边界层气象：脉动量预报方程展开 | 湍流脉动速度方差预报方程 | 平均湍流动能收支方程推导
写成分量形式
原始式子：
∂ u i ′ ∂ t + u ‾ j ∂ u i ′ ∂ x j + u j ′ ∂ u ‾ i ∂ x j + u j ′ ∂ u i ′ ∂ x j = − 1 ρ ‾ ⋅ ∂ p ′ ∂ x i + g θ v ′ θ ‾ v δ i 3 + f ϵ i j 3 u j ′ + v ∂ 2 u i ′ ∂ x j 2 + ∂ ( u i ′ u j ′ ‾ ) ∂ x j \begin{align*} \frac{\partial u_i'}{\partial t}+ \overline u_j\frac{\partial u_i'}{\partial x_j}+ u_j'\frac{\partial \overline u_i}{\partial x_j}+u_j'\frac{\partial u_i'}{\partial x_j}= -\frac{1}{\overline \rho}\cdot{ \frac{\partial p'}{\partial x_i}} +g\frac{\theta_v'}{\overline \theta_v}\delta_{i3} +f\epsilon_{ij3}u_j' +v\frac{\partial^2u_i'}{\partial x_j^2} +\frac{\partial (\overline {u_i'u_j'})}{\partial x_j} \end{align*} ∂t∂ui′+uj∂xj∂ui′+uj′∂xj∂ui+uj′∂xj∂ui′=−ρ1⋅∂xi∂p′+gθvθv′δi3+fϵij3uj′+v∂xj2∂2ui′+∂xj∂(ui′uj′)
x 方向：
∂ u ′ ∂ t + ( u ‾ ∂ u ′ ∂ x + v ‾ ∂ u ′ ∂ y + w ‾ ∂ u ′ ∂ z ) + ( u ′ ∂ u ‾ ∂ x + v ′ ∂ u ‾ ∂ y + w ′ ∂ u ‾ ∂ z ) + ( u ′ ∂ u ′ ∂ x + v ′ ∂ u ′ ∂ y + w ′ ∂ u ′ ∂ z ) = − 1 ρ ‾ ⋅ ∂ p ′ ∂ x + f v ′ + v ∂ 2 u ′ ∂ x 2 + v ∂ 2 u ′ ∂ y 2 + v ∂ 2 u ′ ∂ z 2 + ∂ ( u ′ u ′ ‾ ) ∂ x + ∂ ( u ′ v ′ ‾ ) ∂ y + ∂ ( u ′ w ′ ‾ ) ∂ z \begin{align*} \frac{\partial u'}{\partial t}+ (\overline u\frac{\partial u'}{\partial x}+ \overline v\frac{\partial u'}{\partial y}+ \overline w\frac{\partial u'}{\partial z})+ ( u'\frac{\partial \overline u}{\partial x}+ v'\frac{\partial \overline u}{\partial y}+ w'\frac{\partial \overline u}{\partial z} )+ ( u'\frac{\partial u'}{\partial x}+ v'\frac{\partial u'}{\partial y}+ w'\frac{\partial u'}{\partial z} )=\\ -\frac{1}{\overline \rho}\cdot{ \frac{\partial p'}{\partial x}}\\ +fv'\\ +v\frac{\partial^2u'}{\partial x^2}+ v\frac{\partial^2u'}{\partial y^2}+ v\frac{\partial^2u'}{\partial z^2}\\ +\frac{\partial (\overline {u'u'})}{\partial x}+ \frac{\partial (\overline {u'v'})}{\partial y}+ \frac{\partial (\overline {u'w'})}{\partial z} \end{align*} ∂t∂u′+(u∂x∂u′+v∂y∂u′+w∂z∂u′)+(u′∂x∂u+v′∂y∂u+w′∂z∂u)+(u′∂x∂u′+v′∂y∂u′+w′∂z∂u′)=−ρ1⋅∂x∂p′+fv′+v∂x2∂2u′+v∂y2∂2u′+v∂z2∂2u′+∂x∂(u′u′)+∂y∂(u′v′)+∂z∂(u′w′)
y 方向：
