Streamwise Perturbations Research Articles

Small perturbation evolution in compressible Poiseuille flow: pressure–velocity interactions and obliqueness effects

Small perturbation evolution in compressible Poiseuille flow is contrasted against the incompressible case using direct simulations and non-modal linear analysis. The onset of compressibility effects leads to a profound change in the behaviour of pressure and its interaction with the velocity field. Linear analysis shows that the most significant compressibility outcome is the harmonic coupling between pressure and wall-normal velocity perturbations. Oscillations in normal perturbations can lead to periods of negative production causing suppression of perturbation growth. The extent of the influence of compressibility can be characterized in terms of an effective gradient Mach number ($M_{g}^{e}$). Analysis shows that $M_{g}^{e}$ diminishes as the angle of the perturbation increases with respect to the shear plane. Direct numerical simulations show that streamwise perturbations, which would lead to Tollmien–Schlichting instability in the incompressible case, are completely suppressed in the compressible case and experience the highest $M_{g}^{e}$. At the other extreme, computations reveal that spanwise perturbations, which experience negligible $M_{g}^{e}$, are entirely unaltered from the incompressible case. Perturbation behaviour at intermediate obliqueness angles is established. Moreover, the underlying pressure–velocity interactions are explicated.

A statistical state dynamics-based study of the structure and mechanism of large-scale motions in plane Poiseuille flow

The perspective of statistical state dynamics (SSD) has recently been applied to the study of mechanisms underlying turbulence in a variety of physical systems. An SSD is a dynamical system that evolves a representation of the statistical state of the system. An example of an SSD is the second-order cumulant closure referred to as stochastic structural stability theory (S3T), which has provided insight into the dynamics of wall turbulence, and specifically the emergence and maintenance of the roll/streak structure. S3T comprises a coupled set of equations for the streamwise mean and perturbation covariance, in which nonlinear interactions among the perturbations has been removed, restricting nonlinearity in the dynamics to that of the mean equation and the interaction between the mean and perturbation covariance. In this work, this quasi-linear restriction of the dynamics is used to study the structure and dynamics of turbulence in plane Poiseuille flow at moderately high Reynolds numbers in a closely related dynamical system, referred to as the restricted nonlinear (RNL) system. Simulations using this RNL system reveal that the essential features of wall-turbulence dynamics are retained. Consistent with previous analyses based on the S3T version of SSD, the RNL system spontaneously limits the support of its turbulence to a small set of streamwise Fourier components, giving rise to a naturally minimal representation of its turbulence dynamics. Although greatly simplified, this RNL turbulence exhibits natural-looking structures and statistics, albeit with quantitative differences from those in direct numerical simulations (DNS) of the full equations. Surprisingly, even when further truncation of the perturbation support to a single streamwise component is imposed, the RNL system continues to self-sustain turbulence with qualitatively realistic structure and dynamic properties. RNL turbulence at the Reynolds numbers studied is dominated by the roll/streak structure in the buffer layer and similar very large-scale structure (VLSM) in the outer layer. In this work, diagnostics of the structure, spectrum and energetics of RNL and DNS turbulence are used to demonstrate that the roll/streak dynamics supporting the turbulence in the buffer and logarithmic layer is essentially similar in RNL and DNS.

On the Nonlinear Stability of Plane Parallel Shear Flow in a Coplanar Magnetic Field

Lyapunov direct method has been used to study the nonlinear stability of laminar flow between two parallel planes in the presence of a coplanar magnetic field for streamwise perturbations with stress-free boundary planes. Two Lyapunov functions are defined. By means of the first, it is proved that the transverse components of the perturbations decay unconditionally and asymptotically to zero for all Reynolds numbers and magnetic Reynolds numbers. By means of the second, it is showed that the other components of the perturbations decay conditionally and exponentially to zero for all Reynolds numbers and the magnetic Reynolds numbers below \(\frac{\pi ^2}{2M}\), where M is the maximum of the absolute value of the velocity field of the laminar flow.

