Plane Couette Flow Research Articles

High-speed shear-driven dynamos. Part 2. Numerical analysis

This paper aims to numerically verify the large Reynolds number asymptotic theory of magneto-hydrodynamic (MHD) flows proposed in the companion paper Deguchi (J. Fluid Mech., vol. 868, 2019, pp. 176–211). To avoid any complexity associated with the chaotic nature of turbulence and flow geometry, nonlinear steady solutions of the viscous resistive MHD equations in plane Couette flow have been utilised. Two classes of nonlinear MHD states, which convert kinematic energy to magnetic energy effectively, have been determined. The first class of nonlinear states can be obtained when a small spanwise uniform magnetic field is applied to the known hydrodynamic solution branch of plane Couette flow. The nonlinear states are characterised by the hydrodynamic/magnetic roll–streak and the resonant layer at which strong vorticity and current sheets are observed. These flow features, and the induced strong streamwise magnetic field, are fully consistent with the vortex/Alfvén wave interaction theory proposed in the companion paper. When the spanwise uniform magnetic field is switched off, the solutions become purely hydrodynamic. However, the second class of ‘self-sustained shear-driven dynamos’ at the zero external magnetic field limit can be found by homotopy via the forced states subject to a spanwise uniform current field. The discovery of the dynamo states has motivated the corresponding large Reynolds number matched asymptotic analysis in the companion paper. Here, the reduced equations derived by the asymptotic theory have been solved numerically. The asymptotic solution provides remarkably good predictions for the finite Reynolds number dynamo solutions.

Direct numerical simulation and statistical analysis of stress-driven turbulent Couette flow with a free-slip boundary

The effects of free-slip boundary on shear turbulence are studied numerically using the direct numerical simulation (DNS) approach. The flow considered in this study is a stress-driven turbulent Couette flow between two flat boundaries. The top boundary has an imposed shear stress in the streamwise direction and a free-slip condition for the streamwise velocity fluctuation and the spanwise velocity, and the bottom boundary satisfies the no-slip condition. This type of flow has a mean flow pattern similar to the turbulent plane Couette flow between a stationary flat plate and a moving flat plate but exhibits considerable differences in turbulence statistics due to the effects of the free-slip boundary. Statistical analysis based on the DNS data and theoretical derivation based on Taylor series expansion show that near the free-slip surface the turbulence variances of the three velocity components vary as quadratic functions of the vertical distance from the boundary while the Reynolds shear stress exhibits a linear behavior, which are very different from the counterparts near the no-slip boundary. The free-slip surface condition also leads to zero horizontal vorticities at the surface but allows nonzero vertical vorticity in the meantime, leading to considerable differences in the near-boundary statistics of vorticities and coherent vortex structures. Comparison of three DNS runs with different grid resolutions shows that smoothly resolving the more energetic turbulent flow structures near the free-slip boundary requires a higher horizontal grid resolution than that used for resolving the structures near the no-slip boundary.

Flow-induced resonant shear-wave instability between a viscoelastic fluid and an elastic solid

