Session Z10: Flow Instability: Transition to Turbulence (12:15pm - 1:00pm CST)

Influence of the inlet velocity profile on the flow stability in a 2D symmetric channel expansion

In a channel flow with a sudden expansion, whether for 3D tubular flow, for 3D channel flow and for 2D planar flow, it is known that increasing the Reynolds number beyond a critical value $Re_c$ induces a symmetry breaking Pitchfork bifurcation. The linear stability analysis of the symmetric steady solution enables to determine efficiently the $Re_c$ and thus explore the influence of the expansion ratio (ER), the ratio between upstream and downstream diameter regarding the expansion. In this study, using linear stability analysis and direct numerical simulations, we investigate the behaviour of the flow after 2D sudden expansions while varying the ER and the inlet flow profile, e.g. transition profiles between plug and Poiseuille flow that could be reached for flow after a sudden constriction. Results demonstrate that imposing a plug flow at the inlet gives a higher $Re_c$ than other profiles and that recirculation zones are shorter for a plug flow than for other profiles. We, then, show that these results can be rationalized using basic convection-diffusion arguments.   

Evolution of Conditionally-Averaged Second Order Structure Functions in a Transitional Boundary Layer

We consider boundary layer bypass transition and compute the evolution of the second-order structure function, $\langle du^2\rangle(\vec{x},\vec{r}) $ using DNS. In order to separate the contributions from laminar and turbulent events, we apply conditional sampling based on the local instantaneous intermittency. We then define and calculate two-point intermittencies, $\gamma^{(TT)}$, $\gamma^{(LL)}$ and $\gamma^{(TL)}$ which physically represent the probabilities that both points are in turbulent or laminar patches, or one in turbulent and the other in a laminar patch, respectively. Similarly, we define and calculate the conditionally-averaged structure functions, $\langle du^2 \rangle ^{(TT)}$, $\langle du^2 \rangle ^{(LL)}$ and $\langle du^2 \rangle ^{(TL)}$. It is found that in the transition region, laminar streaky structures maintain their geometrical characteristics in the physical and scale space well inside the transition region, even after the initial break down to form turbulent spots. Analysis of the $\langle du^2 \rangle ^{(TT)}$ fields reveal that the outer mode is the dominant secondary instability mechanism. Further analysis reveals how turbulence spots penetrate the boundary layer, approach the wall and grow in the streamwise and spanwise directions.   

Modal and non-modal stability of plane Couette-Poiseuille flow of a viscoplastic fluid overlying a porous layer

Hydrodynamic stability of Couette-Poiseuille flow of a viscoplastic fluid overlying a porous layer is attempted in the current study. Modal analysis reveal non-existence of any unstable eigenvalue, analogous to what is observed for viscoplastic fluid flow in a non-porous configuration. Therefore, we resort to non-modal analysis to identify possible transient amplifications of the flow system. The principal objective is to understand the intricate interaction between the Couette component and the porous layer parameters in shaping the characteristics of flow transition. The results obtained in the present work are quite different as compared to the corresponding ones reported in the literature pertaining to Newtonian flow. The primary cause for this is the contribution of yield stress, represented by the Bingham number, and its intricate interplay with the upper plate motion and the parameters of the porous layer (depth, permeability, anisotropy, inhomogeneity, etc.). We have also attempted to explore the possible physical mechanism that dictates the non-modal, short-lived amplifications owing to the interplay between the disturbance waves and the mean shear flow.   

The Orr mechanism in transition of parallel shear flow

The precise role of the Orr mechanism in transition of parallel shear flow is studied using the linear optimal perturbation for spanwise velocity. We find that the spanwise velocity of a small-amplitude perturbation can be mostly amplified via a lift-up effect induced by the Orr mechanism at the streamwise wavelength comparable to the given spanwise wavelength. The optimal perturbation for spanwise velocity is subsequently introduced into the plane Couette flow together with the optimal perturbation for all velocity components, and two transition scenarios are found via varying the amplitudes of the two optimal perturbations. The first scenario is the oblique transition, where the optimal spanwise velocity perturbation amplified in the very early stage of transition via the linear growth process mediates both streak amplification and breakdown. However, the role of the Orr mechanism is limited to the streak breakdown in the second transition scenario, streak transition. The oblique transition is more energetically efficient than the streak transition. Many similarities can be observed between the oblique transition and the transition triggered by the minimal seed \footnote{R. Kerswell, \textbf{Annual Review of Fluid Mechanics 50,} 319, 2018}.   

