This paper gives a new unified formula for the Newtonian fluids valid for all pipe flow regimes from laminar to the fully rough turbulent. It includes laminar, unstable sharp jump from laminar to turbulent, and all types of the turbulent regimes: smooth turbulent regime, partial non-fully developed turbulent and fully developed rough turbulent regime. The formula follows the inflectional form of curves as suggested in Nikuradse's experiment rather than monotonic shape proposed by Colebrook and White. The composition of the proposed unified formula consists of switching functions and of the interchangeable formulas for laminar, smooth turbulent and fully rough turbulent flow. The proposed switching functions provide a smooth and a computationally cheap transition among hydraulic regimes. Thus, the here presented formulation represents a coherent hydraulic model suitable for engineering use. The model is compared to existing literature models, and shows smooth and computationally cheap transitions among hydraulic regimes.
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.
The interaction between a fluid and a wall is described with a certain boundary condition for the fluid velocity at the wall. To understand how fluids behave near a rough wall, the fluid velocity at every point of the rough surface may be provided. This approach requires detailed knowledge of, and likely depends strongly on the roughness. Another approach of modeling the boundary conditions of a rough wall is to coarse grain and extract a penetration depth over which on average the fluid penetrates into the roughness. In this work we show that for a broad range of periodic roughness patterns a…
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…
The aim of this Letter is to characterize the flow regimes of suspensions of finite-size rigid particles in a viscous fluid at finite inertia. We explore the system behavior as function of the particle volume fraction and the Reynolds number (the ratio of flow and particle inertia to viscous forces). Unlike single phase flows where a clear distinction exists between the laminar and the turbulent states, three different regimes can be identified in the presence of a particulate phase, with smooth transitions between them. At low volume fractions, the flow becomes turbulent when increasing the R…
Liquid-infused surfaces (LIS) can reduce friction drag in both laminar and turbulent flows. However, the heat transfer properties of such multi-phase surfaces have still not been investigated to a large extent. We use numerical simulations to study conjugate heat transfer of liquid-filled grooves. It is shown that heat transfer can increase for both laminar and turbulent liquid flows due to recirculation in the surface texture. For the increase to be substantial, the thermal conductivity of the solid must be similar to the thermal conductivity of the fluids, and the recirculation in the groove…
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
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.
We present numerical simulations of laminar and turbulent channel flow of an elastoviscoplastic fluid. The non-Newtonian flow is simulated by solving the full incompressible Navier-Stokes equations coupled with the evolution equation for the elastoviscoplastic stress tensor. The laminar simulations are carried out for a wide range of Reynolds numbers, Bingham numbers and ratios of the fluid and total viscosity, while the turbulent flow simulations are performed at a fixed bulk Reynolds number equal to 2800 and weak elasticity. We show that in the laminar flow regime the friction factor increas…
The correct description of the surface water dynamics in the model of shallow water requires accounting for friction. To simulate a channel flow in the Chezy model the constant Manning roughness coefficient is frequently used. The Manning coefficient n M is an integral parameter which accounts for a large number of physical factors determining the flow braking. We used computational simulations in a shallow water model to determine the relationship between the Manning coefficient and the parameters of small-scale perturbations of a bottom in a long channel. Comparing the transverse water veloc…
The friction f is the property of wall-bounded flows that sets the pumping cost of a pipeline, the draining capacity of a river, and other variables of practical relevance. For highly turbulent rough-walled pipe flows, f depends solely on the roughness length scale r, and the f -- r relation may be expressed by the Strickler empirical scaling f $\propto$ r$^{\frac{1}{3}}$. Here, we show experimentally that for soap film flows that are the two-dimensional (2D) equivalent of highly turbulent rough-walled pipe flows, f $\propto$ r and the f -- r relation is not the same in 2D as in 3D. Our findin…