Sensitivity of two-dimensional flow past transversely oscillating cylinder to streamwise cylinder oscillations

In this paper we study the sensitivity of flow past a transversely oscillating cylinder to streamwise harmonic perturbations. The value of the Reynolds number is equal to 150, for which the flow is two-dimensional. We start with a transversely oscillating cylinder and then impose a small streamwise (in-line) perturbation with a frequency equal to twice the transverse oscillation frequency. The cylinder is thus following an eight-shaped trajectory, which can be traversed in a counter-clockwise or clockwise direction. For low values of the in-line amplitude, we find for the counter-clockwise mode that the power transfer from the fluid to the structure increases with the amplitude of oscillation in the streamwise direction, even though the magnitude of the fluctuations of the forces is decreased. For the clockwise mode of motion we observe the reverse trend, i.e., the power transfer from the fluid to the structure is decreased, even though the magnitude of the fluctuations of the forces is increased. It is shown that the variation of the power transfer in both types of motion is due primarily to the variation of the phase between the transverse oscillation of the cylinder and the vortex lift force as a result of the in-line oscillation. For higher values of the streamwise amplitude, the contribution of the fluctuating drag to the power transfer also becomes important, especially for the counter-clockwise mode. Both modes of oscillation are characterized by the presence of the third harmonic in the spectrum of the lift force as the in-line oscillation is increased and by the appearance of a combination of single vortices on the one side of the wake, and vortex pairs on the other side.

A minimal model of self-sustaining turbulence

In this work, we examine the turbulence maintained in a Restricted Nonlinear (RNL) model of plane Couette flow. This model is a computationally efficient approximation of the second order statistical state dynamics obtained by partitioning the flow into a streamwise averaged mean flow and perturbations about that mean, a closure referred to herein as the RNL∞ model. The RNL model investigated here employs a single member of the infinite ensemble that comprises the covariance of the RNL∞ dynamics. The RNL system has previously been shown to support self-sustaining turbulence with a mean flow and structural features that are consistent with direct numerical simulations (DNS). Regardless of the number of streamwise Fourier components used in the simulation, the RNL system’s self-sustaining turbulent state is supported by a small number of streamwise varying modes. Remarkably, further truncation of the RNL system’s support to as few as one streamwise varying mode can suffice to sustain the turbulent state. The close correspondence between RNL simulations and DNS that has been previously demonstrated along with the results presented here suggest that the fundamental mechanisms underlying wall-turbulence can be analyzed using these highly simplified RNL systems.

Stabilizing action of pressure in homogeneous compressible shear flows: effect of Mach number and perturbation obliqueness

AbstractCompressibility exerts a stabilizing influence on a variety of high-speed shear flows such as turbulent mixing layers, transitioning boundary layers and homogeneously sheared turbulence. An important stabilizing feature that is common amongst all shear flows is the velocity–pressure interaction dynamics. In this study, velocity–pressure interactions of individual perturbation or fluctuation modes are investigated using direct numerical simulations and linear analysis in high-Mach-number homogeneous shear flow. For a given perturbation wave mode, the action of pressure is shown to depend on two important factors: the orientation of the perturbation wavevector with respect to the shear plane and the Mach number. It is shown that the streamwise perturbation wave mode rapidly develops a high level of kinetic energy but is self-limiting owing to the action of pressure. On the other hand, the energy of spanwise perturbation wave modes grows unaffected by pressure or Mach number. Oblique modes combine spanwise and streamwise characteristics and are shown to be chiefly responsible for stabilizing effects seen in shear flows. Three regimes of obliqueness of different linear stability characteristics are identified. The critical role of perturbation obliqueness on stabilization is established.

