Abstract
An analysis of existing and newly derived fast-projection methods for the numerical integration of incompressible Navier–Stokes equations is proposed. Fast-projection methods are based on the explicit time integration of the semi-discretized Navier–Stokes equations with a Runge–Kutta (RK) method, in which only one Pressure Poisson Equation is solved at each time step. The methods are based on a class of interpolation formulas for the pseudo-pressure computed inside the stages of the RK procedure to enforce the divergence-free constraint on the velocity field. The procedure is independent of the particular multi-stage method, and numerical tests are performed on some of the most commonly employed RK schemes. The proposed methodology includes, as special cases, some fast-projection schemes already presented in the literature. An order-of-accuracy analysis of the family of interpolations here presented reveals that the method generally has second-order accuracy, though it is able to attain third-order accuracy only for specific interpolation schemes. Applications to wall-bounded 2D (driven cavity) and 3D (turbulent channel flow) cases are presented to assess the performances of the schemes in more realistic configurations.
Highlights
Numerical discretization of the incompressible Navier–Stokes (NS) equations is nowadays a common practice for the analysis of a variety of fluid-flow phenomena, encompassing both fundamental research and industrial applications
A common approach to integrate this set of differential–algebraic equations (DAEs) in time is by means of so-called projection methods: The momentum equation is time-advanced without satisfying the incompressibility constraint, a correction is applied to the provisional velocity field to project it onto a divergence-free space without changing its rotational component [6]
The approach, called the FPJ method, is able to decrease the number of Pressure Poisson Equation (PPE) to be solved within a time step from s to only one, significantly alleviating the required computational cost
Summary
Numerical discretization of the incompressible Navier–Stokes (NS) equations is nowadays a common practice for the analysis of a variety of fluid-flow phenomena, encompassing both fundamental research and industrial applications. A typical example is the direct simulation of turbulent flows, where the number of degrees of freedom required to accurately describe the entire range of spatial and temporal scales remains prohibitive for problems of engineering interest [1] This situation warrants the continuous quest for novel numerical tools and efficient implementations that are able to reduce the amount of computational work needed to solve the discrete flow equations. An early attempt to circumvent this drawback was presented by Le and Moin [9], who proposed the enforcement of the divergence-free constraint approximately at intermediate sub-stages by using an explicit estimate of the pressure, whereas the exact projection of the velocity field was performed only at the final step. We build on previous work to analyze a broad class of FPJ methods based on a general family of linear interpolations of the pressure inside the RK stages, and we test their accuracy on benchmarks of increasing complexity
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