Pipe networks are critical infrastructure systems for long-distance transportation of oil and gas, urban water supply, heating systems, and other essential services. In the design, operation scheduling, and optimization of pipe networks, it is important to quickly and accurately obtain the distributions of flow rate (velocity) and pressure. The solution of flow and pressure typically relies on pressure-velocity coupling algorithms, and the SIMPLE algorithm is a widely used one. This algorithm assumes no initial relationship between pressure and velocity while neglecting the influence of neighboring nodes' correction velocities on the current node. These two assumptions lead to pressure-velocity mismatch, resulting in an unstable solving process, accompanied by low computational efficiency and slow convergence. To address these issues, Sun and Tao proposed the IDEAL algorithm, which overcomes these assumptions through inner iterations of the pressure equation twice, achieving significant acceleration in solving fluid flow in two- and three-dimensional problems. However, this algorithm has not yet been applied to solving one-dimensional fluid flow in the pipe network, and its implementation faces three key challenges:
(1) Underdetermined velocity equation system in the IDEAL algorithm:
The momentum equation discretization yields fewer independent velocity equations than unknown variables. Using a conventional staggered grid, velocity equations can be derived from internal grid nodes, velocity-specified inlet and outlet boundaries, and mass continuity equations at connection nodes. However, due to the presence of connection nodes among different pipe components, the total number of discrete velocity equations is insufficient to uniquely determine all velocity variables. For example, when four pipes connect at one node, there is only one mass continuity equation but four unknown velocity variables at that junction. Additionally, the inlet and outlet velocity equations need to be supplemented when only pressure boundary conditions exist.
(2) Underdetermined pressure equation system in the IDEAL algorithm:
Similarly, the pressure equations only can be established for internal gird nodes, pressure-specified inlet and outlet boundaries, and pressure equality conditions at connection nodes. The discretization of continuity and momentum equations produces fewer pressure equations than unknown pressure variables at connection nodes and velocity boundary nodes, the system remains underdetermined.
(3) Empirical dependency of the IDEAL algorithm:
The IDEAL algorithm mitigates SIMPLE’s assumptions via dual inner iterations of the pressure equation. However, the iterations require empirical determination: Too few iterations slow convergence, while excessive iterations introduce computational overhead. Determining the optimal iteration number for different network configurations remains nontrivial.
Besides, due to the interconnection and coupling effects among different pipe components, the algebraic equations obtained from IDEAL algorithm discretization form a sparse matrix. This leads to slow convergence in iterative solving and potential solution failure, constraining the rapid computation of pressure-velocity coupling in pipe networks.
To address the aforementioned challenges, an allocation method of the control volume for the one-dimensional staggered grid is proposed in this study. Under this staggered grid, pipe endpoints are enabled to possess their own control volumes and derive corresponding momentum equations, thereby supplementing the velocity equations at connection points of pipe networks. Furthermore, by utilizing mass continuity at connection points, additional velocity and pressure relationships between connected pipe components are established, respectively. Consequently, the equation system for the IDEAL algorithm in solving the fluid flow in the pipe networks becomes well-posed with complete governing equations for all unknown velocity and pressure degrees of freedom, overcoming the abovementioned challenges 1 and 2.
Furthermore, the above implementation method of IDEAL algorithm incorporates with pipe components decoupling strategy (predictor-corrector scheme): boundary values for each pipe component are initially predicted and then iteratively corrected. This ensures effective pressure and velocity information exchange while allowing each pipe component to be solved independently using the TDMA (Tridiagonal Matrix Algorithm) method, greatly improving computational efficiency. Additionally, this predictor-corrector approach demonstrates adaptive capabilities for different pressure or velocity boundary conditions. During the application of the predictor-corrector scheme, the initial predicted boundary conditions for the pipe components are consistent with the converged solution from the previous iteration steps. This ensures that the pressure-velocity field variations remain smooth, leading to a more stable solving process. Consequently, the IDEAL algorithm requires only a single inner iteration, thereby overcoming its empirical limitations.
The numerical tests demonstrate significant advantages of the proposed method:
(1) Compared to the SIMPLE algorithm, the IDEAL algorithm achieves a relaxation factor near 1.0, with over 100% efficiency enhancement.
(2) When integrated with the IDEAL algorithm, the proposed pipe components decoupling strategy exhibits a three-order-of-magnitude speedup over conventional Gauss-Seidel iteration methods.

IDEAL algorithm,fluid flow simulation,pipe networks