∂ v ′ ∂ t + ( u ‾ ∂ v ′ ∂ x + v ‾ ∂ v ′ ∂ y + w ‾ ∂ v ′ ∂ z ) + ( u ′ ∂ v ‾ ∂ x + v ′ ∂ v ‾ ∂ y + w ′ ∂ v ‾ ∂ z ) + ( u ′ ∂ v ′ ∂ x + v ′ ∂ v ′ ∂ y + w ′ ∂ v ′ ∂ z ) = − 1 ρ ‾ ⋅ ∂ p ′ ∂ y − f u ′ + v ∂ 2 v ′ ∂ x 2 + v ∂ 2 v ′ ∂ y 2 + v ∂ 2 v ′ ∂ z 2 + ∂ ( v ′ u ′ ‾ ) ∂ x + ∂ ( v ′ v ′ ‾ ) ∂ y + ∂ ( v ′ w ′ ‾ ) ∂ z \begin{align*} \frac{\partial v'}{\partial t}+ (\overline u\frac{\partial v'}{\partial x}+ \overline v\frac{\partial v'}{\partial y}+ \overline w\frac{\partial v'}{\partial z})+ ( u'\frac{\partial \overline v}{\partial x}+ v'\frac{\partial \overline v}{\partial y}+ w'\frac{\partial \overline v}{\partial z} )+ ( u'\frac{\partial v'}{\partial x}+ v'\frac{\partial v'}{\partial y}+ w'\frac{\partial v'}{\partial z} )=\\ -\frac{1}{\overline \rho}\cdot{ \frac{\partial p'}{\partial y}}\\ -fu'\\ +v\frac{\partial^2v'}{\partial x^2}+ v\frac{\partial^2v'}{\partial y^2}+ v\frac{\partial^2v'}{\partial z^2}\\ +\frac{\partial (\overline {v'u'})}{\partial x}+ \frac{\partial (\overline {v'v'})}{\partial y}+ \frac{\partial (\overline {v'w'})}{\partial z} \end{align*} ∂t∂v′+(u∂x∂v′+v∂y∂v′+w∂z∂v′)+(u′∂x∂v+v′∂y∂v+w′∂z∂v)+(u′∂x∂v′+v′∂y∂v′+w′∂z∂v′)=−ρ1⋅∂y∂p′−fu′+v∂x2∂2v′+v∂y2∂2v′+v∂z2∂2v′+∂x∂(v′u′)+∂y∂(v′v′)+∂z∂(v′w′)
z 方向：
∂ w ′ ∂ t + ( u ‾ ∂ w ′ ∂ x + v ‾ ∂ w ′ ∂ y + w ‾ ∂ w ′ ∂ z ) + ( u ′ ∂ w ‾ ∂ x + v ′ ∂ w ‾ ∂ y + w ′ ∂ w ‾ ∂ z ) + ( u ′ ∂ w ′ ∂ x + v ′ ∂ w ′ ∂ y + w ′ ∂ w ′ ∂ z ) = − 1 ρ ‾ ⋅ ∂ p ′ ∂ z + g θ v ′ θ ‾ v + v ∂ 2 w ′ ∂ x 2 + v ∂ 2 w ′ ∂ y 2 + v ∂ 2 w ′ ∂ z 2 + ∂ ( w ′ u ′ ‾ ) ∂ x + ∂ ( w ′ v ′ ‾ ) ∂ y + ∂ ( w ′ w ′ ‾ ) ∂ z \begin{align*} \frac{\partial w'}{\partial t}+ (\overline u\frac{\partial w'}{\partial x}+ \overline v\frac{\partial w'}{\partial y}+ \overline w\frac{\partial w'}{\partial z})+ ( u'\frac{\partial \overline w}{\partial x}+ v'\frac{\partial \overline w}{\partial y}+ w'\frac{\partial \overline w}{\partial z} )+ ( u'\frac{\partial w'}{\partial x}+ v'\frac{\partial w'}{\partial y}+ w'\frac{\partial w'}{\partial z} )=\\ -\frac{1}{\overline \rho}\cdot{ \frac{\partial p'}{\partial z}}\\ +g\frac{\theta_v'}{\overline \theta_v}\\ +v\frac{\partial^2w'}{\partial x^2}+ v\frac{\partial^2w'}{\partial y^2}+ v\frac{\partial^2w'}{\partial z^2}\\ +\frac{\partial (\overline {w'u'})}{\partial x}+ \frac{\partial (\overline {w'v'})}{\partial y}+ \frac{\partial (\overline {w'w'})}{\partial z} \end{align*} ∂t∂w′+(u∂x∂w′+v∂y∂w′+w∂z∂w′)+(u′∂x∂w+v′∂y∂w+w′∂z∂w)+(u′∂x∂w′+v′∂y∂w′+w′∂z∂w′)=−ρ1⋅∂z∂p′+gθvθv′+v∂x2∂2w′+v∂y2∂2w′+v∂z2∂2w′+∂x∂(w′u′)+∂y∂(w′v′)+∂z∂(w′w′)
