Unsteady Base Flow Research Articles

Instability of the optimal edge trajectory in the Blasius boundary layer

In the context of linear stability analysis, considering unsteady base flows is notoriously difficult. A generalisation of modal linear stability analysis, allowing for arbitrarily unsteady base flows over a finite time, is therefore required. The recently developed optimally time-dependent (OTD) modes form a projection basis for the tangent space. They capture the leading amplification directions in state space under the constraint that they form an orthonormal basis at all times. The present numerical study illustrates the possibility to describe a complex flow case using the leading OTD modes. The flow under investigation is an unsteady case of the Blasius boundary layer, featuring streamwise streaks of finite length and relevant to bypass transition. It corresponds to the state space trajectory initiated by the minimal seed; such a trajectory is unsteady, free from any spatial symmetry and shadows the laminar–turbulent separatrix for a finite time only. The finite-time instability of this unsteady base flow is investigated using the 8 leading OTD modes. The analysis includes the computation of finite-time Lyapunov exponents as well as instantaneous eigenvalues, and of the associated flow structures. The reconstructed instantaneous eigenmodes are all of outer type. They map unambiguously the spatial regions of largest instantaneous growth. Other flow structures, previously reported as secondary, are identified with this method as relevant to streak switching and to streamwise vortical ejections. The dynamics inside the tangent space features both modal and non-modal amplification. Non-normality within the reduced tangent subspace, quantified by the instantaneous numerical abscissa, emerges only as the unsteadiness of the base flow is reduced.

High-Order-Accurate Incompressible Flow Solver for High-Fidelity Numerical Simulations and Linear Stability Analysis

The development of high-fidelity versatile incompressible Navier–Stokes solver that is applicable for both direct numerical simulations (DNSs) and linear global stability investigations is presented and discussed. The solver is based on a vorticity–velocity formulation of the Navier–Stokes equations for orthogonal curvilinear grids. A rigorous and efficient approach is proposed that guarantees the divergence-free condition for the velocity and vorticity fields. The code incorporates advanced numerical algorithms including compact finite differences in combination with a pseudo-spectral method, together with an efficient and high-order-accurate Poisson solver. Linear stability modules based on the linearized Navier–Stokes equations (LNSEs) tailored for primary and secondary instability analyses are developed and incorporated into the code, such that the solver can switch from DNS to LNSE in a consistent manner by turning off the nonlinear terms and vice versa. The linear framework includes all the nonparallel effects with respect to the steady/unsteady baseflow and the form of disturbances, and because it is formulated as an initial value problem, both convective and absolute/global instability mechanisms can be investigated. An additional significant advantage of the linear module is that, for the secondary stability analysis, time-periodic assumption is not required and/or the unsteady baseflow does not have to be computed a priori. In the present paper, results obtained with the new Navier–Stokes solver are compared with benchmark solutions for the flow past a static and oscillating circular cylinder. The new code then was employed for a three-dimensional DNS of the flow for a wing section with a modified NACA 6 series airfoil at a chord Reynolds number of . Furthermore, the capability of the linear stability modules was demonstrated by investigating the primary and secondary (convective and absolute) instability mechanisms for boundary layers and the flow past a circular cylinder.

Transient instability in long, tilted water columns with fast-settling, particle-laden layers

We study the stability of unsteady particle-laden flows in long, tilted water columns in batch settling mode, where the quasi-steady assumption of base flow no longer holds for the fast settling of particles. For this purpose, we introduce a settling time scale in the momentum and transport equations to solve the unsteady base flow, and utilise non-modal analysis to examine the stability of the disturbance flow field. The base flow increases in magnitude as the settling speed decreases and attains its maximum value when the settling speed becomes infinitesimal. The time evolution of the disturbance flow energy experiences an algebraic growth caused by the lift-up mechanism of the wall-normal disturbance, followed by an exponential growth owing to the shear instability of the base flow. The streamwise and spanwise wavenumbers corresponding to the peak energy gain are identified for both stages. In particular, the flow instability is enhanced as the Prandtl number increases, which is attributed to the sharpening of the particle-laden interface. On the other hand, the flow instability is suppressed by the increase in settling speed, because less disturbance energy can be extracted from the base flow. There exists an optimal tilted angle for efficient sedimentation, where the particle-laden flow is relatively stable and is accompanied by a smaller energy gain of the disturbance.

