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 to lower-branch solutions of shear flows. We give implementation details of the GMRES-hookstep and Arnoldi iterations used for computing these solutions and their spectra, while pointing out the new aspects of our method.
A three-layer asymptotic structure for turbulent pipe flow is proposed, revealing in terms of intermediate variables, the existence of a Reynolds-number invariant logarithmic region. It provides a theoretical foundation for addressing important questions in the scaling of the streamwise mean velocity and variance. The key insight emerging from the analysis is that the scale separation between two adjacent layers is proportional to $\sqrt{Re_τ}$, rather than $Re_τ$. This suggests that, in order to realise Reynolds-number asymptotic invariance, much higher Reynolds numbers may be necessary to ac…
We investigated the propagation of turbulent fronts in pipe flow at high Reynolds numbers by direct numerical simulation. We used a technique combining a moving frame of reference and an artificial damping to isolate the fronts in short periodic pipes, which enables us to explore the bulk Reynolds number up to Re = $10^5$ with affordable computation power. We measured the propagation speed of the downstream front and observed that a fit of $1.971-(Re/1925)^{-0.825}$ (in unit of bulk speed) well captures this speed above $Re\simeq 5000$. The speed increases monotonically as Re increases, in sta…
We study the global, i.e. radially averaged, high Reynolds number (asymptotic) scaling of streamwise turbulence intensity squared defined as ${I^2=\overline{u^2}/U^2}$, where $u$ and $U$ are the fluctuating and mean velocities, respectively (overbar is time averaging). The investigation is based on the mathematical abstraction that the logarithmic region in wall turbulence extends across the entire inner and outer layers. Results are matched to spatially integrated Princeton Superpipe measurements [Hultmark M, Vallikivi M, Bailey SCC and Smits AJ. Logarithmic scaling of turbulence in smooth- a…
In this Letter we suggest a simple and physically transparent analytical model of the pressure driven turbulent wall-bounded flows at high but finite Reynolds numbers Re. The model gives accurate qualitative description of the profiles of the mean-velocity and Reynolds-stresses (second order correlations of velocity fluctuations) throughout the entire channel or pipe in the wide range of Re, using only three Re-independent parameters. The model sheds light on the long-standing controversy between supporters of the century-old log-law theory of von-Kàrmàn and Prandtl and proposers of a newer th…
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
New families of three-dimensional nonlinear travelling waves are discovered in pipe flow. In contrast to known waves (Faisst & Eckhardt Phys. Rev. Lett. 91, 224502 (2003), Wedin & Kerswell, J. Fluid Mech. 508, 333 (2004)), they possess no rotational symmetry and exist at much lower Reynolds numbers. Particularly striking is an `asymmetric mode' which has one slow streak sandwiched between two fast streaks located preferentially to one side of the pipe. This family originates in a pitchfork bifurcation from a mirror-symmetric travelling wave which can be traced down to a Reynolds number of 773.…
Spin hydrodynamic coupling is a recently discovered method to directly generate electricity from an electrically conducting fluid flow in the absence of Lorentz forces. This method relies on a collective coupling of electron spins - the internal quantum mechanical angular momentum of the electrons - with the local vorticity of a fluid flow. In this work, we experimentally investigate the spin hydrodynamic coupling in circular and non-circular capillary pipe flows and extend a previously obtained range of Reynolds numbers to smaller and larger values, 20<Re<21,500, using the conducting liquid m…
Several experimental and numerical studies have shown that turbulent motions in circular pipe flow near transitional Reynolds numbers may not persist forever, but may decay. We study the properties of these decaying states within direct numerical simulations for Reynolds numbers up to 2200 and in pipes with lengths equal to 5, 9 and 15 times the diameter. We show that the choice of the ensemble of initial conditions affects the short time parts of lifetime distributions, but does not change the characteristic decay rate for long times. Comparing lifetimes for pipes of different length we notic…
Relaminarization of wall-bounded turbulent flows by means of external static magnetic fields is a long-known phenomenon in the physics of electrically conducting fluids at low magnetic Reynolds numbers. Despite the large literature on the subject, it is not yet completely clear what combination of the Hartmann ($M$) and the Reynolds number has to be used to predict the laminar-turbulent transition in channel or pipe flows fed by upstream turbulent flows free of magnetic perturbations. Relying upon standard phenomenological approaches related to mixing length and structural concepts, we put for…
