A continuum one-dimensional model of the plane Couette-Poiseuille flow is developed to describe the pressure distribution in a drag stage of molecular pump of either the Gaede or Holweck type. In spite of its simplicity and approximate nature the model provides a good qualitative representation of the drag pump operation in the whole range of the regimes from the continuum to free molecular ones.
The normal-mode analysis of the Reynolds-Orr energy equation governing the stability of viscous motion for general three-dimensional disturbances has been revisited. The energy equation has been solved as an unconstrained minimization problem for the Couette-Poiseuille flow. The minimum Reynolds number for every Couette-Poiseuille velocity profile has been computed and compared with those available in the literature. For fully three-dimensional disturbances, it is shown that the minimum Reynolds number is in general smaller than the corresponding two-dimensional counterpart for all the Couette…
We present a new experimental set-up that creates a shear flow with zero mean advection velocity achieved by counterbalancing the nonzero streamwise pressure gradient by moving boundaries, which generates plane Couette-Poiseuille flow. We carry out the first experimental results in the transitional regime for this flow. Using flow visualization we characterize the subcritical transition to turbulence in Couette-Poiseuille flow and show the existence of turbulent spots generated by a permanent perturbation. Due to the zero mean advection velocity of the base profile, these turbulent structures …
The stability of two-dimensional Poiseuille flow and plane Couette flow for concentrated suspensions is investigated. Linear stability analysis of the two-phase flow model for both flow geometries shows the existence of a convectively driven instability with increasing growth rates of the unstable modes as the particle volume fraction of the suspension increases. In addition it is shown that there exists a bound for the particle phase viscosity below which the two-phase flow model may become ill-posed as the particle phase approaches its maximum packing fraction. The case of two-dimensional Po…
The linear amplification mechanisms leading to streamwise-constant large-scale structures in laminar and turbulent channel flows are considered. A key feature of the analysis is that the Orr--Sommerfeld and Squire operators are each considered separately. Physically this corresponds to considering two separate processes: (i) the response of wall-normal velocity fluctuations to external forcing; and (ii) the response of streamwise velocity fluctuations to wall-normal velocity fluctuations. In this way we exploit the fact that, for streamwise-constant fluctuations, the dynamics governing the wal…
We show possibility of the Plane Couette (PC) flow instability for Reynolds number Re>Reth=140. This new result of the linear hydrodynamic stability theory is obtained on the base of refusal from the traditionally used assumption on longitudinal periodicity of the disturbances along the direction of the fluid flow. We found that earlier existing understanding on the linear stability of this flow for any arbitrary large Reynolds number is directly related with an assumption on the separation of the variables of the spatial variability for the disturbance field and their periodicity in linear th…
In the above paper by Bechtel, Cai, Rooney and Wang, Physics of Fluids, 2004, 16, 3955-3974 six different theories of a Newtonian viscous fluid are investigated and compared, namely, the theory of a compressible Newtonian fluid, and five constitutive limits of this theory: the incompressible theory, the limit where density changes only due to changes in temperature, the limit where density changes only with changes in entropy, the limit where pressure is a function only of temperature, and the limit of pressure a function only of entropy. The six theories are compared through their ability to …
Waviness and self-sustained turbulence in plane Couette-Poiseuille flow
Universidad de Buenos Aires, Facultad de Ciencias Exactas y Naturales（布宜诺斯艾利斯大学，理学院）；CONICET - Universidad de Buenos Aires, Instituto de Física Interdisciplinaria y Aplicada (INFINA)（阿根廷国家科学研究委员会-布宜诺斯艾利斯大学，交叉与应用物理研究所 (INFINA)）；Institut Franco-Argentin de Dynamique des Fluides pour l’Environnement, IRL 2027, CNRS, Universidad de Buenos Aires, CONICET（法阿环境流体力学研究所，IRL 2027，法国国家科学研究中心，布宜诺斯艾利斯大学，阿根廷国家科学研究委员会）；Université Savoie Mont Blanc, CNRS UMR 5271, LOCIE（萨瓦-蒙布朗大学，法国国家科学研究中心 UMR 5271，LOCIE）；Physique et Mécanique des Milieux Hétérogènes (PMMH UMR 7636) CNRS, ESPCI Paris, Université PSL, Sorbonne Université, Université Paris Cité（异质介质物理与力学实验室 (PMMH UMR 7636)，法国国家科学研究中心，巴黎高等物理化工国立学校，巴黎文理研究大学，索邦大学，巴黎西岱大学）
We explore the effect of forcing on the linear shear flow or plane Couette flow, which is also the background flow in the very small region of the Keplerian accretion disk. We show that depending on the strength of forcing and boundary conditions suitable for the systems under consideration, the background plane shear flow and, hence, accretion disk velocity profile modifies to parabolic flow, which is plane Poiseuille flow or Couette-Poiseuille flow, depending on the frame of reference. In the presence of rotation, plane Poiseuille flow becomes unstable at a smaller Reynolds number under pure…
