#### Journal of Fluid Mechanics

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ISSN / EISSN : 0022-1120 / 1469-7645
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Published: 21 October 2021
Journal of Fluid Mechanics, Volume 929; https://doi.org/10.1017/jfm.2021.815

Abstract:
This paper presents an extension of Kolmogorov's local similarity hypotheses of turbulence to include the influence of mean shear on the statistics of the fluctuating velocity in the dissipation range of turbulent shear flow. According to the extension, the moments of the fluctuating velocity gradients are determined by the local mean rate of the turbulent energy dissipation $\left \langle \epsilon \right \rangle$ per unit mass, kinematic viscosity $\nu$ and parameter $\gamma \equiv S (\nu /\left \langle \epsilon \right \rangle )^{1/2}$ , provided that $\gamma$ is small in an appropriate sense, where $S$ is an appropriate norm of the local gradients of the mean flow. The statistics of the moments are nearly isotropic for sufficiently small $\gamma$ , and the anisotropy of moments decreases approximately in proportion to $\gamma$ . This paper also presents a report on the second-order moments of the fluctuating velocity gradients in direct numerical simulations (DNSs) of turbulent channel flow (TCF) with the friction Reynolds number $Re_\tau$ up to $\approx 8000$ . In the TCF, there is a range $y$ where $\gamma$ scales approximately $\propto y^ {-1/2}$ , and the anisotropy of the moments of the gradients decreases with $y$ nearly in proportion to $y^ {-1/2}$ , where $y$ is the distance from the wall. The theoretical conjectures proposed in the first part are in good agreement with the DNS results.
Published: 21 October 2021
Journal of Fluid Mechanics, Volume 929; https://doi.org/10.1017/jfm.2021.821

Abstract:
A scalar emanating from a point source in a turbulent boundary layer does not mix homogeneously, but is organized in large regions with little variation of the concentration: uniform concentration zones. We measure scalar concentration using laser-induced fluorescence and, simultaneously, the three-dimensional velocity field using tomographic particle image velocimetry in a water tunnel boundary layer. We identify uniform concentration zones using both a simple histogram technique, and more advanced cluster analysis. From the complete information on the turbulent velocity field, we compute two candidate velocity structures that may form the boundaries between two uniform concentration zones. One of these structures is related to the rate of point separation along Lagrangian trajectories and the other one involves the magnitude of strong shear in snapshots of the velocity field. Therefore, the first method allows for the history of the flow field to be monitored, while the second method only looks at a snapshot. The separation of fluid parcels in time was measured in two ways: the exponential growth of the separation as time progresses (related to finite-time Lyapunov exponents and unstable manifolds in the theory of dynamical systems), and the exponential growth as time moves backward (stable manifolds). Of these two, a correlation with the edges of uniform concentration zones was found for the past Lyapunov field but not with the time-forward future field. The magnitude of the correlation is comparable to that of the regions of strong shear in the instantaneous velocity field.
Published: 21 October 2021
Journal of Fluid Mechanics, Volume 929; https://doi.org/10.1017/jfm.2021.867

Abstract:
By coupling direct numerical simulation of homogeneous isotropic turbulence with a localised solution of the convection–diffusion equation, we model the rate of transfer of a solute (mass transfer) from the surface of small, neutrally buoyant, axisymmetric, ellipsoidal particles (spheroids) in dilute suspension within a turbulent fluid at large Péclet number, $\textit {Pe}$ . We observe that, at $\textit {Pe} = O(10)$ , the average transfer rate for prolate spheroids is larger than that of spheres with equivalent surface area, whereas oblate spheroids experience a lower average transfer rate. However, as the Péclet number is increased, oblate spheroids can experience an enhancement in mass transfer relative to spheres near an optimal aspect ratio $\lambda \approx 1/4$ . Furthermore, we observe that, for spherical particles, the Sherwood number $\textit {Sh}$ scales approximately as $\textit {Pe}^{0.26}$ over $\textit {Pe} = 1.4\times 10^{1}$ to $1.4\times 10^{4}$ , which is below the $\textit {Pe}^{1/3}$ scaling observed for inertial particles but consistent with available experimental data for tracer-like particles. The discrepancy is attributed to the diffusion-limited temporal response of the concentration boundary layer to turbulent strain fluctuations. A simple model, the quasi-steady flux model, captures both of these phenomena and shows good quantitative agreement with our numerical simulations.
Dongdong Wan, Guangrui Sun,
Published: 21 October 2021
Journal of Fluid Mechanics, Volume 929; https://doi.org/10.1017/jfm.2021.852

