In many instances the sensitivity of buoyancy-driven enclosure flows can be linked to the presence of multiple bifurcation points that yield laminar thermal convective processes that transition from steady to various modes of unsteady flow . This behavior is brought to light by a problem as `simple' as a differentially-heated tall rectangular cavity (8:1 height/width aspect ratio) filled with a Boussinesq fluid with - which defines, at least partially, the focus of this special session. For our purposes, the differentially-heated cavity provides a virtual fluid dynamics laboratory as pointed out by Le Quéré :
``In conclusion let us emphasize that the differentially-heated cavity, in addition to its relevance as a model of convective heat transfer, turns out to be a real fluid mechanics laboratory in itself. The spatial structure of the flow is made of vertical and horizontal boundary layers, of corner structures, of a stratified core ... which depend very sensitively on the aspect ratio, Prandtl number and thermal boundary conditions (even a fly-wheel structure can be found at low ). All these features cooperate to give rise to very complex time behaviors resulting from several instability mechanisms, traveling waves in the vertical boundary layers, thermal instabilities along the horizontal walls in particular, which can interact strongly with internal wave dynamics.''
In the 8:1 cavity, the spectrum of the Jacobian of the Navier-Stokes equations about a steady-state solution is characterized by an infinite number of eigenvalues, either real or complex conjugates. For increasing Rayleigh number, some of the eigenvalues can cross the imaginary axis indicating bifurcation points (i.e., steady flow becomes unsteady, á la Hopf). Preliminary computations in the air filled 8:1 cavity (Xin and Le Quéré) have indicated that two pairs of complex conjugate eigenvalues cross the imaginary axis in the vicinity of . One of the corresponding eigenmodes has the skew-symmetry property of the base flow, while the other, the first unstable mode, is not skew-symmetric. This suggests that there will be significant sensitivity to the choice of boundary and initial conditions. That is, the choice of skew-symmetric conditions can promote the saturation of the second unstable mode which is skew-symmetric. In contrast, a random perturbation of the temperature field around the mean can promote the growth of the first unstable mode which is not skew-symmetric - at least for a finite period of time. Due, in part, to the presence of multiple unstable modes with a relatively small separation in , the apparently simple differentially-heated cavity problem is not as simple as one might initially believe.
Additionally, the simulation of this buoyancy-driven flow is remarkably susceptible to the deleterious effects of numerical damping and/or dispersion introduced by commonly used `tricks-of-the-trade', thus making it surprisingly challenging. For example, numerical tests have demonstrated that the damping/dispersion artifacts from the simplest time-marching advection treatment with balancing tensor diffusivity can destroy the delicate thermal convective processes present in this enclosure when close to the critical Rayleigh number. In fact, coarse-grid computations may exhibit steady-state solutions even though the true solution is unsteady - requiring higher resolution grids than may be initially thought. Even when dissipation and dispersion have seemingly been minimized, computational experiments have shown that the amplitude of the periodic temperature oscillations can vary by as much as an order of magnitude depending on the specifics of the spatial discretization, grid resolution, stopping criteria for iterative solvers, and even the use of advective vs. conservative forms of the governing equations.
Ultimately the sensitivity of this class of flow problem to initial and boundary conditions, formulation details, and numerical procedures raises at least the following questions: What is the critical Rayleigh number, above which the flow will be unsteady, for the 8:1 enclosure? What is the behavior of the flow field at Rayleigh numbers slightly above critical? What is the role of linear stability analyses in predicting unstable modes? Can non-linear dynamics provide any insight into the behavior of the 8:1 cavity? What can be said about the relationship between (unstable) steady-state solutions and time-averaged periodic solutions? What is the best formulation and associated numerical procedure to use in order to ameliorate the sensitivities observed in practice and raise the level of accurate predictability? Which `other' numerical methods, i.e., discretization, time integrator, stabilization, preconditioned iterative technique, and `tricks-of-the-trade' are at least viable - and which are not?
In order to answer these questions and more, a special session is being organized for the First MIT Conference on Computational Fluid and Solid Mechanics. The session organizers are soliciting the contribution of solutions to the 8:1 differentially-heated cavity problem for near-critical Rayleigh numbers, and hope to see finite difference, finite volume, finite element, and spectral methods applied to this seemingly-simple 2-D problem. The application of commercial CFD codes is also highly encouraged.
The 8:1 buoyancy driven enclosure flow problem is based upon the geometrical configuration shown in Figure 1 where is the width and the height of the enclosure. The enclosure aspect ratio is and takes on the value . The gravity vector is directed in the negative -coordinate direction, and the Boussinesq approximation for the buoyancy forces is assumed to be valid; i.e., only small temperature excursions from the mean temperature are admitted.
The non-dimensional governing equations for the time-dependent thermal convection problem are the incompressible Navier-Stokes equations, conservation of mass, and the energy equation written in terms of temperature:
The Rayleigh number is
The enclosure boundary conditions are simple and consist of no-slip walls, insulated (zero heat flux) horizontal walls, and constant-temperature vertical walls. The no-slip and no-penetration conditions are prescribed as on all walls, the left-wall is held at a constant `hot' temperature, while the right-wall is held at a constant `cold' temperature. The non-dimensional boundary conditions are summarized in Table 1.
