Solution For Channel Flow Research Articles

Streamwise localization of traveling wave solutions in channel flow.

Channel flow of an incompressible fluid at Reynolds numbers above 2400 possesses a number of different spatially localized solutions that approach laminar flow far upstream and downstream. We use one such relative time-periodic solution, which corresponds to a spatially localized version of a Tollmien-Schlichting wave, to illustrate how the upstream and downstream asymptotics can be computed analytically. In particular, we show that for these spanwise uniform states the asymptotics predict the exponential localization that has been observed for numerically computed solutions of several canonical shear flows but never properly understood theoretically.

Thermal-flow analysis of a simple LTD (Low-Temperature-Differential) heat engine

A combined thermal and flow analysis was carried out to study the behavior and performance of a small, commercial LTD (Low-Temperature-Differential) heat engine. Laminar-flow solutions for annulus and channel flows were employed to estimate the viscous drags on the piston and the displacer and the pressure difference across the displacer. Temperature correction factors were introduced to account for the departure from the ideal heat transfer processes. The analysis results indicate that the work required to overcome the viscous drags on engine moving parts and to move the displacer is much smaller than the moving-boundary work produced by the power piston for temperature differentials in the neighborhood of <TEX>$20^{\circ}C$</TEX> and engine speeds below 10 RPS. A comparison with experimental data reveals large degradations from the ideal heat transfer processes. Thus, heat-transfer devices inside the displacer cylinder are recommended.

Shear banding of semidilute polymer solutions in pressure-driven channel flow

Shear banding is observed in many soft materials. We recently developed a two-fluid model for semidilute entangled polymer solutions by using the generalized bracket approach of nonequilibrium thermodynamics. This model assumes that Fickian diffusion and stress-induced migration generate a nontrivial velocity difference between the polymers and the solution, thereby resulting in shear band formation. A straightforward implementation of the slip boundary conditions is possible because the differential velocity is treated as a state variable. Numerical calculations showed obvious shear banding in the polymer concentration profile. Increasing the pressure gradient reduces the inhomogeneity of the concentration profile and moves the transition region toward the center of the channel. Moreover, the velocity deviates from the typical parabolic form and shows a plug-like profile with a low shear band near the center and a high one near the walls. The lack of hysteresis in the profiles of the volumetric flow rate calculated with the increasing and decreasing pressure gradient demonstrates the uniqueness of the solution. In addition, the flow rate exhibits a spurt at a critical pressure gradient, as experimentally observed for shear banding materials. The simplicity of the new model encourages us to analyze it in more complicated flows.

Exact coherent states in channel flow

Three spatially extended travelling wave exact coherent states, together with one spanwise-localised state, are presented for channel flow. Two of the extended flows are derived by homotopy from solutions to the problem of channel flow subject to a spanwise rotation investigated by Wall &amp; Nagata (J. Fluid Mech., vol. 727, 2013, pp. 523–581). Both these flows are asymmetric with respect to the channel centreplane, and feature streaky structures in streamwise velocity flanked by staggered vortical structures. One of these flows features two streak/vortex systems per spanwise wavelength, while the other features one such system. The former substantially reduces the value of the lowest Reynolds number at which channel flow solutions ,other than the basic flow, are known to exist down to 665. The third flow has, in contrast, half-turn rotational symmetry about a streamwise axis through a point on the channel centreplane, and is found to be the flow from which one of the asymmetric flows bifurcates in a symmetry-breaking bifurcation. This flow is found to exist on an isolated bifurcation branch, whose upper and lower branches both lie on the boundary basin separating initial conditions that lead to turbulent events, and those that directly decay back to laminar flow. The structure of this flow, in which the disturbance to the basic flow is concentrated in a core region in a spanwise period, allowed the derivation of a corresponding spanwise-localised flow, which is also discussed.

