scholarly journals Critical slope for laminar transcritical shallow-water flows

2015 ◽  
Vol 783 ◽  
Author(s):  
O. Thual ◽  
L. Lacaze ◽  
M. Mouzouri ◽  
B. Boutkhamouine
Keyword(s):  
Stokes Equations ◽  
Laminar Flows ◽  
Navier Stokes ◽  
Critical Surface ◽  
Steady Flows ◽  
Surface Slope ◽  
Critical Height ◽  
Navier Stokes Equations ◽  
Shallow Water Flows ◽  
Critical Slope

Backwater curves denote the depth profiles of steady flows in a shallow open channel. The classification of these curves for turbulent regimes is commonly used in hydraulics. When the bottom slope $I$ is increased, they can describe the transition from fluvial to torrential regimes. In the case of an infinitely wide channel, we show that laminar flows have the same critical height $h_{c}$ as that in the turbulent case. This feature is due to the existence of surface slope singularities associated to plug-like velocity profiles with vanishing boundary-layer thickness. We also provide the expression of the critical surface slope as a function of the bottom curvature at the critical location. These results validate a similarity model to approximate the asymptotic Navier–Stokes equations for small slopes $I$ with Reynolds number $Re$ such that $Re\,I$ is of order 1.

scholarly journals Subcritical transition in channel flows

2002 ◽  
Vol 451 ◽  
pp. 35-97 ◽  
Author(s):  
S. JONATHAN CHAPMAN
Keyword(s):  
Poiseuille Flow ◽  
Stokes Equations ◽  
Domain Of Attraction ◽  
Reynolds Numbers ◽  
Linear Instability ◽  
Laminar Flows ◽  
Navier Stokes ◽  
Plane Poiseuille Flow ◽  
Navier Stokes Equations ◽  
Laminar State

Certain laminar flows are known to be linearly stable at all Reynolds numbers, R, although in practice they always become turbulent for sufficiently large R. Other flows typically become turbulent well before the critical Reynolds number of linear instability. One resolution of these paradoxes is that the domain of attraction for the laminar state shrinks for large R (as Rγ say, with γ < 0), so that small but finite perturbations lead to transition. Trefethen et al. (1993) conjectured that in fact γ <−1. Subsequent numerical experiments by Lundbladh, Henningson & Reddy (1994) indicated that for streamwise initial perturbations γ =−1 and −7/4 for plane Couette and plane Poiseuille flow respectively (using subcritical Reynolds numbers for plane Poiseuille flow), while for oblique initial perturbations γ =−5/4 and −7/4 Here, through a formal asymptotic analysis of the Navier–Stokes equations, it is found that for streamwise initial perturbations γ =−1 and −3/2 for plane Couette and plane Poiseuille flow respectively (factoring out the unstable modes for plane Poiseuille flow), while for oblique initial perturbations γ =−1 and −5/4. Furthermore it is shown why the numerically determined threshold exponents are not the true asymptotic values.

scholarly journals Dynamics of laminar flows with coaxial oppositely-rotating layers

Vestnik MGSU ◽  
2019 ◽  
pp. 332-346
Author(s):  
Andrey L. Zuikov
Keyword(s):  
Reynolds Number ◽  
Active Zone ◽  
Vortex Flow ◽  
Stokes Equations ◽  
Laminar Flows ◽  
Radial Velocities ◽  
Navier Stokes ◽  
Recirculation Region ◽  
Vortex Flows ◽  
Navier Stokes Equations

Introduction. The work relates to the scientific foundations of hydraulic and energy construction and is devoted to the study of laminar flows with coaxial oppositely-rotating layers. In the literature, such flows are called counter-vortex. In the turbulent range, counter-vortex flows are characterized by intensive mixing of the medium, which is widely used in the technologies of mixing non-natural and multi-phase media in thermal and atomic energy, in systems of mass- and heat transfer, in chemistry and microbiology, ecology, engine and rocket production. The aim of the theoretical study is to study the physical laws of the hydrodynamics of counter-vortex flows. Research methods. The theoretical Navier-Stokes equations and continuity equation are the basis of the theoretical model of the laminar counter-vortex flow. Results. Assuming the radial velocities are much less than the azimuthal and axial velocities and taking the Oseen approximation, the solution of the Navier - Stokes equations is obtained as Fourier - Bessel series or products of Fourier - Bessel series. In particular, the following were obtained: formulas for calculating the radial-longitudinal distributions of the normalized azimuthal, axial and radial velocities in the flow under study, the velocities are presented graphically in the form of radial profiles; equations for the calculation of current lines and viscous vortex fields, which are also presented in the form of graphs, were obtained. The two-layer and four-layer counter-vortex flows are considered. The analysis of the obtained theoretical results is performed. Conclusions. On the axis at the beginning of the active zone, the formation of a return flow with significant negative velocities is characteristic. This leads to the formation of a recirculation region, the mass exchange between which and the external flow is absent. Cascades of concentric vortexes of such high intensity that are not found in streams of a different nature are generated in the active zone. Calculation formulas include exp (-λ2x/Re) exponent multiplied by Reynolds number in degree b = 0 or b = -1, therefore increasing Reynolds number when b = 0 leads to proportional transfer of arbitrary characteristic counter-vortex flow down the pipe; and at b = -1, the bias of characteristic is accompanied by a proportional decrease in its scale.