In this section, we describe one set of initial conditions that may be
used for a transient simulation. Here, the fluid is isothermal and
initially at rest:
We encourage the use of alternative initial conditions as a means to test the sensitivity of this problem and to further probe the space of solutions. For example, a random perturbation of the initial constant temperature is also an acceptable initial condition - as are any that are not skew-symmetric.
Due to limited space for the publications, the authors should not repeat the introductory material or problem definition in their submissions. Instead, authors are asked to simply cite this document for the problem definition and present only their compulsory results as discussed below.
Submissions for the special session are due by October 15, 2000 and should be sent directly to:
Mark A. Christon LSTC 7209 Aztec Rd., NE Albuquerque, NM 87110Authors will be informed prior to December 1, 2000 regarding acceptance of their contribution. In addition to the conference proceedings, we anticipate that the final results presented at the special session will be published in summary form in the International Journal for Numerical Methods in Fluids.
We realize that complete results for this problem may not be available by October 15th, and note that submissions with only preliminary results - particularly all or most of the tabulated results outlined below - along with a description of anticipated additional results are acceptable.
The following sections detail the parameter space and components of an `ideal' submission and include both compulsory and optional components.
This section outlines the quantities of primary interest that are to be reported, and suggests some additional quantities that contributors may wish to report. The compulsory data should be viewed as highly desirable for the sake of performing meaningful comparisons rather than an absolute requirement for paper submission.
The compulsory results for the special session are to be presented in 3 tables and two plots as outlined below. In the ensuing description, the compulsory data for the transient and steady-state computations are categorized according to the date type, i.e., point, wall and global data. We encourage and hope to see results from steady-state, transient, and stability computations.
In this problem, the primary physical parameters consist of the Prandtl number and a super-critical Rayleigh number which is fixed at . Due to the amount of data that is required for this problem and the fact that computations on multiple grids will be necessary, only one Rayleigh number is being specified. There is, of course, no restriction on submission of results for additional Rayleigh numbers and we encourage participants to include their best prediction of the critical Rayleigh number, : the transition from steady to time-periodic behavior.
There are no strict requirements on grid resolution, however, all participants are asked to produce results that they believe are sufficiently accurate. In order to provide some guidance on grid resolution, we report here some preliminary results. Using a second-order method, a smoothly graded `coarse-grid' resolution consisting of grid points has been used to successfully compute time-dependent results for . Doubling the grid resolution, e.g., grid points, has shown that there is still significant sensitivity in (at least) the amplitude of temperature oscillations. Interestingly, the amplitude of the temperature oscillations is also dependent on the specifics of the Navier-Stokes formulation.
Some guidance on the vertical-wall boundary layer thickness may be found in Gill. Note that graded meshes are suggested since the resolution for a uniform grid may make the computations prohibitively time consuming. We suggest using grids with approximately a 1:5 x-to-y ratio of grid points starting with a coarse grid of . Increasing grid resolution should be obtained by grid doubling, e.g., a medium grid of and a fine grid of .
A series of physical data should be recorded at the compulsory time-history points from Table 2. The coordinates that are identified in Table 2 are non-dimensional and shown in Figure 1.
The time history data for point 1 should be tabulated as shown in Table 3. In addition, a plot of the oscillatory variation in temperature at point 1, , should accompany the tabulated results. A second plot showing the skewness, , should be included only if the skewness is found to be non-zero during the periodic phase. For all time-dependent computations, the average value and oscillation amplitude are to be reported, along with the period of oscillation. The average should be taken over one or more complete periods where the amplitude and period should be constant. The use of an FFT to obtain the amplitude and period is encouraged, but not mandatory. Where steady-state computations are concerned, only the point-values are required and may be tabulated in a similar format.
The computation of the cycle average is based upon achieving a
statistically stationary state where the period and amplitude are
constant. For a generic variable,
average may be computed as
Using the average value, perturbation fields may be computed as
The vorticity is defined as
In Table 3, the skewness provides a measure of the
skew-symmetry of the temperature field. Using time-history points 1
and 2, the skewness is computed as
The pressure differences in Table 3 are defined as
The wall Nusselt numbers are also a component of the compulsory
results and are defined as
In addition to the point and wall data, a number of relevant global
quantities are required as delineated in Table 5. For
our purposes, the kinetic energy and enstrophy provide useful metrics;
e.g., the square-root of kinetic energy provides a measure of the
average velocity in the enclosure. That is, our velocity metric is
A summary of the method used to solve the problem is required and should provide a concise description of the following items:
A summary of the computational resources is required and should provide a concise description of the following items:
Extensions of the results beyond the compulsory data is encouraged. However, due to space limitations, the optional results should not be reported in the paper submission. Several possibilities for optional results include:
|October 15, 2000||Deadline for submissions|
|December 15, 2001||Notification of acceptance|
|June 12-14, 2001||First M.I.T. Conference|
Questions regarding the special session may be sent via e-mail to any of the organizers.
|Mark A. Christon||Philip M. Gresho||Steven B. Sutton|
|LSTC/Univ. of New Mexico||LLNL||LLNL|
|e-mail: firstname.lastname@example.org||e-mail: email@example.com||e-mail: firstname.lastname@example.org|
|Ph: (505) 875-0746||Ph: (925) 422-1812||Ph: (925) 422-0322|
|Fx: (505) 875-0430||Fx: (925) 423-5167||Fx: (925) 423-6195|
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