In Computational Fluid Dynamics (CFD), resolving boundary layers and high-gradient shear regions requires non-uniform grid clustering near solid walls while keeping cell counts manageable in free-stream regions. Here is a Fortran routine to generate a geometrically stretched, symmetric 2D rectangular grid.
Domain: Rectangular 2D domain \([a, b] imes [c, d]\) • Method: Geometric progression expansion clustering toward boundaries
Mathematical Formulation
Given domain limits \(x \in [a, b]\) with \(n_x\) divisions and expansion ratio \(r_x\), the initial wall-adjacent step size \(\eta_x\) is computed from the geometric sum across the symmetric half-domain:
eta_x = (b - a) * (1 - exp_x) / (2 * (1 - exp_x**(n_x / 2)))
eta_y = (d - c) * (1 - exp_y) / (2 * (1 - exp_y**(n_y / 2)))Fortran Source Code
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! 2D Non-Uniform Symmetric Rectangular Grid Generator
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exp_x = 1.05 ! x-axis expansion factor
exp_y = 1.10 ! y-axis expansion factor
a = 0.0
b = 1.0
c = 0.0
d = 1.0
! Uniform step sizes for reference
del_x = (b - a) / n_x
del_y = (d - c) / n_y
! Compute first wall-adjacent step sizes (eta_x, eta_y)
eta_x = (b - a) * (1.0 - exp_x) / (2.0 * (1.0 - exp_x**(n_x / 2)))
eta_y = (d - c) * (1.0 - exp_y) / (2.0 * (1.0 - exp_y**(n_y / 2)))
x(1) = a
y(1) = c
! --- X-Direction Mesh Generation ---
! Half 1: Wall to Center (Expanding)
DO i = 2, n_x / 2 + 1
x(i) = x(i - 1) + eta_x * exp_x**(i - 2)
END DO
! Half 2: Center to Opposite Wall (Contracting)
DO i = n_x / 2 + 2, n_x + 1
x(i) = x(i - 1) + eta_x * exp_x**(n_x + 1 - i)
END DO
! --- Y-Direction Mesh Generation ---
! Half 1: Wall to Center (Expanding)
DO j = 2, n_y / 2 + 1
y(j) = y(j - 1) + eta_y * exp_y**(j - 2)
END DO
! Half 2: Center to Opposite Wall (Contracting)
DO j = n_y / 2 + 2, n_y + 1
y(j) = y(j - 1) + eta_y * exp_y**(n_y + 1 - j)
END DO
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