Turbulence in the highly restricted dynamics of a closure at second order: comparison with DNS

S3T (Stochastic Structural Stability Theory) employs a closure at second order to obtain the dynamics of the statistical mean turbulent state. When S3T is implemented as a coupled set of equations for the streamwise mean and perturbation states, nonlinearity in the dynamics is restricted to interaction between the mean and perturbations. The S3T statistical mean state dynamics can be approximately implemented by similarly restricting the dynamics used in a direct numerical simulation (DNS) of the full Navier-Stokes equations (referred to as the NS system). Although this restricted nonlinear system (referred to as the RNL system) is greatly simplified in its dynamics in comparison to the associated NS, it nevertheless self-sustains a turbulent state in wall-bounded shear flow with structures and dynamics comparable to those observed in turbulence. Moreover, RNL turbulence can be analysed effectively using theoretical methods developed to study the closely related S3T system. In order to better understand RNL turbulence and its relation to NS turbulence, an extensive comparison is made of diagnostics of structure and dynamics in these systems. Although quantitative differences are found, the results show that turbulence in the RNL system closely parallels that in NS and suggest that the S3T/RNL system provides a promising reduced complexity model for studying turbulence in wall-bounded shear flows.

Conditional sampling of transitional boundary layers in pressure gradients

AbstractStatistical analysis of transitional boundary layers in pressure gradients is performed using the flow fields from direct numerical simulations of bypass transition. Laminar–turbulent discrimination separates the streaky laminar flow from turbulent regions. Individual streaks are identified and tracked in the flow field in order to obtain statistics of the amplitude of the streak population. An extreme value model is proposed for the distribution of streak amplitudes. It is also possible to differentiate those streaks which break down into turbulent spots from innocuous events. It is shown that turbulence onset is due to high-amplitude streaks, with streamwise perturbation velocity exceeding 20 % of the free stream speed. The resulting turbulent spots are tracked downstream. The current analysis allows for the measurement of the lateral spreading angles of individual spots and their spatial extent and volumes. It is demonstrated that the volumetric growth rate of turbulent spots is insensitive to pressure gradient.

Turbulent structure of air flow over wind-induced gravity waves

This is the second paper in a group of three that reports the systematic measurements of wind-generated water waves in a wind tunnel experiment. Here, the structure of the boundary layer on the air side of the water–air interface was analysed and compared with the boundary layer over a smooth plane rigid wall. The contribution of the wave-induced Reynolds stress was detected through filtering the spectrum of velocity fluctuations. Wave-induced Reynolds stresses became negligible for z > 5 H rms. The intermittency factor in the boundary layer over water waves was similar to that in a boundary layer over a rigid plane wall, with several differences near the interface. Here, the presence/absence of water damps out the turbulence. The quadrant analyses revealed that ejection and sweep events were dominant and more concentrated. At small fetches, the large-amplitude negative streamwise perturbations were preferentially lifted. Turbulence energy production peaked at z/δ = 0.2 and had a distribution similar to that observed for a self-preserving boundary layer with a strong adverse gradient pressure. The quadrant analysis contribution to the energy production revealed that ejections still dominated the balance and that the production was spatially modulated in the wind direction with a couple of cells and with a minimum in the area of the free surface wave height reduction.

An experimental investigation of the recirculation zone formed downstream of a forward facing step

An experimental investigation of the recirculation zone formed downstream of a forward facing step immersed in a turbulent boundary layer has been undertaken using particle image velocimetry. Bluff body flow is observed with the fixed separation point located at the leading edge of the step. The recirculation region dimensions are characterised over a range of Reynolds numbers (1400–19 000), with Re h based on the step height and the free stream velocity. Turbulent perturbations are produced in the free shear layer which develops between the recirculating flow close to the step and the free stream flow. Contour maps of amplification factor, streamwise perturbation velocity and Reynolds stresses are constructed, providing insight into optimal placement of structures within such topographical features. The mechanisms affecting the reattachment distance, namely the turbulent mixing within the boundary layer and the velocity deficit in the boundary layer, are discussed.