Linear stability analysis of plane Couette flow of a viscoelastic, upper-convected Maxwell (UCM) fluid past a deformable elastic solid is carried out in the low Reynolds number limit using both numerical and asymptotic techniques. The UCM fluid is characterized by its viscosity η, density ρ, and relaxation time τR, whereas the deformable solid is considered to be a linear elastic solid of shear modulus G. The asymptotic analysis is performed in the Re ≪ 1 limit, where Re = ρVR/μ is the Reynolds number, V is the top plate velocity, and R is the thickness of the fluid. Both asymptotic and numerical approaches are used to understand the effect of solid elasticity, represented by the dimensionless parameter Γ, and fluid elasticity, characterized by the Weissenberg number W, on the growth rate of a class of modes with high frequencies (compared to the imposed shear rate, termed high-frequency Gorodtsov-Leonov, or “HFGL” modes) in the Re ≪ 1 limit. Here, the dimensionless groups are defined as W = τRV/R and Γ = ηV/GR. The results obtained from the numerical analysis show that there is an interaction between the shear waves in the fluid and the elastic solid, which are coupled via the continuity conditions at the interface. The interaction is particularly pronounced when W = Γ, strongly reminiscent of resonance. The resonance-induced interaction leads to shear waves in the coupled system with a decay rate of ci = −1/[2k(W + Γ)]. In this case, it is not possible to differentiate the fluid and solid shear waves individually and the coupled fluid-solid system behaves as a single composite material. The leading order asymptotic analysis suggests that the growth rate of the HFGL modes is proportional to W2 for W ≪ 1. The asymptotic analysis, up to first correction, shows an oscillating behavior of ci with an increase in Γ, in agreement with the results from our numerical approach. In addition, we also carry out an asymptotic analysis in the no-flow, but nonzero inertia limits to illustrate the role played by the imposed flow in the instability of the shear waves. It is found that, at the leading order, the wave speed for the coupled fluid-solid problem is neutrally stable in the absence of flow. Thus, the unstable resonant modes in the coupled fluid-solid system are shown to be driven by the imposed flow.

Spectral link and macroscopic non-universality in turbulent plane Couette flow

In fully developed turbulent plane Couette flow the classical experimental data of frictional factor f versus Reynolds number Re display well-known disparities which have long remained unexplained. These disparities are accompanied by previously unnoticed disparities in the wakes, and only in the wakes, of the attendant mean-velocity profiles (MVPs). To help explain these experimental data, we apply the model of the ‘spectral link’ for MVPs to turbulent plane Couette flows. The model links the dissipative range, the inertial range, and the energetic range of the standard phenomenological model of the spectrum of turbulent kinetic energy to, respectively, the buffer layer, the log layer, and the wake of the MVPs. By assessing the empirical data using the spectral model, we argue for the existence, in plane Couette flow, of multiple states of turbulence which differ from one another only at the largest lengthscales in the flow, corresponding to the energetic range, where the spectrum is subject to finite-domain effects. Thus, the multiplicity of turbulent states is entirely consistent with small-scale universality, and the experimental data on plane Couette flow pose no challenge to the phenomenological theory of turbulence. Our findings might apply more broadly to a general class of flows engendered by moving boundaries.

Nonlinear stability results for plane Couette and Poiseuille flows.

We prove that the plane Couette and Poiseuille flows are nonlinearly stable if the Reynolds number is less than Re_{Orr}(2π/(λsinθ))/sinθ when a perturbation is a tilted perturbation in the direction x^{'} which forms an angle θ∈(0,π/2] with the direction i of the basic motion and does not depend on x^{'}. Re_{Orr} is the critical Orr-Reynolds number for spanwise perturbations which is computed for wave number 2π/(λsinθ), with λ being any positive wavelength. By taking the minimum with respect to λ, we obtain the critical energy Reynolds number for a fixed inclination angle and any wavelength: for plane Couette flow, it is Re_{Orr}=44.3/sinθ, and for plane Poiseuille flow, it is Re_{Orr}=87.6/sinθ (in particular, for θ=π/2 we have the classical values Re_{Orr}=44.3 for plane Couette flow and Re_{Orr}=87.6 for plane Poiseuille flow). Here the nondimensional interval between the planes bounding the channel is [-1,1]. In particular, these results improve those obtained by Joseph, who found for streamwise perturbations a critical nonlinear value of 20.65 in the plane Couette case, and those obtained by Joseph and Carmi who found the value 49.55 for plane Poiseuille flow for streamwise perturbations. If we fix some wavelengths from the experimental data and the numerical simulations, the critical Reynolds numbers that we obtain are in a very good agreement both with the the experiments and the numerical simulation. These results partially solve the Couette-Sommerfeld paradox.

