DNS of the shear flows between two counter-rotating cylinders, radius ratio eta = 0.9

For over 100 years the Taylor-Couette flow (TCF), the flow between two independently rotating coaxial cylinders, has provided a paradigm to analyse shear fluid motions and centrifugal fluid motions, which commonly occur in the nature and technology. For example, the Taylor-Couette flows have allowed for the analysis of the geophysical and astrophysical flows, helped in interpretation of the transitional processes in such configurations as a circular pipe or a rotating channel. Currently, the emphasis is put on the study of turbulent rotating flows, and on the subsequent stages leading from the laminar to turbulent flows. Two main paths leading to turbulence can be distinguished: supercritical and subcritical, both occur in the Taylor-Couette (TC) device.

In the supercritical path the transition process begins with a gradual linear increase of disturbances, followed by a series of subsequent bifurcations. The subcritical transition is the result of the rapid, nonlinear process. In the Taylor-Couette flows this basic division has led to numerous studies performed for varying values of geometrical and physical parameters.

The Taylor-Couette flows are traditionally characterized by two physical parameters, i.e. Reynolds numbers of the inner and outer cylinders Re₁ = Ω₁R₁(R₂-R₁)/ν, Re₂ = Ω₂R₂(R₂-R₁)/ν, and by two geometrical parameters: aspect ratio Γ = H/(R₂-R₁) and radius ratio η = R₁/R₂, where Ω₁, Ω₂ are angular velocities of the inner and outer cylinders (α = Ω₂/Ω₁, R₁, R₂ are radii of cylinders, H denotes the distance between the end-walls and ν is kinematic viscosity). The combination of these two Reynolds numbers and radius ratio η creates two new parameters which perfectly capture the features of the rotating fluids, i.e. shear Reynolds number (the ratio of inertial to viscous forces) Re = 2|ηRe₂-Re₁|/(1 + η) and the ratio of Coriolis to inertial forces (named rotational number) R_Ω = (1-η)(Re₁ + Re₂)/(ηRe₂-Re₁). Based on the value of R_Ω, the basic division of the vortex motion into cyclone (R_Ω > 0) and anticyclone (R_Ω < 0) has been made. Additionally, shear Reynolds number Re and rotation number R_Ω are relevant to characterize the flow when the curvature is small, η → 1 (Brauckmann, Salewski, Eckhard [1], see Fig. 1).

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The instability processes in the TC flows are analysed mostly in the (Re₂, Re₁) plane. If the TC flow is stable to infinitely small disturbances, the fluid elements follow circular paths (the so called circular Couette flow). The circular Couette flow is linearly (centrifugally) unstable above the neutral stability line—the example neutral line in the (Re₂, Re₁) plane is presented in Fig. 1a (the black dashed line). In the classic Taylor-Couette flow (with Ω₂ = 0, Re₂ = 0), when Ω₁ exceeds a certain critical value, the flow undergoes the first bifurcation which breaks the flow axial-symmetry, and a system of axisymmetric, counter-rotating vortices appears in the meridian section (known as the Taylor-Couette vortices). With further increase of Ω₁, the Taylor-Couette vortices become unstable themselves, leading to more complex states, and finally to turbulence, [4‐10]. Below the neutral line the TC flow is linearly stable, however, in the linearly stable area two different regimes can occur. In cyclonic area (R_Ω > 0) the subcritical transition to turbulence can occur. The stable flow cases with R_Ω < 0 (the anti-cyclonic, named also the quasi-Keplerian flows) are discussed mostly due to the potential relevance of this regime to the accretion discs, [11‐14]. The numerical investigation of the subcritical flow in the counter-rotating Taylor-Couette configuration using DNS (direct numerical simulation) method is the main object of interest here.

