Stability bounds on turbulent Poiseuille flow

1988 ◽  
Vol 187 ◽  
pp. 435-449 ◽  
Author(s):  
G. R. Ierley ◽  
W. V. R. Malkus
Keyword(s):  
Reynolds Number ◽  
Reynolds Stress ◽  
Poiseuille Flow ◽  
Numerical Solutions ◽  
Numerical Study ◽  
Linear Instability ◽  
Mean Flow ◽  
Starting Point ◽  
The Mean ◽  
The Stability

For steady-state turbulent flows with unique mean properties, we determine a sense in which the mean velocity is linearly supercritical. The shear-turbulence literature on this point is ambiguous. As an example, we reassess the stability of mean profiles in turbulent Poiseuille flow. The Reynolds & Tiederman (1967) numerical study is used as a starting point. They had constructed a class of one-dimensional flows which included, within experimental error, the observed profile. Their numerical solutions of the resulting Orr-Sommerfeld problems led them to conclude that the Reynolds number for neutral infinitesimal disturbances was twenty-five times the Reynolds number characterizing the observed mean flow. They found also that the first nonlinear corrections were stabilizing. In the realized flow, this latter conclusion appears incompatible with the former. Hence, we have sought a more complete set of velocity profiles which could exhibit linear instability, retaining the requirement that the observed velocity profile is included in the set. We have added two dynamically generated modifications of the mean. The first addition is a fluctuation in the curvature of the mean flow generated by a Reynolds stress whose form is determined by the neutrally stable Orr-Sommerfeld solution. We find that this can reduce the stability of the observed flow by as much as a factor of two. The second addition is the zero-average downstream wave associated with the above Reynolds stress. The three-dimensional linear instability of this modification can even render the observed flow unstable. Those wave amplitudes that just barely will ensure instability of the observed flow are determined. The relation of these particular amplitudes to the limiting conditions admitted by an absolute stability criterion for disturbances on the mean flow is found. These quantitative results from stability theory lie in the observationally determined Reynolds-Tiederman similarity scheme, and hence are insensitive to changes in Reynolds number.

scholarly journals Divergent versus Nondivergent Instabilities of Piecewise Uniform Shear Flows on the f Plane

2009 ◽  
Vol 39 (7) ◽  
pp. 1685-1699
Author(s):  
Nathan Paldor ◽  
Yona Dvorkin ◽  
Eyal Heifetz
Keyword(s):  
Numerical Solutions ◽  
Delta Function ◽  
Relative Vorticity ◽  
Linear Instability ◽  
Large Radius ◽  
Mean Flow ◽  
Uniform Shear ◽  
Linear Shear ◽  
The Mean ◽  
Inner Region

Abstract The linear instability of a piecewise uniform shear flow is classically formulated for nondivergent perturbations on a 2D barotropic mean flow with linear shear, bounded on both sides by semi-infinite half-planes where the mean flows are uniform. The problem remains unchanged on the f plane because for nondivergent perturbations the instability is driven by vorticity gradient at the edges of the inner, linear shear region, whereas the vorticity itself does not affect it. The instability of the unbounded case is recovered when the outer regions of uniform velocity are bounded, provided that these regions are at least twice as wide as the inner region of nonzero shear. The numerical calculations demonstrate that this simple scenario is greatly modified when the perturbations’ divergence and the variation of the mean height (which geostrophically balances the mean flow) are retained in the governing equations. Although a finite deformation radius exists on the shallow water f plane, the mean vorticity gradient that governs the instability in the nondivergent case remains unchanged, so it is not obvious how the instability is modified by the inclusion of divergence in the numerical solutions of the equations. The results here show that the longwave instability of nondivergent flows is recovered by the numerical solution for divergent flows only when the radius of deformation is at least one order of magnitude larger than the width of the inner uniform shear region. Nevertheless, even at this large radius of deformation both the amplitude of the velocity eigenfunction and the distribution of vorticity and divergence differ significantly from those of nondivergent perturbations and vary strongly in the cross-stream direction. Whereas for nondivergent flows the vorticity and divergence both have a delta-function structure located at the boundaries of the inner region, in divergent flows they are spread out and attain their maximum away from the boundaries (either in the inner region or in the outer regions) in some range of the mean shear. In contrast to nondivergent flows for which the mean shear is merely a multiplicative factor of the growth rates, in divergent flows new unstable modes exist for sufficiently large mean shear with no shortwave cutoff. This unstable mode is strongly affected by the sign of the mean shear (i.e., the sign of the mean relative vorticity).

