JFM Papers
Similarity and structure of wall turbulence with lateral wall shear stress variations
- D. Chung, J. P. Monty, N. Hutchins
-
- Published online by Cambridge University Press:
- 23 May 2018, pp. 591-613
-
- Article
- Export citation
-
Wall-bounded turbulence, where it occurs in engineering or nature, is commonly subjected to spatial variations in wall shear stress. A prime example is spatially varying roughness. Here, we investigate the configuration where the wall shear stress varies only in the lateral direction. The investigation is idealised in order to focus on one aspect, namely, the similarity and structure of turbulent inertial motion over an imposed scale of stress variation. To this end, we analyse data from direct numerical simulation (DNS) of pressure-driven turbulent flow through a channel bounded by walls of laterally alternating patches of high and low wall shear stress. The wall shear stress is imposed as a Neumann boundary condition such that the wall shear stress ratio is fixed at 3 while the lateral spacing $s$ of the uniform-stress patches is varied from 0.39 to 6.28 of the half-channel height $\unicode[STIX]{x1D6FF}$. We find that global outer-layer similarity is maintained when $s$ is less than approximately $0.39\unicode[STIX]{x1D6FF}$ while local outer-layer similarity is recovered when $s$ is greater than approximately $6.28\unicode[STIX]{x1D6FF}$. However, the transition between the two regimes through $s\approx \unicode[STIX]{x1D6FF}$ is not monotonic owing to the presence of secondary roll motions that extend across the whole cross-section of the flow. Importantly, these secondary roll motions are associated with an amplified skin-friction coefficient relative to both the small- and large-$s/\unicode[STIX]{x1D6FF}$ limits. It is found that the relationship between the secondary roll motions and the mean isovels is reversed through this transition from low longitudinal velocity over low stress at small $s/\unicode[STIX]{x1D6FF}$ to high longitudinal velocity over low stress at large $s/\unicode[STIX]{x1D6FF}$.
Spontaneous imbalance in the non-hydrostatic Boussinesq equations
- Hossein A. Kafiabad, Peter Bartello
-
- Published online by Cambridge University Press:
- 23 May 2018, pp. 614-643
-
- Article
- Export citation
-
Whereas high-frequency waves are valid solutions to the Boussinesq equations in certain limits, their amplitudes are generally observed to be small in large-scale atmospheric and oceanic data. Traditionally, this has led to the development of balance models, reducing the dynamics to only the slow subset. Their solutions, however, can spontaneously generate imbalance in the context of the full equations. To quantify this, we calculate how much energy is transferred from the balanced to the unbalanced part of a turbulent rotating stratified flow that has been initialised to remove high frequencies. We lay out an approach to derive the time evolution of the balanced modes in which their interactions with unbalanced modes are taken into account. This enables us to calculate the budget of balanced (and unbalanced) energy. Our results show that imbalance generation occurs at scales where the Froude and Rossby numbers are still small and the energy spectrum is steep. We find that the scale at which maximum imbalance is generated depends on the peak of the energy spectrum and is invariant to the strength of rotation over the range examined. The unbalanced energy, after being transferred from the balanced component of the flow at larger scales, is cascaded forward and forms a shallow energy spectrum. The steep balanced subrange of the energy spectrum and the shallow subrange cross and form a kink in the total energy spectrum consistent with observed atmospheric and oceanic data. A frequency analysis at different wavenumbers shows that the separation of time scales breaks down at wavenumbers larger than those of maximum imbalance generation, but smaller than the kink of the energy spectrum. Below these scales, there is a single turbulent distribution of frequencies.
