Sensitivity of the thresholds to unsteadiness upon confinement shape in deformed driven cavity

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A. Redondo, G. Kasperski and G. Labrosse



The lid driven cavity problem in non orthogonal geometries is considered with two purposes: one is to develop numerical tools for obtaining spectrally accurate flows in such geometries and then to point out the sensitivity upon the boundary deformations of the transition thresholds to unsteadiness. Accurate solutions of the Navier Stokes equations are indeed obtained thanks to a spectral Chebyshev discretization [2] preconditioned with Finite Volumes [3]. Then, the influence of the deformation upon flow topology and stability properties is shown.

Numerical Method

The Navier-Stokes equations

 \scriptstyle \frac{\partial \boldsymbol{v}}{\partial t}+{(\boldsymbol{v}}.\nabla)\boldsymbol{v}=\frac{1}{Re}\nabla^2\boldsymbol{v}-\nabla p,

 \scriptstyle \nabla. \boldsymbol{v}=0

are solved for the incompressible flow of a Newtonian fluid (of kinematic viscosity \scriptstyle \nu and density \scriptstyle \rho) confined in deformed driven cavities of height \scriptstyle 2L. Taking \scriptstyle L and \scriptstyle U as length and velocity scales, the Reynolds number is defined as  \scriptstyle Re=\frac{UL}{\nu}, \scriptstyle U being the velocity amplitude of the moving lid. In order to suppress the singularities at the upper corners, the velocity of the lid is regularized according to  \scriptstyle U_{Lid}=(1-\left(\frac{2x}{D}\right)^2)^2 where  \scriptstyle D is the length of the upper lid and \scriptstyle x\in [-D/2:D/2], the coordinate along the upper lid. The lateral deformations are of parabolic type, described by \scriptstyle f(y)=\pm (1+A_d(1-y^2)) where \scriptstyle A_d is the amplitude (positive or negative) of the deformation.

Temporal discretization is based on second-order explicit Adams Bashforth scheme for the non linear terms and an implicit backward Euler for linear terms.

A Projection-Diffusion algorithm is used to uncouple the pressure and velocity fields [1,4], which leads to solve:

a Poisson equation for the pressure,

  \scriptstyle \nabla^2p=\nabla.\boldsymbol{F},    (1)

with Neumann boundary conditions:

\boldsymbol{n}.\nabla p=\boldsymbol{n}.(-\frac{\partial \boldsymbol{v}_0}{\partial t}+\boldsymbol{F}-\frac{1}{Re}\nabla\times\nabla\times\boldsymbol{v}),

where \scriptstyle \boldsymbol{F} stands for the nonlinear term  \scriptstyle (\boldsymbol{v}.\nabla)\boldsymbol{v} assumed to be explicitely evaluated in time.

An Helmholtz problem is then solved for each velocity component :

\frac{1}{Re}\nabla^2-\frac{\partial{}}{\partial t}\right]\boldsymbol{v}=\nabla p - \boldsymbol{F},

with Dirichlet boundary conditions.

The remaining Poisson and Helmholtz problems are treated using spectral Chebyshev spatial approximation.

Spatial derivatives are evaluated using a Gauss-Lobatto collocation method (\scriptstyle 71 \times 71 ), and the time step is fixed to \scriptstyle 10^{-3}. For each fixed deformation, the Reynolds number is recursively incremented. The criterion for steadiness, based on the vorticity field \scriptstyle \omega, is \scriptstyle \| \ \omega^{n+1}-\omega^n
\| \ {\infty} / (\| \ \omega^{n+1}\| \ {\infty} . \delta t)\leq
10^{-6} where \scriptstyle \omega^n=\omega (n\delta t).

Determination of transition thresholds

These computations led to estimates of the transition thresholds from steady to oscillatory flows for boundary deformations \scriptstyle A_d  \in [-0.4:0.4] . Results are reported on Fig.1 : transition thresholds lie in between a lower limit corresponding to a steady state and an upper limit for which the flow is oscillatory.

These first results show the sensitivity of the steady flow stability to boundary deformations and a strongly non monotonic behaviour of the thresholds.

The graphic can be read considering three different zones, approximatively defined by \scriptstyle A_d \in [-0.4:-0.1] , \scriptstyle A_d \in [-0.05:0.1] and  \scriptstyle A_d \in
[0.2:0.4] .

The first zone corresponds to the pinched configuration wherein the critical Reynolds number value is significantly lowered, from 10000 for a straight cavity to 5000. In the second zone, corresponding to straight or slightly deformed cavities, the thresholds are only weakly modified. Refining their determination is necessary to characterize the local behaviour about  A_d=0 . The inflated deformation also significantly reduces the threshold values, from 10000 to 7000. In both first and third zones, a non monotonic sensitivity of the thresholds upon \scriptstyle A_d is observed, quite clearly for the inflated cavities.

Figure 1: Critical Reynolds of transition to unsteadiness as a function of Deformation Amplitude  A_d

Flow topology

A correlation exists between what has just been described and the flow topologies. The streamfunction is presented for the last observed steady flows, before transition, in Fig.2

The straight cavity is taken as a reference. The flow is made of a big roll driven by the upper lid, two counter-rotating cells in the left-down (LD) and right-down (RD) corners, a counter rotating cell close to the left-up (LU) corner and recirculating Moffatt corner cells.

The pinching deformations tend to stretch the RD cell until it merges the LD cell. The \scriptstyle A_d value associated with this junction is located between points C and D of Fig.1 where there is an abrupt evolution of the thresholds.

In a same way, inflating the cavity tends to stretch the LD cell until it reaches the LU cell. The junction is located between points F and G of Fig.1 and seems to be linked to the second abrupt behaviour of the thresholds.

Thus, each zone of Fig.1 corresponds to a different topology of the flow, mainly composed of horizontally organized cells for \scriptstyle A_d<0 , and vertically organized ones for  \scriptstyle A_d>0. A non-monotonic behaviour in each configuration seems linked to the appearance of new cells (e.g. in cases  \scriptstyle A_d=-0.4 and  \scriptstyle A_d=0.4).

Fig.2 : Evolution of the flow topology with boundary deformation
Ad=-0.4 - (A)
Ad=-0.3 - (B)
Ad=-0.1 - (C)
Ad=-0.05 - (D)
Ad=0 - (E)
Ad=+0.1 - (F)
Ad=+0.2 - (G)
Ad=+0.3 - (H)
I: Ad=+0.4


[1] A. Batoul, H. Khallouf, and G. Labrosse. Une méthode de Résolution Directe (Pseudo-Spectrale) du problème de Stokes 2D/3D Instationnaire. Application à la Cavité Entraînée Carrée. C.R. Acad. Sci. Paris, 319(I):1455-1461, 1994.

[2] C. Canuto, M.Y. Hussaini, A. Quarteroni, and T.A. Zang. Spectral Methods un Fluid Dynamics. Springer Series in Computational Physics. Springer-Verlag, New York, 1988.

[3] V. Delgado, G. Kasperski, and G. Labrosse. Spectrally accurate numerical incompressible flows with deformed boundaries: a finite volume preconditioning method. J. Comput. and Applied Math., 168:145-154,2004.

[4] E. Leriche and G. Labrosse. High-order direct Stokes solvers with or without temporal splitting : numerical investigations of their comparative properties. SIAM J. Scient. Comput., 22(4):1386-1410,2000.