Linear temporal and spatio-temporal stability analysis of a binary liquid film flowing down an inclined uniformly heated plate

Temporal and spatio-temporal instabilities of binary liquid films flowing down an inclined uniformly heated plate with Soret effect are investigated by using the Chebyshev collocation method to solve the full system of linear stability equations. Seven dimensionless parameters, i.e. the Kapitza, Galileo, Prandtl, Lewis, Soret, Marangoni, and Biot numbers (Ka,G,Pr,L, χ,M,B), as well as the inclination angle (β) are used to control the flow system. In the case of pure spanwise perturbations, thermocapillary S- and P-modes are obtained. It is found that the most dangerous modes are stationary for positive Soret numbers (χ≥0), and oscillatory for χ<0. Moreover, the P-mode which is short-wave unstable for χ=0 remains so for χ<0, but becomes long-wave unstable for χ>0 and even merges with the long-wave S-mode. In the case of streamwise perturbations, a long-wave surface mode (H-mode) is also obtained. From the neutral curves, it is found that larger Soret numbers make the film flow more unstable as do larger Marangoni numbers. The increase of these parameters leads to the merging of the long-wave H- and S-modes, making the situation long-wave unstable for any Galileo number. It also strongly influences the short-wave P-mode which becomes the most critical for large enough Galileo numbers. Furthermore, from the boundary curves between absolute and convective instabilities (AI/CI) calculated for both the long-wave instability (S- and H-modes) and the short-wave instability (P-mode), it is shown that for small Galileo numbers the AI/CI boundary curves are determined by the long-wave instability, while for large Galileo numbers they are determined by the short-wave instability.

Symmetric vortex shedding in the near wake of a circular cylinder due to streamwise perturbations

Symmetric perturbations imposed on cylinder wakes may result in a modification of the vortex shedding mode from its natural antisymmetric, or alternating, to a symmetric one where twin vortices are simultaneously shed from both sides of the cylinder. In this paper, the symmetric mode in the wake of a circular cylinder is induced by periodic perturbations imposed on the in-flow velocity. The wake field is examined by PIV and LDV for Reynolds numbers about 1200 and for a range of perturbation frequencies between three and four times the natural shedding frequency of the unperturbed wake. In this range, a strong competition between symmetric and antisymmetric vortex shedding occurs for the perturbation amplitudes employed. The results show that symmetric formation of twin vortices occurs close to the cylinder synchronized with the oscillatory component of the flow. The symmetric mode rapidly breaks down and gives rise to an antisymmetric arrangement of vortex structures further downstream. The downstream wake may or may not be phase-locked to the imposed oscillation. The number of cycles for which the symmetric vortices persist in the near wake is a probabilistic function of the perturbation frequency and amplitude. Finally, it is shown that symmetric shedding is associated with positive energy transfer from the fluid to the cylinder due to the fluctuating drag.

The role of streamwise perturbations in pipe flow transition

The phenomenon of subcritical transition in Hagen-Poiseuille or pipe flow is explored for a wide range of Reynolds numbers within the interval Re∊[2.5×103,1.26×104] by means of a computational method that numerically resolves the transitional dynamics with nearly 3.5×104 degrees of freedom on a medium aspect-ratio domain of length 32π∕5. The aim of this exploration is to provide a theoretical characterization of the basin of attraction of the basic regime by measuring the minimal amplitude of an initial global perturbation leading to transition. The analysis is based on a particular theoretical scenario that considers streamwise-independent finite amplitude initial vortical perturbations that trigger global transition via optimal inflectional instabilities of streamwise-dependent modes with selected axial wave numbers. Disturbances consisting of 1, 2, and 3 pairs of vortices are investigated. Special attention is given to relaminarization phenomena that is frequently observed for low Reynolds numbers. Long lasting turbulent regimes and relaminarized flows are distinguished by means of time integrations of suitable length between Tmin=600 and Tmax=1000 advective time units. Some transitional runs are specifically analyzed to exemplify the transition scenario under investigation and its independence of pipe length is verified with a few computations on a longer pipe of length 32π (1.4×105 degrees of freedom). For large values of the Reynolds number, a theoretical scaling law for the threshold amplitude of a perturbation required to trigger transition is provided. Different types of perturbations seem to respond to different scaling laws.

