There is no deterministic solution for many fluid problems but by applying analytical solutions many of them are approximated. In this study an implicit finite difference method presented which solves the potential function and further expanded to drive out the velocity components in 2D-space by applying a point-by-point swiping approach. The results showed the rotational behavior of both potential function as well as velocity components while encountering central obstacle.
Finite difference method is a strong tool which has been developed by Euler in 1768 and extensively used in computational fluid dynamics problems. For instance, rate of transported particles in a river or coastal area depends on stream power and current direction. Stream power defined as shear stress multiplied by velocity. Hence velocity field calculations are essential element in derivation of sediment pattern in coastal areas to protect and sustainably manage coast structures such as breakwaters . It can also be used in floodplain delineation as well as sediment velocity calculations. This study focused on a point-by-point implicit application to develop a MATLAB code which can derivative fluid velocity from fluid potential function.
Finite difference method is a derivation of Taylor’s polynomial series expansion. The error in this method is defined as the difference between the approximation and exact analytical solution. It can be used in computational fluid dynamic modeling, heat transfer problems in both explicit and implicit forms. For instance, in explicit method, a forward difference in time and a second order central difference in the space (FTCS) approach may be applied and in its implicit form one can use a backward difference in time and second-order central difference in space (BTCS) to complete the simulation process [2-6]. Many applications in chemical engineering, fluid mechanics and geology involve interaction of particles and fluid which applies Navier-Stokes simulation by finite difference approach in a particle-laden flows [7, 8]. The goal of this study was to calculate the velocity components of U and V in x and y directions, respectively.
Potential function has been driven in a 20 × 20 meters box as in figure 1, including a central obstacle at the middle by an opening at each direction. Central box has a dimension of four meters. The potential function is defined by equation 1 below.
To obtain an acceptable accuracy domain has been discretized.
The entire domain consists of mesh size of 1m times 1m including
400 smaller boxes. Each one meter also has been divided into 6
segments (k=6) which finally resulted in 120 × 120 boxes. Figure
2(a, b) showed a schematic view of side segmentation [15, 16].
In equation 13 in velocity calculation, velocity in y direction,
v is a constant. In terms of simplicity of calculations, considered
to be zero.
Without having central obstacle line by line swiping provides
the potential function for the entire area as presented in figure
Case 2: With central obstacle
By considering small box at the center the potential function
behavior and velocity distribution changed into two groups.
One group moves from upper part of obstacle and the other one
redirected toward beneath path, but both meets each other after
the central box. Consequently, x and y components of the velocity
can be driven out of potential function for each case which is
provided in figure 5.
The novelty of this study is to find a finite difference numerical
solution to approximate fluid potential function. Physical
interpretation of the problem is a key component to understand
the behavior of the fluid in such examples and converts the
abstract problem an into a more comprehendible and pragmatic
problem-solving tool. Ψ is the Potential function and as soon as
the water reaches the central obstacle it starts rotating around
to find its path towards the exit. This trend is observable within
both potential function as well as velocity component path.
Convergence and divergence of flow patterns could be observed
both in the experimental visualization fluid dynamics lab as well
as computational dynamic modeling CFDs .
The author declares that there is no conflict of interest.
They have no known competing financial interests or personal
relationships that could have appeared to influence the work
reported in this paper.