We study compressible turbulent flow in a circular pipe, at computationally high Reynolds number. Classical related issues are addressed and discussed in light of the DNS data, including validity of compressibility transformations, velocity/temperature relations, passive scalar statistics, and size of turbulent eddies.Regarding velocity statistics, we find that Huang's transformation yields excellent universality of the scaled Reynolds stresses distributions, whereas the transformation proposed by Trettel and Larsson (2016) yields better representation of the effects of strong variation of den…
The impact of wall roughness on fully developed laminar pipe flow is investigated numerically. The roughness is comprised of square bars of varying size and pitch. Results show that the inverse relation between the friction factor and the Reynolds number in smooth pipes still persists in rough pipes, regardless of the rib height and pitch. At a given Reynolds number, the friction factor varies quadratically with roughness height and linearly with roughness pitch. We propose a single correlation for the friction factor that successfully collapses the data.
Turbulence intensity profiles are compared for smooth- and rough-wall pipe flow measurements made in the Princeton Superpipe. The profile development in the transition from hydraulically smooth to fully rough flow displays a propagating sequence from the pipe wall towards the pipe axis. The scaling of turbulence intensity with Reynolds number shows that the smooth- and rough wall level deviates with increasing Reynolds number. We quantify the correspondence between turbulence intensity and the friction factor.
It has hitherto been widely considered that a mixing layer is unstable at all Reynolds numbers. However this is untenable from energy considerations, which demand that there must exist a non-zero Reynolds number below which disturbances cannot extract energy from the mean flow. It is shown here that a linear stability analysis of similarity solutions of the plane mixing layer, including the effects of flow non-parallelism, using the minimal composite theory and the properties of adjoints following Govindarajan & Narasimha (2005), resolves the issue by yielding non-zero critical Reynolds number…
A zonal similarity analysis of friction factors in pipe flow of power law fluids is presented. It uses the critical Reynolds number and friction factor at transition as estimates of the normalised velocity and distance between the wall and log-law layers and successfully collapses both Newtonian and non-Newtonian data for laminar, transition and turbulent flows. Key words: zonal similarity analysis, friction factor, Reynolds number, transition
Almost 50 years ago Bob Brodkey and his student, Corino, conceived of and carried out a visualization experiment for the very near wall region of a turbulent pipe flow that, together with the turbulent boundary layer visualization of Kline et al., excited the turbulence research community. Using a high speed movie camera mounted on a lathe bed that recorded magnified images in a frame of reference moving with the flow, they observed the motions of sub-micron particles in the sub-, buffer- and log-layers. Surprisingly, these motions were not nearly so locally random as was the general view of t…
Experiments have been conducted to assess the sizes and energy fractions of structure in fully developed turbulent pipe flow regime in two pipe facilities, ColaPipe at BTU Cottbus-Senftenberg, and CICLoPE at University of Bologna, for shear Reynolds number in the range $2.5\cdot{10^3}\le{\mathrm{Re_τ}}\le{3.7\cdot{10^4}}$, utilizing a single hot-wire probe. Considerations are given to the spectra of the streamwise velocity fluctuations, and to large scale motions and their energy contents from the pipe near-wall to centerline. The analysis of the velocity fluctuations revealed a Reynolds-numbe…
Experiments in an extraordinary turbulent boundary layer called the sink flow, displaying a perfect streamwise invariance, show a wide extent of logarithmic scaling for moments of streamwise velocity up to order 12, even at moderate Reynolds numbers. This is in striking contrast to canonical constant-pressure turbulent boundary layers that show such comprehensive log scaling only at ultra-high Reynolds numbers. This demonstrates that ultra-high Reynolds number is not necessary for comprehensive log scaling to occur; while specific details of the sink flow are special, the relevance to general …
The modified Townsend-Perry attached eddy model of Vassilicos et al (2015) combines the outer peak/plateau behaviour of rms streamwise turbulence velocity profiles and the Townsend-Perry log-decay of these profiles at higher distances from the wall. This model was validated by these authors for high Reynolds number tur- bulent pipe flow data and is shown here to describe equally well and with about the same parameter values turbulent boundary layer flow data from four different facilities and a wide range of Reynolds numbers. The model has predictive value as, when extrapolated to the extremel…
This is a comment on arXiv:physics/0608292, subsequently published in Phys Rev Lett, 98, 014501 (2007), in which Willis and Kerswell study in direct numerical simulations the statistics of turbulent lifetimes in pipe flow. They suggest that their data for six different Reynolds numbers support a divergence of the lifetime near a Reynolds number of about 1870. However, their conclusion is not compelling: a re-analysis of their data shows that it is also compatible with an exponentially increasing lifetime.