The steady state of a dilute gas enclosed between two infinite parallel plates in relative motion and under the action of a uniform body force parallel to the plates is considered. The Bhatnagar-Gross-Krook model kinetic equation is analytically solved for this Couette-Poiseuille flow to first order in the force and for arbitrary values of the Knudsen number associated with the shear rate. This allows us to investigate the influence of the external force on the non-Newtonian properties of the Couette flow. Moreover, the Couette-Poiseuille flow is analyzed when the shear-rate Knudsen number and…
We study the nonlinear stability of plane Couette and Poiseuille flows with the Lyapunov second method by using the classical L2-energy. We prove that the streamwise perturbations are L2-energy stable for any Reynolds number. This contradicts the results of Joseph [10], Joseph and Carmi [12] and Busse [4], and allows us to prove that the critical nonlinear Reynolds numbers are obtained along two-dimensional perturbations, the spanwise perturbations, as Orr [16] had supposed. This conclusion combined with recent results by Falsaperla et al. [8] on the stability with respect to tilted rolls, pro…
We present a detailed study of the linear stability of plane Couette-Poiseuille flow in the presence of a cross-flow. The base flow is characterised by the cross flow Reynolds number, $R_{inj}$ and the dimensionless wall velocity, $k$. Squire's transformation may be applied to the linear stability equations and we therefore consider 2D (spanwise-independent) perturbations. Corresponding to each dimensionless wall velocity, $k\in[0,1]$, two ranges of $R_{inj}$ exist where unconditional stability is observed. In the lower range of $R_{inj}$, for modest $k$ we have a stabilisation of long wavelen…
Here we report some interesting new features of the spatio-temporal instability of the incompressible plane Couette-Poiseuille flow (CPF). First of all, this flow represents the first instance of a "non-inflectional" absolute instability, within constant-viscosity formulation, which is triggered when one of the plates moves opposite to the bulk motion. More strikingly, with further increase in the negative plate motion, the absolute instability ($\textrm{AI}$) transitions to an "upstream" convective instability ($\textrm{CI}^-$), wherein an unstable wave packet moves opposite to the direction …
An investigation of the effect of a destabilizing cross-stream temperature gradient on the transient growth phenomenon of plane Poiseuille flow and plane Couette flow is presented. Only the streamwise-uniform and nearly streamwise-uniform disturbances are highly influenced by the Rayleigh number Ra and Prandtl number Pr. The maximum optimal transient growth G max of streamwise-uniform disturbances increases slowly with increasing Ra and decreasing Pr. For all Ra and Pr, at moderately large Reynolds numbersRe, the supremum of G max is always attained for streamwise-uniform perturbations (or nea…
Exact solutions are obtained for the steady flow of a power-law fluid between parallel plates with partial slip conditions and uniform cross flow. The problem is properly formulated and similarities are exploited. The exact solutions are obtained in terms of integrals which can be performed, in closed form, in special cases of the power-law index n. Solutions to cases of n=1/2, 1, and 2; representing a pseudo-plastic, a Newtonian, and a dilatant fluid, respectively, are presented. Tendencies to corresponding degenerate cases in the literature are demonstrated. Depending on the strength of the …
We report the results of an experimental investigation into the decay of turbulence in plane Couette-Poiseuille flow using 'quench' experiments where the flow laminarises after a sudden reduction in Reynolds number $Re$. Specifically, we study the velocity field in the streamwise-spanwise plane. We show that the spanwise velocity containing rolls, decays faster than the streamwise velocity, which displays elongated regions of higher or lower velocity called streaks. At final Reynolds numbers above 425, the decay of streaks displays two stages: first a slow decay when rolls are present and seco…
In this article we prove, choosing an appropriately weighted $L_2$-energy equivalent to the classical energy, that the plane Couette and Poiseuille flows are nonlinearly stable with respect to streamwise perturbations for any Reynolds number. In this case the coefficient of time-decay of the energy is $π^2/(2 {\rm Re})$, and it is a bound from above of the time-decay of streamwise perturbations of linearized equations. We also prove that the plane Couette and Poiseuille flows are nonlinearly stable if the Reynolds number is less then ${\rm Re}_{Orr}/\sin φ$ when the perturbation is a tilted pe…
