tied to the decrease in the energy transferred from the mean flow (E1) in the same limit. The sharp decay of the viscous eigenfunctions with increasing Mach number is responsible for the decrease of E1 for the present mean flow.
Linear stability and the nonmodal transient energy growth in compressible plane Couette flow are investigated for the uniform shear flow with constant viscosity. For a given M, the critical Reynolds number (Re), the dominant instability (over all streamwise wavenumbers, α) of each mean flow belongs different modes for a range of supersonic M. An analysis of perturbation energy reveals that the instability is primarily caused by an excess transfer of energy from meanflow to perturbations. It is shown that the energytransfer from meanflow occurs close to the moving topwall for “mode I” instability, whereas it occurs in the bulk of the flow domain for “mode II”.For the Nonmodal transient growth anlaysis, it is shown that the maximum temporal amplification of perturbation energy, G max,, and the corresponding timescale are significantly larger for the uniform shear case compared to those for its nonuniform counterpart. For α = 0, the linear stability operator can be partitioned into L ~ L ¯ L +Re2Lp is shown to have a negligibly small contribution to perturbation energy which is responsible for the validity of the wellknown quadraticscaling law in uniform shear flow: G(t/Re) ~ Re2 . In contrast , the dominance of Lp is responsible for the invalidity of this scalinglaw in nonuniform shear flow. An inviscid reduced model, based on EllignsenPalmtype solution, has been shown to capture all salient features of transient energy growth of full viscous problem. For both modal and nonmodal instability, the viscositystratification of the underlying mean flow would lead to a delayed transition in compressible Couette flow.
Modal and nonmodal spatial growths of perturbations in compressible plane Couette flow are studied. The modal instability at a chosen set of parameters is caused by the scond leastdecaying mode in the otherwise stable parameter setting. The eigenfunction is accurately computed using a threedomain spectral collocation method, and an anlysis of the energy contained in the leastdecaying mode reveals that the instability is due to the work by the pressure fluctuations and an increased transfer of energy from mean flow. In the case of oblique modes the stability at higher spanwise wave number is due to higher thermal diffusion rate. At high frequency range there are disjoint regions of instability at chosen Reynolds number and Mach number. The stability characteristics in the inviscid limit is also presented. The increase in Mach number and frequency is found to further destabilize the unstable modes for the case of twodimensional(2D) perturbations. The behaviors of the noninflexional neutral modes are found to be similar to that of compressible boundary layer. A leading order viscous correction to the inviscid solution reveals that the neutral and unstable modes are destabilized by the noslip enforced by viscosity. The viscosity has a dual role on the stable inviscid mode. A spatial transient growth studies have been performed and it is found that the transient amplification is of the order of Reynolds number for a superposition of stationary modes. The optimal perturbations are similar to the streamwise invariant perturbations in the temporal setting. Ellignsen & Palm solution for the spatial algebraic growth of stationary inviscid perturbation has been derived and found to agree well with the transient growth of viscous counterpart. This inviscid solution captures the features of streamwise vortices and streaks, which are observed as optimal viscous perturbations.
The temporal secondary instability of mostunstable primary wave is also studied. The secondary growthrate is many fold higher when compared with that of primary wave and found to be phaselocked. The fundamental mode is more unstable than subharmonic or detuned modes. The secondary growth is studied by varying the parameters such as β, Re, M and the detuning parameter. 
URI:  http://hdl.handle.net/2005/758 
Appears in Collections:  Aerospace Engineering (aero)

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