Here we report the measurements of two-dimensional (2-D) spectra of the streamwise velocity ($u$) in a high Reynolds number turbulent boundary layer. A novel experiment employing multiple hot-wire probes was carried out at friction Reynolds numbers ranging from 2400 to 26000. Taylor's frozen turbulence hypothesis is used to convert temporal-spanwise information into a 2-D spatial spectrum which shows the contribution of streamwise ($λ_x$) and spanwise ($λ_y$) length scales to the streamwise variance at a given wall height ($z$). At low Reynolds numbers, the shape of the 2-D spectra at a consta…
In this paper we present axisymmetric nonlinear simulations of magnetized Ekman and Stewartson layers in a magnetized Taylor-Couette flow with a centrifugally stable angular-momemtum profile and with a magnetic Reynolds number below the threshold of magnetorotational instability. The magnetic field is found to inhibit the Ekman suction. The width of the Ekman layer is reduced with increased magnetic field normal to the end plate. A uniformly-rotating region forms near the outer cylinder. A strong magnetic field leads to a steady Stewartson layer emanating from the junction between differential…
At sufficiently high Reynolds numbers, shear-flow turbulence close to a wall acquires universal properties. When length and velocity are rescaled by appropriate characteristic scales of the turbulent flow and thereby measured in \emph{inner units}, the statistical properties of the flow become independent of the Reynolds number. We demonstrate the existence of a wall-attached non-chaotic exact invariant solution of the fully nonlinear 3D Navier-Stokes equations for a parallel boundary layer that captures the characteristic self-similar scaling of near-wall turbulent structures. The branch of t…
We present measurements of quasi-Keplerian flows in a Taylor-Couette device that identify the boundary conditions required to generate near-ideal flows that exhibit self-similarity under scaling of the Reynolds number. These experiments are contrasted with alternate boundary configurations that result in flows that progressively deviate from ideal Couette rotation as the Reynolds number is increased. These behaviors are quantitatively explained in terms of the tendency to generate global Ekman circulation and the balance of angular momentum fluxes through the axial and radial boundary layers.