Transitions in Poiseuille flow of nematic liquid crystal

Recent experiments by Sengupta et al. (Phys. Rev. Lett. 2013) [9] revealed interesting transitions that can occur in flow of nematic liquid crystal under carefully controlled conditions within a long microfluidic channel of width much larger than height, and homeotropic anchoring at the walls. At low flow rates the director field of the nematic adopts a configuration that is dominated by the surface anchoring, being nearly parallel to the channel height direction over most of the cross-section; but at high flow rates there is a transition to a flow-dominated state, where the director configuration at the channel centerline is aligned with the flow (perpendicular to the channel height direction). We analyze simple channel-flow solutions to the Leslie–Ericksen model for nematics. We demonstrate that two solutions exist, at all flow rates, but that there is a transition between the elastic free energies of these solutions: the anchoring-dominated solution has the lowest energy at low flow rates, and the flow-dominated solution has lowest energy at high flow rates.

Analytical and numerical solution of three-dimensional channel flow in presence of a sinusoidal fluid injection and a chemical reaction

Abstract Modeling of three-dimensional channel flow in a chemically-reacting fluid between two long vertical parallel flat plates in the presence of a transverse magnetic field is presented. The stationary plate is subjected to a transverse sinusoidal injection velocity distribution while the uniformly moving plate is subjected to a constant suction and slip boundary conditions. Due to this type of injection velocity, the flow becomes three dimensional. Comparisons with previously published work are performed and the results are found to be in excellent agreement. An increase in the permeability/magnetic parameter is found to escalate the velocity near the plate in motion. Growing Reynolds number or magnetic parameter enhances the x -component and reduces the z -component of the skin-friction at the wall at rest. The acquired knowledge in our study can be used by designers to control MHD flow as suitable for certain applications which include laminar magneto-aerodynamics, materials processing and MHD propulsion thermo-fluid dynamics.

Spanwise-localized solutions of planar shear flows

AbstractWe present several new spanwise-localized equilibrium and travelling-wave solutions of plane Couette and channel flows. The solutions exhibit concentrated regions of vorticity that are centred over low-speed streaks and flanked on either side by high-speed streaks. For several travelling-wave solutions of channel flow, the vortex structures are concentrated near the walls and form particularly isolated and elemental versions of coherent structures in the near-wall region of shear flows. One travelling wave appears to be the invariant solution corresponding to a near-wall coherent structure educed from simulation data by Jeong et al. (J. Fluid Mech., vol. 332, 1997, pp. 185–214) and analysed in terms of transient growth modes of streaky flow by Schoppa &amp; Hussain (J. Fluid Mech., vol. 453, 2002, pp. 57–108). The solutions are constructed by a variety of methods: application of windowing functions to previously known spatially periodic solutions, continuation from plane Couette to channel flow conditions, and from initial guesses obtained from turbulent simulation data. We show how the symmetries of localized solutions derive from the symmetries of their periodic counterparts, analyse the exponential decay of their tails, examine the scale separation and scaling of their streamwise Fourier modes, and show that they develop critical layers for large Reynolds numbers.

The projection method for the incompressible Navier–Stokes equations: The pressure near a no-slip wall

An explicit staggered projection method for the incompressible Navier–Stokes equations with no-slip walls is analyzed and used in simulations to address several issues related to the pressure boundary condition required when the continuity equation is replaced by the standard pressure Poisson equation (PPE), ∇2p=∇⋅(−u⋅∇u+f). First, it is shown that a PPE system supplemented with a Neumann pressure boundary condition derived from the momentum equation can be made consistent with the Navier–Stokes equations if it is extended with the requirement that ∇⋅∇2u=0 is zero near the wall and the solution is sufficiently smooth. This implies that it is possible to formulate a boundary condition for the standard PPE without the necessity to resort to Green's functions, which is interesting for theoretical reasons. Second, the equivalence is shown between the staggered projection method and the staggered discretization of above PPE system. The derivation of the equivalence sheds light upon the so-called PPE paradox and leads to an approximation of the wall value of ∂p/∂n, which is not required but implied by the staggered projection method. Third, the (near-wall) regularity of a solution of the Navier–Stokes equations is numerically analyzed by means of Direct Numerical Simulation of turbulent channel flow performed with the staggered projection method. From the numerical inspection of all terms of the momentum equation in the near-wall region, it is concluded that the three components of the momentum equation are satisfied on the wall for t>0 (for short times, but also in the turbulent regime). In the limit t→0, the pressure gradient is observed to converge to the initial pressure gradient in the L2-norm, which confirms a disputed theoretical result in literature. Even in the maximum norm, the pressure gradient appears to converge to the initial pressure gradient. The only discontinuities observed in the simulations are the discontinuities of the tangential viscous terms and the time derivatives of the tangential velocities on the wall at t=0. Thus the numerical results indicate that the regularity of the solution for turbulent channel flow is stronger than claimed by existing theory.