湍流脉动速度方差得预报方程推导
对于原始式子
∂ u i ′ ∂ t + u ‾ j ∂ u i ′ ∂ x j + u j ′ ∂ u ‾ i ∂ x j + u j ′ ∂ u i ′ ∂ x j = − 1 ρ ‾ ⋅ ∂ p ′ ∂ x i + g θ v ′ θ ‾ v δ i 3 + f ϵ i j 3 u j ′ + v ∂ 2 u i ′ ∂ x j 2 + ∂ ( u i ′ u j ′ ‾ ) ∂ x j \begin{align*} \frac{\partial u_i'}{\partial t}+ \overline u_j\frac{\partial u_i'}{\partial x_j}+ u_j'\frac{\partial \overline u_i}{\partial x_j}+u_j'\frac{\partial u_i'}{\partial x_j}= -\frac{1}{\overline \rho}\cdot{ \frac{\partial p'}{\partial x_i}} +g\frac{\theta_v'}{\overline \theta_v}\delta_{i3} +f\epsilon_{ij3}u_j' +v\frac{\partial^2u_i'}{\partial x_j^2} +\frac{\partial (\overline {u_i'u_j'})}{\partial x_j} \end{align*} ∂t∂ui′+uj∂xj∂ui′+uj′∂xj∂ui+uj′∂xj∂ui′=−ρ1⋅∂xi∂p′+gθvθv′δi3+fϵij3uj′+v∂xj2∂2ui′+∂xj∂(ui′uj′)
在等式两边同时乘以 u i ′ u_i' ui′，根据微分的性质进行结合，类似
2 u i ′ ∂ u i ′ ∂ t = ∂ u i ′ 2 ∂ t 2u_i'\frac{\partial u_i'}{\partial t}=\frac{\partial u_i'^2}{\partial t} 2ui′∂t∂ui′=∂t∂ui′2
得到
∂ u i ′ 2 ∂ t + u ‾ j ∂ u ′ 2 ∂ x j + 2 u i ′ u j ′ ∂ u ‾ i ∂ x j + 2 u i ′ u j ′ ∂ u i ′ ∂ x j = − 2 u i ′ ρ ‾ ⋅ ∂ p ′ ∂ x i + 2 g θ v ′ θ ‾ v δ i j 3 u j ′ + 2 v u i ′ ∂ 2 u i ′ ∂ x j 2 + 2 u i ′ u i ′ u j ′ ‾ ∂ x j \begin{align*} \frac{\partial u_i'^2}{\partial t}+ \overline u_j\frac{\partial u'^2}{\partial x_j}+ 2u_i'u_j'\frac{\partial \overline u_i}{\partial x_j}+ 2u_i'u_j'\frac{\partial u_i'}{\partial x_j}=\\ -\frac{2u_i'}{\overline \rho}\cdot{ \frac{\partial p'}{\partial x_i}}\\ +2g\frac{\theta_v'}{\overline \theta_v}\delta_{ij3}u_j'\\ +2vu_i'\frac{\partial^2u_i'}{\partial x_j^2}\\ +2u_i'\frac{\overline{u_i'u_j'}}{\partial x_j} \end{align*} ∂t∂ui′2+uj∂xj∂u′2+2ui′uj′∂xj∂ui+2ui′uj′∂xj∂ui′=−ρ2ui′⋅∂xi∂p′+2gθvθv′δij3uj′+2vui′∂xj2∂2ui′+2ui′∂xjui′uj′
其中，左侧第四项可以表示为，其中右侧第二项的右半边为连续方程，其值为 0