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…
Laminar flows through pipes driven at steady, pulsatile or oscillatory rates undergo a sub-critical transition to turbulence. We carry out an extensive linear non-modal stability analysis of these flows and show that for sufficiently high pulsation amplitudes the stream-wise vortices of the classic lift-up mechanism are outperformed by helical disturbances exhibiting an Orr-like mechanism. In oscillatory flow, the energy amplification depends solely on the Reynolds number based on the Stokes-layer thickness and for sufficiently high oscillation frequency and Reynolds number, axisymmetric distu…
The fluctuations of turbulence intensity in a pipe flow around the critical Reynolds number is difficult to study but important because they are related to turbulent-laminar transitions. We here propose a rare-event sampling method to study such fluctuations in order to measure the time-scale of the transition efficiently. The method is composed of two parts: (i) the measurement of typical fluctuations (the bulk part of an accumulative probability function) and (ii) the measurement of rare fluctuations (the tail part of the probability function) by employing dynamics where a feedback control o…
Computational fluid dynamics (CFD) is a useful tool for prediction of turbulence in aerodynamic and biomedical applications. The choice of appropriate turbulence models is key to reaching accurate predictions. The present investigation concentrated on the comparison of different turbulence models for predicting the flow field downstream of a constricted pipe. This geometry is relevant to arterial stenosis in patients with vascular diseases. More specifically, the results of Unsteady Reynolds-Averaged Navier-Stokes (URANS) and scale resolving simulation (SRS) turbulence models such as Large Edd…
Its is a well known fact that Turbulence exhibits non-locality, however, modeling has largely received local treatment following the work of Prandl over mixing-length model. Thus, in this article we report our findings by formulating a non-local closure model for Reynolds-averaged Navier-Stokes (RANS) equation using Fractional Calculus. Two model formulations are studied, namely one- and two-sided for Channel, Pipe and Couette flow, where the results shown have less 1% error. The motivation of two-sided model lies in recognising the fact that non-locality at a given spatial location is an aggr…
A modal stability analysis shows that pressure-driven pipe flow of an Oldroyd-B fluid is linearly unstable to axisymmetric perturbations, in stark contrast to its Newtonian counterpart which is linearly stable at all Reynolds numbers. The dimensionless groups that govern stability are the Reynolds number, the elasticity number, and the ratio of solvent to solution viscosity. The unstable mode has a phase speed close to the base-state maximum over the entire unstable region in the relevant parameter space, implying that the unstable mode belongs to a class of viscoelastic center modes. Unlike t…
A theoretical mechanism of laminar-turbulent transition originated from the deceleration of fluid streams on the walls of the channel or pipe is proposed. For Poiseuille flow an analytical expression relating the critical Reynolds number with the degree of disturbance of the flow is derived.
We study turbulent flows in a smooth straight pipe of circular cross--section up to $Re_τ \approx 6000$ using direct--numerical-simulation (DNS) of the Navier--Stokes equations. The DNS results highlight systematic deviations from Prandtl friction law, amounting to about $2\%$, which would extrapolate to about $4\%$ at extreme Reynolds numbers. Data fitting of the DNS friction coefficient yields an estimated von Kármán constant $k \approx 0.387$, which nicely fits the mean velocity profile, and which supports universality of canonical wall-bounded flows. The same constant also applies to the p…
We study the energy convergence of the Karhunen-Loève decomposition of the turbulent velocity field in a high-Reynolds-number pressure-driven boundary layer as a function of the number of modes. An energy-optimal Karhunen-Loève (KL) decomposition is obtained from wall-modeled large-eddy simulations at "infinite" Reynolds number. By explicitly using Fourier modes for the horizontal homogeneous directions, we are able to construct a basis of full rank, and we demonstrate that our results have reached statistical convergence. The KL dimension, corresponding to the number of modes per unit volume …
Although the critical Reynolds number for linear instability of the laminar flow in a straight pipe is infinite, we show that it is finite for a divergent pipe, and approaches infinity as the inverse of the divergence angle. The velocity profile at the threshold of inviscid stability is obtained. A non-parallel analysis yields linear instability at surprisingly low Reynolds numbers, of about 150 for a divergence of 3 degrees, which would suggest a role for such instabilities in the transition to turbulence. A multigrid Poisson equation solver is employed for the basic flow, and an extended eig…
Pipe flow often traverses a regime where laminar and turbulent flow co-exist. Prandtl and Tietjens explained this intermittency as a feedback between the fluctuations of the internal flow resistance and the constant pressure drop driving the flow. However, because the focus has moved towards studying intermittency without flow fluctuations near the universal critical Reynolds number, their explanation has largely disappeared. Here we refine the mechanism, which has never been put to a quantitative test, to develop a model that agrees with experiments at higher Reynolds numbers, enabling us to …
Extreme dissipation events in turbulent flows are rare, but they can be orders of magnitude stronger than the mean dissipation rate. Despite its importance in many small-scale physical processes, there is presently no accurate theory or model for predicting the extrema as a function of the Reynolds number. Here, we introduce a new model for the dissipation PDF based on the concept of significant shear layers, which are thin regions of elevated local mean dissipation. At very high Reynolds numbers these significant shear layers develop layered substructures. The flow domain is divided into the …
We study streamwise turbulence intensity definitions using smooth- and rough-wall pipe flow measurements made in the Princeton Superpipe. Scaling of turbulence intensity with the bulk (and friction) Reynolds number is provided for the definitions. The turbulence intensity scales with the friction factor for both smooth- and rough-wall pipe flow. Turbulence intensity definitions providing the best description of the measurements are identified. A procedure to calculate the turbulence intensity based on the bulk Reynolds number (and the sand-grain roughness for rough-wall pipe flow) is outlined.