A Numerical Study of Vortex Breakdown in Turbulent Swirling Flows

1999 ◽  
Vol 122 (1) ◽  
pp. 179-183 ◽  
Author(s):  
Robert E. Spall ◽  
Blake M. Ashby
Keyword(s):  
Reynolds Stress ◽  
Stokes Equations ◽  
Vortex Breakdown ◽  
Numerical Study ◽  
Reynolds Stress Model ◽  
Navier Stokes ◽  
Stress Model ◽  
Mean Flow ◽  
Navier Stokes Equations ◽  
The Mean

Solutions to the incompressible Reynolds-averaged Navier–Stokes equations have been obtained for turbulent vortex breakdown within a slightly diverging tube. Inlet boundary conditions were derived from available experimental data for the mean flow and turbulence kinetic energy. The performance of both two-equation and full differential Reynolds stress models was evaluated. Axisymmetric results revealed that the initiation of vortex breakdown was reasonably well predicted by the differential Reynolds stress model. However, the standard K-ε model failed to predict the occurrence of breakdown. The differential Reynolds stress model also predicted satisfactorily the mean azimuthal and axial velocity profiles downstream of the breakdown, whereas results using the K-ε model were unsatisfactory. [S0098-2202(00)01601-1]

Stability of fluid flow in a flexible tube to non-axisymmetric disturbances

2000 ◽  
Vol 407 ◽  
pp. 291-314 ◽  
Author(s):  
V. SHANKAR ◽  
V. KUMARAN
Keyword(s):  
Reynolds Number ◽  
Fluid Flow ◽  
Phase Velocity ◽  
Poiseuille Flow ◽  
Inviscid Limit ◽  
Mean Flow ◽  
Flexible Tube ◽  
Moderate Reynolds Number ◽  
The Stability ◽  
Axisymmetric Disturbances

The stability of fluid flow in a flexible tube to non-axisymmetric perturbations is analysed in this paper. In the first part of the paper, the equivalents of classical theorems of hydrodynamic stability are derived for inviscid flow in a flexible tube subjected to arbitrary non-axisymmetric disturbances. Perturbations of the form vi = v˜i exp [ik(x − ct) + inθ] are imposed on a steady axisymmetric mean flow U(r) in a flexible tube, and the stability of mean flow velocity profiles and bounds for the phase velocity of the unstable modes are determined for arbitrary values of azimuthal wavenumber n. Here r, θ and x are respectively the radial, azimuthal and axial coordinates, and k and c are the axial wavenumber and phase velocity of disturbances. The flexible wall is represented by a standard constitutive relation which contains inertial, elastic and dissipative terms. The general results indicate that the fluid flow in a flexible tube is stable in the inviscid limit if the quantity Ud[Gscr ]/dr [ges ] 0, and could be unstable for Ud[Gscr ]/dr < 0, where [Gscr ] ≡ rU′/(n2 + k2r2). For the case of Hagen–Poiseuille flow, the general result implies that the flow is stable to axisymmetric disturbances (n = 0), but could be unstable to non-axisymmetric disturbances with any non-zero azimuthal wavenumber (n ≠ 0). This is in marked contrast to plane parallel flows where two-dimensional disturbances are always more unstable than three-dimensional ones (Squire theorem). Some new bounds are derived which place restrictions on the real and imaginary parts of the phase velocity for arbitrary non-axisymmetric disturbances.In the second part of this paper, the stability of the Hagen–Poiseuille flow in a flexible tube to non-axisymmetric disturbances is analysed in the high Reynolds number regime. An asymptotic analysis reveals that the Hagen–Poiseuille flow in a flexible tube is unstable to non-axisymmetric disturbances even in the inviscid limit, and this agrees with the general results derived in this paper. The asymptotic results are extended numerically to the moderate Reynolds number regime. The numerical results reveal that the critical Reynolds number obtained for inviscid instability to non-axisymmetric disturbances is much lower than the critical Reynolds numbers obtained in the previous studies for viscous instability to axisymmetric disturbances when the dimensionless parameter Σ = ρGR2/η2 is large. Here G is the shear modulus of the elastic medium, ρ is the density of the fluid, R is the radius of the tube and η is the viscosity of the fluid. The viscosity of the wall medium is found to have a stabilizing effect on this instability.

