Abstract
The benchmark is devoted to convection of low-Pr fluids subject to buoyancy forces in a long rectangular cavity whose vertical walls are maintained at constant temperatures: T1 and T2. The cavity has a rigid bottom wall, and two types of conditions are considered for its upper surface: (i) rigid (denoted R-R) and (ii) shear-stress free (denoted R-F). Experiments [1] and [2] establish that the buoyancy-driven flow which ensues as soon as T1 ≠ T2 becomes oscillatory beyond a certain critical value, Grosc, of the Grashof number which is defined as Gr=gβγH4/v2, where γ=ΔT/L and AT=T2-T1. Here H and L denote the height and the length of the cavity, and A (=L/H) denotes the aspect ratio(see Fig. 1).
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Abbreviations
- x, y, z:
-
non-dimension cartesian coordinates
- u, v, w:
-
non-dimensional velocity components
- U :
-
two-dimensional velocity vector (u = (u,v))
- vref :
-
reference velocity (vref = vGr0.5/H)
- U*:
-
maximum of I u(y) I, at x-1/2
- V*:
-
maximum of I v(x) I, at y=A/4 in R-R cases, and x=1 in R-F cases
- A:
-
aspect ratio (A=L/H)
- Ay:
-
transverse aspect ratio (Ay=W/H)
- H:
-
height (refence length)
- L:
-
length
- W:
-
width
- T:
-
temperature
- Tref :
-
reference temperature (Tref = ΔT / A)
- p:
-
pressure
- t:
-
time
- tref :
-
reference time (tref = H2/v)
- f:
-
frequency (inverse of the non-dimensional period)
- Gr:
-
Grashof number (Gr= Gr= g βγ H4/ v2)
- Pr:
-
v / κ
- Vi Vd, Vb :
-
constants in Navier-Stokes equations (1) and (3)
- Ti, Td :
-
constants in energy equations (4)
- β:
-
coefficient of thermal expansion
- γ:
-
external temperature gradient (γ = AT /L)
- ΔT:
-
external temperature difference (ΔT=T2-T1)
- θ:
-
non-dimensional temperature (T-T1)/ Tref
- κ:
-
thermal diffusivity
- v:
-
kinematic viscosity
- Ψ:
-
streamfunction
- Ψ*:
-
maximum of | Ψ |
- Ω:
-
vorticity
References
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HURLE D.T.J., JAKEMAN E. and JOHNSON C.P. (1974), J. Fluid Mech. 64, 565–576.
OSTRACH S. (1976), Proc. Secd. European Symp. on Material Sciences in Space, ESA-SP 114.
de VAHL DAVIS G. (1986), Heat Transfer 86. Hemisphere, Washington,101–109.
WINTERS K.H. (1987), Proc. 5th Int. Conf. Num. Meth.Therm. Prob., Pineridge Press.Swansea.
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© 1990 Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig
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Roux, B. (1990). Benchmark Definition. In: Roux, B. (eds) Numerical Simulation of Oscillatory Convection on Low-Pr Fluids. Notes on Numerical Fluid Mechanics (NNFM), vol 27. Vieweg+Teubner Verlag. https://doi.org/10.1007/978-3-322-87877-9_2
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DOI: https://doi.org/10.1007/978-3-322-87877-9_2
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