Viscoelastic fluid flow simulation using the contravariant deformation formulation

The new formulation for conformation-tensor based viscoelastic fluid models, written in terms of the contravariant deformation tensor as introduced by Hütter et al. (2018), has been approached numerically. It has been implemented in a finite element method framework with the DEVSS-G/SUPG method for stabilisation. A stress-implicit as well as a stress-explicit formulation is presented that allows to integrate the non-linear flow equations, with or without a solvent contribution, forward in time with second-order accuracy. The time-dependent stability in shear flow is maintained using the contravariant deformation, which is tested by solving a perturbed planar Couette flow of an upper-convected Maxwell (UCM) fluid. The stability and accuracy of our implementation has furthermore been tested by solving the flow around a cylinder with confining walls. The new formulation turns out to be more stable as compared to the formulation in terms of the conformation tensor. It enables to simulate the flow of a Giesekus fluid with non-linear parameter α=0.01 and viscosity ratio ν=ηs/η0=0.59 way beyond the Weissenberg number at which the High Weissenberg Number Problem manifests itself using the standard formulation. The stability appears to be similar to the log-conformation representation. The computed stress profiles up to a Weissenberg number of Wi=0.6 for an Oldroyd-B fluid with viscosity ratio ν=0.59, compare well with benchmark results from other studies, including other finite element, finite volume and spectral element methods.

A framework for input–output analysis of wall-bounded shear flows

We propose a new framework to evaluate input–output amplification properties of nonlinear models of wall-bounded shear flows, subject to both square integrable and persistent disturbances. We focus on flows that are spatially invariant in one direction and whose base flow can be described by a polynomial, e.g. streamwise-constant channel, Couette and pipe flows. Our methodology is based on the notion of dissipation inequalities in control theory and provides a single unified approach for examining flow properties such as energy growth, worst-case disturbance amplification and stability to persistent excitations (i.e. input-to-state stability). It also enables direct analysis of the nonlinear partial differential equation rather than of a discretized form of the equations, thereby removing the possibility of truncation errors. We demonstrate how to numerically compute the input–output properties of the flow as the solution of a (convex) optimization problem. We apply our theoretical and computational tools to plane Couette, channel and pipe flows. Our results demonstrate that the proposed framework leads to results that are consistent with theoretical and experimental amplification scalings obtained in the literature.

Effect of elastic walls on suspension flow

We study suspensions of rigid particles in a plane Couette flow with deformable elastic walls. We find that, in the limit of vanishing inertia, the elastic walls induce shear thinning of the suspen ...

Fractal sets of neutral curves for stably stratified plane Couette flow

The linear stability of plane Couette flow is investigated when the plates are horizontal, and the fluid is stably stratified with a cubic basic density profile. The disturbances are treated as inviscid and diffusion of the density field is neglected. Previous studies have shown that this density profile can develop multiple neutral curves, despite the stable stratification, and the fact that plane Couette flow of homogeneous fluid is stable. It is shown that when the neutral curves are plotted with wave angle on one axis, and location of the density inflexion point on the other axis, they produce a self-similar fractal pattern. The repetition on smaller and smaller scales occurs in the limit when the waves are highly oblique, i.e. longitudinal vortices almost aligned with the flow; the corresponding limit for two-dimensional waves is that of strong buoyancy/weak flow. The fractal set of neutral curves also represents a fractal of bifurcation points at which nonlinear solutions can be continued from the trivial state, and these may be helpful for understanding turbulent states. This may be the first example of a fractal generated by a linear ordinary differential equation.