The classic subcritical transition occurs in the cavity of the rotating outer cylinder Ω₂ ≠ 0 and the steady inner cylinder Ω₁ = 0. Burin, Czarnocki [15] observed in the cavity of Γ = 92, η = 0.97 the successive appearance of different flow structures with the increase of Re₂, and the gradual disappearance of all these flow structures during the process of reducing Re₂. The authors found a strong hysteresis which is an essential feature of the subcritical transition. In the Taylor-Couette flows the subcritical transition is also observed between the counter-rotating cylinders, if the distance between cylinders is small enough, and if the outer cylinder rotates faster than the inner one, [2‐7, 9, 14, 15‐17]. For example, Meseguer et al. [16] performed numerical investigations in the counter-rotating cavity of Γ = 29.9, η = 0.883 with fixed Re₂ = − 1200. Starting from the turbulent flow they gradually reduced Re₁ and observed the following flow structures: spiral turbulence (SPT, near the inner cylinder), intermittency (INT, characterized by localized turbulent spots), interpenetrating spirals (ISP, transition to laminar flow) and 2D Couette structure (laminar flow). The turbulent spirals were observed, among others, by Andereck et al. [5] in the cavity of η = 0.883, Γ = 30. The authors found that turbulent spirals rotate at angular velocity of Ω_S ≈ (Ω₁ +Ω₂)/2. Berghout et al. [2] investigated spiral turbulence in the TC configuration of η = 0.91, 42 >  Γ > 125, − 2000 < Re₂ < − 1000 using DNS. Crowley et al. [3] studied experimentally (using particle image velocimetry, PIV) the subcritical transition in the short counter-rotating configuration of η = 0.905, Γ = 5.26. With the increase of Re₁ they observed aperiodic flows with ISP, intermittency areas, and with the further increase of Re₁ quick transition to turbulence. Crowley et al. [3] emphasized the similarity of phenomena observed in the counter-rotating TC flows to other shear flows, for example to the flows in rotating channel or in circular pipes, [18‐20]. The experimental study of such processes in the channels requires gigantic length scale in the main flow direction. In the Taylor-Couette flow such investigation seems to be more realistic due to the natural periodic boundary conditions, but the investigation must be carried out in a configuration of large aspect ratio Γ, and with η close to one (see Γ = 750, η = 0.9968, [17]). The influence of these main geometrical (Γ, η) and physical (Re₁, Re₂) parameters, and the end-wall boundary conditions on the Taylor-Couette fluid dynamics has been studied by many authors, [2‐17, 21‐32]. The present research also falls within area of this issue.

The aim of research here is to characterize (using DNS) the shear flow occurring in the short (Γ = 4.7) counter-rotating Taylor-Couette configuration of η = 0.9 with the end-walls attached to the inner cylinder, and to compare the results with the data of [3] obtained in the short cavity (Γ = 5.26, η = 0.905), but with the end-walls attached to the outer cylinder. The present results obtained for η = 0.9 (Re₂ from − 1000 to − 500) are also discussed in the light of the present data obtained for the wide gap cavity of η = 0.8 (Re₂ from − 1500 to − 500). Such range of parameters makes it possible to show the general trends in the flow organization, in the profiles of statistical parameters and in the PSD distributions depending on α, η, Re₁ and R₂. The comparison of the present critical lines presented in the (Re₂, Re₁) plane, with those presented in [3] shows the great influence of the end-wall boundary conditions on the bifurcation processes. The obtained flow structures are discussed in the light of the results known from literature [2, 3, 8, 16, 21]. The presented results complement the existing literature.

The outline of the paper is as follows: the considered problem is defined and the 3D DNS algorithm, based on the spectral collocation method, is described shortly in Sect. 2. In Sect. 3.1 the general trends observed in the narrow gap flow cases are discussed. The critical bifurcation lines obtained for the flow cases of η = 0.9 and the figures illustrating the flow structures are presented in Sect. 3.2. The radial profiles of the averaged azimuthal velocity component and the PSD distributions are presented in Sect. 3.3. The radial profiles of Reynolds stress tensor components obtained for η = 0.9 are presented and discussed in the light of the data from literature in Sect. 3.4. The TC wide gap characteristics (η = 0.8) are compared with the results obtained for the narrow gap flow case (η = 0.9) in Sect. 3.5. The research is summarized in Sect. 4.

In the paper the Taylor-Couette flow is investigated by solving the incompressible Navier–Stokes equations using the 3D DNS method. The DNS code is based on a pseudo-spectral Chebyshev- Fourier approximation described in [22‐28]. The inner and outer cylinders rotate with the angular velocities Ω₁ [rad^.s⁻¹] and Ω₂ [rad^.s⁻¹], respectively. The end-walls are attached to the inner cylinder. The dimensionless axial and radial coordinates are denoted as follows z = Z/(H/2), z \(\in\)[-1, 1], r = [2R – (R₂ + R₁)]/(R₂-R₁), r \(\in\)[-1, 1]. The velocity components in the radial (U), azimuthal (V) and axial (W) directions are normalized by Ω₁R₂ (the dimensionless components are denoted by u, v, w). Time is normalized by Ω₁⁻¹.

The governing equations are approximated in time using the second-order semi-implicit scheme: an implicit scheme is used for the diffusive terms and an explicit Adams–Bashforth scheme is used for the non-linear convective terms. In the algorithm, the predictor/corrector method is used. The predicted velocity field (u^p, v^p, w^p, p^p obtained by solving the Helmholtz equation with the appropriate boundary conditions) is corrected by taking into account the pressure gradient at the new time section t⁽ⁱ⁺¹⁾. The new correction parameter is introduced ϕ = 2 δt(pⁱ⁺¹-p^p)/3, which is computed from the following equation and boundary condition:where n is the normal vector. All flow variables Ψ = [u^p, v^p, w^p, p^p, ϕ]^T are obtained by solving the Helmholtz equation which can be written in the following form:where q and S depend on Re₁, Γ, η and δt (increment of time).