scholarly journals Instability of supersonic compression ramp flow

2014 ◽  
Vol 372 (2020) ◽  
pp. 20130342
Author(s):  
R. P. Logue ◽  
J. S. B. Gajjar ◽  
A. I. Ruban
Keyword(s):  
Reynolds Number ◽  
Initial Conditions ◽  
Grid Size ◽  
Numerical Results ◽  
Mean Flow ◽  
Governing Equations ◽  
Global Modes ◽  
The Mean ◽  
Compression Ramp ◽  
The Stability

The instability of supersonic compression ramp flow is investigated. It is assumed that the Reynolds number is large and that the governing equations are the unsteady triple-deck equations. The mean flow is first calculated by solving the steady equations for various scaled ramp angles α , and the numerical results suggest that there is no singularity for increasing ramp angles. The stability of the flow is investigated using two approaches, first by solving the linearized unsteady equations and looking for global modes proportional to e λ t . In the second approach, the linearized unsteady equations are solved numerically with various initial conditions. Whereas no globally unsteady modes could be found for the range of ramp angles studied, the numerical simulations show the formation of wavepacket type disturbances which grow and convect and reach large amplitudes. However, the numerical results show large variations with grid size even on very fine grids.

On the stability of attachment-line boundary layers. Part 2. The effect of leading-edge curvature

1997 ◽  
Vol 333 ◽  
pp. 125-137 ◽  
Author(s):  
RAY-SING LIN ◽  
MUJEEB R. MALIK
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Critical Reynolds Number ◽  
Leading Edge ◽  
Mean Flow ◽  
Eigenvalue Approach ◽  
Curvature Effects ◽  
The Mean ◽  
The Stability ◽  
Attachment Line

The stability of the incompressible attachment-line boundary layer has been studied by Hall, Malik & Poll (1984) and more recently by Lin & Malik (1996). These studies, however, ignored the effect of leading-edge curvature. In this paper, we investigate this effect. The second-order boundary-layer theory is used to account for the curvature effects on the mean flow and then a two-dimensional eigenvalue approach is applied to solve the linear stability equations which fully account for the effects of non-parallelism and leading-edge curvature. The results show that the leading-edge curvature has a stabilizing influence on the attachment-line boundary layer and that the inclusion of curvature in both the mean-flow and stability equations contributes to this stabilizing effect. The effect of curvature can be characterized by the Reynolds number Ra (based on the leading-edge radius). For Ra = 104, the critical Reynolds number R (based on the attachment-line boundary-layer length scale, see §2.2) for the onset of instability is about 637; however, when Ra increases to about 106 the critical Reynolds number approaches the value obtained earlier without curvature effect.

On scaling the mean momentum balance and its solutions in turbulent Couette–Poiseuille flow

2007 ◽  
Vol 573 ◽  
pp. 371-398 ◽  
Author(s):  
TIE WEI ◽  
PAUL FIFE ◽  
JOSEPH KLEWICKI
Keyword(s):  
Reynolds Stress ◽  
Poiseuille Flow ◽  
Stress Gradient ◽  
Statistical Properties ◽  
Relevant Parameter ◽  
Viscous Stress ◽  
Mean Flow ◽  
Theoretical Approaches ◽  
Mean Pressure ◽  
The Mean

