Petrov–Galerkin computation of nonlinear waves in pipe flow of shear-thinning fluids: First theoretical evidences for a delayed transition
Introduction
The study of the transition to turbulence in pipe flow of Newtonian and non-Newtonian fluids is an active area of research. The transition to turbulence has an impact on the head losses, which increase: delaying the transition could lead to a reduction of power consumption. From a more fundamental point of view, the interest of the scientific community in this subject could be explained by the fact that there are still mysteries and controversies running, see e.g. the recent study of the lifetime of turbulence in [1]. The difficulties with these studies, as far as modelling approaches are concerned, stem from the fact that the Hagen–Poiseuille base flow is linearly stable for all Reynolds number, see e.g. [2] and references therein. Hence standard stability methods cannot be used.
The same theoretical difficulty exists for non-Newtonian fluids, which are ubiquitous in nature and industry: blood, muds, paints, cement, polymer solutions are various examples of such fluids. Most of them exhibit two rather different types of non-Newtonian effects. Firstly, they are often shear-thinning: many of their flow properties can be described with a viscous model where the viscosity depends on the rate of strain (more precisely on the second invariant of the rate-of-strain tensor, see Eq. (5) below), decreasing when the rate of strain increases. Secondly, they may also display an elastic response to strain.
Experimentally, one can define the onset of the transition to turbulence in pipe flow as the Reynolds number at which the relative level IT of fluctuations of the axial velocity vz (the time-averaged value of , with 〈vz〉t the time-averaged value of vz), measured close to the wall, starts to increase significantly and quit the ‘noise’ value obtained in the laminar regime (IT ≃ 2 − 5% depending on the setup and on the measurement method). This increase is connected with the appearance, in an intermittent manner, of ‘puffs’. In setups where no special care is taken to reduce perturbations, and when a Newtonian fluid is used, this Reynolds number (based on the mean flow speed 〈vz〉rθ and the pipe diameter) is of the order of 2000. On these topics, see e.g. [3], [4], and references therein. In non-Newtonian fluids, the relevant Reynolds number is the wall-viscosity Reynolds number , defined in Eq. (32) below, and based on the viscosity at the wall deduced from the wall shear-stress. The onset of the transition to turbulence measured in the same way appears to be larger than 2000. The first ones to mention this delayed transition are, to our knowledge, [5]. In their article, p. 210, one can read that ‘the onset of transition is slightly but progressively delayed in the sequence’ of fluids ‘by a factor of about two in Reynolds number’, i.e. in some fluids transition comes in only for . The comparison of the curves IT vs. in the Fig. 3b, for a Newtonian fluid, and Fig. 5b, for a non-Newtonian fluid, of [6] also shows a clear delay to the transition in the latter case. A recent, spectacular example of delayed transition is given by the case of 0.125% PAA (an aqueous solution of polyacrylamide, of concentration 0.125% w/w) in the Fig. 4b of [7]: in this fluid the level of fluctuations IT increases abruptly only for .
The interpretation of this delayed transition is not straightforward, since all fluids are both, to certain extent, shear-thinning and viscoelastic. Therefore it appears interesting to try to model this delayed transition with, for instance, a purely viscous constitutive law, in order to focus on the influence of the shear-thinning effects in the absence of elastic response. This is the aim of the work presented here.
As stated hereabove, one serious difficulty encountered in the modelling of the transition to turbulence in pipe flow is the fact that the laminar base flow is linearly stable. One could use direct numerical simulations to attack the problem. At this stage we should mention the work of [8], who focused, however, on the transitional or turbulent regimes at large . We want to use here an alternate approach, which has emerged recently in the Newtonian studies.
Following the ideas of [9], developed for plane shear flows, it has been shown that nonlinear traveling wave solutions of the Navier–Stokes equation exist above a critical Reynolds number in pipe flow [10]. If (r, θ, z) designate the cylindrical coordinates with z the axial direction of the pipe, these solutions are invariant under the rotation θ ↦ θ + 2π/m0 with m0 the fundamental azimuthal wavenumber, and under the translation z ↦ z + 2π/q0 with q0 the fundamental axial wavenumber. Moreover they are invariant under the spatio-temporal translations (z, t) ↦ (z + δz, t + δt) provided that δz = cδt with c the axial phase velocity. The Reynolds number at which these wave solutions appear, through a saddle-node bifurcation, could be viewed as a lower bound for the transition Reynolds number. Indeed, the transient turbulent states at intermediate Reynolds numbers, in the transitional regime, i.e., the ‘puffs’, would ‘live’ upon the manifold of such nonlinear wave solutions, see e.g. [11], [12]. The relevant wave solutions have a fundamental azimuthal wavenumber m0 = 2, 3, 4, … and appear at the critical Reynolds numbers when defined with the mean flow speed 〈vz〉rθ and the pipe diameter. The corresponding Reynolds numbers defined with the centerline velocity of the base flow and the pipe radius are Rec2 = 1663, Rec3 = 1631, Rec4 = 2280, … For each azimuthal wavenumber, the axial wavenumber has been determined by minimizing the critical Reynolds number ; for instance q0(m0 = 2) = 1.55, q0(m0 = 3) = 2.44 [13]. More recently, nonlinear wave solutions with m0 = 1, which appear at lower Reynolds numbers and , and present either two or one ‘shift-and-reflect’ symmetries (like the one defined in Eq. (28) below), have also been found [14], [15]. The role of these new solutions is not already quite clear. Some of them may support the ‘boundary’ between laminar and turbulent flow, see e.g. [16].
