Three‐dimensional flow in a shear‐driven cube

The flow in a cubic cavity is studied when a constant shear stress is imposed on one of its square faces. The three‐dimensional basic flow undergoes a first steady, symmetry‐breaking, pitchfork bifurcation. On an increase of the Reynolds number the symmetry‐broken flow becomes time‐dependent via a Hopf bifurcation. Even though the basic flow is similar to the one in the lid‐driven cube, the sequence of bifurcations differs significantly.


Introduction
The flow of an incompressible fluid with kinematic viscosity ν and density ρ confined to a cubic cavity with edge length L is considered. The flow is driven by imposing a constant shear stress τ on one of its faces. The system represents a basic model for thermocapillary flow in a differentially heated cavity in the limit of small Prandtl numbers [1] and it is related to the shear flow over an open cavity [2]. Interest in the shear-driven cubic cavity (SDC) also derives from its similarity to the well-known lid-driven cubic cavity (LDC) which has been thoroughly investigated during the last decade [3][4][5][6]. Thus one may inquire about similarities of and differences between both systems.
The Navier-Stokes equations are made dimensionless using the scales L, L 2 /ν, ν/L, ρν 2 /L 2 for length, time, velocity and pressure, respectively. The stress-based Reynolds number is defined as Re τ = L 2 τ /ρν 2 . To compute the steady flow in the cube V = [−1/2, 1/2] 3 the BoostConv algorithm [7] is employed. It can be implemented around any unsteady Navier-Stokes solver. The algorithm is based on a smart recombination of residuals to accelerate convergence to steady state by erasing the most amplified linear perturbation modes. In particular, the method allows to recover an unstable steady solution even if the most dangerous perturbation modes are not oscillatory. Here, the spectral element solver NEK5000 is employed, using a spatial discretization of 12 3 elements of order 6 and a third-order Adams-Bashforth scheme (BDF-3) in time.

Results
As the problem is symmetric with respect to the midplane z = 0 (Fig. 1a), a three-dimensional flow exists at low Reynolds numbers exhibiting the same symmetry, i.e. w(z = 0) = 0. However at Re τ = 53500, the symmetric steady basic state is not stable anymore and the dynamical system settles on another steady state with broken symmetry for which w(z = 0) = 0 as shown in The loss of stability is also recovered by a linear stability analysis: a real eigenvalue of the linear stability problem crosses the imaginary axis at Re τ,c = 53486 ( in Fig. 2a). As the Reynolds number increases, the amplitude of the symmetrybreaking flow, measured by w(x p ) at the monitoring point x p = (−0.4, 0, 0) on the midplane (cross in Fig. 1b), increases with the square root of the distance from the critical point ∼ (Re τ − Re τ,c ) 1/2 , the footprint of a pitchfork bifurcation. The spanwise velocity squared w 2 (x p ) is shown in Fig. 2(a) as function of Re τ ( ) together with a linear fit (line). The extrapolation to w 2 (x p ) = 0 yields a critical Reynolds number Re τ,c = 53485 which agrees with the value obtained by the linear stability analysis.
On a further increase of Re τ , the linear stability analysis shows that the unstable symmetric basic flow also becomes unstable to an oscillatory mode at Re τ,c2 = 54100 with frequency ω c2 = 689.4 ( in Fig. 2a). On the other hand, the finiteamplitude symmetry-breaking solution bifurcating from the basic state at Re τ,c1 becomes unstable to an oscillatory flow at Re a τ,c = 55700 ( in Fig. 2a) with ω = 764.16, where the superscript 'a' indicates the asymmetric flow. This frequency is close to ω c2 , despite of the different flows from which the oscillatory modes bifurcate. The energy budgets of the oscillatory neutral modes, as defined in [8], is shown in Fig. 2b. The integral contributions to the different destabilization processes [8] is nearly the same for both modes. This suggests the Hopf bifurcations from the symmetric and from the asymmetric solution branches are caused by the same type of instability mechanism with the flow asymmetry delaying the oscillatory instability.

Discussion and Conclusion
The symmetric basic flows in the SDC and the LDC become unstable at about the same flow strengths (measured by Re). However, the bifurcation scenario is different: In the LDC the basic flow becomes linearly unstable at Re LDC = 1921 [3] to a backward-bifurcating oscillatory but symmetric mode. The symmetric oscillations are nearly always unstable (saddle limit cycle) and the system is driven to another solution branch which, however, is unstable with respect to symmetry breaking perturbations. This leads to an intermittent chaotic behavior with periods of symmetric and asymmetric oscillations [3,5]. On the other hand, for the present SDC both the first and the secondbifurcation from the basic flow are symmetry breaking. Thus the symmetry is broken before the flow becomes oscillatory. Since the LDC and SDC flows are similar, it is concluded that the sequence of bifurcations is rather sensitive to details of the flow structure. Also the different boundary conditions for perturbations on the driving face of the cube (no-slip versus free-slip) may contribute to the differences found. The transition to chaotic flow in the SDC remains to be investigated.