The statistical properties of fully developed planar turbulent Couette–Poiseuille flow result from the simultaneous imposition of a mean wall shear force together with a mean pressure force. Despite the fact that pure Poiseuille flow and pure Couette flow are the two extremes of Couette–Poiseuille flow, the statistical properties of the latter have proved resistant to scaling approaches that coherently extend traditional wall flow theory. For this reason, Couette–Poiseuille flow constitutes an interesting test case by which to explore the efficacy of alternative theoretical approaches, along with their physical/mathematical ramifications. Within this context, the present effort extends the recently developed scaling framework of Wei et al. (2005a) and associated multiscaling ideas of Fife et al. (2005a, b) to fully developed planar turbulent Couette–Poiseuille flow. Like Poiseuille flow, and contrary to the structure hypothesized by the traditional inner/outer/overlap-based framework, with increasing distance from the wall, the present flow is shown in some cases to undergo a balance breaking and balance exchange process as the mean dynamics transition from a layer characterized by a balance between the Reynolds stress gradient and viscous stress gradient, to a layer characterized by a balance between the Reynolds stress gradient (more precisely, the sum of Reynolds and viscous stress gradients) and mean pressure gradient. Multiscale analyses of the mean momentum equation are used to predict (in order of magnitude) the wall-normal positions of the maxima of the Reynolds shear stress, as well as to provide an explicit mesoscaling for the profiles near those positions. The analysis reveals a close relationship between the mean flow structure of Couette–Poiseuille flow and two internal scale hierarchies admitted by the mean flow equations. The averaged profiles of interest have, at essentially each point in the channel, a characteristic length that increases as a well-defined ‘outer region’ is approached from either the bottom or the top of the channel. The continuous deformation of this scaling structure as the relevant parameter varies from the Poiseuille case to the Couette case is studied and clarified.

Linear instability of annular Poiseuille flow

2008 ◽  
Vol 610 ◽  
pp. 391-406 ◽  
Author(s):  
C. J. HEATON
Keyword(s):  
Reynolds Number ◽  
Poiseuille Flow ◽  
Critical Reynolds Number ◽  
Reynolds Numbers ◽  
Linear Instability ◽  
Circular Cylinders ◽  
Flow Profile ◽  
Mean Flow ◽  
Annular Pipe ◽  
Axisymmetric Disturbances

The linear stability of flow along an annular pipe formed by two coaxial circular cylinders is considered. We find that the flow is unstable above a critical Reynolds number for all 0 < η ≤ 1, where η is the ratio between the radii of the inner and outer cylinders. This contradicts a recent claim that the flow is stable at all Reynolds numbers for radius ratio η less than a finite critical value. We find that non-axisymmetric disturbances become stable at all Reynolds numbers for η < 0.11686215, and we are able to study this ‘bifurcation from infinity’ asymptotically. However, axisymmetric disturbances remain unstable, with critical Reynolds number tending to infinity as η → 0. A second asymptotic analysis is performed to show that the critical Reynolds number Rec ∝ η−1 log(η−1) as η → 0, with the form of the mean flow profile causing the appearance of the logarithm. The stability of Hagen–Poiseuille flow (η = 0) at all Reynolds numbers is therefore interpreted as a limit result, and there are no annular pipe flows which share this stability.

scholarly journals Generation of Reverse Meniscus Flow by Applying An Electromagnetic Brake

2021 ◽  
Author(s):  
Alexander Vakhrushev ◽  
Abdellah Kharicha ◽  
Ebrahim Karimi-Sibaki ◽  
Menghuai Wu ◽  
Andreas Ludwig ◽  
...  
Keyword(s):  
Magnetic Field ◽  
Lorentz Force ◽  
Applied Magnetic Field ◽  
Numerical Study ◽  
Flow Direction ◽  
Experimental Studies ◽  
Submerged Entry Nozzle ◽  
Mean Flow ◽  
Slab Caster ◽  
The Mean

AbstractA numerical study is presented that deals with the flow in the mold of a continuous slab caster under the influence of a DC magnetic field (electromagnetic brakes (EMBrs)). The arrangement and geometry investigated here is based on a series of previous experimental studies carried out at the mini-LIMMCAST facility at the Helmholtz-Zentrum Dresden-Rossendorf (HZDR). The magnetic field models a ruler-type EMBr and is installed in the region of the ports of the submerged entry nozzle (SEN). The current article considers magnet field strengths up to 441 mT, corresponding to a Hartmann number of about 600, and takes the electrical conductivity of the solidified shell into account. The numerical model of the turbulent flow under the applied magnetic field is implemented using the open-source CFD package OpenFOAM®. Our numerical results reveal that a growing magnitude of the applied magnetic field may cause a reversal of the flow direction at the meniscus surface, which is related the formation of a “multiroll” flow pattern in the mold. This phenomenon can be explained as a classical magnetohydrodynamics (MHD) effect: (1) the closure of the induced electric current results not primarily in a braking Lorentz force inside the jet but in an acceleration in regions of previously weak velocities, which initiates the formation of an opposite vortex (OV) close to the mean jet; (2) this vortex develops in size at the expense of the main vortex until it reaches the meniscus surface, where it becomes clearly visible. We also show that an acceleration of the meniscus flow must be expected when the applied magnetic field is smaller than a critical value. This acceleration is due to the transfer of kinetic energy from smaller turbulent structures into the mean flow. A further increase in the EMBr intensity leads to the expected damping of the mean flow and, consequently, to a reduction in the size of the upper roll. These investigations show that the Lorentz force cannot be reduced to a simple damping effect; depending on the field strength, its action is found to be topologically complex.