Capillary waves with surface viscosity
- Li Shen, Fabian Denner, Neal Morgan, Berend van Wachem, Daniele Dini
-
- Published online by Cambridge University Press:
- 25 May 2018, pp. 644-663
-
- Article
-
- You have access Access
- Open access
- HTML
- Export citation
-
Experiments over the last 50 years have suggested a tentative correlation between the surface (shear) viscosity and the stability of a foam or emulsion. We examine this link theoretically using small-amplitude capillary waves in the presence of a surfactant solution of dilute concentration, where the associated Marangoni and surface viscosity effects are modelled via the Boussinesq–Scriven formulation. The resulting integro-differential initial value problem is solved analytically, and surface viscosity is found to contribute an overall damping effect to the amplitude of the capillary wave with varying degree depending on the length scale of the system. Numerically, we find that the critical damping wavelength increases for increasing surface concentration but the rate of increase remains different for both the surface viscosity and the Marangoni effect.
Experimental investigation of in-line flow-induced vibration of a rotating circular cylinder
- J. Zhao, D. Lo Jacono, J. Sheridan, K. Hourigan, M. C. Thompson
-
- Published online by Cambridge University Press:
- 25 May 2018, pp. 664-699
-
- Article
- Export citation
-
This study experimentally investigates the in-line flow-induced vibration (FIV) of an elastically mounted circular cylinder under forced axial rotation in a free stream. The present experiments characterise the structural vibration, fluid forces and wake structure of the fluid–structure system at a low mass ratio (the ratio of the total mass to the displaced fluid mass) over a wide parameter space spanning the reduced velocity range $5\leqslant U^{\ast }\leqslant 32$ and the rotation rate range $0\leqslant \unicode[STIX]{x1D6FC}\leqslant 3.5$ , where $U^{\ast }=U/(\,f_{nw}D)$ and $\unicode[STIX]{x1D6FC}=|\unicode[STIX]{x1D6FA}|D/(2U)$ , with $U$ the free-stream velocity, $D$ the cylinder outer diameter, $f_{nw}$ the natural frequency of the system in quiescent water and $|\unicode[STIX]{x1D6FA}|$ the angular velocity of the cylinder rotation. The corresponding Reynolds number (defined by $Re=UD/\unicode[STIX]{x1D708}$ , with $\unicode[STIX]{x1D708}$ the kinematic viscosity of the fluid) was varied over the interval $1349\leqslant Re\leqslant 8624$ , where it is expected that the FIV response is likely to be relatively insensitive to the Reynolds number. The fluid–structure system was modelled using a low-friction air-bearing system in conjunction with a free-surface water-channel facility. Three vibration regions that exhibited vortex-induced vibration (VIV) synchronisation, rotation-induced galloping and desynchronised responses were observed. In both the VIV synchronisation and rotation-induced galloping regions, significant cylinder vibration was found to be correlated with wake–body synchronisation within the rotation rate range $2.20\lesssim \unicode[STIX]{x1D6FC}\lesssim 3.15$ . Of significant interest, the frequency of the streamwise fluid force could be modulated by the imposed rotation to match that of the transverse lift force, resulting in harmonic synchronisation. Measurements using the particle image velocimetry (PIV) technique were performed to identify the wake structure. Interestingly, the imposed rotation can cause regular vortex shedding in in-line FIV at rotation rates that see suppression of the Bénard–von-Kármán vortex shedding in the case of a rigidly mounted cylinder ( $\unicode[STIX]{x1D6FC}\gtrsim 1.75$ ). There is a monotonic increase in the drag coefficient with rotation rate beyond $\unicode[STIX]{x1D6FC}=2$ for a non-oscillating rotating cylinder. This suggests that the mechanism for sustaining the large rotation-induced galloping oscillations at higher $\unicode[STIX]{x1D6FC}$ is due to a combination of aerodynamic forcing from the locked induced vortex shedding associated with the oscillations, assisted by aerodynamic forcing, evaluated using quasi-steady theory.