We present a physical model for turbulent friction on rough surfaces with regularly distributed roughness elements. Wall shear stresses are expressed as functions of physical quantities. Surfaces with varying roughness densities and roughness elements with different aspect ratios are considered. We propose a straight forward method based on the conservation of momentum to deduce the drag on elements by expressing it as functions of the maximum drag and drag reductions ratios, as the drag on individual elements decreases as packing density increases. A drag reduction effect of momentum redistri…
The flow of viscous incompressible fluid over a periodically corrugated surface is investigated numerically by solving the Navier-Stokes equation with the local slip and no-slip boundary conditions. We consider the effective slip length which is defined with respect to the level of the mean height of the surface roughness. With increasing corrugation amplitude the effective no-slip boundary plane is shifted towards the bulk of the fluid, which implies a negative effective slip length. The analysis of the wall shear stress indicates that a flow circulation is developed in the grooves of the rou…
As everyone knows who has opened a kitchen faucet, pipe flow is laminar at low flow velocities and turbulent at high flow velocities. At intermediate velocities there is a transition wherein plugs of laminar flow alternate along the pipe with "flashes" of a type of fluctuating, non-laminar flow which remains poorly known. We show experimentally that the fluid friction of flash flow is diagnostic of turbulence. We also show that the statistics of flash flow are in keeping with Kolmogorov's phenomenological theory of turbulence (so that, e.g., the energy spectra of both flash flow and turbulent …
Drawing on an analogy to critical phenomena, it was shown that the Nikuradse turbulent friction factor ($f_t$) measurements in pipes of radius $R$ and wall roughness $r$ can be collapsed onto a one-dimensional curve expressed as a conveyance law $f_t Re^{1/4}=g_o(χ)$, where $Re$ is a bulk Reynolds number, $χ=Re^{3/4}\left({r}/{R}\right)$. The implicit function $g_o(.)$ was conjectured based on matching two asymptotic limits of $f_t$. However, the connection between $g_o(.)$ and the phenomenon it proclaims to represent - turbulent eddies - remains lacking. Using models for the wall-normal veloc…
Modeling and simulating overland flow fed by rainfall is a common issue in watershed surface hydrology. Modelers have to choose among various friction models when defining their simulation framework. The purpose of this work is to compare the simulation quality for the Manning, Darcy-Weisbach, and Poiseuille friction models on the simple case of a constant rain on a thin experimental flume. Results show that the usual friction law of Manning is not suitable for this type of flow. The Poiseuille friction model gave the best results both on the flux at the outlet and the velocity and depth profi…
This study considers the linear stability of Poiseuille-Rayleigh-Bénard flows, subjected to a transverse magnetic field to understand the instabilities that arise from the complex interaction between the effects of shear, thermal stratification and magnetic damping. This fundamental study is motivated in part by the desire to enhance heat transfer in the blanket ducts of nuclear fusion reactors. In pure MHD flows, the imposed transverse magnetic field causes the flow to become quasi-2D and exhibit disturbances that are localised to the horizontal walls. However, the vertical temperature strati…
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…
Wall-roughness induces extra drag in wall-bounded turbulent flows. Mapping any given roughness geometry to its fluid dynamic behaviour has been hampered by the lack of accurate and direct measurements of skin-friction drag. Here the Taylor-Couette (TC) system provides an opportunity as it is a closed system and allows to directly and reliably measure the skin-friction. However, the wall-curvature potentially complicates the connection between the wall friction and the wall roughness characteristics. Here we investigate the effects of a hydrodynamically fully rough surface on highly turbulent, …
We consider a laminar Blasius boundary-layer flow above a slightly heated horizontal plate and study the effect of polymer additives on the heat transport. We show that the action of the polymers can be understood as a space-dependent effective viscosity that first increases from the zero-shear value then decreases exponentially back to the zero-shear value as one moves away from the boundary. We find that with such an effective viscosity, both the horizontal and vertical velocities near the plate are decreased thus leading to an increase in the friction drag and a decrease in the heat transpo…