The flow instability and further transition to turbulence in a toroidal pipe (torus) with curvature (tube-to-coiling diameter) 0.049 is investigated experimentally. The flow inside the toroidal pipe is driven by a steel sphere fitted to the inner pipe diameter. The sphere is moved with constant azimuthal velocity from outside the torus by a moving magnet. The experiment is designed to investigate curved pipe flow by optical measurement techniques. Using stereoscopic particle image velocimetry, laser Doppler velocimetry and pressure drop measurements, the flow is measured for Reynolds numbers r…
A scaling relation for the high frequency regime of streamwise energy spectra (in frequency domain) in the near-wall region is proposed. This is based on the dimensional analysis approach of \cite{perry1977} and \cite{Zamalloa2014} together with the hypothesis that the small-scale fluctuations in the near-wall region should only depend on the viscous scales, analogous to the Prandtl's law-of-the-wall for the mean flow. This allows us to examine the lower bound for the high frequency regime where law-of-the-wall in spectra would hold. Observations in high-Reynolds-number turbulent boundary laye…
In the present treatise, a stability analysis of the bottom boundary layer under solitary waves based on energy bounds and nonmodal theory is performed. The instability mechanism of this flow consists of a competition between streamwise streaks and two- dimensional perturbations. For lower Reynolds numbers and early times, streamwise streaks display larger amplification due to their quadratic dependence on the Reynolds number, whereas two-dimensional perturbations become dominant for larger Reynolds numbers and later times in the deceleration region of this flow, as the maximum amplification o…
A recently developed so-called explicit algebraic Reynolds-stress (EARS) model is applied to a transitioning atmospheric boundary layer (ABL). The simulation describes a diurnal cycle with a deep convective ABL during daytime and an extremely thin and stably stratified ABL during nighttime. The predictions of the EARS model are compared to large-eddy simulations (LES) of Couvreux \emph{et al.}~ (Bound Layer Meteorol 176:369-400, 2020). The model simulation is extended in time in order to study several consecutive diurnal cycles. The EARS model uses the same parametrization and model coefficien…
The laminar-turbulent transition in the circular pipe flow has been tested experimentally. The critical Reynolds numbers for the flows of different gases (He, Ne, Ar, Kr, Xe, N2, CO2, SF6) and liquids (H2O, D2O, C2H5OH) have been compared. The difference up to 40% was observed. The possible reasons of non-universality of critical Reynolds number are discussed. The conclusion is the statistical approach is needed to penetrate into the nature of observed phenomenon.
We investigate the instabilities and transition mechanisms of Boussinesq stratified boundary layers on sloping boundaries when subjected to oscillatory body forcing parallel to the slope. Such conditions are typical of the boundary layers generated by low wavenumber internal tides sloshing up and down adiabatic abyssal slopes in the absence of mean flows, high wavenumber internal tides, and resonant tide-bathymetry interactions. We examine flows within a region of non-dimensional parameter space typical of the mid- to low-latitude oceanic $M_2$ tides on hydraulically smooth abyssal slopes by d…