Onset of absolutely unstable behaviour in the Stokes layer: a Floquet approach to the Briggs method

2021 ◽  
Vol 928 ◽  
Author(s):  
Alexander Pretty ◽  
Christopher Davies ◽  
Christian Thomas
Keyword(s):  
Growth Rate ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Steady Flows ◽  
Navier Stokes Equations ◽  
Electron Stream ◽  
Stokes Layer ◽  
Symmetry Constraints ◽  
Spatio Temporal ◽  
Spatial Locations

For steady flows, the Briggs (Electron-Stream Interaction with Plasmas. MIT Press, 1964) method is a well-established approach for classifying disturbances as either convectively or absolutely unstable. Here, the framework of the Briggs method is adapted to temporally periodic flows, with Floquet theory utilised to account for the time periodicity of the Stokes layer. As a consequence of the antiperiodicity of the flow, symmetry constraints are established that are used to describe the pointwise evolution of the disturbance, with the behaviour governed by harmonic and subharmonics modes. On coupling the symmetry constraints with a cusp-map analysis, multiple harmonic and subharmonic cusps are found for each Reynolds number of the flow. Therefore, linear disturbances experience subharmonic growth about fixed spatial locations. Moreover, the growth rate associated with the pointwise development of the disturbance matches the growth rate of the disturbance maximum. Thus, the onset of the Floquet instability (Blennerhassett & Bassom, J. Fluid Mech., vol. 464, 2002, pp. 393–410) coincides with the onset of absolutely unstable behaviour. Stability characteristics are consistent with the spatio-temporal disturbance development of the family-tree structure that has hitherto only been observed numerically via simulations of the linearised Navier–Stokes equations (Thomas et al., J. Fluid Mech., vol. 752, 2014, pp. 543–571; Ramage et al., Phys. Rev. Fluids, vol. 5, 2020, 103901).

scholarly journals Modelling of the flow in front of the jet obstacle at variation of oncoming flow parameters

2021 ◽  
pp. 1-14
Author(s):  
Alexander Evgenyevich Bondarev ◽  
Artyom Evgenyevich Kuvshinnikov ◽  
Tatiana Nikolaevna Mikhailova ◽  
Irina Gennadievna Ryzhova ◽  
Lev Zalmanovich Shapiro
Keyword(s):  
Software Package ◽  
Boundary Layer Thickness ◽  
Stokes Equations ◽  
Laminar Flows ◽  
Navier Stokes ◽  
Incoming Flow ◽  
Oncoming Flow ◽  
Input Flow ◽  
Navier Stokes Equations ◽  
Flow Parameters

The results of numerical simulation of the problem of interaction of supersonic flow with a jet obstacle under variation of input flow parameters are considered. The problem is solved in the system of Navier-Stokes equations. Laminar flows are considered. The qualitative flow pattern has been studied under the variation of incoming flow velocity and boundary layer thickness in the incoming flow. The calculations were performed using the OpenFOAM software package.

scholarly journals Stability of bedforms in laminar flows with free surface: from bars to ripples

2009 ◽  
Vol 642 ◽  
pp. 329-348 ◽  
Author(s):  
O. DEVAUCHELLE ◽  
L. MALVERTI ◽  
É. LAJEUNESSE ◽  
P.-Y. LAGRÉE ◽  
C. JOSSERAND ◽  
...  
Keyword(s):  
Free Surface ◽  
Stability Analysis ◽  
Stokes Equations ◽  
Transport Model ◽  
Bedload Transport ◽  
Laminar Flows ◽  
Three Dimensions ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
The Stability

The present paper is devoted to the formation of sand patterns by laminar flows. It focuses on the rhomboid beach pattern, formed during the backswash. A recent bedload transport model, based on a moving-grains balance, is generalized in three dimensions for viscous flows. The water flow is modelled by the full incompressible Navier–Stokes equations with a free surface. A linear stability analysis then shows the simultaneous existence of two distinct instabilities, namely ripples and bars. The comparison of the bar instability characteristics with laboratory rhomboid patterns indicates that the latter could result from the nonlinear evolution of unstable bars. This result, together with the sensibility of the stability analysis with respect to the parameters of the transport law, suggests that the rhomboid pattern could help improving sediment transport models, so critical to geomorphologists.