Numerical simulations of the motion of ellipsoids in planar Couette flow of Giesekus viscoelastic fluids

The motion of neutrally buoyant ellipsoids in a planar Couette flow of Giesekus viscoelastic fluids between two narrowly set plates is numerically simulated with a fictitious domain method. The aspect ratio of the ellipsoid is 4 (i.e., prolate spheroids) and the Deborah number (De) ranges from 0 to 4.0. For a single ellipsoid initially placed in the mid-plane between the two plates, the ellipsoid major axis rotates around the vorticity axis in a kayaking mode at relatively low Deborah numbers, and is tilted in the flow-vorticity plane when the Deborah number exceeds a critical value, with the orientation being closer to the flow direction for a larger De. For a single ellipsoid initially not placed in the mid-plane, the ellipsoid undergoes lateral migration toward the nearby wall, and it is interesting that the ellipsoid turns its orientation to the vorticity axis at relatively small De and a direction close to the vorticity axis at large De (above 3.0), in contrast to the ellipsoid placed in the mid-plane without lateral migration, whose terminal orientation exhibits a kayaking motion at relatively small De and is close to the flow direction for De > 3. As a result, for the multiple-ellipsoid case, there exists a transient stage where the average orientation of the ellipsoids turns toward the vorticity axis for all nonzero Deborah numbers studied, and the orientation close to the vorticity axis can be often observed for the isolated ellipsoids. Both the particle interactions and the wall effect promote the ellipsoids to align with the flow direction. Particle aggregation and the dynamic aligning structures are observed at large Deborah numbers.

The Use of Wall Functions for Simulating the Turbulent Thermal Boundary Layer

An important problem in the numerical simulation of turbulent heat exchange in fluids is accurate prediction of hydrodynamic characteristics of the flow in the boundary layer, which requires a fine grid near rigid surfaces. In applications, it is not always possible to have a fine grid and the use of a coarser grid results in significant loss of accuracy. A well-known approach to improving the accuracy of the numerical simulation of the boundary layer is the use of universal wall functions for computing the friction and thermal flux. In this paper, we consider the known wall functions for computing the thermal flux. The accuracy of these functions in problems of turbulent nonisothermal flow of fluid is studied. These are the flow in a plane channel, Couette flow, and flow along a heated plate. Each of these problems is solved on grids with various near wall resolutions. The results of solving these problems provide a basis for estimating the accuracy of the wall functions used for solving them. It is shown that the wall functions considered in this study yield nonmonotonic convergence of the results as the grid is refined.

Exact invariant solution reveals the origin of self-organized oblique turbulent-laminar stripes

Wall-bounded shear flows transitioning to turbulence may self-organize into alternating turbulent and laminar regions forming a stripe pattern with non-trivial oblique orientation. Different experiments and flow simulations identify oblique stripe patterns as the preferred solution of the well-known Navier-Stokes equations, but the origin of stripes and their oblique orientation remains unexplained. In concluding his lectures, Feynman highlights the unexplained stripe pattern hidden in the solution space of the Navier-Stokes equations as an example demonstrating the need for improved theoretical tools to analyze the fluid flow equations. Here we exploit dynamical systems methods and demonstrate the existence of an exact equilibrium solution of the fully nonlinear 3D Navier-Stokes equations that resembles oblique stripe patterns in plane Couette flow. The stripe equilibrium emerges from the well-studied Nagata equilibrium and exists only for a limited range of pattern angles. This suggests a mechanism selecting the non-trivial oblique orientation angle of turbulent-laminar stripes.

On the integration of the SPH equations for a highly viscous fluid

This paper shows how the combination of Strang splitting and the exact integration of SPH pair-wise viscous interactions, enables highly viscous flows such as lava or magma, to be integrated efficiently, even when the typical time scale of the dissipation terms is much less than the time scale arising from other constraints such as the Courant condition. We first apply the algorithm to the simulation of a planar Couette flow in two dimensions and find it is stable up to the highest viscosity coefficient used which, in SI units is μ=105 which is equivalent to 108 times the viscosity of water. The accuracy is excellent up to μ=800 but for larger values of μ the errors are larger until the flow is close to the final state. The second application is to the two dimensional flow of a viscous fluid under gravity over a rigid surface and with a free upper surface. The agreement with the similarity solution of Huppert (1982) [3] is very satisfactory even when the time step is ∼600 times the stable time step for an explicit integration.