Shear reversal in dense suspensions: the challenge to fabric evolution models from simulation data
- Rahul N. Chacko, Romain Mari, Suzanne M. Fielding, Michael E. Cates
-
- Published online by Cambridge University Press:
- 29 May 2018, pp. 700-734
-
- Article
- Export citation
-
Dense suspensions of hard particles are important as industrial or environmental materials (e.g. fresh concrete, food, paint or mud). To date, most constitutive models developed to describe them are, explicitly or effectively, ‘fabric evolution models’ based on: (i) a stress rule connecting the macroscopic stress to a second-rank microstructural fabric tensor $\unicode[STIX]{x1D64C}$; and (ii) a closed time-evolution equation for $\unicode[STIX]{x1D64C}$. In dense suspensions, most of the stress comes from short-ranged pairwise steric or lubrication interactions at near-contacts (suitably defined), so a natural choice for $\unicode[STIX]{x1D64C}$ is the deviatoric second moment of the distribution $P(\boldsymbol{p})$ of the near-contact orientations $\boldsymbol{p}$. Here we test directly whether a closed time-evolution equation for such a $\unicode[STIX]{x1D64C}$ can exist, for the case of inertialess non-Brownian hard spheres in a Newtonian solvent. We perform extensive numerical simulations accessing high levels of detail for the evolution of $P(\boldsymbol{p})$ under shear reversal, providing a stringent test for fabric evolution models. We consider a generic class of these models as defined by Hand (J. Fluid Mech., vol. 13, 1962, pp. 33–46) that assumes little as to the micromechanical behaviour of the suspension and is only constrained by frame indifference. Motivated by the smallness of microstructural anisotropies in the dense regime, we start with linear models in this class and successively consider those increasingly nonlinear in $\unicode[STIX]{x1D64C}$. Based on these results, we suggest that no closed fabric evolution model properly describes the dynamics of the fabric tensor under reversal. We attribute this to the fact that, while a second-rank tensor captures reasonably well the microstructure in steady flows, it gives a poor description during significant parts of the microstructural evolution following shear reversal. Specifically, the truncation of $P(\boldsymbol{p})$ at second spherical harmonic (or second-rank tensor) level describes ellipsoidal distributions of near-contact orientations, whereas on reversal we observe distributions that are markedly four-lobed; moreover, ${\dot{P}}(\boldsymbol{p})$ has oblique axes, not collinear with those of $\unicode[STIX]{x1D64C}$ in the shear plane. This structure probably precludes any adequate closure at second-rank level. Instead, our numerical data suggest that closures involving the coupled evolution of both a fabric tensor and a fourth-rank tensor might be reasonably accurate.
Koopman analysis of the long-term evolution in a turbulent convection cell
- Dimitrios Giannakis, Anastasiya Kolchinskaya, Dmitry Krasnov, Jörg Schumacher
-
- Published online by Cambridge University Press:
- 29 May 2018, pp. 735-767
-
- Article
- Export citation
-
We analyse the long-time evolution of the three-dimensional flow in a closed cubic turbulent Rayleigh–Bénard convection cell via a Koopman eigenfunction analysis. A data-driven basis derived from diffusion kernels known in machine learning is employed here to represent a regularized generator of the unitary Koopman group in the sense of a Galerkin approximation. The resulting Koopman eigenfunctions can be grouped into subsets in accordance with the discrete symmetries in a cubic box. In particular, a projection of the velocity field onto the first group of eigenfunctions reveals the four stable large-scale circulation (LSC) states in the convection cell. We recapture the preferential circulation rolls in diagonal corners and the short-term switching through roll states parallel to the side faces which have also been seen in other simulations and experiments. The diagonal macroscopic flow states can last as long as 1000 convective free-fall time units. In addition, we find that specific pairs of Koopman eigenfunctions in the secondary subset obey enhanced oscillatory fluctuations for particular stable diagonal states of the LSC. The corresponding velocity-field structures, such as corner vortices and swirls in the midplane, are also discussed via spatiotemporal reconstructions.
