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SLOSHING IN A T-BAFFLED RECTANGULAR STORAGE TANK 17
NUMERICAL STUDY FOR 2–D PROBLEMS

u1, j  0 (6)
v1, j  v2, j
for all j

2.2 Numerical Stability and Accuracy

Numerical calculations can have quantities that develop large, high frequency
oscillations in space or time or both of them. This behaviour is usually referred to as a
numerical instability. To prevent this type of numerical instability or inaccuracy,
certain restrictions must be observed in defining the mesh increments xi and yj, the
time increment t and the upstream differencing parameter.

For accuracy, the mesh increments must be chosen small enough to resolve the expected
spatial variations in all dependent variables. Once a mesh has been chosen, the choice
of the time increment necessary for stability is governed by two restrictions. First,
material cannot move through more than one cell in one time step, because the
difference equations assume fluxes only between adjacent cells. Therefore, the time
increment must satisfy the inequality,

t  (x, y) or, (7)
U i, j

 Min  xi , yj  (8)

t  ui, j vi, j 

where the minimum is with respect to every cell in the mesh. When a non-zero value of
kinematic viscosity, momentum must not diffuse more than one cell in one time step. A
linear stability analysis shows that this limitation implies,

1 (9)
 t 
2  1/ xi 2 1/  y j 2

In this study, t is automatically chosen to satisfy the above inequalities. In order to
insure the numerical stability, the parameter  is,

    ui, j  t vi, j  t  (10)
M ax x y 
1 , 


2.3 Tank configuration

Fig. 1 denotes the 2D-rectangular tank with T-shape baffle and the locations of the transducers
to obtain the pressure distributions with time (Akyildiz and Unal, 2006; Chen et al., 2009). For
all cases, the fluid depth (h) is 75% of the tank height. The baffles are assumed to be rigid and

Sayı 1, 2014 GiDB|DERGi