A Local Pressure Boundary Condition Spectral Collocation Scheme for the Three-Dimensional Navier–Stokes Equations

Abstract

A spectral collocation scheme for the three-dimensional incompressible $({\varvec{u}},p)$ formulation of the Navier–Stokes equations, in domains $\varOmega$ with a non-periodic boundary condition, is described. The key feature is the high order approximation, by means of a local Hermite interpolant, of a Neumann boundary condition for use in the numerical solution of the pressure Poisson system. The time updates of the velocity ${\varvec{u}}$ and pressure $p$ are decoupled as a result of treating the pressure gradient in the momentum equation explicitly in time. The pressure update is computed from a pressure Poisson equation. Extension of the overall methodology to the Boussinesq system is also described. The uncoupling of the pressure and velocity time updates results in a highly efficient scheme that is simple to implement and well suited for simulating moderate to high Reynolds and Rayleigh number flows. Accuracy checks are presented, along with simulations of the lid-driven cavity flow and a differentially heated cavity flow, to demonstrate the scheme produces accurate three-dimensional results at a reasonable computational cost.

Appendix: A Derivative Formulas for Lagrange Interpolation Basis Function

Given K, let $\{l_k(y)\}_{k=0}^K$ denote the $K\!+\!1$ Lagrange interpolation basis functions based on the $K\!+\!1$ distinct points $\{y_j\}_{j=0}^K$. The functions are defined by $l_k(y) = \displaystyle { \mathop {\prod _{i=0}^{K}}_{ i \ne k} \frac{y-y_i}{y_k-y_i}},$ and satisfy $l_k(y_j)=\delta _{k,j}$ for $0 \le k,j \le K$. A direct calculation gives

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and

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