The persistence of viscous effects in the overlap region, and the mean velocity in turbulent pipe and channel flows
Katepalli R. Sreenivasan, Anupam Sahay
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15 pages, Latex, 7 eps figures, label positions will need adjustment if the text height or width is changed. To appear in `Self-Sustaining Mechanisms of Wall Turbulence', ed. R. L. Panton, Comp. Mech. Pub., 1997
We argue that important elements of the dynamics of wall-bounded flows reside at the wall-normal position $y_p^+$ corresponding to the peak of the Reynolds shear stress. Specializing to pipe and channel flows, we show that the mean momentum balance in the neighborhood of $y_p^+$ is distinct in character from those in the classical inner and outer layers. We revisit empirical data to confirm that $y_p^+ = O(R^{1/2})$ and show that, in a neighborhood of order $R^{1/2}$ around $y_p^+$, only the viscous effects balance pressure-gradient terms. Here, R is the Reynolds number based on friction velocity and pipe radius (or channel half-width). This observation provides a mechanism by which viscous effects play an important role in regions traditionally thought to be inviscid or inertial; in particular, it throws doubt on the validity of the classical matching principle. Even so, it is shown that the classical semi-logarithmic behavior for the mean velocity distribution can be a useful approximation. It is argued that the recently advanced power-law profiles possess a rich underlying structure, and could be good approximations to the data over an extended region (but they too are unlikely to be exact).
We extend the procedure outlined in [Basse, "Scaling of global properties of fluctuating and mean streamwise velocities in pipe flow: Characterization of a high Reynolds number transition region," Phys. Fluids Vol. 33, 065127 (2021)] to study global, i.e. radially averaged, scaling of streamwise velocity fluctuations. A viscous term is added to the log-law scaling which leads to the existence of a mathematical abstraction which we call the "global peak". The position and amplitude of this global peak are characterized and compared to the inner and outer peaks. A transition at a friction Reynol…
A detailed experimental and theoretical study has been performed about a phenomenon, not previously reported in the literature, occurring in highly viscous liquids: the formation of a definite pipe structure induced by the passage of a heavy body, this structure lasting for quite a long time. A very rich phenomenology (including mechanical, optical and structural effects) associated with the formation of the pipe has been observed in different liquids. Actually, the peculiar dynamical evolution of that structure does not appear as a trivial manifestation of standard relaxation or spurious effe…
We present convincing evidence of a direct connection between the pressure drop, viscous dissipation, and the turbulent energy spectrum. We use this finding to explain Nikuradse's experimental results of pressure drop for turbulent flows in rough pipes, in terms of a modified Kolmogorov length scale that varies with the surface roughness. Furthermore, we use Laufer's measurements of turbulent energy spectra in pipe flow to calculate -- to a good approximation -- the turbulent component of the pressure drop directly from an averaged turbulent energy spectrum. We also show that the incompressibi…
We study the viscous dissipation in pipe flows in long channels with porous or semipermeable walls, taking into account both the dissipation in the bulk of the channel and in the pores. We give simple closed form expressions for the dissipation in terms of the axially varying flow rate $Q(x)$ and the pressure $p(x)$, generalizing the well known expression $\dot W=Q\,Δp$ for the case of impenetrable walls with constant $Q$ and a pressure difference $Δp$ between the ends of the pipe. When the pressure $p_0$ outside the pipe is constant, the result is the straightforward generalization $\dot W=Δ\…
An accurate experimental and theoretical study has been performed about a phenomenon, not previously reported in the literature, occurring in highly viscous liquids: the formation of a definite pipe structure induced by the passage of a heavy body, this structure lasting for quite a long time. A very rich phenomenology (including mechanical, optical and structural effects) associated with the formation of the pipe has been observed in different liquids. Actually, the peculiar dynamical evolution of that structure does not appear as a trivial manifestation of standard relaxation or spurious eff…
Laboratory experiments in which synthetic, partially molten rock is subjected to forced deformation provide a context for testing hypotheses about the dynamics and rheology of the mantle. Here our hypothesis is that the aggregate viscosity of partially molten mantle is anisotropic, and that this anisotropy arises from deviatoric stresses in the rock matrix. We formulate a model of pipe Poiseuille flow based on theory by Takei and Holtzman [2009a] and Takei and Katz [2013]. Pipe Poiseuille is a configuration that is accessible to laboratory experimentation but for which there are no published r…