2 u i ′ u j ′ ∂ u i ′ ∂ x j = ∂ ( u i ′ 2 u j ′ ) ∂ x j − u i ′ 2 ∂ u j ′ ∂ x j = ∂ ( u i ′ 2 u j ′ ) ∂ x j 【由连续方程 ∂ u j ′ ∂ x j = 0 】 2u_i'u_j'\frac{\partial u_i'}{\partial x_j}= \frac{\partial({{u_i'^2u_j'}})}{\partial x_j}- {u_i'}^2\frac{\partial u_j'}{\partial x_j}= \frac{\partial({{u_i'^2u_j'}})}{\partial x_j}【由连续方程\frac{\partial u_j'}{\partial x_j}=0】 2ui′uj′∂xj∂ui′=∂xj∂(ui′2uj′)−ui′2∂xj∂uj′=∂xj∂(ui′2uj′)【由连续方程∂xj∂uj′=0】
对于右侧第一项
− 2 u i ′ ρ ‾ ⋅ ∂ p ′ ∂ x i = − 2 ρ ‾ ( ∂ ( u i ′ p ′ ) ∂ x i − ∂ u i ′ ∂ x i p ′ ) , 其中右侧第二项为连续方程，为 0 -\frac{2u_i'}{\overline \rho}\cdot{ \frac{\partial p'}{\partial x_i}}=-\frac{2}{\overline \rho}(\frac{\partial (u_i'p')}{\partial x_i}-\frac{\partial u_i'}{\partial x_i}p'),其中右侧第二项为连续方程，为0 −ρ2ui′⋅∂xi∂p′=−ρ2(∂xi∂(ui′p′)−∂xi∂ui′p′),其中右侧第二项为连续方程，为0
因此有
− 2 u i ′ ρ ‾ ⋅ ∂ p ′ ∂ x i = − 2 ρ ‾ ⋅ ∂ ( u i ′ p ′ ) ∂ x i -\frac{2u_i'}{\overline \rho}\cdot{ \frac{\partial p'}{\partial x_i}}=-\frac{2}{\overline \rho}\cdot{ \frac{\partial (u_i'p')}{\partial x_i}} −ρ2ui′⋅∂xi∂p′=−ρ2⋅∂xi∂(ui′p′)
对于右侧第三项有
2 v u i ′ ∂ 2 u i ′ ∂ x j 2 = v ∂ 2 u i ′ 2 ∂ x j 2 − 2 v ( ∂ u i ′ ∂ x j ) 2 = v ∂ 2 u i ′ 2 ∂ x j 2 − 2 ϵ 2vu_i'\frac{\partial^2u_i'}{\partial x_j^2}= v\frac{\partial^2u_i'^2}{\partial x_j^2}-2v(\frac{\partial u_i'}{\partial x_j})^2=v\frac{\partial^2u_i'^2}{\partial x_j^2}-2\epsilon 2vui′∂xj2∂2ui′=v∂xj2∂2ui′2−2v(∂xj∂ui′)2=v∂xj2∂2ui′2−2ϵ
将上述式子代入后对取雷诺平均，同时忽略两个小项（ v ∂ 2 u i ′ 2 ‾ ∂ x j 2 , 2 u i ′ ∂ ( u i ′ u j ′ ) ‾ ∂ x j ‾ v\frac{\partial^2\overline{u_i^{'2}}}{\partial x_j^2},2\overline{u_i'\frac{\partial \overline{(u_i'u_j')}}{\partial x_j}} v∂xj2∂2ui′2,2ui′∂xj∂(ui′uj′)），得到
∂ u i ′ 2 ‾ ∂ t + u ‾ j ∂ u ′ 2 ‾ ∂ x j = − 2 ρ ‾ ⋅ ∂ ( u i ′ p ′ ) ‾ ∂ x i + 2 g θ ‾ v ( u i ′ θ v ′ ) ‾ δ i j 3 − 2 u i ′ u j ′ ‾ ∂ u i ‾ ∂ x j − ∂ ( u i ′ 2 u j ′ ‾ ) ∂ x j − 2 ϵ \begin{align*} \frac{\partial \overline {u_i'^2}}{\partial t}+ \overline u_j\frac{\partial \overline{u'^2}}{\partial x_j} =\\ -\frac{2}{\overline \rho}\cdot{ \frac{\partial\overline{(u_i'p')}}{\partial x_i}}\\ +2\frac{g}{\overline \theta_v}\overline{(u_i'\theta_v')}\delta_{ij3}\\ -2\overline{u_i'u_j'}\frac{\partial \overline{u_i}}{\partial x_j}\\ -\frac{\partial(\overline{{u_i'^2u_j'}})}{\partial x_j}\\ -2\epsilon \end{align*} ∂t∂ui′2+uj∂xj∂u′2=−ρ2⋅∂xi∂(ui′p′)+2θvg(ui′θv′)δij3−2ui′uj′∂xj∂ui−∂xj∂(ui′2uj′)−2ϵ