Abstract:
Axisymmetric viscoelastic pipe flow of Oldroyd-B fluids has been recently found to be linearly unstable by Garg et al. (Phys. Rev. Lett., vol. 121, 2018, 024502). From a nonlinear point of view, this means that the flow can transition to turbulence supercritically, in contrast to the subcritical Newtonian pipe flows. Experimental evidence of subcritical and supercritical bifurcations of viscoelastic pipe flows have been reported, but these nonlinear phenomena have not been examined theoretically. In this work, we study the weakly nonlinear stability of this flow by performing a multiple-scale expansion of the disturbance around linear critical conditions. The perturbed parameter is the Reynolds number with the others being unperturbed. A third-order Ginzburg–Landau equation is derived with its coefficient indicating the bifurcation type of the flow. After exploring a large parameter space, we found that polymer concentration plays an important role: at high polymer concentrations (or small solvent-to-solution viscosity ratio $\beta \lessapprox 0.785$ ), the nonlinearity stabilizes the flow, indicating that the flow will bifurcate supercritically, while at low polymer concentrations ( $\beta \gtrapprox 0.785$ ), the flow bifurcation is subcritical. The results agree qualitatively with experimental observations where critical $\beta \approx 0.855$ . The pipe flow of upper convected Maxwell fluids can be linearly unstable and its bifurcation type is also supercritical. At a fixed value of $\beta$ , the Landau coefficient scales with the inverse of the Weissenberg number ( $Wi$ ) when $Wi$ is sufficiently large. The present analysis provides a theoretical understanding of the recent studies on the supercritical and subcritical routes to the elasto-inertial turbulence in viscoelastic pipe flows.
Published: 21 October 2021
Journal of Fluid Mechanics, Volume 929; https://doi.org/10.1017/jfm.2021.839

Abstract:
The present study concerns a temporally evolving turbulent natural convection boundary layer (NCBL) adjacent to an isothermally heated vertical wall, with Prandtl number 0.71. Three-dimensional direct numerical simulations (DNS) are carried out to investigate the turbulent flow up to $\textit {Gr}_\delta =1.21\times 10^8$ , where $\textit {Gr}_\delta$ is the Grashof number based on the boundary layer thickness $\delta$ . In the near-wall region, there exists a constant heat flux layer, similar to previous studies for the spatially developing flows (e.g. George & Capp, Intl J. Heat Mass Transfer, vol. 22, 1979, pp. 813–826). Beyond a wall-normal distance $\delta _i$ , the NCBL can be characterised as a plume-like region. We find that this region is well described by a self-similar integral model with profile coefficients (cf. van Reeuwijk & Craske, J. Fluid Mech., vol. 782, 2015, pp. 333–355) which are $\textit {Gr}_\delta$ -independent after $\textit {Gr}_\delta =10^7$ . In this Grashof number range both the outer plume-like region and the near-wall boundary layer are turbulent, indicating the beginning of the so-called ultimate turbulent regime (Grossmann & Lohse, J. Fluid Mech., vol. 407, 2000, pp. 27–56; Grossmann & Lohse, Phys. Fluids, vol. 23, 2011, 045108). Solutions to the self-similar integral model are analytically obtained by solving ordinary differential equations with profile coefficients empirically obtained from the DNS results. In the present study, we have found the wall heat transfer of the NCBL is directly related to the top-hat scales which characterise the plume-like region. The Nusselt number is found to follow $\textit {Nu}_\delta \propto \textit {Gr}_\delta ^{0.381}$ , slightly higher than the empirical $1/3$ -power-law correlation reported for spatially developing NCBLs at lower $\textit {Gr}_\delta$ , but is shown to be consistent with the ultimate heat transfer regime with a logarithmic correction suggested by Grossmann & Lohse (Phys. Fluids, vol. 23, 2011, 045108). By modelling the near-wall buoyancy force, we show that the wall shear stress would scale with the bulk velocity only at asymptotically large Grashof numbers.
Published: 21 October 2021
Journal of Fluid Mechanics, Volume 929; https://doi.org/10.1017/jfm.2021.866