Numerical solution of the FENE-CR model in complex flows

A finite difference technique for solving the FENE-CR (Finite Extendable Non-linear Elastic – Chilcott and Rallison) closure constitutive model in complex flows has been developed and tested. The governing equations are solved using a Marker-and-Cell type method on a staggered grid. The momentum equation is integrated employing an implicit method while the FENE-CR constitutive equation is approximated by a second-order Runge–Kutta scheme. To demonstrate that the numerical technique can cope with complex flows governed by the FENE-CR model, three flow problems were analysed: the fully-developed channel flow, the 2D cross-slot flow and the impacting drop problem. The analytic solution for fully-developed channel flow of FENE-CR fluids with a solvent viscosity is also presented for validation purposes. This flow problem is used to verify the numerical method and to quantify its accuracy by comparing numerical results of fully-developed channel flow with the analytic solution. The second flow is employed to assess whether the numerical methodology is capable of capturing the purely-elastic instabilities predicted in the literature for 2D cross-slot confined flows. Additionally, the complex free surface flow corresponding to the filling of a 2D cross geometry has also been investigated. The last problem concerns the flow dynamics of a FENE-CR fluid drop impacting on a rigid surface, which allows the assessment of the capability of the model to deal with free surfaces. The effects of varying the Reynolds number, the Weissenberg number and the finite extensibility of the polymer molecules (L2) on the resulting flow patterns are analysed.

Analytical solutions for channel flows of Phan-Thien–Tanner and Giesekus fluids under slip

Analytical and semi-analytical solutions are presented for the cases of channel and pipe flows with wall slip for viscoelastic fluids described by the simplified PTT (using both the exponential and the linearized kernel) and the Giesekus models. The slip laws used are the linear and nonlinear Navier, the Hatzikiriakos and the asymptotic models. For the nonlinear Navier slip only natural numbers can be used for the exponent of the tangent stress in order to obtain analytical solutions. For other values of the exponent and other nonlinear laws a numerical scheme is required, and thus, the solution is semi-analytical. For these cases the intervals containing the solution and the corresponding proof for the existence and uniqueness are also presented. For the Giesekus model the influence of the wall slip on the restrictions of the slip models is also investigated.

Application of analytical solutions in trapezoidal compound channel flow

AbstractThe accuracy of predictions using an analytical model for a lateral depth varying open channel flow can be affected by the friction factor and the dimensionless eddy viscosity. In this paper, different approaches to determine these two parameters are discussed by comparing analytical results with experimental data obtained from a symmetrical trapezoidal compound open channel. To reflect different hydraulic characteristics along the wetted perimeter, the trapezoidal cross‐section was divided into several sub‐sections, and the friction factor in each sub‐section was calculated with Manning's formula by adopting local hydraulic parameters. The wetted perimeters of main channel and side slope included the interface between panels to reflect the effects of the lateral momentum exchange. This approach is shown to be effective once the roughness coefficients and hydraulic radius in different sub‐regions have been determined. The results show that the friction factor is the major parameter affecting the accuracy of the analytical solution. The effect of dimensionless eddy viscosity on the precision of predictions is much smaller, which means that for practical applications, the dimensionless eddy viscosity can be determined empirically. Inclusion of a term for secondary flows in the model can improve the prediction, especially in the transition from main channel to the floodplain. A reasonable prediction of the lateral distribution of depth mean Reynolds' stress in this region is obtained. Copyright © 2010 John Wiley &amp; Sons, Ltd.

An element for modeling potential channel flows between impervious underground structures in an urban regional groundwater flow field

In an urban regional groundwater flow field, the presence of thousands of channels between impervious structures makes the field difficult to simulate using the finite-element method (FEM), because the scale of the field is usually several orders of magnitude larger than that of the channels. To overcome this problem, a simple element for the simulation of potential channel flows has been developed. This element works with linear triangular elements and can be easily implemented in a finite-element code to simulate the channel flows with a sparse mesh without a loss of global accuracy. The transmissivity matrix of this element is deduced from the analytical solution for channel flow. The application of the element is discussed, and the accuracy of the element is assessed. The element makes it easy to merge small structures that are close to each other into a larger one that can be modeled with a sparse mesh.