The effect of slip surfaces on the laminar-turbulent separatrix of plane Poiseuille flow is studied by direct numerical simulation. Turbulence lifetimes, the likelihood that turbulence is sustained, is investigated for transitional flows with various slip lengths. Slip surfaces decrease the likelihood of sustained turbulence compared to the no-slip case, and likelihood is further decreased as slip length is increased. A deterministic analysis of the effects of slip surfaces on transition to turbulence is performed using nonlinear traveling wave solutions to the Navier-Stokes equations, also kn…
In the present study, we use a fifth-order ordinary differential equation, a generalization of the Blasius equation derived from the Leray-alpha model, to examine the transition to turbulence in rough-wall boundary-layer flows. This equation, combined with a weaker formulation of the von Karman log law modified to include the effects of surface roughness, provides a family of turbulent velocity profiles with two free parameters: the momentum thickness based Reynolds number and the roughness function. As a result, a skin-friction correlation is obtained and predictions of the transitional Reyno…
Complex liquids flow is known to be drastically affected by the roughness condition at the interfaces. We combined stresses measurements and observations of the flow during the motion of different rough surfaces in dry liquid foams. We visually show that three distinct friction regimes exists: slippage, stick-slip motion, and anchored soap films. Our stress measurements are validated for slippage and anchored regimes based on existing models, and we propose a leverage rule to describe the stresses during the stick-slip regime. We find that the occurrence of the stick-slip or anchored regimes i…
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 classical experiments on turbulent friction in rough pipes were performed by J. Nikuradse in the 1930's. Seventy years later, they continue to defy theory. Here we model Nikuradse's experiments using the phenomenological theory of Kolmogórov, a theory that is widely thought to be applicable only to highly idealized flows. Our results include both the empirical scalings of Blasius and Strickler, and are otherwise in minute qualitative agreement with the experiments; they suggest that the phenomenological theory may be relevant to other flows of practical interest; and they unveil the existe…
We show that the main results of the analysis of the friction factor for turbulent pipe flow reported in G. Gioia and P. Chakraborty (GC), Phys. Rev. Lett. 96, 044502 (1996) can be recovered by assuming the Heisenberg closure hypothesis for the turbulent spectrum. This highlights the structural features of the turbulent spectrum underlying GC's analysis.
We use momentum transfer arguments to predict the friction factor $f$ in two-dimensional turbulent soap-film flows with rough boundaries (an analogue of three-dimensional pipe flow) as a function of Reynolds number Re and roughness $r$, considering separately the inverse energy cascade and the forward enstrophy cascade. At intermediate Re, we predict a Blasius-like friction factor scaling of $f\propto\textrm{Re}^{-1/2}$ in flows dominated by the enstrophy cascade, distinct from the energy cascade scaling of $\textrm{Re}^{-1/4}$. For large Re, $f \sim r$ in the enstrophy-dominated case. We use …
Results of numerical simulations obtained by a staggered finite difference scheme together with an efficient immersed boundary method are presented to understand the effects of the shape of three-dimensional obstacles on the transition of a boundary layer from a laminar to a turbulent regime. Fully resolved Direct Numerical Simulations (DNS), highlight that the closer to the obstacle the symmetry is disrupted the smaller is the transitional Reynolds number. It has been also found that the transition can not be related to the critical roughness Reynolds number used in the past. The simulations …
A numerical method based upon the immersed boundary technique for the fluid-solid coupling and on a soft-sphere approach for solid-solid contact is used to perform direct numerical simulation of the flow-induced motion of a thick bed of spherical particles in a horizontal plane channel. The collision model features a normal force component with a spring and a damper, as well as a damping tangential component, limited by a Coulomb friction law. The standard test case of a single particle colliding perpendicularly with a horizontal wall in a viscous fluid is simulated over a broad range of Stoke…
Direct numerical simulations (DNS) are performed for two wall-bounded flow configurations: laminar Couette flow at $Re=740$ and turbulent channel flow at $Re_τ=180$, where $τ$ is the shear stress at the wall. The top wall is smooth and the bottom wall is a realistically rough superhydrophobic surface (SHS), generated from a three-dimensional surface profile measurement. The air-water interface, which is assumed to be flat, is simulated using the volume-of-fluid (VOF) approach. The laminar Couette flow is studied with varying interface heights $h$ to understand the effect on slip and drag reduc…