Laminar channel flow driven by accelerating walls

1991 ◽  
Vol 2 (4) ◽  
pp. 359-385 ◽  
Author(s):  
P. Watson ◽  
W. H. H. Banks ◽  
M. B. Zaturska ◽  
P. G. Drazin
Keyword(s):  
Stokes Equations ◽  
Navier Stokes ◽  
Steady Flows ◽  
Navier Stokes Equations ◽  
Partial Differential System ◽  
Laminar Channel Flow ◽  
Two Dimensional Flow ◽  
Pitchfork Bifurcations ◽  
Route To Chaos ◽  
Physical Result

The two-dimensional flow of a viscous incompressible fluid in a channel with accelerating walls is analysed by use of Hiemenz's similarity solution. Steady flows and their instabilities are calculated, and unsteady flows are computed by solving the initial-value problem for the governing partial-differential system. Thereby, these exact solutions of the Navier–Stokes equations are found to exhibit turning points, pitchfork bifurcations, Hopf bifurcations and Takens–Bogdanov bifurcations along the route to chaos. The substantial physical result is that the chaos previously found for flows with symmetrically accelerating walls is easily destroyed by a little asymmetry.

I. Eigenmotion analysis

1965 ◽  
Vol 285 (1402) ◽  
pp. 382-399 ◽  
Keyword(s):  
Dihedral Angle ◽  
Stokes Equations ◽  
Slow Motion ◽  
Laminar Flows ◽  
Navier Stokes ◽  
Flow Properties ◽  
Navier Stokes Equations ◽  
Weakly Singular ◽  
Complex Eigenvalues ◽  
Slow Motions

General wedge and comer problems lead to the introduction of complex Navier-Stokes equations of complex laminar motions the real parts of which describe real laminar flows. Under the non-slip condition at the surface of a dihedral angle the general solution of the complex Navier-Stokes equations is established on the basis of the corresponding integral of the Stokes equations of slow motions. The latter integration is accomplished in terms of slow-motion eigenfunctions with real and complex eigenvalues. The results render valuable information about the flow properties at the leading or trailing edge of a dihedral angle. Weakly singular solutions, which are characterized by an infinite pressure and a finite velocity at the edge of the dihedral angle, are shown to exist for all wedges under asymmetric attack and for sharp wedges under symmetric attack. The existence of critical and branching eigenvalues exhibits the non-analyticity of laminar flows around dihedral angles in their dependence upon the corresponding wedge or corner angles.

A resonant wave theory

1999 ◽  
Vol 395 ◽  
pp. 237-251 ◽  
Author(s):  
LUN-SHIN YAO
Keyword(s):  
Stokes Equations ◽  
Flow Instability ◽  
Initial Conditions ◽  
Laminar Flows ◽  
Navier Stokes ◽  
Real Interval ◽  
Mean Flow ◽  
Navier Stokes Equations ◽  
The Mean ◽  
The Waves

Analysis is used to show that a solution of the Navier–Stokes equations can be computed in terms of wave-like series, which are referred to as waves below. The mean flow is a wave of infinitely long wavelength and period; laminar flows contain only one wave, i.e. the mean flow. With a supercritical instability, there are a mean flow, a dominant wave and its harmonics. Under this scenario, the amplitude of the waves is determined by linear and nonlinear terms. The linear case is the target of flow-instability studies. The nonlinear case involves energy transfer among the waves satisfying resonance conditions so that the wavenumbers are discrete, form a denumerable set, and are homeomorphic to Cantor's set of rational numbers. Since an infinite number of these sets can exist over a finite real interval, nonlinear Navier–Stokes equations have multiple solutions and the initial conditions determine which particular set will be excited. Consequently, the influence of initial conditions can persist forever. This phenomenon has been observed for Couette–Taylor instability, turbulent mixing layers, wakes, jets, pipe flows, etc. This is a commonly known property of chaos.

Sign in / Sign up

Export Citation Format

Share Document