Shock-wave solutions in two-layer channel flow. II. Linear and nonlinear stability

We investigate the flow of two immiscible fluids in an inclined channel, building on the work presented in Part I of this study. In this paper, we examine the stability of the flow to spanwise perturbations in both the linear and nonlinear regimes. The evolution equation governing the interfacial dynamics, derived using lubrication theory in Part I, is linearised to study the effect of system parameters on the linear stability of the interface. A transient growth analysis of the linearised equation is carried out with no-flux conditions in the spanwise direction. The results of this analysis reveal that increasing the density and/or the viscosity of the upper layer, and/or increasing the counter-current nature of the flow configuration exerts a stabilising influence. Inspection of the flow profiles indicates that single Lax-shocks and the trailing Lax-shocks in Lax-undercompressive double-shocks are unstable to finger formation; undercompressive shocks and rarefaction waves are stable. In unstably stratified cases, increasing the channel inclination away from verticality, such that a denser upper layer overhangs a less dense lower one, is found to be destabilising. These results are used to guide our transient numerical simulations aimed at studying the nonlinear development of fingering phenomena.

ADCP 계기 부근에서 발생하는 관측 오차의 실험 및 수치모의에 의한 고찰

본 논문은 ADCP 계측기기 부근에서 음향학적인 그리고 ADCP 계기로 인해 발생하는 흐름의 교란에 기인한 속도 오차에 대한 원인들을 실험 및 수치모의를 통해 고찰한다. 실험실에서의 연구는 선박에 탑재되지 않은 독립된 ADCP에 대해 수행하였고, 수평 및 수직면에서 ADCP에 의해 유발되는 흐름 교란은 ADV를 이용하여 관측하였다. 그리고 ADCP와 ADV의 동시적 측정이 ADCP계측에 있어 추가적인 계측기기 부근의 영향들을 고려하기 위해 수행되었다. 수치모의는 ADCP가 선박에 탑재되었을 때 ADCP 계기 부근에 발생하는 잠재적인 오차에 대해 연구하기 위해 설계되었다. 수치해석 기법 사용된 LES (Large Eddy Simulation)는 선박에 탑재된 ADCP에 작용하는 항력과 계기와 선박의 막음효과에 의해 발생하는 흐름의 교란의 크기와 범위를 모의하였다. 결과로 ADCP에 의해 관측된 속도는 계측기기 하단의 제한된 범위 내에서 일정 정도 오차를 보였고 교란되지 않은 하천의 흐름 조건에 따라 오차가 실질적으로 다르게 나타났다. This paper reports results of a joint experimental and numerical investigation of the causes of near-transducer errors due to the combined effect of acoustic and ADCP-induced flow disturbance near the ADCP transducer. The laboratory study focused on an isolated ADCP (deployment without boat). Measurements of the flow disturbance produced by the ADCP in vertical and horizontal planes were obtained acquiring measurements with an Acoustic Doppler Velocimeter (ADV). Concurrent measurements with ADCP and ADV were made to infer additional near-transducer effects in the ADCP measurements. The numerical investigation was designed to extend the inquiry on the near-transducer potential errors when the ADCP is deployed from a boat. Large Eddy Simulation (LES) was conducted to obtain the extent and magnitude of the disturbances induced by the drag acting on a boat-mounted ADCP and by the blockage effect of the instrument and boat. It is found the velocities measured by the ADCP are biased low and differ substantially from the undisturbed channel flow solution within a limited layer beneath the instrument.

Lattice Boltzmann formulation for Braginskii magnetohydrodynamics

Braginskii magnetohydrodynamics is a single-fluid description of large-scale motions in a strongly magnetised plasma, one in which the ion gyroradius is much smaller than the mean free path between collisions in an unmagnetised plasma of the same density and temperature. The relation between stress and strain rate then becomes highly anisotropic, with the viscous stress being predominantly aligned parallel to the magnetic field lines. We modify an existing lattice Boltzmann formulation of isotropic resistive magnetohydrodynamics to simulate Braginskii magnetohydrodynamics by introducing an anisotropic collision operator for the hydrodynamic distribution functions. This collision operator applies a different relaxation time to the component of stress directed parallel to the magnetic field. The method is illustrated by comparison with independent numerical solutions for steady planar channel flows.