Large eddy simulations of spatially growing subsonic and supersonic turbulent round jets

This paper presents an application of large eddy simulations (LESs) using the filtered structure function model to spatially developing compressible round jets issuing from a perturbed upstream velocity profile close to a top hat. For centreline Mach number M = 0.9 and Reynolds number Re = 3600, the numerical solution compares satisfactorily against a forced-jet direct numerical simulation (DNS) and experimental data, both previously reported. High Reynolds number (Re = 36 000) ‘free’ jets at Mach 0.7 (case 1) and 1.4 (case 2) are studied. Here, an isotropic random white-noise perturbation is superposed on the upstream velocity. The Mach 0.7 jet has a convective Mach number of 0.35, and is weakly affected by compressibility. In this case, axisymmetric vortex rings are first shed from the nozzle and undergo alternate pairing further downstream. Then turbulence develops. The centreline velocity decay and some other statistical quantities are, in the self-similarity region, in very good agreement with previous incompressible experiments. At Mach 1.4, an impressive upstream reduction of the jet spreading rate is observed, due to an important delay of Kelvin–Helmholtz instability due to compressibility effects. Alternate pairing occurs immediately, and vortices are much more elongated in the flow direction. Further downstream, the jet becomes subsonic, develops into turbulence and spreads out again at a rate comparable with its subsonic counterpart. The potential-core length is increased by 27% from the subsonic to the supersonic case. This is in agreement with several laboratory experiments. Finally, the effects of Mach number increase upon various statistical quantities such as Reynolds stresses and radial lengthscale are studied. Results compare favourably against some experiments and temporal DNSs. From the point of view of Lumley's anisotropy invariant map evaluated on the whole physical domain, the Mach 0.7 jet is dominated by axisymmetric structures and the Mach 1.4 jet by streamwise perturbations.

Free-Surface–Vorticity Interactions in an Open Channel Flow

The interactions between vortical structures, free surface, and background shear in an open channel flow is experimentally investigated. Deterministic structures are generated by a vertical injection from the channel bottom and a number of disturbances characterized by different jet amplitudes and Reynolds number are considered. Three cases are selected as representatives of the whole phenomenology and their evolution is discussed in detail. The streamwise perturbation is visualized by fluorescein while the streamwise component of velocity is measured by a laser Doppler anemometer. The perturbation in the cross stream plane is analyzed quantitatively and the velocity field is reconstructed on the symmetry plane, to extract the traces of the individual vortical structures. Depending on the intensity of the external perturbation, the structures either evolve as in the absence of the interface (small intensity) or they approach the free surface to undergo a topological change due to their reconnection with the free surface. The process originating in the so-called blockage and viscous layers is analyzed in detail. As a major conclusion, the interaction with the free surface is found to deplete the velocity gradients generated by the liftup of the quasi-streamwise vortices. This effect is suggested to be a major cause of stabilization of the open channel flow, with respect to a closed channel at the same Reynolds number.

Optimal linear growth in the asymptotic suction boundary layer

A variational technique in the temporal framework is used to study initial configurations of disturbance velocity which maximize perturbation kinetic energy in the asymptotic suction boundary layer (ASBL). These optimal perturbations (OP) excite significant and remarkably persistent transient growth, on the order of that which occurs in the Blasius boundary layer (BBL). In contrast, classical modal analysis of the ASBL predicts a critical Reynolds number two orders of magnitude larger than that for the BBL. As in other two-dimensional boundary layer flows, disturbances undergoing maximum amplification are infinitely elongated in the direction of the flow and take the form of streamwise-oriented vortices which induce strong variations in the streamwise perturbation velocity (streaks). The Reynolds number dependence of the maximum growth, and the best choice of scaling for the spanwise wavenumber of the perturbation causing it, are elucidated. There is good agreement between the streak resulting from OP and disturbances measured in experiments in which the asymptotic suction boundary layer is subject to free stream turbulence (FST). This agreement is shown to improve as the level of FST increases.