TKE 方程推导
将湍流脉动速度方差展开
写成分量形式 (将湍能耗散项放在了 x 方向，确保下方求和后只有 2 ϵ 2\epsilon 2ϵ)：
∂ u ′ 2 ‾ ∂ t + ∂ v ′ 2 ‾ ∂ t + ∂ w ′ 2 ‾ ∂ t + ( u ‾ ∂ u ′ 2 ‾ ∂ x + v ‾ ∂ u ′ 2 ‾ ∂ y + w ‾ ∂ u ′ 2 ‾ ∂ y ) + ( u ‾ ∂ v ′ 2 ‾ ∂ x + v ‾ ∂ v ′ 2 ‾ ∂ y + w ‾ ∂ v ′ 2 ‾ ∂ y ) + ( u ‾ ∂ v ′ 2 ‾ ∂ x + v ‾ ∂ v ′ 2 ‾ ∂ y + w ‾ ∂ v ′ 2 ‾ ∂ y ) = − 2 ρ ‾ ⋅ [ ∂ ( u ′ p ′ ) ‾ ∂ x + ∂ ( v ′ p ′ ) ‾ ∂ y + ∂ ( w ′ p ′ ) ‾ ∂ z ] + 2 g θ ‾ v ( w ′ θ v ′ ‾ ) − 2 ( u ′ u ′ ‾ ∂ u ‾ ∂ x + u ′ v ′ ‾ ∂ u ‾ ∂ y + u ′ w ′ ‾ ∂ u ‾ ∂ x + v ′ u ′ ‾ ∂ v ‾ ∂ z + v ′ v ′ ‾ ∂ v ‾ ∂ y + v ′ w ′ ‾ ∂ v ‾ ∂ z + w ′ u ′ ‾ ∂ w ‾ ∂ z + w ′ v ′ ‾ ∂ w ‾ ∂ y + w ′ w ′ ‾ ∂ w ‾ ∂ z ) − [ ∂ ( u ′ 2 u ′ ) ‾ ∂ x + ∂ ( u ′ 2 v ′ ‾ ) ∂ y + ∂ ( u ′ 2 w ′ ‾ ) ∂ z + ∂ ( v ′ 2 u ′ ) ‾ ∂ x + ∂ ( v ′ 2 v ′ ‾ ) ∂ y + ∂ ( v ′ 2 w ′ ‾ ) ∂ z + ∂ ( w ′ 2 u ′ ) ‾ ∂ x + ∂ ( w ′ 2 v ′ ‾ ) ∂ y + ∂ ( w ′ 2 w ′ ‾ ) ∂ z ] − 2 ϵ \begin{align*} \frac{\partial \overline{{u^{'}}^2}}{\partial t}+ \frac{\partial \overline{{v^{'}}^2}}{\partial t}+ \frac{\partial \overline{{w^{'}}^2}}{\partial t} \\ +(\overline u\frac{\partial \overline{{u^{'}}^2}}{\partial x}+ \overline v\frac{\partial \overline{{u^{'}}^2}}{\partial y}+ \overline w\frac{\partial \overline{{u^{'}}^2}}{\partial y})+(\overline u\frac{\partial \overline{{v^{'}}^2}}{\partial x}+ \overline v\frac{\partial \overline{{v^{'}}^2}}{\partial y}+ \overline w\frac{\partial \overline{{v^{'}}^2}}{\partial y})+ (\overline u\frac{\partial \overline{{v^{'}}^2}}{\partial x}+ \overline v\frac{\partial \overline{{v^{'}}^2}}{\partial y}+ \overline w\frac{\partial \overline{{v^{'}}^2}}{\partial y}) =\\ -\frac{2}{\overline \rho}\cdot{[ \frac{\partial\overline{(u'p')}}{\partial x}+ \frac{\partial\overline{(v'p')}}{\partial y}+ \frac{\partial\overline{(w'p')}}{\partial z}]}\\ +2\frac{g}{\overline \theta_v}(\overline {w'\theta_v^{'}})\\ -2(\overline{u'u'}\frac{\partial \overline u}{\partial x}+ \overline{u'v'}\frac{\partial \overline u}{\partial y}+ \overline{u'w'}\frac{\partial \overline