Laminar flow over a bubble mattress is expected to experience a significant reduction in friction since the individual surfaces of the bubbles are shear-free. However, if the bubbles are sufficiently curved, their protrusion into the fluid and along the flow direction can lead to an increase in friction as was recently demonstrated experimentally and computationally. We provide in this paper a simple model for this result. We consider a shear flow at low Reynolds number past a two-dimensional array of bubbles, and calculate analytically the effective slip length of the surface as function of t…
Direct Numerical Simulations are used to solve turbulent flow and heat transfer over a variety of rough walls in a channel. The wall geometries are exactly resolved in the simulations. The aim is to understand the effect of roughness morphology and its scaling on the augmentation of heat transfer relative to that of skin friction. A number of realistic rough surface maps obtained from the scanning of gas turbine blades and internal combustion engines as well as several artificially generated rough surfaces are examined. In the first part of the paper, effects of statistical surface properties,…
Classical scaling relationships for rheological quantities such as the $μ(J)$-rheology have become increasingly popular for closures of two-phase flow modeling. However, these frameworks have been derived for monodisperse particles. We aim to extend these considerations to sediment transport modeling by using a more realistic sediment composition. We investigate the rheological behavior of sheared sediment beds composed of polydisperse spherical particles in a laminar Couette-type shear flow. The sediment beds consist of particles with a diameter size ratio of up to ten, which corresponds to g…
A numerical study of two-dimensional flow past a confined circular cylinder with slip wall is performed. A dimensionless number, Knudsen number ($Kn$) is used to describe the slip length of cylinder wall. The Reynolds number ($Re$) and Knudsen number ($Kn$) ranges considered are $Re = [1, 180]$ and $Kn = [0, \infty)$, respectively. Time-averaged flow separation angle ($\bar{θ_s}$), dimensionless recirculation length ($\bar{L_s}$) and the tangential velocity ($\bar{u_τ}$) distributed on the cylinder's wall, drag coefficient ($\bar{C_d}$) and drag reduction ($DR$) are investigated. The time-aver…
The movement of subaqueous sediment in laminar shearing flow is numerically investigated by the coupled lattice Boltzmann and discrete element methods. First, the numerical method is validated by comparing the phase diagram proposed by Ouriemi et al. ({\it J. Fluid Mech}., vol. 636, 2009, pp. 321-336). Second, a detailed study on sediment movement is performed for sediment with varying solid volume fractions, and a nonlinear relationship between the normalised thickness of the mobile layer and the normalised fluid flow rate is observed for a densely-packed sediment. Third, an independent inves…
Direct numerical simulations of turbulent channels with rough walls are conducted in the transitionally rough regime. The effect that roughness produces on the overlying turbulence is studied using a modified triple decomposition of the flow. This decomposition separates the roughness-induced contribution from the background turbulence, with the latter essentially free of any texture footprint. For small roughness, the background turbulence is not significantly altered, but merely displaced closer to the roughness crests, with the change in drag being proportional to this displacement. As the …
Liquid-infused surfaces (LIS) are a promising technique for reducing friction, fouling and icing in both laminar and turbulent flows. Previous work has demonstrated that these surfaces are susceptible to shear-driven drainage. Here, we report a different failure mode using direct numerical simulations of a turbulent channel flow with liquid-infused longitudinal grooves. When the liquid-liquid surface tension is small and/or grooves are wide, we observe traveling-wave perturbations on the interface with amplitudes larger than the viscous sublayer of the turbulent flow. These capillary waves ind…
In this paper we present electro-osmotic (EO) flow within a more traditional fluid mechanics framework. Specifically, the modified Bernoulli equation (viz. the energy equation, the mechanical energy equation, the pipe flow equation, etc.) is shown to be applicable to EO flows if an electrical potential energy term is also included. The form of the loss term in the modified Bernoulli equation is unaffected by the presence of an electric field; i.e., the loss term still represents the effect of wall shear stress, which can be represented via a friction factor. We show that that the friction fact…