A Smooth Dissipative Particle Dynamics method for nonisothermal liquid and gas flows in bounded domains

A Smoothed Dissipative Particle Dynamics formulation with dynamic virtual particle allocation (SDPD-DV) is developed for the simulation of mesoscale nonisothermal flows with non-slip, constant temperature and constant heat flux boundary conditions. For liquids, the particle internal energy is evaluated through a first-order expansion from an initial volume and entropy state. For liquids, the particle pressure and temperature are evaluated using the coefficients of adiabatic compressibility, volumetric thermal expansion and heat capacity. For monatomic and diatomic gases, the fluid particle internal energy is evaluated through a Sackur-Tetrode type of equation. For both liquids and gases, the viscosity and thermal conductivity are modeled by a power law. Integration of position and momentum equations is performed with a velocity-Verlet method and bounce-forward reflection. Integration of the entropy equation is performed with a Runge-Kutta method. Verification of SDPD-DV is performed for planar isothermal Couette, planar Fourier, and planar nonisothermal Couette flows of liquid H2O and gaseous N2, and for a nonisothermal planar Poiseuille flow of a fluid with an exponential temperature-depended viscosity. SDPD-DV simulations for liquid H2O, gaseous Ar and N2 systems under thermal equilibrium assess the impact of the ratio of particle mass (mi) over the real molecular mass and the size of smoothing length (h) on fluid properties, hydrodynamic fluctuations and transport coefficients. For liquid systems the density fluctuations exhibit scatter about the analytical estimates and the temperature fluctuations follow the analytical estimates for mass ratios smaller than 10,000. The velocity fluctuations are in excellent agreement with theoretical estimates. For gases, the density fluctuations exhibit scatter about the analytical estimates and the pressure fluctuations agree with the analytical estimates for mass ratios smaller than 10,000. The temperature and velocity fluctuations are in excellent agreement with the analytical estimates. The self-diffusivity evaluated from SDPD-DV simulations for liquid and gases is shown to increase with smoothing length for a given mass ratio, scales with h2/mi and is in excellent agreement with analytical estimates. The shear viscosity evaluated from SDPD-DV simulations for liquid and gases exhibits scatter around the experimental values and is shown to be scale free.

Hysteresis behavior in spanwise rotating plane Couette flow with varying rotation rates

In direct numerical simulations of spanwise rotating plane Couette flow, hysteresis is found between a 2-pair-roll-cell as the rotation increases to a 3-pair-roll-cell when the rotation decreases.

Finite Difference Analysis of Plane Couette Flow using MATLAB

Internal flow is a flow for which the fluid is confined by a surface. Hence, the boundary layer is unable to develop without eventually being constrained. An internal flow is constrained by the bounding walls, and the viscous effects will grow and meet and permeate the entire flow. Some examples of internal flows are: flow between two parallel horizontal plates, Couette flow, plane Poiseuille flow, flow through pipes, etc. In the present study, a plane Couette flow has been analyzed by a classical method (exact solution of Navier-Stokes equation) as well as by an approximate method using central difference scheme (numerical solution of Navier-Stokes equation). The flow domain has been divided into various nodes and the velocities are obtained at different nodes for various time intervals. The stability condition for the convergence of solution has been determined and further the convergence of solution has been obtained using a MATLAB program for Couette flow using Crank-Nicolson scheme.

