Session A27: Flow Instability: Boundary Layers and Transition to Turbulence I

From aging to memoryless turbulent stripes in channel flow

Puffs, stripes, and spots that appear during the transition in wall-bounded shear flows have finite lifetimes, and as demonstrated in many studies, their decay is typically memoryless, i.e., they do not age, and the decay probability is constant in time. As shown recently, this standard result unexpectedly does not apply to turbulent stripes in channel flow. Their decay probability increases with time in sufficiently large channels where stripes fully localize. As it will be shown, the associated lifetime distributions correspond to what is known in population ecology as a type 1 aging process. In the present study, we explore the aspect ratio (width/height) dependence of the decay process of channel stripes. Starting from the classic planar channel case and approaching the square duct limit, the nature of the decay and the corresponding lifetime distributions change from aging to non-aging (i.e., memoryless).   

Properties of the Generalized Airy Functions of W.H. Reid

In a paper titled Composite Solutions to the Orr-Sommerfeld Equation (Studies in Appl Math., 51, pp 361-368, 1972) W.H. Reid introduced a family of functions that he called generalized Airy functions. Reid and his co-workers demonstrated the usefulness of these functions in a number of papers and, especially, in the treatise he co-authored with P.G. Drazin (Hydrodynamic Stability, Cambridge University Press, first edition 1981, second edition 2004) an appendix to which contains the fullest exposition of the theory of these functions. Two families of Reid functions are A_k(z,p,q) and B_k(z,p,q), in which k∈{1,2,3}, z and p are complex numbers, and q is a natural number. Reid denotes the special cases A_k(z,p,0) and B_k(z,p,0) in the two-argument form A_k(z,p) and B_k(z,p). Reid's definitions include that of one more function, B₀(z,p). §9.13(ii) of the National Institute of Standards and Technology Digital Library of Mathematical Functions (DLMF) includes the two-argument forms of Reid's functions. Reid's definitions of his functions are in the form of integral representations of Laplace type. The appendix to Hydrodynamic Stability includes a great deal of additional information including asymptotic expansions, recursion relations, and values at the origin z=0. Although this exposition was satisfactory for Reid's purposes it does not include Maclaurin series expansions about z=0. The present work fills this lacuna and, in the process expresses A_k(z,p,0) and B_k(z,p,0) in terms of the G-function of C.S. Meijer.   

Numerical investigation of laminar-turbulent transition for hypersonic blunt bodies with distributed roughness

Transitional and turbulent heating play a crucial role in the design and analysis of hypersonic vehicles. Distributed roughness alters the mean flow, e.g., generating locally separated wake regions depending on the roughness amplitude, which may enhance and/or introduce new boundary layer instability mechanisms. Previous numerical and experimental work have focused on flat plates and (blunt) cones, with recent experiments also exploring the effect of roughness on boundary layer transition in blunt body flows. However, there has been limited computational studies for boundary layer transition over hypersonic blunt bodies with roughness. In this work, we use direct numerical simulation (DNS) to study the effect of distributed surface roughness on the boundary layer laminar-to-turbulent transition process for a cylinder in hypersonic flow. We apply adaptive mesh refinement to achieve the necessary grid resolution required to capture transition to turbulence over stagnating blunt bodies at a tractable computational cost. We also investigate the impact of roughness amplitude and phasing on transition and compare results between two Navier-Stokes solvers, US3D and CHAMPS.   

Statistical Modelling of Laminar-to-Turbulent Transition in a Blasius boundary layer

Transient, non-modal growth of a perturbation can cause significant energy amplification and ultimately trigger transition to turbulence, even when the LNS operator is stable as defined by its eigenvalues. Transient growth is typically characterized by the optimal energy growth and initial condition.  However, Frame & Towne (2024) demonstrated via statistical analysis that, within a temporal stability framework, the optimal energy growth is generally much larger than the realized energy growth.  Our current work seeks to expand this statistical framework to spatial stability within a boundary layer. Using a compressible Blasius boundary layer as a proof of concept, we apply the statistical framework and show a disparity between the downstream responses of the optimal inlet condition and an ensemble of realizations of inlet conditions drawn from a PDF with a physically informed correlation length.  The downstream response is simulated using a linear One-Way Navier-Stokes solver and used to estimate the mean and PDF of the perturbation energy amplification. Given some transition threshold energy, we can then predict the likelihood of laminar-to-turbulent transition as a function of the streamwise direction.

 

Frame, P. & Towne, A. (2024). Beyond optimal disturbances: a statistical framework for transient growth. Journal of Fluid Mechanics, 983, A2.   

Temporal & Spatial Symmetry Breaking in K-type Boundary Layer Transition

We uncover the key symmetry breaking mechanisms in time and space within deterministic K-type boundary layer transition using a suite of techniques from modal decomposition and dynamical systems. Specifically, we employ space-time proper orthogonal decomposition (STPOD), spectral POD (SPOD), and D1 symmetry decomposition to identify coherent space-time structures with distinct optimality properties. Our analysis of the early transitional regime identifies a prototypical transition scenario characterized by dynamics on a periodic limit cycle, where all early pre-turbulence mechanisms are reproducible by a varying number of modes at the fundamental frequency and its harmonics. Our investigation in the late transitional and turbulent regimes beyond the skin friction maximum reveals the emergence of periodic and non-periodic, as well as symmetric and anti-symmetric, structures. We extract modes associated with the variance and instability of the fundamental limit cycle and demonstrate that it is their phase space dynamics and streamwise locations that promote the overall flow dynamics to develop increasing degrees of chaos and, ultimately, broadband turbulence. Our findings suggest that the modes responsible for symmetry breaking are identifiable as hydrodynamic instability mechanisms around the limit cycle of the fundamental harmonic coherent structure.   