Wall distance influences on the stability and transition to turbulence of free shear layers separating at low Reynolds number

Using viscous linear stability calculations and experiments, we study the effects of wall proximity, represented by the shape factor $H$, on the laminar-to-turbulent transition of flows separating at Reynolds number 5-300 (momentum thickness based). The occurrence of transition depends on the disturbance growth rate in the separated shear layer which is mainly a function of the local shape factor and Reynolds number. For shape factors larger than 15, shear layers show large growth rates even at Re as low as 5. For smaller shape factors, however, instabilities are greatly reduced or eliminated, and the stabilizing effect of wall proximity increases as the Re number decreases. Transition was characterized experimentally at Reynolds number 20, using a thin flat plate with a thick half-ellipse leading edge. At $H_s$ = 22, the experimental growth of velocity disturbances is well predicted by linear stability theory up to turbulent flow. Simulations, however, show a laminar reattachment process beginning right after separation. In a second experiment with separation closer to the wall, $H_s$ = 9, laminar reattachment inhibited disturbance growth and transition to turbulence.   

On a new class of unstable skewed 3D modes for plane canonical shear flows

We conduct a linear stability analysis of the Asymptotic Suction Boundary Layer. For this case the resulting Orr-Sommerfeld equation allows for an exact analytic solution in terms of hypergeometric functions and, in turn, the related boundary conditions give rise to an algebraic eigenvalue problem (EVP), which has to be solved numerically. A remarkable feature was uncovered when expanding the EVP for asymptotically small streamwise wave numbers and large Reynolds number, where the expanded EVP yields solutions only in the distinguished limit when the product of the latter numbers is finite. This finding may provide an analytical link to the emergence of large-scale turbulent structures in plane shear flows at the presence of large Reynolds numbers. We further present the existence of novel ``skewed'' 3D stability modes in plane shear flows. The methodology to obtain these modes is obtained by extending the classical Squire transformation of 2D to 3D modes by assuming not only stream- but also spanwise spatial growth/decay. We can show that a new class of skewed unstable 3D modes are obtained even for canonical flows which are known to be modally stable, e.g. plane Couette flow.   

Experiments for characterizing the influence of free-stream turbulence length scales and intensity on boundary layer transition - EXCALIBUR

Since the thirties of the last century, a significant number of studies have been performed on free-stream turbulence (FST) induced boundary layer transition. Despite this, the basic problem in modelling this transition scenario is the lack of physical understanding. The primary intent of the study is to understand the influence of FST conditions namely, the integral length scale and turbulence intensity (Tu) on the boundary layer receptivity of the FST (low to high levels as much as 7{\%}). A new approach is investigated against the need to establish a transition detection method using electret sub-miniature microphones mounted in a cavity behind a pinhole on a flat plate. The current study relies on the information extracted from both single-point and two-point correlation measurements using hot-wire anemometers and a total of 300 microphones. Main results to be presented from this huge experimental campaign include data obtained at various free-stream velocities and initial FST conditions. The results show that, there is a clear dependence of FST conditions on the spanwise scale of elongated unsteady streamwise streaks. Further, a semi-empirical transition prediction model, that includes the effect of integral length scale and Tu will be presented.   

Stability of gravity-driven free-surface flow past a deformable solid: The role of depth-dependent modulus

The linear stability of a Newtonian liquid layer flowing down an inclined plane lined with a deformable linear elastic solid characterized by a continuously varying modulus is analyzed in this work. A low-$k$ asymptotic analysis is performed to obtain an expression for the wavespeed, which shows striking similarity with the earlier results of Sahu and Shankar [Phys. Rev. E $\textbf{94}, 013111 (2016)$] for gravity-driven flow of Newtonian fluid past solid bilayer having constant shear modulus in each layer. This shows that a deformable solid layer having a continuously varying shear modulus can be treated as a generalization of a system having multiple solid layers of constant shear modulus. Also, in the low-$k$ limit, we show that the stability of the free surface is governed by the value of effective shear modulus $G_{eff}$ and not by the detailed spatial variation of the modulus. For finite-wavenumbers, we analyzed different configurations of the modulus function that have the same spatially-averaged modulus but have different values at the interface and found the systems having higher shear modulus at the liquid-solid interface are more stable as compared to other configurations. Thus, the depth-dependent modulus offers more control to passively manipulate the instabilities.   