Numerical study of plane Couette flow: turbulence statistics and the structure of pressure–strain correlations

AbstractThe paper presents the results of direct numerical simulation of turbulent plane Couette flow. The calculations were performed for Reynolds numbers Re =U0H/ν(His the height of the channel, ±U0/2 is the motion velocities of the lower and upper walls, respectively,νis the kinematic viscosity) from 5200 (where viscous effects significantly affect the flow structure) to 80000 (where a logarithmic layer is clearly observed). Estimates of terms of the equation for the balance of turbulent Reynolds stresses are obtained, which indicate the importance of the kinetic energy transport by velocity fluctuations. The vertical transport of the turbulent momentum flux is less important and partly compensated by the transport of pressure fluctuations. It is shown that in the logarithmic layer the normal components of the ‘pressure–strain rate’ correlation tensor are described in the framework of the ‘isotropization of production’ model, and in the central part of the channel they are described by the linear Rotta model [29]. The additive model considering both the interaction of the velocity field fluctuation and the influence of the mean velocity gradient is a good approximation only for the off-diagonal component of the tensor entering the balance equation for the turbulent momentum flux.

A trigonometric series expansion method for the Orr-Sommerfeld equation

A trigonometric series expansion method and two similar modified methods for the Orr-Sommerfeld equation are presented. These methods use the trigonometric series expansion with an auxiliary function added to the highest order derivative of the unknown function and generate the lower order derivatives through successive integrations. The proposed methods are easy to implement because of the simplicity of the chosen basis functions. By solving the plane Poiseuille flow (PPF), plane Couette flow (PCF), and Blasius boundary layer flow with several homogeneous boundary conditions, it is shown that these methods yield results with the same accuracy as that given by the conventional Chebyshev collocation method but with better robustness, and that obtained by the finite difference method but with fewer modal number.

A new wall model for slip boundary conditions in dissipative particle dynamics

SummaryDissipative particle dynamics (DPD) for modeling complex flows requires proper boundary conditions to simulate the behavior of a wall‐bounded flow. Although most DPD simulations use the no‐slip boundary condition, slip should be considered for a more complete and realistic description of such flows. The present work suggests a new wall model, via a wall‐particle virtual velocity approach, to impose tunable partial‐slip boundary conditions while structuring a modified reflection mechanism to enforce the no‐slip boundary condition. The new wall model is validated through simulating the planar Poiseuille and Couette flows and implemented to investigate the wall slip status affected by the wall virtual velocity.

Nonlinear exact coherent structures in pipe flow and their instabilities

In this paper, we present computational results of some two-fold azimuthally symmetric travelling waves and their stability. Calculations over a range of Reynolds numbers ($Re$) reveal connections between a class of solutions computed by Wedin & Kerswell (J. Fluid Mech., vol. 508, 2004, pp. 333–371) (henceforth called the WK solution) and the $Re\rightarrow \infty$ vortex–wave interaction theory of Hall & Smith (J. Fluid Mech., vol. 227, 1991, pp. 641–666) and Hall & Sherwin (J. Fluid Mech., vol. 661, 2010, pp. 178–205). In particular, the continuation of the WK solutions to larger values of $Re$ shows that the WK solution bifurcates from a shift-and-rotate symmetric solution, which we call the WK2 state. The WK2 solution computed for $Re\leqslant 1.19\times 10^{6}$ shows excellent agreement with the theoretical $Re^{-5/6}$, $Re^{-1}$ and $O(1)$ scalings of the waves, rolls and streaks respectively. Furthermore, these states are found to have only two unstable modes in the large $Re$ regime, with growth rates estimated to be $O(Re^{-0.42})$ and $O(Re^{-0.92})$, close to the theoretical $O(Re^{-1/2})$ and $O(Re^{-1})$ asymptotic results for edge and sinuous instability modes of vortex–wave interaction states (Deguchi & Hall, J. Fluid Mech., vol. 802, 2016, pp. 634–666) in plane Couette flow. For the nonlinear viscous core states (Ozcakir et al., J. Fluid Mech., vol. 791, 2016, pp. 284–328), characterized by spatial a shrinking of the wave and roll structure towards the pipe centre with increasing $Re$, we continued the solution to $Re\leqslant 8\times 10^{6}$ and we find only one unstable mode in the large Reynolds number regime, with growth rate scaling as $Re^{-0.46}$ within the class of symmetry-preserving disturbances.