Direct measurement of the Sears function in turbulent flow
- Mingshui Li, Yang Yang, Ming Li, Haili Liao
-
- Published online by Cambridge University Press:
- 29 May 2018, pp. 768-785
-
- Article
- Export citation
-
The applicability of the strip assumption in the estimation of the unsteady lift response of a two-dimensional wing in turbulent flow is investigated. The ratio between the lift spectrum calculated from the two-wavenumber analysis and the lift spectrum calculated from the strip assumption is used to evaluate the accuracy of the strip assumption. It is shown that the accuracy of the strip assumption is controlled by the ratio of the turbulence integral scale to the chord and the aspect ratio. With an increase of these two parameters, the ratio for evaluating the accuracy of the strip assumption increases, the one-wavenumber transfer function obtained from the strip assumption approaches the Sears function gradually. When these two parameters take suitable values, the strip assumption could be applicable to the calculation of the unsteady lift on a wing in turbulent flow. Here, the term aspect ratio refers to the ratio of the specified span (an finite spanwise length of the two-dimensional wing) to the chord, the unsteady lift is calculated over this specified spanwise length. The theoretical analysis is verified by means of force measurement experiments conducted in a wind tunnel. In the experiment, a square passive grid is installed downstream of the entrance of the test section to generate approximately homogeneous and isotropic turbulence. Three rectangular wings with different aspect ratios ($\unicode[STIX]{x1D703}=3$, 5 and 7) are used. These wing models have an NACA 0015 profile cross-section and a fixed chord length $c=0.16~\text{m}$. The testing results show that, at a fixed ratio of turbulence integral scale to chord, the deviation between the experimental one-wavenumber transfer function obtained from the strip assumption and the Sears function is reduced with increasing aspect ratio, as expected by the theoretical predictions. However, due to the effect of thickness, the experimental values at high frequencies cannot be captured by the Sears function which is derived based on the flat plate assumption. In practical applications, the effect of thickness on the transfer function should be considered.
Vortex-induced vibration of a transversely rotating sphere
- Methma M. Rajamuni, Mark C. Thompson, Kerry Hourigan
-
- Published online by Cambridge University Press:
- 29 May 2018, pp. 786-820
-
- Article
- Export citation
-
The effects of transverse rotation on the vortex-induced vibration (VIV) of a sphere in a uniform flow are investigated numerically. The one degree-of-freedom sphere motion is constrained to the cross-stream direction, with the rotation axis orthogonal to flow and vibration directions. For the current simulations, the Reynolds number of the flow, $Re=UD/\unicode[STIX]{x1D708}$, and the mass ratio of the sphere, $m^{\ast }=\unicode[STIX]{x1D70C}_{s}/\unicode[STIX]{x1D70C}_{f}$, were fixed at 300 and 2.865, respectively, while the reduced velocity of the flow was varied over the range $3.5\leqslant U^{\ast }~(\equiv U/(f_{n}D))\leqslant 11$, where, $U$ is the upstream velocity of the flow, $D$ is the sphere diameter, $\unicode[STIX]{x1D708}$ is the fluid viscosity, $f_{n}$ is the system natural frequency and $\unicode[STIX]{x1D70C}_{s}$ and $\unicode[STIX]{x1D70C}_{f}$ are solid and fluid densities, respectively. The effect of sphere rotation on VIV was studied over a wide range of non-dimensional rotation rates: $0\leqslant \unicode[STIX]{x1D6FC}~(\equiv \unicode[STIX]{x1D714}D/(2U))\leqslant 2.5$, with $\unicode[STIX]{x1D714}$ the angular velocity. The flow satisfied the incompressible Navier–Stokes equations while the coupled sphere motion was modelled by a spring–mass–damper system, under zero damping. For zero rotation, the sphere oscillated symmetrically through its initial position with a maximum amplitude of approximately 0.4 diameters. Under forced rotation, it oscillated about a new time-mean position. Rotation also resulted in a decreased oscillation amplitude and a narrowed synchronisation range. VIV was suppressed completely for $\unicode[STIX]{x1D6FC}>1.3$. Within the $U^{\ast }$ synchronisation range for each rotation rate, the drag force coefficient increased while the lift force coefficient decreased from their respective pre-oscillatory values. The increment of the drag force coefficient and the decrement of the lift force coefficient reduced with increasing reduced velocity as well as with increasing rotation rate. In terms of wake dynamics, in the synchronisation range at zero rotation, two equal-strength trails of interlaced hairpin-type vortex loops were formed behind the sphere. Under rotation, the streamwise vorticity trail on the advancing side of the sphere became stronger than the trail in the retreating side, consistent with wake deflection due to the Magnus effect. This symmetry breaking appears to be associated with the reduction in the observed amplitude response and the narrowing of the synchronisation range. In terms of variation with Reynolds number, the sphere oscillation amplitude was found to increase over the range $Re\in [300,1200]$ at $U^{\ast }=6$ for each of $\unicode[STIX]{x1D6FC}=0.15$, 0.75 and 1.5. The VIV response depends strongly on Reynolds number, with predictions indicating that VIV will persist for higher rotation rates at higher Reynolds numbers.
Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis
- Aaron Towne, Oliver T. Schmidt, Tim Colonius
-
- Published online by Cambridge University Press:
- 29 May 2018, pp. 821-867
-
- Article
- Export citation
-
We consider the frequency domain form of proper orthogonal decomposition (POD), called spectral proper orthogonal decomposition (SPOD). Spectral POD is derived from a space–time POD problem for statistically stationary flows and leads to modes that each oscillate at a single frequency. This form of POD goes back to the original work of Lumley (Stochastic Tools in Turbulence, Academic Press, 1970), but has been overshadowed by a space-only form of POD since the 1990s. We clarify the relationship between these two forms of POD and show that SPOD modes represent structures that evolve coherently in space and time, while space-only POD modes in general do not. We also establish a relationship between SPOD and dynamic mode decomposition (DMD); we show that SPOD modes are in fact optimally averaged DMD modes obtained from an ensemble DMD problem for stationary flows. Accordingly, SPOD modes represent structures that are dynamic in the same sense as DMD modes but also optimally account for the statistical variability of turbulent flows. Finally, we establish a connection between SPOD and resolvent analysis. The key observation is that the resolvent-mode expansion coefficients must be regarded as statistical quantities to ensure convergent approximations of the flow statistics. When the expansion coefficients are uncorrelated, we show that SPOD and resolvent modes are identical. Our theoretical results and the overall utility of SPOD are demonstrated using two example problems: the complex Ginzburg–Landau equation and a turbulent jet.
The modified Myers boundary condition for swirling flow
- James R. Mathews, Vianney Masson, Stéphane Moreau, Hélène Posson
-
- Published online by Cambridge University Press:
- 29 May 2018, pp. 868-906
-
- Article
- Export citation
-
This paper gives a modified Myers boundary condition in swirling inviscid flow, which differs from the standard Myers boundary condition by assuming a small but non-zero boundary layer thickness. The new boundary condition is derived and is shown to have the correct quadratic error behaviour with boundary layer thickness and also to agree with previous results when the swirl is set to zero. The boundary condition is initially derived for swirling flow with constant azimuthal velocity, but easily extends to radially varying swirling flow, with terms depending on the boundary layer model. The modified Myers boundary condition is then given in the time domain rather than in the frequency domain. The effect of the boundary layer profile is then considered, and shown to be small compared to the boundary layer thickness. The boundary condition is then used to analyse the eigenmodes and Green’s function in a realistic flow. Modelling the thickness of the boundary layer properly is shown to be essential in order to get accurate results.