We report on the modification of drag by neutrally buoyant spherical particles in highly turbulent Taylor-Couette flow. These particles can be used to disentangle the effects of size, deformability, and volume fraction on the drag, when contrasted with the drag for bubbly flows. We find that rigid spheres hardly change the drag of the system beyond the trivial viscosity effects caused by replacing the working fluid with particles. The size of the particle has a marginal effect on the drag, with smaller diameter particles showing only slightly lower drag. Increasing the particle volume fraction…
Using the Taylor--Couette geometry we experimentally investigate the effect of salt on drag reduction caused by bubbles present in the flow. We combine torque measurements with optical high-speed imaging to relate the bubble size to the drag experienced by the flow. Previous findings have shown that a small percentage of air (4%) can lead to dramatic drag reduction (40%). In contrast to previous laboratory experiments, which mainly used fresh water, we will vary the salinity from fresh water to the average salinity of ocean water. We find that the drag reduction is increasingly more inhibited …
We create a highly controlled lab environment-accessible to both global and local monitoring-to analyse turbulent boiling flows and in particular their shear stress in a statistically stationary state. Namely, by precisely monitoring the drag of strongly turbulent Taylor-Couette flow (the flow in between two co-axially rotating cylinders, Reynolds number $\textrm{Re}\approx 10^6$) during its transition from non-boiling to boiling, we show that the intuitive expectation, namely that a few volume percent of vapor bubbles would correspondingly change the global drag by a few percent, is wrong. Ra…
The problem of two-dimensional steady nonlinear dynamics in plane Couette flow is revisited using homotopy from either plane Poiseuille flow or from plane Couette flow perturbed by a small symmetry-preserving identity operator. Our results show that it is not possible to obtain the nonlinear plane Couette flow solutions reported by Cherhabili and Ehrenstein [Eur. J. Mech. B/Fluids, 14, 667 (1995)] using their Poiseuille-Couette homotopy. We also demonstrate that the steady solutions obtained by Mehta and Healey [Phys. Fluids, 17, 4108 (2005)] for small symmetry-preserving perturbations are inf…
The phenomenon of drag reduction induced by injection of bubbles into a turbulent carrier fluid has been known for a long time; the governing control parameters and underlying physics is however not well understood. In this paper, we use three dimensional numerical simulations to uncover the effect of deformability of bubbles injected in a turbulent Taylor-Couette flow on the overall drag experienced by the system. We consider two different Reynolds numbers for the carrier flow, i.e. $Re_i=5\times 10^3$ and $Re_i=2\times 10^4$; the deformability of the bubbles is controlled through the Weber n…
In this study we experimentally investigate bubbly drag reduction in a highly turbulent flow of water with dispersed air at $5.0 \times 10^{5} \leq \text{Re} \leq 1.7 \times 10^{6}$ over a non-wetting surface containing micro-scale roughness. To do so, the Taylor-Couette geometry is used, allowing for both accurate global drag and local flow measurements. The inner cylinder - coated with a rough, hydrophobic material - is rotating, whereas the smooth outer cylinder is kept stationary. The crucial control parameter is the air volume fraction $α$ present in the working fluid. For small volume fr…
We experimentally study the influence of wall roughness on bubble drag reduction in turbulent Taylor-Couette flow, i.e.\ the flow between two concentric, independently rotating cylinders. We measure the drag in the system for the cases with and without air, and add roughness by installing transverse ribs on either one or both of the cylinders. For the smooth wall case (no ribs) and the case of ribs on the inner cylinder only, we observe strong drag reduction up to $DR=33\%$ and $DR=23\%$, respectively, for a void fraction of $α=6\%$. However, with ribs mounted on both cylinders or on the outer…
Air cavities, i.e. air layers developed behind cavitators, are seen as a promising drag reducing method in the maritime industry. Here we utilize the Taylor-Couette (TC) geometry, i.e. the flow between two concentric, independently rotating cylinders, to study the effect of air cavities in this closed setup, which is well-accessible for drag measurements and optical flow visualizations. We show that stable air cavities can be formed, and that the cavity size increases with Reynolds number and void fraction. The streamwise cavity length strongly depends on the axial position due to buoyancy for…
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 study the homogeneous and the spatially periodic instabilities in a nematic liquid crystal layer subjected to steady plane {\em Couette} or {\em Poiseuille} flow. The initial director orientation is perpendicular to the flow plane. Weak anchoring at the confining plates and the influence of the external {\em electric} and/or {\em magnetic} field are taken into account. Approximate expressions for the critical shear rate are presented and compared with semi-analytical solutions in case of Couette flow and numerical solutions of the full set of nematodynamic equations for Poiseuille flow. In …
