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The ultimate goal of a sound theory of turbulence in fluids is to close in a rational way the Reynolds equations, namely to express the time averaged turbulent stress tensor as a function of the time averaged velocity field. This closure problem is a deep and unsolved problem of statistical physics whose solution requires to go beyond the assumption of a homogeneous and isotropic state, as fluctuations in turbulent flows are strongly related to the geometry of this flow. This links the dissipation to the space dependence of the average velocity field. Based on the idea that dissipation in fully developed turbulence is by singular events resulting from an evolution described by the Euler equations, it has been recently observed that the closure problem is strongly restricted, and that it implies that the turbulent stress is a non local function (in space) of the average velocity field, an extension of classical Boussinesq theory of turbulent viscosity. The resulting equations for the turbulent stress are derived here in one of the simplest possible physical situation, the turbulent Poiseuille flow between two parallel plates. In this case the integral kernel giving the turbulent stress, as function of the averaged velocity field, takes a simple form leading to a full analysis of the averaged turbulent flow in the limit of a very large Reynolds number. In this limit one has to match a viscous boundary layer, near the walls bounding the flow, and an outer solution in the bulk of the flow. This asymptotic analysis is non trivial because one has to match solution with logarithms. A non trivial and somewhat unexpected feature of this solution is that, besides the boundary layers close to the walls, there is another inner boundary layer near the center plane of the flow.
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A modal stability analysis shows that plane Poiseuille flow of an Oldroyd-B fluid becomes unstable to a `center mode with phase speed close to the maximum base-flow velocity, $U_{max}$. The governing dimensionless groups are the Reynolds number $Re = rho U_{max} H/eta$, the elasticity number $E = lambda eta/(H^2rho)$, and the ratio of solvent to solution viscosity $eta_s/eta$; here, $lambda$ is the polymer relaxation time, $H$ is the channel half-width, and $rho$ is the fluid density. For experimentally relevant values (e.g., $E sim 0.1$ and $beta sim 0.9$), the predicted critical Reynolds number, $Re_c$, for the center-mode instability is around $200$, with the associated eigenmodes being spread out across the channel. In the asymptotic limit of $E(1 -beta) ll 1$, with $E$ fixed, corresponding to strongly elastic dilute polymer solutions, $Re_c propto (E(1-beta))^{-frac{3}{2}}$ and the critical wavenumber $k_c propto (E(1-beta))^{-frac{1}{2}}$. The unstable eigenmode in this limit is confined in a thin layer near the channel centerline. The above features are largely analogous to the center-mode instability in viscoelastic pipe flow (Garg et al., Phys. Rev. Lett., 121, 024502 (2018)), and suggest a universal linear mechanism underlying the onset of turbulence in both channel and pipe flows of suffciently elastic dilute polymer solutions.
Turbulence structure in the quasi-linear restricted nonlinear (RNL) model is analyzed and compared with DNS of turbulent Poiseuille flow at Reynolds number R=1650. The turbulence structure is obtained by POD analysis of the two components of the flow partition used in formulating the RNL model: the streamwise-mean flow and the associated perturbations. The dominant structures are found to be similar in RNL simulations and DNS despite the neglect of perturbation-perturbation nonlinearity in the RNL formulation. POD analysis of the streamwise-mean flow indicates that the dominant structure in both RNL and DNS is a coherent roll-streak structure in which the roll is collocated with the streak in a manner configured to reinforce the streak by the lift-up process. This mechanism of roll-streak maintenance accords with analytical predictions made using the second order statistical state dynamics (SSD) model, referred to as S3T, which shares with RNL the dynamical restriction of neglecting the perturbation-perturbation nonlinearity. POD analysis of perturbations from the streamwise-mean streak reveals that similar structures characterize these perturbations in both RNL and DNS. The perturbation to the low-speed streak POD are shown to have the form of oblique waves collocated with the streak that can be identified with optimally growing structures on the streak. Given that the mechanism sustaining turbulence in RNL has been analytically characterized, this close correspondence between the streamwise-mean and perturbation structures in RNL and DNS supports the conclusion that the self-sustaining mechanism in DNS is the same as that in RNL.
The convection velocity of localized wave packet in plane-Poiseuille flow is found to be determined by a solitary wave at the centerline of a downstream vortex dipole in its mean field after deducting the basic flow. The fluctuation component following the vortex dipole oscillates with a global frequency selected by the upstream marginal absolute instability, and propagates obeying the local dispersion relation of the mean flow. By applying localized initial disturbances, a nonzero wave-packet density is achieved at the threshold state, suggesting a first order transition.
Recently, detailed experiments on visco-elastic channel flow have provided convincing evidence for a nonlinear instability scenario which we had argued for based on calculations for visco-elastic Couette flow. Motivated by these experiments we extend the previous calculations to the case of visco-elastic Poiseuille flow, using the Oldroyd-B constitutive model. Our results confirm that the subcritical instability scenario is similar for both types of flow, and that the nonlinear transition occurs for Weissenberg numbers somewhat larger than one. We provide detailed results for the convergence of our expansion and for the spatial structure of the mode that drives the instability. This also gives insight into possible similarities with the mechanism of the transition to turbulence in Newtonian pipe flow.
In wall-bounded flows, the laminar regime remain linearly stable up to large values of the Reynolds number while competing with nonlinear turbulent solutions issued from finite amplitude perturbations. The transition to turbulence of plane channel flow (plane Poiseuille flow) is more specifically considered via numerical simulations. Previous conflicting observations are reconciled by noting that the two-dimensional directed percolation scenario expected for the decay of turbulence may be interrupted by a symmetry-breaking bifurcation favoring localized turbulent bands. At the other end of the transitional range, a preliminary study suggests that the laminar-turbulent pattern leaves room to a featureless regime beyond a well defined threshold to be determined with precision.
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