Corrigendum
The structure and budget of turbulent kinetic energy in front of a wall-mounted cylinder – CORRIGENDUM
- Wolfgang Schanderl, Ulrich Jenssen, Claudia Strobl, Michael Manhart
-
- Published online by Cambridge University Press:
- 29 May 2018, pp. 907-911
-
- Article
-
- You have access Access
- HTML
- Export citation
JFM Rapids
On the role of secondary motions in turbulent square duct flow
- Davide Modesti, Sergio Pirozzoli, Paolo Orlandi, Francesco Grasso
-
- Published online by Cambridge University Press:
- 24 May 2018, R1
-
- Article
- Export citation
-
We use a direct numerical simulations (DNS) database for turbulent flow in a square duct up to bulk Reynolds number $Re_{b}=40\,000$ to quantitatively analyse the role of secondary motions on the mean flow structure. For that purpose we derive a generalized form of the identity of Fukagata, Iwamoto and Kasagi (FIK), which allows one to quantify the effect of cross-stream convection on the mean streamwise velocity, wall shear stress and bulk friction coefficient. Secondary motions are found to contribute approximately 6 % of the total friction, and to act as a self-regulating mechanism of turbulence whereby wall shear stress non-uniformities induced by corners are equalized, and universality of the wall-normal velocity profiles is established. We also carry out numerical experiments whereby the secondary motions are artificially suppressed, in which case their equalizing role is partially taken by the turbulent stresses.
Acceleration statistics of tracer particles in filtered turbulent fields
- Cristian C. Lalescu, Michael Wilczek
-
- Published online by Cambridge University Press:
- 29 May 2018, R2
-
- Article
- Export citation
-
We present results from direct numerical simulations of tracer particles advected in filtered velocity fields to quantify the impact of the scales of turbulence on Lagrangian acceleration statistics. Systematically removing spatial scales reduces the frequency of extreme acceleration events, consistent with the notion that they are rooted in the small-scale structure of turbulence. We also find that acceleration variance and flatness as a function of filter scale closely resemble experimental results of neutrally buoyant, finite-sized particles, corroborating the picture that particle size determines the scale on which turbulent fluctuations are sampled.
Towards simulating natural transition in hypersonic boundary layers via random inflow disturbances
- Christoph Hader, Hermann F. Fasel
-
- Published online by Cambridge University Press:
- 29 May 2018, R3
-
- Article
- Export citation
-
A random forcing approach was implemented into a high-order accurate finite-difference code in order to investigate ‘natural’ laminar–turbulent transition in hypersonic boundary layers. In hypersonic transition wind-tunnel experiments, transition is caused ‘naturally’, by free-stream disturbances even when so-called quiet tunnels are employed such as the Boeing/AFOSR Mach 6 Quiet Tunnel (BAM6QT) at Purdue University. The nature and composition of the free-stream disturbance environment in high-speed transition experiments is difficult to assess and therefore largely unknown. Consequently, in the direct numerical simulations (DNS) presented here, the free-stream disturbance environment is simply modelled by random pressure (acoustic) disturbances with a broad spectrum of frequencies and a wide range of azimuthal wavenumbers. Results of a high-resolution DNS for a flared cone at Mach 6, using the random forcing approach, are presented and compared to a fundamental breakdown simulation using a ‘controlled’ disturbance input (with a specified frequency and azimuthal wavenumber). The DNS results with random forcing clearly exhibit the ‘primary’ and ‘secondary’ streak pattern, which has previously been observed in our ‘controlled’ breakdown simulations and the experiments in the BAM6QT. In particular, the spanwise spacing of the ‘primary’ streaks for the random forcing case is identical to the spacing obtained from the ‘controlled’ fundamental breakdown simulation. A comparison of the wall pressure disturbance signals between the random forcing DNS and experimental data shows remarkable agreement. The random forcing approach seems to be a promising strategy to investigate nonlinear breakdown in hypersonic boundary layers without introducing any bias towards a distinct nonlinear breakdown mechanism and/or the selection of specific frequencies or wavenumbers that is required in the ‘controlled’ breakdown simulations.
Front Cover (OFC, IFC) and matter
FLM volume 847 Cover and Front matter
-
- Published online by Cambridge University Press:
- 11 June 2018, pp. f1-f4
-
- Article
-
- You have access Access
- Export citation
Back Cover (OBC, IBC) and matter
FLM volume 847 Cover and Back matter
-
- Published online by Cambridge University Press:
- 11 June 2018, pp. b1-b9
-
- Article
-
- You have access Access
- Export citation