The
lid-driven cavity is an important fluid mechanical system serving
as a benchmark for testing numerical methods and for studying fundamental
aspects of incompressible flows in confined volumes which are driven
by the tangential motion of a bounding wall.
Let us now see how Flow
Simulation predicts lid-driven
(i.e., shear-driven) 2D recirculating flows in closed 2D triangular
and trapezoidal cavities with one or two moving walls (lids) in
comparison with the calculations performed in Ref. 16 and
Ref. 6.
These two cavities
are shown in Figure 1. The triangular
cavity has a moving top wall, the trapezoidal cavity has a moving
top wall also, whereas its bottom wall is considered in two versions:
as motionless and as moving at the top wall velocity. The no-slip
conditions are specified on all the walls.
Figure 1. The 2D triangular (left) and trapezoidal
(right) cavities with the moving walls (the motionless walls are
shown with dashes).
As shown in Ref. 16 and
Ref. 6, the
shear-driven recirculating flows in these cavities are fully governed
by their Reynolds numbers Re = ρ·Uw·h/μ,
where ρ is
the fluid density, μ is
the fluid dynamic viscosity, Uw is
the moving wall velocity, h is the cavity height. So, we can specify
the height of the triangular cavity h = 4 m,
the height of the trapezoidal cavity h = 1 m, Uw = 1 m/s for
all cases under consideration, the fluid density ρ = 1 kg/m3,
the fluid dynamic viscosity μ = 0.005 Pa·s
in the triangular cavity produces a Re = 800,
and μ = 0.01,
0.0025, 0.001 Pa·s in the trapezoidal cavity produces a Re = 100,
400, 1000, respectively.
The cavities’ models
are shown in Figure 2. The Flow
Simulation calculation of flow
in the triangular cavity has been performed on the 48x96 computational
mesh. The results in comparison with those from Ref. 16 are
presented in Figure 3 (streamlines)
and in Figure 4 (the fluid
velocity X-component along the central vertical bisector shown by
a green line in Figure 2). A good
agreement of these calculations is clearly seen.
Figure 2. The models for calculating the lid-driven
2D flows in the triangular (left) and trapezoidal (right) cavities
with Flow
Simulation, where MW - moving
walls.
Figure 3. The flow trajectories in the
triangular cavity, calculated by Flow
Simulation (right)
and compared to the reference calculation (left).
Figure 4. The triangular cavity’s flow velocity X-component along the central vertical
bisector, calculated by Flow
Simulation (red
line) and compared to the reference calculation (black line with
circlets).
The Flow
Simulation calculations of flows in the trapezoidal
cavity with one and two moving walls at different Re values
have been performed with the 100x50 computational mesh. Their results
in comparison with those from Ref. 6 are
presented in Figure 5 – Figure 10 (streamlines)
and in Figure 11 (the fluid
velocity X-component along the central vertical bisector shown by
a green line in Figure 2). A good
agreement of these calculations is seen.
Figure 5. The flow streamlines
in the trapezoidal cavity with a top only moving wall at Re = 100,
calculated by Flow
Simulation (right)
and compared to the reference calculation (left).
Figure 6. The flow streamlines
in the trapezoidal cavity with a top only moving wall at Re = 400,
calculated by Flow
Simulation (right)
and compared to the reference calculation (left).
Figure 7. The flow streamlines
in the trapezoidal cavity with a top only moving wall at Re = 1000,
calculated by Flow
Simulation (right)
and compared to the reference calculation (left).
Figure 8. The flow streamlines in the trapezoidal
cavity with two moving walls at Re = 100,
calculated by Flow
Simulation (right)
and compared to the reference calculation (left).
Figure 9. The flow streamlines in the trapezoidal
cavity with two moving walls at Re = 400,
calculated by Flow
Simulation (right)
and compared to the reference calculation (left).
Figure 10. The flow streamlines in the trapezoidal
cavity with two moving walls at Re = 1000,
calculated by Flow
Simulation (right)
and compared to the reference calculation (left).
Figure 11. The flow velocity X-component along the central vertical bisector in
the trapezoidal cavity with two moving walls at Re = 400,
calculated by Flow
Simulation (red
line) and compared to the reference calculation (black line).