We experimentally investigate the drag modification by neutrally buoyant finite-size particles with various aspect ratios in a Taylor-Couette (TC) turbulent flow. The current Reynolds number, $Re$, ranges from $6.5\times10^3$ to $2.6\times10^4$, and the particle volume fraction, $Φ$, is up to $10\%$. Particles with three kinds of aspect ratio, $λ$, are used: $λ=1/3$ (oblate), $λ=1$ (spherical) and $λ=3$ (prolate). Unlike the case of bubbly TC flow, we find that the suspended finite-size particles increase the drag of the TC system regardless of their aspect ratios. In addition, the normalized …
Two-phase turbulent Taylor-Couette (TC) flow is simulated using an Euler-Lagrange approach to study the effects of a secondary phase dispersed into a turbulent carrier phase (here bubbles dispersed into water). The dynamics of the carrier phase is computed using Direct Numerical Simulations (DNS) in an Eulerian framework, while the bubbles are tracked in a Lagrangian manner by modelling the effective drag, lift, added mass and buoyancy force acting on them. Two-way coupling is implemented between the dispersed phase and the carrier phase which allows for momentum exchange among both phases and…
We study a large-scale instability in a sheared nonhelical turbulence that causes generation of large-scale vorticity. Three types of the background large-scale flows are considered, i.e., the Couette and Poiseuille flows in a small-scale homogeneous turbulence, and the "log-linear" velocity shear in an inhomogeneous turbulence. It is known that laminar plane Couette flow and antisymmetric mode of laminar plane Poiseuille flow are stable with respect to small perturbations for any Reynolds numbers. We demonstrate that in a small-scale turbulence under certain conditions the large-scale Couette…
A new turbulent Taylor-Couette system consisting of two independently rotating cylinders has been constructed. The gap between the cylinders has a height of 0.927 m, an inner radius of 0.200 m, and a variable outer radius (from 0.279 to 0.220 m). The maximum angular rotation rates of the inner and outer cylinder are 20 and 10 Hz, respectively, resulting in Reynolds numbers up to 3.4 x 10^6 with water as working fluid. With this Taylor-Couette system, the parameter space (Re_i, Re_o, η) extends to (2.0 x 10^6, {\pm}1.4 x 10^6, 0.716-0.909). The system is equipped with bubble injectors, temperat…
We propose a framework to understand input-output amplification properties of non- linear partial differential equation (PDE) models of wall-bounded shear flows, which are spatially invariant in one coordinate (e.g., streamwise-constant plane Couette flow). Our methodology is based on the notion of dissipation inequalities in control theory. In particular, we consider flows with body and other forcings, for which we study the input- to-output properties, including energy growth, worst-case disturbance amplification, and stability to persistent disturbances. The proposed method can be applied t…
We experimentally investigate the influence of alternating rough and smooth walls on bubbly drag reduction (DR). We apply rough sandpaper bands of width $s$ between $48.4\,mm$ and $148.5\,mm$, and roughness height $k = 695\,μm$, around the smooth inner cylinder (IC) of the Twente Turbulent Taylor-Couette facility. Between sandpaper bands, the IC is left uncovered over similar width $s$, resulting in alternating rough and smooth bands, a constant pattern in axial direction. We measure the DR in water that originates from introducing air bubbles to the fluid at (shear) Reynolds numbers $\textit{…
The energy gradient theory is used to study the instability of Taylor-Couette flow between concentric rotating cylinders. This theory has been proposed in our previous works. In our previous studies, the energy gradient theory was demonstrated to be applicable for wall-bounded parallel flows. It was found that the critical value of the energy gradient parameter Kmax at turbulent transition is about 370-389 for wall-bounded parallel flows (which include plane Poiseuille flow, pipe Poiseuille flow and plane Couette flow) below which no turbulence occurs. In this paper, the detailed derivation fo…
A turbulent-laminar banded pattern in plane Couette flow is studied numerically. This pattern is statistically steady, is oriented obliquely to the streamwise direction, and has a very large wavelength relative to the gap. The mean flow, averaged in time and in the homogeneous direction, is analysed. The flow in the quasi-laminar region is not the linear Couette profile, but results from a non-trivial balance between advection and diffusion. This force balance yields a first approximation to the relationship between the Reynolds number, angle, and wavelength of the pattern. Remarkably, the var…
The motion of a viscous drop is investigated when the interface is fully covered with a stagnant layer of surfactant in an arbitrary unsteady Stokes flow for the low surface Péclet number limit. The effect of the interfacial slip coefficient on the behavior of the flow field is also considered. The hydrodynamic problem is solved by the solenoidal decomposition method and the drag force is computed in terms of Faxen's laws using a perturbation ansatz in powers of the surface Péclet number. The analytical expressions for the migration velocity of the drop are also obtained in powers of the surfa…