u}{\partial x}+ \overline{v'u'}\frac{\partial \overline v}{\partial z}+ \overline{v'v'}\frac{\partial \overline v}{\partial y}+ \overline{v'w'}\frac{\partial \overline v}{\partial z}+ \overline{w'u'}\frac{\partial \overline w}{\partial z}+ \overline{w'v'}\frac{\partial \overline w}{\partial y}+ \overline{w'w'}\frac{\partial \overline w}{\partial z})\\ -[\frac{\partial (\overline{{u^{'}}^2u')}}{\partial x}+ \frac{\partial (\overline{{u^{'}}^2v'})}{\partial y}+ \frac{\partial (\overline{{u^{'}}^2w'})}{\partial z}+ \frac{\partial (\overline{{v^{'}}^2u')}}{\partial x}+ \frac{\partial (\overline{{v^{'}}^2v'})}{\partial y}+ \frac{\partial (\overline{{v^{'}}^2w'})}{\partial z}+ \frac{\partial (\overline{{w^{'}}^2u')}}{\partial x}+ \frac{\partial (\overline{{w^{'}}^2v'})}{\partial y}+ \frac{\partial (\overline{{w^{'}}^2w'})}{\partial z}] \\ -2\epsilon \end{align*} ∂t∂u′2+∂t∂v′2+∂t∂w′2+(u∂x∂u′2+v∂y∂u′2+w∂y∂u′2)+(u∂x∂v′2+v∂y∂v′2+w∂y∂v′2)+(u∂x∂v′2+v∂y∂v′2+w∂y∂v′2)=−ρ2⋅[∂x∂(u′p′)+∂y∂(v′p′)+∂z∂(w′p′)]+2θvg(w′θv′)−2(u′u′∂x∂u+u′v′∂y∂u+u′w′∂x∂u+v′u′∂z∂v+v′v′∂y∂v+v′w′∂z∂v+w′u′∂z∂w+w′v′∂y∂w+w′w′∂z∂w)−[∂x∂(u′2u′)+∂y∂(u′2v′)+∂z∂(u′2w′)+∂x∂(v′2u′)+∂y∂(v′2v′)+∂z∂(v′2w′)+∂x∂(w′2u′)+∂y∂(w′2v′)+∂z∂(w′2w′)]−2ϵ
将 (1)除以 1/2，整理后得到 (2)
1 2 [ ∂ ( u ′ 2 ‾ + v ′ 2 ‾ + w ′ 2 ‾ ) ∂ t u ‾ ∂ ( u ′ 2 ‾ + v ′ 2 ‾ + w ′ 2 ‾ ) ∂ x + v ‾ ∂ ( u ′ 2 ‾ + v ′ 2 ‾ + w ′ 2 ‾ ) ∂ y + w ‾ ∂ ( u ′ 2 ‾ + v ′ 2 ‾ + w ′ 2 ‾ ) ∂ z ] = − 1 ρ ‾ ⋅ [ ∂ ( u ′ p ′ ) ‾ ∂ x + ∂ ( v ′ p ′ ) ‾ ∂ y + ∂ ( w ′ p ′ ) ‾ ∂ z ] + g θ ‾ v ( w ′ θ v ′ ‾ ) − ( u ′ u ′ ‾ ∂ u ‾ ∂ x + u ′ v ′ ‾ ∂ u ‾ ∂ y + u ′ w ′ ‾ ∂ u ‾ ∂ x + v ′ u ′ ‾ ∂ v ‾ ∂ z + v ′ v ′ ‾ ∂ v ‾ ∂ y + v ′ w ′ ‾ ∂ v ‾ ∂ z + w ′ u ′ ‾ ∂ w ‾ ∂ z + w ′ v ′ ‾ ∂ w ‾ ∂ y + w ′ w ′ ‾ ∂ w ‾ ∂ z ) − 1 2 [ ∂ u ′ ( u ′ 2 + v ′ 2 + w ′ 2 ) ‾ ∂ x + 1 2 [ ∂ v ′ ( u ′ 2 + v ′ 2 + w ′ 2 ) ‾ ∂ y + 1 2 [ ∂ w ′ ( u ′ 2 + v ′ 2 + w ′ 2 ) ‾ ∂ z ] − ϵ \begin{align*} \frac{1}{2}[ \frac{\partial (\overline {{u'}^2}+\overline {{v'}^2}+\overline {{w'}^2})}{\partial t}\\ \overline{u}\frac{\partial (\overline {{u'}^2}+\overline {{v'}^2}+\overline {{w'}^2})}{\partial x}+ \overline{v}\frac{\partial (\overline {{u'}^2}+\overline {{v'}^2}+\overline {{w'}^2})}{\partial y}+ \overline{w}\frac{\partial (\overline {{u'}^2}+\overline {{v'}^2}+\overline {{w'}^2})}{\partial z} ]=\\ -\frac{1}{\overline \rho}\cdot{[ \frac{\partial\overline{(u'p')}}{\partial x}+ \frac{\partial\overline{(v'p')}}{\partial y}+ \frac{\partial\overline{(w'p')}}{\partial z}]}\\ +\frac{g}{\overline \theta_v}(\overline {w'\theta_v^{'}})\\ -(\overline{u'u'}\frac{\partial \overline u}{\partial x}+ \overline{u'v'}\frac{\partial \overline u}{\partial y}+ \overline{u'w'}\frac{\partial \overline u}{\partial x}+ \overline{v'u'}\frac{\partial \overline v}{\partial z}+ \overline{v'v'}\frac{\partial \overline v}{\partial y}+ \overline{v'w'}\frac{\partial \overline v}{\partial z}+ \overline{w'u'}\frac{\partial \overline w}{\partial z}+ \overline{w'v'}\frac{\partial \overline w}{\partial y}+ \overline{w'w'}\frac{\partial \overline w}{\partial z})-\\ \frac{1}{2}[\frac {\partial \overline{ {u'}({{u'}^2}+ {{v'}^2}+{{w'}^2})}}{\partial x}+ \frac{1}{2}[\frac {\partial \overline{ {v'}({{u'}^2}+ {{v'}^2}+{{w'}^2})}}{\partial y}+ \frac{1}{2}[\frac {\partial \overline{ {w'}({{u'}^2}+ {{v'}^2}+{{w'}^2})}}{\partial z} ]- \epsilon \end{align*} 21[∂t∂(u′2+v′2+w′2)u∂x∂(u′2+v′2+w′2)+v∂y∂(u′2+v′2+w′2)+w∂z∂(u′2+v′2+w′2)]=−ρ1⋅[∂x∂(u′p′)+∂y∂(v′p′)+∂z∂(w′p′)]+θvg(w′θv′)−(u′u′∂x∂u+u′v′∂y∂u+u′w′∂x∂u+v′u′∂z∂v+v′v′∂y∂v+v′w′∂z∂v+w′u′∂z∂w+w′v′∂y∂w+w′w′∂z∂w)−21[∂x∂u′(u′2+v′2+w′2)+21[∂y∂v′(u′2+v′2+w′2)+21[∂z∂w′(u′2+v′2+w′2)]−ϵ 由平均湍流动能表达式 (3)
e ‾ = 1 2 ( u ′ 2 ‾ + v ′ 2 ‾ + w ′ 2 ‾ ) \overline e=\frac{1}{2}(\overline {{u'}^2}+\overline {{v'}^2}+\overline {{w'}^2}) e=21(u′2+v′2+w′2)
由于 u, v, w 三者是独立的，因此对平均湍流动能得偏导数等于 u ′ 2 ‾ , v ′ 2 ‾ , w ′ 2 ‾ \overline {{u'}^2},\overline {{v'}^2},\overline {{w'}^2} u′2,v′2,w′2 各自的偏导数之和除以 2，因此将该式代入 (2) 中得到