Receptivity of the Blunt-Leading-Edge Plate Boundary Layer to Unsteady Vortex Perturbations

The response of the boundary layer on a plate with a blunt leading edge to frozen-in vortex perturbations whose vorticity is normal to the plate surface is found. It is shown that these vortices generate an inhomogeneity of the streamwise velocity component in the boundary layer. This inhomogeneity is analogous to the streaky structure developing as the degree of free-stream turbulence increases. The dependence of the amplitude and shape of the boundary layer inhomogeneity on the distance from the leading edge, the streamwise and spanwise scales, and other parameters is found for periodic and local initial perturbations. It is shown that the receptivity of the boundary layer decreases with increase in the frequency and with decrease in the streamwise perturbation scale.

The role of microstructure in taming the Rayleigh capillary instability of cylindrical jets

It is observed both in nature and in technological processes that filaments with anisotropic molecular-scale structure are less susceptible to breakup due to capillary instability than homogeneous, isotropic fluids in similar filament flows. Here we provide rigorous evidence that the strong coupling of microstructure to the hydrodynamics of cylindrical axisymmetric free surface filaments, indeed fundamentally alters the linearized stability of cylindrical jets. We extend Rayleigh's classical inviscid analysis of cylindrical jets to the three-dimensional (3D), macroscopic flow-orientation equations derived from the Doi kinetic theory for liquid crystalline polymers (LCPs). These equations assume rigid rod-like molecules and incorporate LCP effects of molecular relaxation, anisotropic drag, polymer kinetic energy, LCP density, and an intermolecular potential which couple orientation dynamics to standard free surface fluid equations. Depending on the LCP density, there are between one and three flow-independent orientation equilibria which persist in a constant-velocity, cylindrical free surface flow: an isotropic phase exists at all concentrations, whereas two anisotropic phases exist at sufficiently high LCP density. These equilibrium LCP cylindrical jets have two independent sources of instability, hydrodynamic and orientational, each identified within the coupled flow/orientation free surface equations. For this paper we restrict to equilibria free of orientational instabilities. All streamwise perturbations of wavelength greater than the jet circumference are unstable to capillary instability; only the strength of the instability and most dominant wavelength are affected by LCP microstructure. The degree to which microstructure reduces the capillary instability depends on two critical scaling parameters: an LCP capillary number Ca lcp (a ratio of LCP-induced surface stress to interfacial capillary stress); and the anisotropic drag/friction parameter σ. The most striking result is: for sufficiently large Ca lcp and highly anisotropic drag ( σ ∼ 0) the capillary growthrate can be uniformly lowered, arbitrarily close to zero. For sufficiently small Ca lcp, all capillary-dominated growthrates are reduced, but are bounded below in terms of an explicit, sharp estimate and bounded above by the Rayleigh formula. The upshot is: inviscid LCP jets are predicted to yield bigger drops which form on longer timescales than an inviscid isotropic fluid with the same surface tension.

Linear and nonlinear evolution of boundary layer instabilities generated by acoustic-receptivity mechanisms

Experimental results are presented concerning the evolution of instabilities generated by the interaction between low-level broad-band acoustic waves and small two-dimensional roughness elements. Streamwise perturbation velocity spectra are measured and it is found that on the smooth plate, naturally-occurring Tollmien-Schlicting (T-S) waves grow in a manner consistent with a resonant subharmonic wave interaction. However, in the presence of small two-dimensional roughness elements, a strong primary T-S mode is forced by an interaction with the background acoustic field. This leads to a K-type of nonlinear interaction characterized by the generation of harmonics (up to six harmonics are observed) at the expense of the subharmonic. The scaling of the T-S waves and their harmonics with the number and amplitude of the wall roughness is also considered.

On the Disintegration of the Jet in a Plunging Breaker

Abstract An inviscid mechanism is proposed for the breakup of the jet in a plunging surface wave. Streamwise perturbations of the original surface are shown to grow rapidly owing to stretching of the thin jet and to drastic reduction in the normal pressure gradient. This converts transverse gravity waves into almost pure capillary waves. Conservation of wave action for the perturbations then implies a strong increase in the perturbation amplitude.