In this work, a novel numerical method for solving the unsteady incompressible Navier-Stokes equations with primitive variables in two dimensions that can be easily extendable to three dimensions. The methodology is based on Projection Methods using a mixed approach through the Classical Integral Transform Technique (CITT). Knowing that the main bottleneck of the classical pressure-correction approach is the solution of the pressure Poisson equation (PPE), this work proposes a different approach using the classic Projection Method for advancing in time the Navier-Stokes equations and CITT to find pressure dependence on the discrete velocity field in a semi-analytical manner, using the previous time-step pressure field as a filter. For comparison purposes, the Finite Volume Method (FVM) was also used, where the PPE was solved with a Gauss-Seidel iterative procedure. Two variations of the method were proposed: Single Transformation (CITT-ST) and Double Transformation (CITT-DT).

A novel methodology for solving unsteady convective heat transfer problems via the generalized integral transform technique has been developed. The proposed scheme is based on writing the unknown potential in term of eigenfunction expansions; however, rather than 10 transforming advection terms, an upwind approximation is used prior to the integral transformation. The formal application of the methodology is demonstrated for a general multidimensional problem, and numerical results for a one-dimensional test case are calculated. With the approximation, numerical diffusion is introduced, which is shown to reduce unwanted oscillations that arise at higher Pèclet values and stronger nonlinear effects, reducing mean square errors.

  • Numerical Methods
  • Computational Fluid Dynamics
  • Hybrid Methods for Solving PDEs
  • Ingral Transformation Technique
  • Numerical Advection-Diffusion Problems
  • Heat and Mass Transfer
  • Symbolic and Numerical Computation
  • Thermodynamics