Optimal perturbations in viscous round jets subject to Kelvin–Helmholtz instability

Abstract

Stability of a temporally evolving natural convection boundary layer on an isothermal wall

The stability properties of a natural convection boundary layer adjacent to an isothermally heated vertical wall, with Prandtl number 0.71, are numerically investigated in the configuration of a temporally evolving parallel flow. The instantaneous linear stability of the flow is first investigated by solving the eigenvalue problem with a quasi-steady assumption, whereby the unsteady base flow is frozen in time. Temporal responses of the discrete perturbation modes are numerically obtained by solving the two-dimensional linearized disturbance equations using a ‘frozen’ base flow as an initial-value problem at various $Gr_{\unicode[STIX]{x1D6FF}}$, where $Gr_{\unicode[STIX]{x1D6FF}}$ is the Grashof number based on the velocity integral boundary layer thickness $\unicode[STIX]{x1D6FF}$. The resultant amplification rates of the discrete modes are compared with the quasi-steady eigenvalue analysis, and both two-dimensional and three-dimensional direct numerical simulations (DNS) of the temporally evolving flow. The amplification rate predicted by the linear theory compares well with the result of direct numerical simulation up to a transition point. The extent of the linear regime where the perturbations linearly interact with the base flow is thus identified. The value of the transition $Gr_{\unicode[STIX]{x1D6FF}}$, according to the three-dimensional DNS results, is dependent on the initial perturbation amplitude. Beyond the transition point, the DNS results diverge from the linear stability predictions as nonlinear mechanisms become important.

Dynamic linear response of a shock/turbulent-boundary-layer interaction using constrained perturbations

Comprehensive experimental and computational investigations have revealed possible mechanisms underlying low-frequency unsteadiness observed in spanwise homogeneous shock-wave/turbulent-boundary-layer interactions (STBLI). In the present work, we extend this understanding by examining the dynamic linear response of a moderately separated Mach 2.3 STBLI to small perturbations. The statistically stationary linear response is analysed to identify potential time-local and time-mean linear tendencies present in the unsteady base flow: these provide insight into the selective amplification properties of the flow at various points in the limit cycle, as well as asymmetry and restoring mechanisms in the dynamics of the separation bubble. The numerical technique uses the synchronized large-eddy simulation method, previously developed for free shear flows, significantly extended to include a linear constraint necessary for wall-bounded flows. The results demonstrate that the STBLI fosters a global absolute linear instability corresponding to a time-mean linear tendency for upstream shock motion. The absolute instability is maintained through constructive feedback of perturbations through the recirculation: it is self-sustaining and insensitive to external forcing. The dynamics are characterized for key frequency bands corresponding to high–mid-frequency Kelvin–Helmholtz shedding along the separated shear layer$(St_{L}\sim 0.5)$, low–mid-frequency oscillations of the separation bubble$(St_{L}\sim 0.1)$and low-frequency large-scale bubble breathing and shock motion$(St_{L}\sim 0.03)$, where the Strouhal number is based on the nominal length of the separation bubble,$L$:$St_{L}=fL/U_{\infty }$. A band-pass filtering decomposition isolates the dynamic flow features and linear responses associated with these mechanisms. For example, in the low-frequency band, extreme shock displacements are shown to correlate with time-local linear tendencies toward more moderate displacements, indicating a restoring mechanism in the linear dynamics. However, a disparity between the linearly stable shock position and the mean shock position leads to an observed asymmetry in the low-frequency shock motion cycle, in which upstream motion occurs more rapidly than downstream motion. This is explained through competing linear and nonlinear (mass depletion through shedding) mechanisms and discussed in the context of an oscillator model. The analysis successfully illustrates how time-local linear dynamics sustain several key unsteady broadband flow features in a causal manner.

