Smooth-Wall Boundary Conditions for Energy-Dissipation Turbulence Models

TL;DR

This paper identifies that traditional smooth-wall boundary conditions in dissipation-based turbulence models are mathematically incorrect, leading to non-physical solutions, and proposes correct boundary conditions that ensure energy conservation and unique solutions.

Contribution

It demonstrates the incorrectness of common boundary conditions and establishes that setting both k and its gradient to zero at the wall is sufficient and correct.

Findings

Traditional boundary conditions can cause infinite or no solutions.

Correct boundary conditions enforce both k and its gradient to zero.

Proper conditions ensure energy conservation and unique solutions.

Abstract

It is shown that the smooth-wall boundary conditions specified for commonly used dissipation-based turbulence models are mathematically incorrect. It is demonstrated that when these traditional wall boundary conditions are used, the resulting formulations allow either an infinite number of solutions or no solution. Furthermore, these solutions do not enforce energy conservation and they do not properly enforce the no-slip condition at a smooth surface. This is true for all dissipation-based turbulence models, including the k-{\epsilon}, k-{\omega}, and k-{\zeta} models. Physically correct wall boundary conditions must force both k and its gradient to zero at a smooth wall. Enforcing these two boundary conditions on k is sufficient to determine a unique solution to the coupled system of differential transport equations. There is no need to impose any wall boundary condition on {\epsilon}, {\omega}, or {\zeta} at a smooth surface and it is incorrect to do so. The behavior of {\epsilon}, {\omega}, or {\zeta} approaching a smooth surface is that required to satisfy the differential equations and force both k and its gradient to zero at the wall.