The flow-induced deformation of a thin, elastic splitter plate attached to the rear of a circular cylinder and subjected to laminar pulsatile inflow is investigated. The cylinder and elastic splitter plate are contained within a narrow channel and the Reynolds number is mostly restricted to Re = 100, primarily covering the two-dimensional flow regime. An in-house fluid-structure interaction code is employed for simulations, which couples a sharp-interface immersed boundary method for the fluid dynamics with a finite-element method to treat the structural dynamics. The structural solver is impl…
We conducted direct numerical simulations (DNSs) of turbulent flow over three-dimensional sinusoidal roughness in a channel. A passive scalar is present in the flow with Prandtl number $Pr=0.7$, to study heat transfer by forced convection over this rough surface. The minimal channel is used to circumvent the high cost of simulating high Reynolds number flows, which enables a range of rough surfaces to be efficiently simulated. The near-wall temperature profile in the minimal channel agrees well with that of the conventional full-span channel, indicating it can be readily used for heat-transfer…
We show that the Fukagata et al. (2002)'s identity for free-stream boundary layers simplifies to the von Karman momentum integral equation relating the skin-friction coefficient and the momentum thickness when the upper bound in the integrals used to obtain the identity is taken to be asymptotically large. If a finite upper bound is used, the terms of the identity depend spuriously on the bound itself. Differently from channel and pipe flows, the impact of the Reynolds stresses on the wall-shear stress cannot be quantified in the case of free-stream boundary layers because the Reynolds stresse…
Flows in fluid layers are ubiquitous in industry, geophysics and astrophysics. Large-scale flows in thin layers can be considered two-dimensional (2d) with bottom friction added. Here we find that the properties of such flows depend dramatically on the way they are driven. We argue that wall-driven (Couette) flow cannot sustain turbulence at however small viscosity and friction. Direct numerical simulations (DNS) up to the Reynolds number $Re=10^6$ confirm that all perturbations die in a plane Couette flow. On the contrary, for sufficiently small viscosity and friction, we show that finite per…
Super Hydrophobic (SH) coatings are widely used to mitigate drag in various applications. Numerous studies have demonstrated that the beneficial wall-slip effect produced by these materials disappears in laminar flow regimes. The main mechanisms considered to be behind the decrease in performance are Marangoni-induced stresses and air/liquid interface deformation. In the present study, a new mechanism is proposed to explain the loss of performances of SH-surfaces in laminar flow regimes. Here we consider the flow of air inside the plastron and the associated momentum loses induced by roughness…
The lattice Boltzmann method can be used to simulate flow through porous media with full geometrical resolution. With such a direct numerical simulation, it becomes possible to study fundamental effects which are difficult to assess either by developing macroscopic mathematical models or experiments. We first evaluate the lattice Boltzmann method with various boundary handling of the solid-wall and various collision operators to assess their suitability for large scale direct numerical simulation of porous media flow. A periodic pressure drop boundary condition is used to mimic the pressure dr…
We perform a numerical analysis of superfluid turbulence produced by thermal counterflow in He II by using the vortex filament model. Counterflow in a low aspect ratio channel is known to show the transition from laminar flow to the two turbulent states TI and TII. The present understanding is that the TI has the turbulent superfluid and the laminar normal fluid but both fluids are turbulent in the TII state. This work studies the vortex tangle in the TI state. Solid boundary condition is applied to walls of a square channel, and the velocity field of the normal fluid is prescribed to be a lam…
There are different ways for fluid flow to become turbulent, but usually many instabilities take place before that. Among these instabilities the transition from laminar flow to vortex shedding carries significant practical importance. Here we study a flow, where at high enough flow rates, vortices are generated by a fixed obstacle (cylinder), which penetrates a flowing, quasi-two dimensional soap film. We present experimental results that demonstrate that the transition from laminar flow to a periodic Kármán vortex street can be hysteretic, i.e. changing the mean flow rate $\bar V$ vortices c…