The Nonlinear One-Way Navier-Stokes (NOWNS) Approach for High-Speed Boundary-Layer Flows

The Nonlinear One-Way Navier-Stokes (NOWNS) approach has recently been applied to low-speed Blasius boundary-layer flows, where it has been demonstrated to support non-modal and multi-modal effects, as well as strong nonlinearties, which can cause the Nonlinear Parabolized Stability Equations (NPSE) to fail. In NOWNS, a projection operator (based on the linearized Navier-Stokes equations) is applied to the nonlinear equations to remove upstream propagating modes, which results in a set of equations that can be solved efficiently in the frequency domain as a spatial initial-value problem. For hypersonic boundary-layer flows, linear OWNS has been demonstrated to accurately capture complex multi-modal effects where linear PSE fails. Non-modal and multi-modal effects are critical to high-speed boundary-layer transition, and a nonlinear tool that accurately models these effects is of critical importance to the development of optimized aerospace vehicles. Therefore, we seek to demonstrate that NOWNS can capture these effects when performing nonlinear instability and transition analysis of high-speed boundary-layer flows by validating against direct numerical simulation (DNS) results in the literature.   

A Systematic Investigation of the Moody Plot in Transitional Pipe Flow

The transition between smooth laminar and rough turbulent flow dates back over 150 years. Although significant progress has been made on the transitional flow regime from a fundamental perspective, applications in the engineering domain are less well developed. This is exemplified by the famous "Moody diagram", a plot of non-dimensional friction versus Reynolds number within which the transitional regime is usually indeterminate. Here we provide data from a novel method which establishes a systematic dependence for friction within the transitional regime. Specifically, we approach the transitional regime from above by reducing the flow speed from an initially turbulent flow state to try and circumvent the difficulties associated with initial conditions. We find that different gravity-driven pipe flow experiments yield a single curve corresponding to a maximum density of the transitional flow structures. We test the generality of this result using a mass displacement device to drive the flow through the pipe. Our investigation of the flow driven by a syringe produces a different curve, indicating that the method of driving the flow has a significant impact on the final states and the paths leading to them.   

On the analysis of laminar-turbulent transition in viscoelastic channel flows

It is well-known that adding small amounts of long-chain polymers to turbulent flows leads to a significant drag reduction compared to Newtonian flows. Therefore, numerous efforts have been made to understand the drag-reduction mechanisms. However, the study on the transitional behavior of viscoelastic flows remains limited as only early or delayed transition is often reported in the literature. In this study, we perform direct numerical simulations of viscoelastic turbulent channel flows to investigate the dynamics of the laminar-turbulent transition. Disturbance flows are added to perturb the laminar flow to induce the transition. Variables of interest are Reynolds number, polymer concentration, and perturbation magnitude. Preliminary results show that at low Reynolds numbers close to transition, the effect of polymer solutions is imperceptible. The probability of inducing the transition is similar for Newtonian and viscoelastic flows, regardless of the perturbation magnitude. However, at higher Reynolds numbers, viscoelastic flows appear to have a stabilizing effect, especially for higher perturbation magnitudes. At a given perturbation magnitude, polymer solutions show a lower probability of transition in comparison to the Newtonian flow. The effect of polymer concentration will also be discussed.   

Transitional pipe flow of shear-thinning fluids

Shear-thinning fluid properties can substantially delay the onset of turbulence in pipe flow. In Newtonian pipe flow, transition to turbulence is subcritical, resulting from finite amplitude disturbances. While experiments find a symmetry breaking instability of the laminar base flow prior to the onset of turbulence. Simulations using idealized shear-thinning fluids, such as the Carreau-Yasuda model applied in the present computations, do not encounter such instability. This suggests more complex fluid properties in the polymer solutions used in experiments.

The focus of the present study is the effect of the idealized shear-thinning case on the transition to turbulence. We focus on a recent observation that body forces, which generate sequivalent profile shapes as the shear-thinning case, in general suppress spatio-temporal intermittency and that eventually the transition to turbulence encountered does not involve puffs or slugs that dominate the transition regime in ordinary pipe flow. Instead a sharp, discontinuous transition has been suggested for pipe flows subject to body forces. We address the question if despite the different underlying physical mechanism puffs and slugs may also be suppressed for the case of sufficiently strong shear-thinning and if the transition may eventually switch from continuous to discontinuous.    

The effect of wall suction on optimal perturbations in transitional boundary layers

This work applies structured input-output analysis to investigate how asymptotic suction control alters the types of perturbations most likely to trigger transition, i.e., the optimal perturbations. This method is first shown to identify structures consistent with previous analyses of nonlinear optimal perturbations and direct numerical simulations in both Blasius and asymptotic suction boundary layers. A comparison of the results at $Re_{\delta^*}=610$ demonstrates that, as expected, the asymptotic wall suction reduces the flow sensitivity to transition-inducing disturbances and in particular suppresses the Tollmien-Schlichting (TS) waves. Moreover, the structures associated with asymptotic suction boundary layer have larger streamwise and spanwise extent compared to Blasius boundary layer flows.