Linear stability of a viscoelastic liquid flow on an oscillating plane

Linear stability of a viscoelastic liquid on an oscillating plane is studied for disturbances of arbitrary wavenumbers. The main aim is to extend the earlier study of Dandapat & Gupta (J. Fluid Mech., vol. 72, 1975, pp. 425–432) to the finite wavenumber regime, which has not been attempted so far in the literature. The Orr–Sommerfeld boundary value problem is formulated for an unsteady base flow, and it is resolved numerically based on the Chebyshev spectral collocation method along with the Floquet theory. The analytical solution predicts that U-shaped unstable regions appear in the separated bandwidths of the imposed frequency, and the dominant mode of the long-wave instability intensifies in the presence of the viscoelastic parameter. The numerical solution shows that oblique neutral curves come out from the branch points of the U-shaped neutral curves at finite wavenumber and continue with the imposed frequency until the curves cross the next U-shaped neutral curve. As a consequence, in the finite wavenumber regime, no stable bandwidth of the imposed frequency is predicted by the long-wavelength analysis. Further, in some frequency ranges, the finite wavenumber instability is more dangerous than the long-wave instability.

Directivity and intermittency in the nearfield of a Mach 1.3 jet

Local fluctuations in a Mach 1.3 cold jet are tracked to understand the genesis of nearfield directivity and intermittency. A newly developed approach leveraging two synchronized large-eddy simulations is employed to solve the forced Navier–Stokes equations, linearized about the evolving unsteady base flow. The results are summarized by exposing the effect of two acoustically significant turbulent regions: the lip-line and core collapse location. The near-acoustic field displays the clear signature of the two regions. However, for both regions, the nearfield evolution of the perturbation field is characterized by generation of intermittent wavepackets, which propagate into the near-acoustic field and gradually acquire their expected broadband and narrowband characteristics at sideline and downstream angles respectively. The simulations elucidate how higher frequencies are obtained in the sideline directions as lower frequencies are filtered out of the forcing fluctuations. Likewise, shallow-angle acoustic signals arise through filtering of high frequency content in that direction. The directivity and intermittency are connected to the filtering of scales by jet turbulence with empirical mode decomposition. The observations highlight the gradual evolution of seemingly random core turbulence into well-defined intermittent wavepackets in the nearfield of the jet. The manner in which centerline fluctuations are segregated into upstream, sideline, and downstream components is examined through narrowband correlations. A similar analysis for the lipline contribution shows primarily upstream and downstream patterns because of the larger structures in the shear layer.

Three-dimensional instabilities and optimal perturbations of a counter-rotating vortex pair in stratified flows

This paper investigates the three-dimensional instabilities and the optimal perturbations on a pair of horizontal counter-rotating Lamb-Oseen vortices in a vertically stably stratified flow. Two-dimensional (2D) simulations are first performed, showing that while the dipole moves vertically against the stratification the vortex parameters: the radius a∗, the separation distance b∗, and the circulation Γ∗ are solely function of the time rescaled by the Brunt-Väisälä frequency N, independently of the Froude number, when the Reynolds number is large enough. Here, the Froude number is Fr = W0/Nb0 with W0 the initial advection velocity of the dipole and b0 the initial separation distance between the two vortices. For weak and moderate stratifications (large Fr), the stratification acts on a long time scale compared to the advection time of the dipole implying that the 2D flow can be considered as quasi-steady. In that case, when three dimensional instabilities are added, a linear stability analysis of this 2D flow at different instants retrieves the instability peaks corresponding to the Crow instability for the long wavelengths and to the elliptic instability for the short wavelengths showing that the dynamics is almost unaffected by buoyancy effects. The Crow and elliptic instabilities scale with the instantaneous dipole parameters showing in particular that stratification promotes instability by reducing the distance b∗ between vortices [K. K. Nomura et al., “Short-wavelength instability and decay of a vortex pair in stratified fluid,” J. Fluid Mech. 553, 283-322 (2006)]. For strong stratifications (Froude numbers of order unity or smaller), the quasi-steady approximation is not valid, and the question of stability should be formulated in a different way, by, for example, searching for the transient growth of the energy of perturbation that may be computed for steady or unsteady base flow. Then, for each time horizon τ, we should determine the critical perturbation leading to the largest energy growth by the time τ. Presently, we compute the optimal perturbations at two time horizons τ = 4 and τ = 10 dimensionalized by 2πb02/Γ0 with a direct-adjoint technique which takes into account the evolution of the base flow. In the homogeneous case, this technique allows to investigate the effect of the weak unsteadiness of the flow due to viscous diffusion which induces a growth of the vortex core radius a∗. Both Crow and elliptic instabilities are retrieved in the optimal response and in the energy gain curves. Even if very slow, the viscous diffusion is found to increase the gain of the antisymmetric elliptic perturbation compared to the symmetric one. When the fluid is stratified, peaks at small wavenumber and at wavenumber of the order of the vortex core size are found for all Froude numbers with optimal responses strongly resembling, respectively, the Crow and the elliptic modes with optimal gains corresponding to mean growth rates larger than in the homogeneous case for both modes. However, as the strength of stratification increases (Froude numbers smaller than 2), optimal perturbations start departing from their homogeneous counterpart with large perturbation in the wake of the dipole associated with density effects.