Shock-wave solutions in two-layer channel flow. I. One-dimensional flows

We study the dynamics of an interface separating two immiscible layers in an inclined channel. Lubrication theory is used to derive an evolution equation for the interface position that models the two-dimensional flow in both co- and countercurrent configurations. This equation is parameterized by viscosity and density ratios, and a total dimensionless flow rate; the system is further characterized by the height of the interface at the channel inlet and outlet, which are treated as additional parameters. In the present work, which corresponds to part I of a two-part paper, we focus on one-dimensional flows. We use an entropy-flux analysis to delineate the existence of various types of shocklike solutions, which include compressive Lax shocks, pairs of Lax and under-compressive shocks, and rarefaction waves. Flows characterized by unstably stratified layers are accompanied by the formation of propagating, large-amplitude interfacial waves, which are not shocklike in nature. The results of our transient numerical simulations accord with our analytical predictions and elucidate the mechanisms underlying spatio-temporal development of the various types of waves; the stability of these waves to spanwise perturbations is investigated in part II of this work.

Modeling of particle trajectories in an electrostatically charged channel

Modeling and analyses of filtration efficiency in electrostatically charged monolith filters are important for evaluating and designing this class of filters. Unlike traditional fibrous filters which comprise external flow around a fiber, monolith filters are modeled as internal flow through small channels. Analogous to single fiber theory for external flows, single channel theory is used to analyze basic fluid mechanics in monolith filters and predict filtration efficiencies. The model incorporates three forces: hydrodynamic forces, electrostatic forces, and Brownian motion. Fluid velocity within the channels is calculated by using an analytical solution for circular channel flow, within which the slip boundary condition is considered because of small length scales. This velocity field is then used to evaluate the drag force on the particle according to Stokes’s law. For this model, a one-way coupling between the fluid flow and the particle motion is assumed due to the fact that the relaxation time for the particles simulated in this paper is very small compared to the time the particles spend in the channel. The electrostatic field is computed assuming a uniform charge distribution on the inner surface of a cylindrical channel of finite length. Using a Monte Carlo simulation, particles are randomly injected into a single channel to determine the filtration efficiency.

Series solution for MHD channel flow of a Jeffery fluid

This paper is concerned with the magnetohydrodynamic (MHD) flow of a Jeffery fluid in a porous channel. Series solution to the nonlinear problem is constructed by a powerful analytic approach namely the homotopy analysis method (HAM). Convergence of the series solution is established. The obtained solutions are analyzed by plotting graphs.

Nonlinear Mixing Length Model for Prediction of Secondary Currents in Uniform Channel Flows

A nonlinear turbulence model for numerical solution of uniform channel flow is presented. Turbulent stresses are evaluated from a nonlinear mixing length model that relates turbulent stresses to quadratic products of the mean rate of strain and the mean vorticity. The definition of the mixing length, based on a three-dimensional integral measure of boundary proximity, eliminates the need for solution of additional transport equations for the turbulence quantities. Experimental data from the literature for closed and open-channel flows are utilized to validate the model. The model produced the secondary flow vortices successfully. Velocity field and wall shear stresses affected by secondary flow vortices are accurately computed. Bulging of velocity contour lines toward the corners and dipping phenomena of maximum velocity are successfully simulated.

Experimental study on the effects of shear induced structure in a drag-reducing surfactant solution flow

In this paper, The drag reduction characteristics of surfactant solutions have been experimentally studied, as well as, the shear viscosities of turbulent drag-reducing surfactant solution have been measured as a function of concentration, shear rate and temperature by using AG-G2 (TA Instruments, New Castle, USA) rheometer. In comparison the rheological property with the macroscopic behavior of the solutions in turbulent channel flow, a deeper insight into the mechanisms of drag-reducing surfactant solution has been obtained. For no shear induced structure of surfactant solutions they just show features shear thinning, but the drag reduction is very significant phenomenon. Surfactant solution of the shear induced structure is not a surfactant fluid drag reduction of the necessary elements.