Turbulence generally arises in shear flows if velocities and hence, inertial forces are sufficiently large. In striking contrast, viscoelastic fluids can exhibit disordered motion even at vanishing inertia. Intermediate between these cases, a state of chaotic motion, “elastoinertial turbulence” (EIT), has been observed in a narrow Reynolds number interval. We here determine the origin of EIT in experiments and show that characteristic EIT structures can be detected across an unexpectedly wide range of parameters. Close to onset, a pattern of chevron-shaped streaks emerges in qualitative agreement with linear and weakly nonlinear theory. However, in experiments, the dynamics remain weakly chaotic, and the instability can be traced to far lower Reynolds numbers than permitted by theory. For increasing inertia, the flow undergoes a transformation to a wall mode composed of inclined near-wall streaks and shear layers. This mode persists to what is known as the “maximum drag reduction limit,” and overall EIT is found to dominate viscoelastic flows across more than three orders of magnitude in Reynolds number.
The development of complex velocity fields in curved ducts from an initially parabolic profile is studied using a three-dimensional numerical model of the parabolized Navier–Stokes equations. The velocity profiles are influenced strongly by a geometrical parameter Rc (the radius of curvature) and a dynamic parameter Dn (Dean number, Re/(Rc)1/2). For Rc < 10 and Dn up to 200, the velocity fields develop into the previously observed two- and four-cell solutions that are axially invariant and symmetric about the midplane. For Rc =100 and Dn>125 oscillatory solutions develop which are periodic in the axial direction, but are asymmetric about the midplane. Increasing the Dean number over a narrow range results in a significant increase in the frequency of such oscillations. Grid sensitivity tests indicate that such oscillations are not a numerical artifact. Development of oscillatory solutions is delayed with decreasing radius of curvature. Thus for Rc =10, axially invariant two-dimensional solutions that retain the symmetry about the midplane could be obtained for Dn as high as 300. This trend is consistent with one of the earliest observations by Taylor [Proc. R. Soc. London Ser. A. 124, 243 (1929)] that steady, symmetric laminar flows can be observed over a larger range of Dean number in tightly coiled tubes. However, when an asymmetric perturbation is imposed at the inlet, oscillatory solutions develop even for low Rc, indicating that symmetric two-dimensional solutions are not stable to asymmetric perturbations, as indicated by Winters [K. W. Winters and R. C. G. Brindley (private communication)]. Numerical results are also presented for flow through curved ducts with periodic step changes in curvature.
The convective heat transfer in a homogeneous porous duct of rectangular cross section in a horizontal orientation is examined. The flow is modeled by the Darcy equation and an averaged, single-equation model is used for the heat transfer. The aspect ratio of the duct (γ=a/b) appears as the natural geometrical parameter and the Rayleigh number Ra, which is the product of the Grashof number and the Prandtl number, appears as the natural dynamical parameter. Uniqueness of the solution at low values of Ra is demonstrated. The complete structure of the symmetric and asymmetric stationary solutions is traced numerically up to values of Ra of about 10 000, using arclength continuation. The limit points and the symmetry breaking bifurcation points are calculated numerically by using the appropriate extended system formulations. The manner in which these singular points unfold is examined as the aspect ratio is varied over 0.6≤γ≤1.4. Determination of linear stability shows that branches of stationary solutions above a Ra of about 4100 are unstable to arbitrary perturbations. The origin of a curve of Hopf points on one of the fold curves is detected around Ra=6560, γ=1.365.
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