After decades of experimental, theoretical, and numerical research in fluid dynamics, many aspects of turbulence remain poorly understood. The main reason for this is often attributed to the multiscale nature of turbulent flows, which poses a formidable challenge. There are, however, properties of these flows whose roles and inter-connections have never been clarified fully. In this article, we present a new connection between the pressure drop, viscous dissipation, and the turbulent energy spectrum, which, to the best of our knowledge, has never been established prior to our work. We use this…
In this visualisation, the transition from laminar to turbulent flow is characterised by the intermittent ejection of wall fluid into the outer stream. The normalised thickness of the viscous flow layer reaches an asymptotic value but the physical thickness drops exponentially after transition. The critical transition pipe Reynolds number can be obtained simply by equating it with the asymptotic value of the normalised thickness of viscous flow layer. Key words: Transition, critical stability Reynolds number, critical transition Reynolds number, non-Newtonian pipe flow
A commonplace view of pressure-driven turbulence in pipes and channels is as "cascades" of streamwise momentum toward the viscous layer at the wall. We present in this paper an alternative picture of these flows as "inverse cascades" of spanwise vorticity, in the cross-stream direction but away from the viscous sublayer. We show that there is a constant spatial flux of spanwise vorticity, due to vorticity conservation, and that this flux is necessary to produce pressure-drop and energy dissipation. The vorticity transport is shown to be dominated by viscous diffusion at distances closer to the…
Phase-field methods have long been used to model the flow of immiscible fluids. Their ability to naturally capture interface topological changes is widely recognized, but their accuracy in simulating flows of real fluids in practical geometries is not established. We here quantitatively investigate the convergence of the phase-field method to the sharp-interface limit with simulations of two-phase pipe flow. We focus on core-annular flows, in which a highly viscous fluid is lubricated by a less viscous fluid, and validate our simulations with an analytic laminar solution, a formal linear stabi…
This paper presents a new theory of turbulence in time-independent non-Newtonian fluids. The wall layer is modelled in terms of unsteady exchange of viscous momentum between the wall and the main stream, following the classic visualisation of inrush-sweep-ejection/burst. The thickness of the wall layer is found to be the same for Newtonian and purely viscous non-Newtonian fluids, when normalised with the instantaneous wall parameters at the onset of bursting. The results indicate that the mechanisms of turbulence in Newtonian and time-independent fluids are identical when structural similarity…
Using a generalization of vector calculus for the case of non-integer dimensional space we consider a Poiseuille flow of an incompressible viscous fractal fluid in the pipe. Fractal fluid is described as a continuum in non-integer dimensional space. A generalization of the Navier-Stokes equations for non-integer dimensional space, its solution for steady flow of fractal fluid in a pipe and corresponding fractal fluid discharge are suggested.
In the linear theory of hydrodynamic stability up to now there exist examples of flows for which there is full quantitative distinction, as for cylindrical Hagen-Poiseuille (HP) flow in a pipe with round section, between theory conclusions and experimental data on the threshold Reynolds number Reth. In the present work, we show that to get a conclusion of linear instability of the HP flow for finite Reynolds numbers Re, it is necessary to abandon the use of traditional 'normal' form of disturbances which assumes an opportunity of separation of variables describing disturbances variability depe…
The morphological dynamics, instabilities and transitions of elastic filaments in viscous flows underlie a wealth of biophysical processes from flagellar propulsion to intracellular streaming, and are also key to deciphering the rheological behavior of many complex fluids and soft materials. Here, we combine experiments and computational modeling to elucidate the dynamical regimes and morphological transitions of elastic Brownian filaments in a simple shear flow. Actin filaments are employed as an experimental model system and their conformations are investigated through fluorescence microscop…