Effects of finite-rate chemistry and detailed transport on the instability of jet diffusion flames

2014 ◽  
Vol 745 ◽  
pp. 647-681 ◽  
Author(s):  
Yee Chee See ◽  
Matthias Ihme
Keyword(s):  
Transport Properties ◽  
Diffusion Flames ◽  
Diffusive Transport ◽  
Mean Flow ◽  
Stability Behaviour ◽  
Jet Diffusion Flames ◽  
One Step ◽  
The Mean ◽  
Modelling Assumptions ◽  
The Stability

AbstractLocal linear stability analysis has been shown to provide valuable information about the response of jet diffusion flames to flow-field perturbations. However, this analysis commonly relies on several modelling assumptions about the mean flow prescription, the thermo-viscous-diffusive transport properties, and the complexity and representation of the chemical reaction mechanisms. In this work, the effects of these modelling assumptions on the stability behaviour of a jet diffusion flame are systematically investigated. A flamelet formulation is combined with linear stability theory to fully account for the effects of complex transport properties and the detailed reaction chemistry on the perturbation dynamics. The model is applied to a methane–air jet diffusion flame that was experimentally investigated by Füriet al.(Proc. Combust. Inst., vol. 29, 2002, pp. 1653–1661). Detailed simulations are performed to obtain mean flow quantities, about which the stability analysis is performed. Simulation results show that the growth rate of the inviscid instability mode is insensitive to the representation of the transport properties at low frequencies, and exhibits a stronger dependence on the mean flow representation. The effects of the complexity of the reaction chemistry on the stability behaviour are investigated in the context of an adiabatic jet flame configuration. Comparisons with a detailed chemical-kinetics model show that the use of a one-step chemistry representation in combination with a simplified viscous-diffusive transport model can affect the mean flow representation and heat release location, thereby modifying the instability behaviour. This is attributed to the shift in the flame structure predicted by the one-step chemistry model, and is further exacerbated by the representation of the transport properties. A pinch-point analysis is performed to investigate the stability behaviour; it is shown that the shear-layer instability is convectively unstable, while the outer buoyancy-driven instability mode transitions from absolutely to convectively unstable in the nozzle near field, and this transition point is dependent on the Froude number.

scholarly journals On strongly nonlinear gravity waves in a vertically sheared atmosphere

2020 ◽  
Vol 6 (1) ◽  
pp. 63-74
Author(s):  
Mark Schlutow ◽  
Georg S. Voelker
Keyword(s):  
Gravity Waves ◽  
Lower Layer ◽  
Vertical Shear ◽  
Horizontal Wind ◽  
Necessary Condition ◽  
Mean Flow ◽  
Individual Layer ◽  
Strongly Nonlinear ◽  
The Mean ◽  
The Stability

Abstract We investigate strongly nonlinear stationary gravity waves which experience refraction due to a thin vertical shear layer of horizontal background wind. The velocity amplitude of the waves is of the same order of magnitude as the background flow and hence the self-induced mean flow alters the modulation properties to leading order. In this theoretical study, we show that the stability of such a refracted wave depends on the classical modulation stability criterion for each individual layer, above and below the shearing. Additionally, the stability is conditioned by novel instability criteria providing bounds on the mean-flow horizontal wind and the amplitude of the wave. A necessary condition for instability is that the mean-flow horizontal wind in the upper layer is stronger than the wind in the lower layer.

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