Abstract:
Modelling multiscale systems from nanoscale to macroscale requires the use of atomistic and continuum methods and, correspondingly, different computer codes. Here, we develop a seamless method based on DeepONet, which is a composite deep neural network (a branch and a trunk network) for regressing operators. In particular, we consider bubble growth dynamics, and we model tiny bubbles of initial size from 100 nm to 10 $\mathrm {\mu }\textrm {m}$ , modelled by the Rayleigh–Plesset equation in the continuum regime above 1 $\mathrm {\mu }\textrm {m}$ and the dissipative particle dynamics method for bubbles below 1 $\mathrm {\mu }\textrm {m}$ in the atomistic regime. After an offline training based on data from both regimes, DeepONet can make accurate predictions of bubble growth on-the-fly (within a fraction of a second) across four orders of magnitude difference in spatial scales and two orders of magnitude in temporal scales. The framework of DeepONet is general and can be used for unifying physical models of different scales in diverse multiscale applications.
Published: 21 October 2021
Journal of Fluid Mechanics, Volume 928; https://doi.org/10.1017/jfm.2021.827

Abstract:
Direct numerical simulations of oscillatory boundary-layer flows in the transitional regime were performed to explain discrepancies in the literature regarding the phase difference ${\rm \Delta} \phi$ between the bed-shear stress and free-stream velocity maxima. Recent experimental observations in smooth bed oscillatory boundary-layer (OBL) flows, showed a significant change in the widely used ${\rm \Delta} \phi$ diagram (Mier et al., J. Fluid Mech., vol. 922, 2021, A29). However, the limitations of the point-wise measurement technique did not allow us to associate this finding with the turbulent kinetic energy budget and to detect the approach to a ‘near-equilibrium’ condition, defined in a narrow sense herein. Direct numerical simulation results suggest that a phase lag occurs as the result of a delayed and incomplete transition of OBL flows to a stage that mimics the fully turbulent regime. Data from the literature were also used to support the presence of the phase lag and propose a new ${\rm \Delta} \phi$ diagram. Simulations performed for ${\textit {Re}}_{\delta }=671$ confirmed the sensitivity in the development of self-sustained turbulence on the background disturbances ( $\textit{Re}_{\delta}=U_{o}\delta/\nu$ , where $\delta=[2\nu/\omega]^{1/2}$ is the Stokes' length, $U_{o}$ is the maximum free stream velocity of the oscillation, $\nu$ is the kinematic viscosity and $\omega=2{\rm \pi}/T$ is the angular velocity based on the period of the oscillation T). Variations of the mean velocity slope and intersect values for oscillatory flows are also explained in terms of the proximity to near-equilibrium conditions. Relaminarization and transition effects can significantly delay the development of OBL flows, resulting in an incomplete transition. The shape and defect factors are examined as diagnostic parameters for conditions that allow the formation of a logarithmic profile with the universal von Kármán constant and intersect. These findings are of relevance for environmental fluid mechanics and coastal morphodynamics/engineering applications.
Chengjiao Ren, , , , Tingguo Chen
Published: 19 October 2021
Journal of Fluid Mechanics, Volume 929; https://doi.org/10.1017/jfm.2021.832

Abstract:
Bistabilities of two equilibrium states discovered in the coupled side-by-side Kármán wakes are investigated through Floquet analysis and direct numerical simulation (DNS) with different initial conditions over a range of gap-to-diameter ratio ( $g^*= 0.2\text {--}3.5$ ) and Reynolds number ( $Re = 47\text {--}100$ ). Two bistabilities are found in the transitional $g^*-Re$ regions from in-phase (IP) to anti-phase (AP) vortex shedding states. By initialising the flow in DNS with zero initial conditions, the flow in the first bistable region (i.e. bistable IP/FF $_C$ at $g^*= 1.4 \text {--} 2.0$ , where FF $_C$ denotes the conditional flip-flop flow) attains flip-flop (FF) flow, it settles into the IP state by initialising the flow with an IP flow. The second bistability is observed between cylinder-scale IP and AP states at large $g^*$ ( $=$ 2.0–3.5). The transition from the FF $_C$ to IP is dependent on initial conditions and irreversible over the parameter space, meaning that the flow cannot revert back to the FF $_C$ state once it jumps to the IP state irrespective of the direction of $Re$ variations. Its counterpart for the bistable IP/AP state is reversible. We also found that the FF $_C$ flow in the first bistable region is primarily bifurcated from synchronised AP with cluster-scale features, possibly because the cluster-scale AP flow is inherently unstable to FF flow instabilities. It is demonstrated that the irreversible bistability exists in other interacting wakes around multiple cylinders. A good understanding of flow bistabilities is pivotal to flow control applications and the interpretation of desynchronised flow features observed at high $Re$ values.
Vishal Srikanth, Ching-Wei Huang, Timothy S. Su,
Published: 19 October 2021
Journal of Fluid Mechanics, Volume 929; https://doi.org/10.1017/jfm.2021.813