A review of rotating cylinder wake transitions

Recent work on the flow past a rotating cylinder is reviewed and further investigated at low Reynolds numbers. The various two- and three-dimensional transitions that occur as the rotation rate is increased are detailed. Two steady states, steady state I and steady state II, are identified based on the physical characteristics of the wake and the drag force on the body. Steady state I occurs at lower rotation rates, while state steady state II occurs at higher rotation rates. Linear stability analysis shows that two three-dimensional modes become unstable on steady state I and steady state II. Floquet stability analysis of the unsteady base flows that occur at very low rotation rates shows the presence of five three-dimensional modes. The curves of marginal stability are presented, followed by a comparison of numerical simulations to their experimentally obtained counterparts. Furthermore, the spatio-temporal characteristics of each mode and the likely underlying physical mechanisms are briefly discussed.

The stability of the variable-density Kelvin–Helmholtz billow

We perform a three-dimensional stability analysis of the Kelvin–Helmholtz (KH) billow, developing in a shear layer between two fluids with different density. We begin with two-dimensional simulations of the temporally evolving mixing layer, yielding the unsteady base flow fields. The Reynolds number is 1500 while the Schmidt and Froude numbers are infinite. Then exponentially unstable modes are extracted from a linear stability analysis performed at the saturation of the primary mode kinetic energy. The spectrum of the least stable modes exhibits two main classes. The first class comprises three-dimensional core-centred and braid-centred modes already present in the homogeneous case. The baroclinic vorticity concentration in the braid lying on the light side of the KH billow turns the flow into a sharp vorticity ridge holding high shear levels. The hyperbolic modes benefit from the enhanced level of shear in the braid whereas elliptic modes remain quite insensitive to the modifications of the base flow. In the second class, we found typical two-dimensional modes resulting from a shear instability of the curved vorticity-enhanced braid. For a density contrast of 0.5, the wavelength of the two-dimensional instability is about ten times shorter than that of the primary wave. Its amplification rate competes well against those of the hyperbolic three-dimensional modes. The vorticity-enhanced braid thus becomes the preferred location for the development of secondary instabilities. This stands as the key feature of the transition of the variable-density mixing layer. We carry out a fully resolved numerical continuation of the nonlinear development of the two-dimensional braid-mode. Secondary roll-ups due to a small-scale Kelvin–Helmholtz mechanism are promoted by the underlying strain field and develop rapidly in the compression part of the braid. Originally analysed by Reinoud et al. (Phys. Fluids, vol. 12, 2000, p. 2489) from two-dimensional non-viscous numerical simulations, this instability is shown to substantially increase the mixing.

Stability of Transient Natural Convection

A theoretical analysis was carried out to determine the stability properties of unsteady natural convection. In this analysis a perturbation technique was used in conjunction with a special type unsteady base flow for transient natural convection between two infinite parallel vertical plates. Perturbation analysis yields a sequence of ordinary differential equations to describe the coupled time-dependent disturbance behavior. The equations were solved for a number of values of Grashof and wavenumbers and for a wide range of rate of acceleration and deceleration to determine the unsteady base flow stability characteristics. A growth factor was used to determine the stability properties, and it was found that an accelerating (decelerating) base flow is more (less) stable than a steady base flow when compared at common values of the instantaneous Grashof number. This tendency was found to increase rapidly at first and then to slowly decrease with increasing rate of change of the base flow.