We present here a code that has been developed to compute, in the pipe flow of a shear-thinning fluid, nonlinear wave solutions of the first class found historically in the Newtonian case by [10], [13]. The model is presented in Section 2 and the numerical methods in Section 3. In Section 4, we present a validation of the code and a study of its convergence properties, by recovering a forced analytic solution. In Section 5, we present physical results for waves with m0 = 3 and q0 = 2.44, which are the first ones to emerge in this class in the Newtonian case. A concluding Section will follow.
Section snippets
Carreau model: basic equations with dimensional units
Since we focus on shear-thinning fluids without elastic response, the constitutive law is purely viscous: the stress tensorwith p the pressure andthe viscous-stress tensor. In Eq. (2), μ is the viscosity,is the rate-of-strain tensor, with v the velocity field. The constitutive law chosen is the Carreau’s law, which has a firm theoretical base [17]. The viscositywiththe second invariant of the rate-of-strain
Spectral development
The Petrov–Galerkin formulation of [2], [20] has been used to solve Eqs. (10), (27). The solutions are expanded as follows:with the trial fields, of the formgiven in Appendix A. The coefficients almnk obeywith the star designating the complex conjugate, in order for v to be real. This rule and the symmetry (28) impose some restrictions on the coefficients almnk. Denoting by
Validation and convergence tests
In order to test the code, we have performed consistency tests similar to the ones exposed in the Section 4 of [26]. The analytical solutionhas been forced in a case without pressure gradient (G = 0) and base flow (vb = 0). A forcing termhas been calculated by computing firstly the coefficients Xa of the spectral development of va and
Base flows
For a given non-Newtonian fluid and pipe diameter, setting λ given by Eq. (14) amounts to set the characteristic velocityThe Reynolds number (16) is also set toThis Reynolds number is a ‘pressure-gradient Reynolds number’, since, according to the model described in Section 2.3, all flows are controlled by the applied pressure gradientFor the purpose of completeness, we show in Fig. 6 the base flows Wb, the corresponding shear-rates
Concluding discussion
A Petrov–Galerkin code has been developed. It is similar, in his principles, to the code of Meseguer and collaborators, especially to the Newton–Krylov version used in [27], [28]. However, a major difference is the encoding of a nonlinear viscosity that depends on the velocity field according to Eqs. (4), (5). This required a special care at the level of the Newton-GMRES method, as explained in our Section 3.5. This code has been validated and its convergence properties have been studied.
References (28)
- et al.
Linearized pipe flow to Reynolds number 107
J Comput Phys
(2003) - et al.
Drag reduction in the turbulent pipe flow of polymers
J Non-Newtonian Fluid Mech
(1999) - et al.
Observations of asymmetrical flow behaviour in transitional pipe flow of yield-stress and other shear-thinning liquids
J Non-Newtonian Fluid Mech
(2005) - et al.
Asymmetry in transitional pipe flow of drag-reducing polymer solutions
J Non-Newtonian Fluid Mech
(2009) - et al.
Turbulent pipe flow of shear-thinning fluids
J Non-Newtonian Fluid Mech
(2004) - et al.
On a solenoidal Fourier–Chebyshev spectral method for stability analysis of the Hagen–Poiseuille flow
Appl Numer Math
(2007) - et al.
A spectral vanishing viscosity for the LES of turbulent flows within rotating cavities
J Comput Phys
(2007) - et al.
Repeller or attractor? Selecting the dynamical model for the onset of turbulence in pipe flow
Phys Rev Lett
(2008) - et al.
On transition in a pipe. Part 1. The origin of puffs and slugs and the flow in a turbulent slug
J Fluid Mech
(1973) - et al.
On transition in a pipe. Part 2. The equilibrium puff
J Fluid Mech
(1975)
Three-dimensional coherent states in plane shear-flows
Phys Rev Lett
Traveling waves in pipe flow
Phys Rev Lett
Visualizing the dynamics of the onset of turbulence
Science
Experimental observation of nonlinear traveling waves in turbulent pipe flow
Science
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