Abstract:
The focus of this paper is a numerical simulation study of the flow dynamics in a periodic porous medium to analyse the physics of a symmetry-breaking phenomenon, which causes a deviation in the direction of the macroscale flow from that of the applied pressure gradient. The phenomenon is prominent in the range of porosity from 0.43 to 0.72 for circular solid obstacles. It is the result of the flow instabilities formed when the surface forces on the solid obstacles compete with the inertial force of the fluid flow in the turbulent regime. We report the origin and mechanism of the symmetry-breaking phenomenon in periodic porous media. Large-eddy simulation (LES) is used to simulate turbulent flow in a homogeneous porous medium consisting of a periodic, square lattice arrangement of cylindrical solid obstacles. Direct numerical simulation is used to simulate the transient stages during symmetry breakdown and also to validate the LES method. Quantitative and qualitative observations are made from the following approaches: (1) macroscale momentum budget and (2) two- and three-dimensional flow visualization. The phenomenon draws its roots from the amplification of a flow instability that emerges from the vortex shedding process. The symmetry-breaking phenomenon is a pitchfork bifurcation that can exhibit multiple modes depending on the local vortex shedding process. The phenomenon is observed to be sensitive to the porosity, solid obstacle shape and Reynolds number. It is a source of macroscale turbulence anisotropy in porous media for symmetric solid-obstacle geometries. In the macroscale, the principal axis of the Reynolds stress tensor is not aligned with any of the geometric axes of symmetry, nor with the direction of flow. Thus, symmetry breaking in porous media involves unresolved flow physics that should be taken into consideration while modelling flow inhomogeneity in the macroscale.
N. Agastya Balantrapu, Christopher Hickling, W. Nathan Alexander,
Published: 19 October 2021
Journal of Fluid Mechanics, Volume 929; https://doi.org/10.1017/jfm.2021.845

Abstract:
Experiments were performed over a body of revolution at a length-based Reynolds number of 1.9 million. While the lateral curvature parameters are moderate ( $\delta /r_s < 2, r_s^+>500$ , where $\delta$ is the boundary layer thickness and r s is the radius of curvature), the pressure gradient is increasingly adverse ( $\beta _{C} \in [5 \text {--} 18]$ where $\beta_{C}$ is Clauser’s pressure gradient parameter), representative of vehicle-relevant conditions. The mean flow in the outer regions of this fully attached boundary layer displays some properties of a free-shear layer, with the mean-velocity and turbulence intensity profiles attaining self-similarity with the ‘embedded shear layer’ scaling (Schatzman & Thomas, J. Fluid Mech., vol. 815, 2017, pp. 592–642). Spectral analysis of the streamwise turbulence revealed that, as the mean flow decelerates, the large-scale motions energize across the boundary layer, growing proportionally with the boundary layer thickness. When scaled with the shear layer parameters, the distribution of the energy in the low-frequency region is approximately self-similar, emphasizing the role of the embedded shear layer in the large-scale motions. The correlation structure of the boundary layer is discussed at length to supply information towards the development of turbulence and aeroacoustic models. One major finding is that the estimation of integral turbulence length scales from single-point measurements, via Taylor's hypothesis, requires significant corrections to the convection velocity in the inner 50 % of the boundary layer. The apparent convection velocity (estimated from the ratio of integral length scale to the time scale), is approximately 40 % greater than the local mean velocity, suggesting the turbulence is convected much faster than previously thought. Closer to the wall even higher corrections are required.