The spatial approximation of the flow variable Ψ = [u^p, v^p, w^p, p^p, ϕ]^T is given by a series:where r_i and z_j are the Gauss–Lobatto collocation points, and T_n(r_i) and T_m(z_j) are the Chebyshev polynomials of degrees n and m, respectively. The numbers of collocation points in the radial, axial and azimuthal directions are depicted by N, M and K, respectively. The use Gauss–Lobatto collocation points guarantees the high accuracy of the computations. The numerical computations are performed using the following meshes: N = 100, M = 200, K = 50–300. The time increment is from the range: δt = 0.0005–0.001. The computations are carried out until the statistically steady conditions are attained—depending on the considered flow example, this requires between 100000–300000 iterations. Figure 1b contains exemplary time series illustrating the changes of the radial velocity component over the iteration process. In the iterative process the divergence of the velocity field is also monitored. In the considered ranges of Re₁ (up to 1000) and Re₂ (from -1000 to -500), the divergence error for η = 0.9 has been kept in the range between 10^–7 and 10^–6. For the example flow case of η = 0.9, Re₁ = 978, R₂ = -1000, δ t = 0.001, N = 100, M = 200, K = 300 at z = 0 the following values describing the computational precision are obtained: (ΔR⁺)₁ = 0.018, (ΔR⁺)₂ = 0.0217, (ΔZ⁺)₁ = 0.7, (ΔZ⁺)₂ = 0.001.

The λ₂ criterion is used for visualization, Jeong, Hussain [33]. The bifurcation lines are identified by analyzing the averaged (in time and in meridian section) kinetic energy of perturbations as a function of Re₁ and by analyzing the time series (rapid changes of amplitudes and frequencies).

Tables 1 and 2 contain basic information about the considered flow cases.

Studies of the turbulent Taylor-Couette flows in the very long counter-rotating configurations with narrow gap (i.e. η  less equal 0.9) between cylinders, and with sufficiently large |Re₂| have been conducted intensively for two decades by many authors, in [2‐11, 15‐17]. The main goal is to deepen our knowledge on the subcritical lam.-turb. transition using the simple TC configuration.

Unlike most researchers, Crowley et al. [3] conducted the experimental investigations in the short cavity of Γ = 5.26, η = 0.905 with the end-walls attached to the outer cylinder which resulted in a significant stabilization of the flow: the critical lam.-turb. and turb.-lam. lines (both obtained for fixed Re₂) are located above the theoretical neutral line in the (Re₂, Re₁) plane. To show the influence of the end-walls on the transitional flow, in the present paper the DNS computations have been performed in the short counter-rotating Taylor-Couette configuration of Γ = 4.7, η = 0.9 with the end-walls attached to the inner cylinder. For comparison the computations are also performed for the wide gap configuration of Γ = 4.7, η = 0.8. In the investigations Re₁ is increased along the Re₁ =  ηRe₂/α line (with α = − 0.92 for η = 0.9 and α = − 0.8 for η = 0.8) to reach “featureless turbulence” area. Then, starting from the results obtained along these lines, Re₁ is reduced at fixed Re₂ (with Re₂ from -1000 to -500 in the flow case of η = 0.9, and Re₂ from -1500 to -500 in the flow case of η = 0.8). The results show that the turb.-lam. critical line obtained for η = 0.9 is located bellow the critical line of the linear stability theory in the (Re₂, Re₁) plane, in contrast to the Crowley et al. [3] critical lines. This result shows the large destabilizing influence of the discs attached to the inner cylinder. However, there is full agreement with the results of Crowley et al. [3] regarding the flow structures.

The most characteristic radial (< (V’V’/Ω₁²R₁²) > _{t, A( R)})^0.5 profiles obtained for the flow case of η = 0.9 during the reduction of Re₁ at Re₂ = -1000 are presented in Fig. 6. Figure 6 shows that with decreasing Re₁ the value at the peak near the outer cylinder increases, and only after reaching a certain maximum, with the further decrease of Re₁, we observe the decrease of (< (V’V’/Ω₁²R₁²) > _{t, A( R)})^0.5. Figure 7a, b, c and d show that these changes of the (< (V’V’/Ω₁²R₁²) > _{t, A( R)})^0.5 values are related to the changes in the flow structures near the end-walls.

The study shows that the flow structures observed for η = 0.9 and 0.8 are similar, but the radial profiles of the Reynolds stress tensor components (< (V’V’/Ω₁²R₁²) > _{t, A( R)})^0.5 and < (V’U’/Ω₁²R₁²) > _{t, A( R)} are different; the corresponding profiles differ in shape and in the peak values.