Laplace equation solution technique: For three-dimensional potential-flow problems of water (e.g., in a siphon or seepage domain), which method is commonly used to obtain a numerical solution of the Laplacian field?

Introduction / Context:
Potential-flow problems in hydraulics—such as steady incompressible flow in siphons, around structures, or in seepage—are governed by Laplace’s equation. Except for simple geometries, closed-form analytical solutions are not available, and numerical approaches are used to solve the Laplacian field.

Given Data / Assumptions:

• Steady, irrotational, incompressible flow implies the potential satisfies ∇²ϕ = 0.
• Complex three-dimensional domain (e.g., around/within hydraulic structures).
• Boundary conditions specified on solid surfaces and inlets/outlets.

Concept / Approach:

The relaxation (successive over-relaxation or Gauss–Seidel-type) method iteratively updates potential values at grid nodes using the discrete Laplace equation until changes fall below a tolerance. It is a classic and robust approach for Laplacian problems where analytic methods are impractical.

Step-by-Step Solution:

Verification / Alternative check:

Convergence is verified by residual norms or by checking flux balances across boundaries. Benchmark against simpler geometries with known solutions provides confidence.

Why Other Options Are Wrong:

Khosla’s method: specialized analytical charts for seepage under weirs, not general 3D domains.
Unwin’s method: not a standard general solver for 3D Laplace problems.
Pure analytic solution: rarely feasible in complex 3D geometries.
Trial-and-error continuity balancing: non-systematic and not equivalent to solving Laplace’s equation.

Common Pitfalls:

Poor grid resolution, improper boundary conditions, and inadequate stopping criteria can yield inaccurate fields.

Final Answer:

Method of relaxation (iterative numerical solution)