∂ e ‾ ∂ t + u ‾ ∂ e ‾ ∂ x + v ‾ ∂ e ‾ ∂ y + w ‾ ∂ e ‾ ∂ z = − 1 ρ ‾ ⋅ [ ∂ ( u ′ p ′ ) ‾ ∂ x + ∂ ( v ′ p ′ ) ‾ ∂ y + ∂ ( w ′ p ′ ) ‾ ∂ z ] + g θ ‾ v ( w ′ θ v ′ ‾ ) − ( u ′ u ′ ‾ ∂ u ‾ ∂ x + u ′ v ′ ‾ ∂ u ‾ ∂ y + u ′ w ′ ‾ ∂ u ‾ ∂ x + v ′ u ′ ‾ ∂ v ‾ ∂ z + v ′ v ′ ‾ ∂ v ‾ ∂ y + v ′ w ′ ‾ ∂ v ‾ ∂ z + w ′ u ′ ‾ ∂ w ‾ ∂ z + w ′ v ′ ‾ ∂ w ‾ ∂ y + w ′ w ′ ‾ ∂ w ‾ ∂ z ) − [ ∂ ( u ′ e ) ‾ ∂ x + ∂ ( v ′ e ) ‾ ∂ y + ∂ ( w ′ e ) ‾ ∂ z ] − ϵ \begin{align*} \frac{\partial \overline e}{\partial t}+ \overline{u}\frac{\partial \overline e}{\partial x}+ \overline{v}\frac{\partial \overline e}{\partial y}+ \overline{w}\frac{\partial \overline e}{\partial z} =\\ -\frac{1}{\overline \rho}\cdot{[ \frac{\partial\overline{(u'p')}}{\partial x}+ \frac{\partial\overline{(v'p')}}{\partial y}+ \frac{\partial\overline{(w'p')}}{\partial z}]}\\ +\frac{g}{\overline \theta_v}(\overline {w'\theta_v^{'}})\\ -(\overline{u'u'}\frac{\partial \overline u}{\partial x}+ \overline{u'v'}\frac{\partial \overline u}{\partial y}+ \overline{u'w'}\frac{\partial \overline u}{\partial x}+ \overline{v'u'}\frac{\partial \overline v}{\partial z}+ \overline{v'v'}\frac{\partial \overline v}{\partial y}+ \overline{v'w'}\frac{\partial \overline v}{\partial z}+ \overline{w'u'}\frac{\partial \overline w}{\partial z}+ \overline{w'v'}\frac{\partial \overline w}{\partial y}+ \overline{w'w'}\frac{\partial \overline w}{\partial z})\\ -[\frac{\partial \overline {(u' e)}}{\partial x}+ \frac{\partial \overline {(v' e)}}{\partial y}+ \frac{\partial \overline {(w' e)}}{\partial z} ]-\epsilon\\ \end{align*} ∂t∂e+u∂x∂e+v∂y∂e+w∂z∂e=−ρ1⋅[∂x∂(u′p′)+∂y∂(v′p′)+∂z∂(w′p′)]+θvg(w′θv′)−(u′u′∂x∂u+u′v′∂y∂u+u′w′∂x∂u+v′u′∂z∂v+v′v′∂y∂v+v′w′∂z∂v+w′u′∂z∂w+w′v′∂y∂w+w′w′∂z∂w)−[∂x∂(u′e)+∂y∂(v′e)+∂z∂(w′e)]−ϵ
即
∂ e ‾ ∂ t + u j ‾ ∂ e ‾ ∂ x j = − 1 ρ ‾ ⋅ ∂ ( u i ′ p ′ ) ‾ ∂ x i + g θ ‾ v ( u i ′ θ v ′ ‾ ) δ i 3 − u i ′ u j ′ ‾ ∂ u i ‾ ∂ x j − ∂ ( u j ′ e ) ‾ ∂ x j − ϵ \begin{align*} \frac{\partial \overline e}{\partial t}+ \overline{u_j}\frac{\partial \overline e}{\partial x_j} =\\ -\frac{1}{\overline \rho}\cdot{ \frac{\partial\overline{(u_i'p')}}{\partial x_i} }\\ +\frac{g}{\overline \theta_v}(\overline {u_i'\theta_v^{'}})\delta_{i3}\\ -\overline{u_i'u_j'}\frac{\partial \overline {u_i}}{\partial x_j} \\ -\frac{\partial \overline {(u_j' e)}}{\partial x_j}\\ -\epsilon\\ \end{align*} ∂t∂e+uj∂xj∂e=−ρ1⋅∂xi∂(ui′p′)+θvg(ui′θv′)δi3−ui′uj′∂xj∂ui−∂xj∂(uj′e)−ϵ