レイノルズ平均ナビエ-ストークス方程式
∂ u i ̄ ∂ t + u j ̄ ∂ u i ̄ ∂ x j = f i ̄ − 1 ρ ∂ p ̄ ∂ x i + ν ∂ 2 u i ̄ ∂ x j ∂ x j − ∂ u i ′ u j ′ ̄ ∂ x j {\displaystyle {\frac {\partial {\bar {u_{i}}}}{\partial t}}+{\bar {u_{j}}}{\frac {\partial {\bar {u_{i}}}}{\partial x_{j}}}={\bar {f_{i}}}-{\frac {1}{\rho }}{\frac {\partial {\bar {p}}}{\partial x_{i}}}+\nu {\frac {\partial ^{2}{\bar {u_{i}}}}{\partial x_{j}\partial x_{j}}}-{\frac {\partial {\overline {u_{i}^{\prime }u_{j}^{\prime }}}}{\partial x_{j}}}}
さらに変形すると以下のようになる。
ρ ∂ u i ̄ ∂ t + ρ u j ̄ ∂ u i ̄ ∂ x j = ρ f i ̄ + ∂ ∂ x j [ − p ̄ δ i j + 2 μ S i j ̄ − ρ u i ′ u j ′ ̄ ] {\displaystyle \rho {\frac {\partial {\bar {u_{i}}}}{\partial t}}+\rho {\bar {u_{j}}}{\frac {\partial {\bar {u_{i}}}}{\partial x_{j}}}=\rho {\bar {f_{i}}}+{\frac {\partial }{\partial x_{j}}}\left[-{\bar {p}}\delta _{ij}+2\mu {\bar {S_{ij}}}-\rho {\overline {u_{i}^{\prime }u_{j}^{\prime }}}\right]}