Intelligent backpropagated neural networks application on Couette-Poiseuille flow of variable viscosity in a composite porous channel filled with an anisotropic porous layer
In the present note, we show that the uni-directional flows in a rectangular channel and in a circular pipe are exact spatio-temporal solutions of the Navier-Stokes equations over a short time interval. We assert that the classical plane Poiseuille-Couette flow and Hagen-Poiseuille flow are time-independent approximations of the exact solutions if an appropriate initial velocity distribution at starting location is specified. Conceptually, there do not exist absolute steady flows starting from unspecified initial data. The classic experimental measurements by Poiseuille can be explained in ter…
In this work we present numerical simulations of $10^5$ sub-Kolmogorov deformable bubbles dispersed in Taylor-Couette flow (a wall-bounded shear system) with rotating inner cylinder and outer cylinder at rest. We study the effect of deformability of the bubbles on the overall drag induced by the carrier fluid in the two-phase system. We find that an increase in deformability of the bubbles results in enhanced drag reduction due to a more pronounced accumulation of the deformed bubbles near the driving inner wall. This preferential accumulation is induced by an increase in the resistance on the…
The dispersed phase in turbulence can vary from almost inviscid fluid to highly viscous fluid. By changing the viscosity of the dispersed droplet phase, we experimentally investigate how the deformability of dispersed droplets affects the global transport quantity of the turbulent emulsion. Different kinds of silicone oil are employed to result in the viscosity ratio, $ζ$, ranging from $0.53$ to $8.02$. The droplet volume fraction, $ϕ$, is varied from 0\% to 10\% with a spacing of 2\%. The global transport quantity, quantified by the normalized friction coefficient $c_{f,ϕ}/c_{f,ϕ=0}$, shows a…
We investigate the effect of microbubbles on Taylor-Couette flow by means of direct numerical simulations. We employ an Eulerian-Lagrangian approach with a gas-fluid coupling based on the point-force approximation. Added mass, drag, lift, and gravity are taken into account in the modeling of the motion of the individual bubble. We find that very dilute suspensions of small non-deformable bubbles (volume void fraction below 1%, zero Weber number and bubble Reynolds number <10) induce a robust statistically steady drag reduction (up to 20%) in the so called wavy vortex flow regime (Re = 600-2500…
The transition to turbulence in plane Poiseuille flow (PPF) is connected with the presence of exact coherent structures. In contrast to other shear flows, PPF has a number of different coherent states that are relevant for the transition. We here discuss the different states, compare the critical Reynolds numbers and optimal wavelengths for their appearance, and explore the differences between flows operating at constant mass flux or at constant pressure drop. The Reynolds numbers quoted here are based on the mean flow velocity and refer to constant mass flux, the ones for constant pressure dr…
Plane Poiseuille flow shows turbulence at a Reynolds number that is lower than the critical one for the onset of Tollmien-Schlichting waves. The transition to turbulence follows the same route as the by-pass transition in boundary layers, i.e. finite amplitude perturbations are required and the flow is dominated by downstream vortices and streaks in the transitional regime. In order to relate the phenomenology in plane Poiseuille flow to our previous studies of plane Couette flow (Kreilos & Eckhardt, 2012), we study a symmetric subspace of plane Poiseuille flow in which the bifurcation cascade…
Plane Poiseuille flow past a nanoscale cylinder that is arbitrarily confined (i.e., symmetrically or asymmetrically confined) in a slit channel is studied via hydrodynamic lubrication theory and molecular dynamics simulations, considering cases where the cylinder remains static or undergoes thermal motion. Lubrication theory predictions for the drag force and volumetric flow rate are in close agreement with molecular dynamics simulations of flows having molecularly thin lubrication gaps, despite the presence of significant structural forces induced by the crystalline structure of the modeled s…
Recently, detailed experiments on visco-elastic channel flow have provided convincing evidence for a nonlinear instability scenario which we had argued for based on calculations for visco-elastic Couette flow. Motivated by these experiments we extend the previous calculations to the case of visco-elastic Poiseuille flow, using the Oldroyd-B constitutive model. Our results confirm that the subcritical instability scenario is similar for both types of flow, and that the nonlinear transition occurs for Weissenberg numbers somewhat larger than one. We provide detailed results for the convergence o…
