Wave patterns in plane Poiseuille flow created by concentrated disturbances

1989 ◽  
Vol 208 ◽  
pp. 639-656 ◽  
Author(s):  
Fei Li ◽  
Sheila E. Widnall
Keyword(s):  
Laminar Flow ◽  
Reynolds Stress ◽  
Poiseuille Flow ◽  
Fourier Transforms ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Plane Poiseuille Flow ◽  
Turbulent Spot ◽  
Navier Stokes Equations ◽  
Wave Patterns

A model is constructed for perturbations created in the surrounding laminar flow by a turbulent spot in plane Poiseuille flow. The turbulent spot is represented as a distribution of increased Reynolds stress, which travels steadily through the surrounding laminar flow. The Navier-Stokes equations are linearized and are solved by using Fourier transforms in the plane parallel to the channel walls and a finite-difference method in the direction perpendicular to the walls. The travelling Reynolds stress distribution acts as a forcing term in the equations.Numerical results show that a packet of oblique waves are generated around the disturbance when the force is antisymmetric with respect to the channel centreline, whereas no identifiable wave crests are found when the forcing is symmetric. Furthermore, wavelengths of the typical waves composing the packet are insensitive to the size of the region of Reynolds stress. The dependencies of the flow field on Reynolds number and spot speed are investigated. In the case of symmetric forcing, the flow is forced around the disturbance, causing distortions to the basic velocity profiles. These results are in qualitative agreement with experimental observations.

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.

On spatially-growing finite disturbances in plane Poiseuille flow

1962 ◽  
Vol 14 (2) ◽  
pp. 211-221 ◽  
Author(s):  
J. Watson
Keyword(s):  
Poiseuille Flow ◽  
Stokes Equations ◽  
Time Dependent ◽  
Navier Stokes ◽  
Plane Poiseuille Flow ◽  
Navier Stokes Equations

Possible solutions of the Navier-Stokes equations are given representing certain finite disturbances in plane Poiseuille flow which vary with distance parallel to the bounding walls. These solutions are based on infinitesimal disturbances which vary exponentially with distance (upstream or downstream) instead of with time, and they are more closely related to the disturbances investigated experimentally than the corresponding ‘time-dependent’ solutions.

Non-linear theory of unstable plane Poiseuille flow

1969 ◽  
Vol 38 (2) ◽  
pp. 401-414 ◽  
Author(s):  
E. H. Dowell
Keyword(s):  
Linear Theory ◽  
Poiseuille Flow ◽  
Theoretical Study ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Plane Poiseuille Flow ◽  
Mathematical Technique ◽  
Navier Stokes Equations ◽  
Fourier Decomposition ◽  
Non Linear

A theoretical study of plane Poiseuille flow is made using the full non-linear Navier-Stokes equations. The mathematical technique employed is to use a Fourier decomposition in the streamwise spatial variable, a Galerkin expansion in the lateral variable and numerical integration with respect to time. By retaining the non-linear terms, the limit cycle oscillations of an unstable (in a linear sense) flow are obtained. A brief investigation of the possibility of instability due to large (non-linear) disturbances is also made. The results are negative for the cases examined. Comparisons with results previously obtained by others from linear theory illustrate the accuracy and efficacy of the method.

Internal laminar flow

2018 ◽  
Author(s):  
Marcel Escudier
Keyword(s):  
Poiseuille Flow ◽  
Stokes Equations ◽  
Circular Cross Section ◽  
Navier Stokes ◽  
Parallel Plates ◽  
Concentric Cylinder ◽  
Navier Stokes Equations ◽  
Concentric Cylinders ◽  
Constant Viscosity ◽  
Pressure Driven Flow

In this chapter it is shown that solutions to the Navier-Stokes equations can be derived for steady, fully developed flow of a constant-viscosity Newtonian fluid through a cylindrical duct. Such a flow is known as a Poiseuille flow. For a pipe of circular cross section, the term Hagen-Poiseuille flow is used. Solutions are also derived for shear-driven flow within the annular space between two concentric cylinders or in the space between two parallel plates when there is relative tangential movement between the wetted surfaces, termed Couette flows. The concepts of wetted perimeter and hydraulic diameter are introduced. It is shown how the viscometer equations result from the concentric-cylinder solutions. The pressure-driven flow of generalised Newtonian fluids is also discussed.

Stability of Hagen-Poiseuille flow with superimposed rigid rotation

1976 ◽  
Vol 73 (1) ◽  
pp. 153-164 ◽  
Author(s):  
P.-A. Mackrodt
Keyword(s):  
Poiseuille Flow ◽  
Pipe Flow ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Neutral Curve ◽  
Rigid Rotation ◽  
Transition To Turbulence ◽  
Navier Stokes ◽  
Navier Stokes Equations

The linear stability of Hagen-Poiseuille flow (Poiseuille pipe flow) with superimposed rigid rotation against small three-dimensional disturbances is examined at finite and infinite axial Reynolds numbers. The neutral curve, which is obtained by numerical solution of the system of perturbation equations (derived from the Navier-Stokes equations), has been confirmed for finite axial Reynolds numbers by a few simple experiments. The results suggest that, at high axial Reynolds numbers, the amount of rotation required for destabilization could be small enough to have escaped notice in experiments on the transition to turbulence in (nominally) non-rotating pipe flow.

Some numerical experiments on developing laminar flow in circular-sectioned bends

1985 ◽  
Vol 154 ◽  
pp. 357-375 ◽  
Author(s):  
J. A. C. Humphrey ◽  
H. Iacovides ◽  
B. E. Launder
Keyword(s):  
Shear Stress ◽  
Laminar Flow ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Inviscid Flow ◽  
Streamwise Velocity ◽  
Navier Stokes ◽  
Dean Number ◽  
Navier Stokes Equations ◽  
Numerical Predictions

The paper reports numerical solutions to a semi-elliptic truncation of the Navier–Stokes equations for the case of developing laminar flow in circular-sectioned bends over a range of Dean numbers. The ratios of bend radius to pipe radius are 7:1 and 20:1, corresponding with the configurations examined experimentally by Talbot and his co-workers in recent years. The semi-elliptic treatment facilitates a much finer grid than has been possible in earlier studies. Numerical accuracy has been further improved by assuming radial equilibrium over a thin sublayer immediately adjacent to the wall and by re-formulating the boundary conditions at the pipe centre.Streamwise velocity profiles at Dean numbers of 183 and 565 are in excellent agreement with laser-Doppler measurements by Agrawal, Talbot & Gong (1978). Good, albeit less complete, accord is found with the secondary velocities, though the differences that exist may be mainly due to the difficulty of making these measurements. The paper provides new information on the behaviour of the streamwise shear stress around the inner line of symmetry. Upstream of the point of minimum shear stress, our numerical predictions display a progressive shift towards the result of Stewartson, Cebici & Chang (1980) as the Dean number is successively raised. Downstream of the minimum, however, in contrast with the monotonic approach to an asymptotic level reported by Stewartson, the numerical solutions display a damped oscillatory behaviour reminiscent of those from Hawthorne's (1951) inviscid-flow calculations. The amplitude of the oscillation grows as the Dean number is raised.

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