A parametric numerical study of three-dimensional instability of steady flows in a helical pipe of arbitrary curvature and torsion is carried out. The computations are performed by a numerical approach verified against independent experimental and numerical results in a previous study. The results are reported as dependences of the critical Reynolds number, critical wavenumber and the critical frequency on the dimensionless pipe curvature and torsion. A multiplicity of different disturbance modes becoming most unstable at different values of the governing parameters, is observed. Patterns of t…
Numerical and experimental studies of transitional pipe flow have shown the prevalence of coherent flow structures that are dominated by downstream vortices. They attract special attention because they contribute predominantly to the increase of the Reynolds stresses in turbulent flow. In the present study we introduce a convenient detector for these coherent states, calculate the fraction of time the structures appear in the flow, and present a Markov model for the transition between the structures. The fraction of states that show vortical structures exceeds 24% for a Reynolds number of abou…
The velocity and friction properties of laminar pipe flow of a viscoelastic solution are bounded by the corresponding values for two Newtonian fluids, namely, the solvent and a fluid with a viscosity identical to the total viscosity of the solution. The lower friction factor for the flow of the solution when compared to the latter is tracked to an increased strain rate needed to enhance viscous dissipation. Lastly, we show analytically that the effective viscosity varies similarly to the radial diagonal component of the conformation tensor as observed numerically in turbulent flows, and give a…
We report the computation of a family of traveling wave solutions of pipe flow up to $Re=75000$. As in all lower-branch solutions, streaks and rolls feature prominently in these solutions. For large $Re$, these solutions develop a critical layer away from the wall. Although the solutions are linearly unstable, the two unstable eigenvalues approach 0 as $Re\to\infty$ at rates given by $Re^{-0.41}$ and $Re^{-0.87}$ -- surprisingly, the solutions become more stable as the flow becomes less viscous. The formation of the critical layer and other aspects of the $Re\to\infty$ limit could be universal…
We perform fully Eulerian numerical simulations of an initially spherical hyperelastic particle suspended in a Newtonian pressure-driven flow in a cylindrical straight pipe. We study the full particle migration and deformation for different Reynolds numbers and for various levels of particle elasticity, to disentangle the interplay of inertia and elasticity on the particle focusing. We observe that the particle deforms and undergoes a lateral displacement while traveling downstream through the pipe, finally focusing at the pipe centerline. We note that the migration dynamics and the final equi…
We report the results of three-dimensional direct numerical simulations for incompressible viscous fluid in a circular pipe flow with a sudden expansion. At the inlet, a parabolic velocity profile is applied together with a finite amplitude perturbation in the form of a vortex with its axis parallel to the axis of the pipe. At sufficiently high Reynolds numbers the recirculation region breaks into a turbulent patch that changes position axially depending on the strength of the perturbation. This vortex perturbation is believed to produce a less abrupt transition than in previous studies with a…
Viscous fluid exiting a long horizontal circular pipe develops a complex structure comprising a primary jet above and a smaller secondary jet below with a thin fluid curtain connecting them. We present here a combined experimental, theoretical and numerical study of this 'Torricelli's curtain' phenomenon, focusing on the factors that control its morphology. We propose a theoretical model for the curtain in which particle trajectories result from the composition of two motions: a horizontal component corresponding to the evolving axial velocity profile of an axisymmetric viscous jet, and a vert…
In the laminar mode interactions among molecules generate friction between layers of water that slide with respect to each other. This friction triggers the shear stress, which is traditionally presumed to be linearly proportional to the velocity gradient. The proportionality coefficient characterizes the viscosity of water. Remarkably, the standard Navier-Stokes model surmises that materials never fail - the transition to turbulence can only be triggered by some kinematic instability of the flow. This premise is probably the reason why the Navier-Stokes theory fails to explain the so-called s…
Considered here is the derivation of partial differential equations arising in pulsatile flow in pipes with viscoelastic walls. The equations are asymptotic models describing the propagation of long-crested pulses in pipes with cylindrical symmetry. Additional effects due to viscous stresses in bio-fluids are also taken into account. The effects of viscoelasticity of the vessels on the propagation of solitary and periodic waves in a vessel of constant radius are being explored numerically.