Puffs in the self-similar region of a low Reynolds number round jet: a new instability

Jets are ubiquitous in nature and are encountered in wide range of engineering application. We study the laminar to turbulence transition in the self-similar region of low Reynolds number (Re\textless 1000) round jets emanating from a long pipe nozzle through experiments and Linear Stability Theory (LST). For the first time, we observe puffs in the far-field, self-similar region of the jet through flow visualization which is further corroborated through particle image velocimetry measurements. We delineate three regimes: In Regime I (0\textless Re\textless 400) the jet remains stable, in Regime II (400\textless Re\textless 700) the flow is transitional and exhibits puffs and the helical instability and in Regime III (Re\textgreater 700) the flow rapidly transitions to turbulence near the nozzle exit. The helical mode is dominant in the fully developed region and prevails throughout Regime II. In contrast, puffs are less frequent and only observed in 400\textless Re\textless 550. We further show that the formation of puffs is set by a superposition of helical mode pair (n $=$ \textpm 1), predicted to be equally unstable in the fully developed region through LST.   

Emergence of puffs, weak and strong slugs from a stochastic predator-prey model for transitional turbulence with stream-wise shear interactions

In transitional pipe turbulence, a sequence of phases is observed experimentally in the range of Reynolds numbers between 1900 and 5000, passing through the laminar-turbulent transition at Re ~ 2040. These phases are characterized by transient decay of puffs (Re < 2040), puff-splitting and propagation (2040 < Re < 2250), expansion of turbulent regions via “weak slugs” (asymmetric upstream and downstream fronts, 2250 < Re < 4500), and via “strong slugs” (symmetric upstream and downstream fronts, Re > 4500). In earlier work, an intrinsically stochastic model for puff-decay and splitting accounted for the corresponding single-puff super-exponential timescales. This model was focused on the dynamics and fluctuations within a single puff and did not include stream-wise interactions arising through shear. Here we extend this model in a generic way to include these neglected interactions and show that the resulting model recapitulates the full phase diagram of the transition, successfully capturing the weak and strong slug behavior. The model is not restricted to one dimension and is extendable to other transitional shear flows.   

Couette flow transition in 2D directed percolation universality class

In Couette flow turbulence arises despite the linear stability of the laminar flow and the nature of this transition has remained unresolved despite numerous theoretical and experimental efforts. We here report an experimental study where the Couette flow is realized in a very narrow gap (negligible curvature) with azimuthal and axial aspect ratios of about 2000 full gaps. At the same time the periodic boundary conditions allow excessively long observation times. By investigating turbulent stripes at the onset of sustained turbulence we measure the critical exponents that fully characterize this transition. As shown the onset of turbulence in Couette flow falls into the two dimensional directed percolation universality class.   

An experimental approach to directed percolation in pipe ow.

The circumstance that pipe flows are turbulent in practice, while theoretical arguments imply they should remain laminar, has posed a major challenge in fluid mechanics. Recent mounting evidence that the onset of turbulence can be explained as a directed percolation phase transition finally offers a solution to the problem. However the extremely large time scales intrinsic to the transition in pipe flow make direct observations and a characterization of the universality class virtually impossible. We here circumvent these limitations by measuring all processes relevant to turbulence proliferation in experiments and by subsequently implementing them in a simple one dimensional model. The model clearly shows that longer range interactions between turbulent clusters, which had recently been found in experiments, strongly reduce the scaling range. At Reynolds numbers modestly close to the transition point the turbulent puff pattern enters a crystalline state that does not resemble the stochastic characteristics of a directed percolation process. Even closer to the transition point however, when considering excessive spatial and temporal scales, the stochastic nature is recovered. As shown the transition in pipe flow hence indeed falls into the directed percolation universality class.   

Directed percolation and puff crystallization near the transition to pipe turbulence

Both theory and recent experiments in a quasi-one-dimensional Couette cell suggest that the onset of turbulence is a non-equilibrium phase transition in the directed percolation (DP) universality class. However, it is not experimentally clear if this universality class applies to pipe flow, where single-puff time scales vary with Reynolds number in a super-exponential way instead of the expected power-law scaling. To see how puff interactions contribute to the critical behavior, we develop stochastic models of puff dynamics by inputting the interaction function measured in pipe experiments, and calculate the phase diagram and critical phenomena. In agreement with renormalization group predictions, we find strong evidence for critical scaling of the turbulent fraction in the DP universality class, with complex crossovers due to finite size effects and the presence of a crystal-like spatio-temporal pattern which results from the repulsion between puffs.