To date no satisfying model exists to explain the mean velocity profile within the whole turbulent layer of canonical wall bounded flows. We propose a modification of the velocity profile expression that ensues from a recently proposed stochastic representation of fluid flows dynamics. This modeling, called modeling under location uncertainty introduces in a rigorous way a subgrid term generalizing the eddy-viscosity assumption and an eddy-induced advection term resulting from turbulence inhomogeneity. This latter term gives rise to a theoretically well-grounded model for the transitional zone…
We present a study on the nonlinear dynamics of small long-wave disturbances to the laminar state in non-rotating axisymmetric Poiseuille pipe flows. At high Reynolds numbers, the associated Navier-Stokes equations can be reduced to a set of coupled Korteweg-de Vries-type (KdV) equations that support inviscid and smooth travelling waves numerically computed using the Petviashvili method. In physical space they correspond to localized toroidal vortices concentrated near the pipe boundaries (wall vortexons) or that wrap around the pipe axis (centre vortexons), in agreement with the analytical so…
Motivated by the results of recent experiments (Kühnen et al., Flow Turb. Combust., vol. 100, 2018, pp. 919-943), we consider the problem of designing a baffle (an obstacle to the flow) to relaminarise turbulence in pipe flows. Modelling the baffle as a spatial distribution of linear drag $\mathbf{f}(\mathbf{x},t)=-χ(\mathbf{x})\mathbf{u}_{tot}(\mathbf{x},t)$ within the flow ($\mathbf{u}_{tot}$ is the total velocity field and $χ\ge 0$ a scalar field), two different optimisation problems are considered to design $χ$ at a Reynolds number $Re=3000$. In the first, the smallest baffle defined in te…
There is much current interest in modelling suspensions of algae and other micro-organisms for biotechnological exploitation, and many bioreactors are of tubular design. Using generalized Taylor dispersion theory, we develop a population-level swimming-advection-diffusion model for suspensions of micro-organisms in a vertical pipe flow. In particular, a combination of gravitational and viscous torques acting on individual cells can affect their swimming behaviour, which is termed gyrotaxis. This typically leads to local cell drift and diffusion in a suspension of cells. In a flow in a pipe, sm…
Direct numerical simulations (DNS) of particle-laden turbulent flow in straight, mildly curved and strongly bent pipes are performed in which the solid phase is modelled as small heavy spherical particles. A total of seven populations of dilute particles with different Stokes numbers, one-way coupled with their carrier phase, are simulated. The objective is to examine the effect of the curvature on micro-particle transport and accumulation. It is shown that even a slight non-zero curvature in the flow configuration strongly impact the particle concentration map such that the concentration of i…
In this work, a transformation, which maps the mean velocity profiles of compressible wall-bounded turbulent flows to the incompressible law of the wall is proposed. Unlike existing approaches, the proposed transformation successfully collapses, without specific tuning, numerical simulation data from fully developed channel and pipe flows, and boundary layers with or without heat transfer. In all these cases, the transformation is successful across the entire inner layer of the boundary layer (including the viscous sublayer, buffer layer, and logarithmic layer), recovers the asymptotically exa…
Slug flows are extremely interesting multiphase regime phenomena which frequently occur in flexible marine risers used by the petroleum industry in offshore environments and have both a liquid and gaseous phase. This paper describes two simple models of the slug flow regime by means of an equivalent monophase flow with a non-constant density. The slug regime is modelled as a monophase density-varying flow with a sinusoidal density, travelling along the pipe itself towards the top end node of the riser. Starting from the bottom end, it is characterized by adiabatic processes and energy loss alo…
It has long been surmised that the mean-velocity profile (MVP) of a pipe flow is closely related to the spectrum of turbulent energy. Here we perform a spectral analysis to identify the eddies that dominate the production of shear stress via momentum transfer. This analysis allows us to express the MVP as a functional of the spectrum. Each part of the MVP relates to a specific spectral range: the buffer layer to the dissipative range, the log layer to the inertial range, and the wake to the energetic range. The parameters of the spectrum set the thickness of the viscous layer, the amplitude of…
We conduct minimal-channel direct numerical simulations of turbulent flow over two-dimensional rectangular bars aligned in the spanwise direction. This roughness has been often described as $d$-type, as the roughness function $ΔU^+$ is thought to depend only on the outer-layer length scale (pipe diameter, channel half height or boundary layer thickness). This is in contrast to conventional engineering rough surfaces, named $k$-type, for which $ΔU^+$ depends on the roughness height, $k$. The minimal-span rough-wall channel is used to circumvent the high cost of simulating high Reynolds number f…
The motion of colloids in the flow field of a viscous liquid is investigated. The small colloid size compare to the macroscopical scale of the flow allow to calculate their velocity relative to that of the liquid. If the inner colloid density is larger then the density of the liquid the flow field has the domains where the colloid velocity is close to the liquid velocity. But in the domains with a strong braking of the liquid velocity the colloids are accelerated relative to the liquid. This effect is used for the qualitative explanation of the drag reduction in the flow around macroscopical b…
