湍流 气体速度场的大涡模拟。大涡模拟 (Large eddy simulation, LES ) 是用于计算流体动力学 的湍流 数学模型。它最初由约瑟夫·斯玛格林斯基于 1963 年提出，用于模拟大气气流， [ 1] [ 2] 燃烧 、 [ 3] [ 4] [ 5]
通过数值求解Navier-Stokes 方程 来模拟湍流需要解决非常广泛的时间和长度尺度，所有这些都会影响流场。这样的分辨率可以通过直接数值模拟(DNS) 来实现，但 DNS 的计算成本很高，难以模拟具有复杂几何形状或流动配置的实际工程系统，例如湍流喷射、泵、车辆和起落架。
LES 背后的主要思想是通过 Navier-Stokes 方程的低通滤波 忽略最小长度尺度来降低计算成本。这种可以被视为时间和空间平均的低通滤波，有效地从数值解中去除了小尺度信息。然而，这些信息并不是无关紧要的，它对流场的影响必须被建模，由此衍生小尺度可以发挥重要作用的问题的活跃研究领域，例如近壁流， [ 6] [ 7] [ 3] [ 8]
由均匀衰减湍流的直接数值模拟(DNS) 产生的速度场。域大小为 L 3 {\displaystyle L^{3}} 使用盒式过滤器过滤的相同 DNS 速度字段和 Δ = L / 32 {\displaystyle \Delta =L/32} 使用盒式过滤器过滤的相同 DNS 速度字段和 Δ = L / 16 {\displaystyle \Delta =L/16} LES 滤波器可应用于空间和时间场 ϕ ( x , t ) {\displaystyle \phi ({\boldsymbol {x}},t)} [ 9] [ 10]
ϕ ( x , t ) ¯ = ∫ − ∞ ∞ ∫ − ∞ ∞ ϕ ( r , τ ) G ( x − r , t − τ ) d τ d r {\displaystyle {\overline {\phi ({\boldsymbol {x}},t)}}=\displaystyle {\int _{-\infty }^{\infty }}\int _{-\infty }^{\infty }\phi ({\boldsymbol {r}},\tau )G({\boldsymbol {x}}-{\boldsymbol {r}},t-\tau )d\tau d{\boldsymbol {r}}} 其中 G {\displaystyle G}
ϕ ¯ = G ⋆ ϕ . {\displaystyle {\overline {\phi }}=G\star \phi .} 过滤器内核 G {\displaystyle G} Δ {\displaystyle \Delta } τ c {\displaystyle \tau _{c}} ϕ {\displaystyle \phi }
ϕ = ϕ ¯ + ϕ ′ . {\displaystyle \phi ={\bar {\phi }}+\phi ^{\prime }.} 需要注意的是，大涡模拟滤波操作不满足雷诺算子的性质。
LES的控制方程是通过过滤控制流场的偏微分方程 得到的 ρ u ( x , t ) {\displaystyle \rho {\boldsymbol {u}}({\boldsymbol {x}},t)}
对于不可压缩流动，对连续性方程 和 Navier-Stokes 方程进行滤波，得到滤波后的不可压缩连续性方程，
∂ u i ¯ ∂ x i = 0 {\displaystyle {\frac {\partial {\bar {u_{i}}}}{\partial x_{i}}}=0} 和过滤后的 Navier-Stokes 方程，
∂ u i ¯ ∂ t + ∂ ∂ x j ( u i u j ¯ ) = − 1 ρ ∂ p ¯ ∂ x i + ν ∂ ∂ x j ( ∂ u i ¯ ∂ x j + ∂ u j ¯ ∂ x i ) = − 1 ρ ∂ p ¯ ∂ x i + 2 ν ∂ ∂ x j S ¯ i j , {\displaystyle {\frac {\partial {\bar {u_{i}}}}{\partial t}}+{\frac {\partial }{\partial x_{j}}}\left({\overline {u_{i}u_{j}}}\right)=-{\frac {1}{\rho }}{\frac {\partial {\overline {p}}}{\partial x_{i}}}+\nu {\frac {\partial }{\partial x_{j}}}\left({\frac {\partial {\bar {u_{i}}}}{\partial x_{j}}}+{\frac {\partial {\bar {u_{j}}}}{\partial x_{i}}}\right)=-{\frac {1}{\rho }}{\frac {\partial {\overline {p}}}{\partial x_{i}}}+2\nu {\frac {\partial }{\partial x_{j}}}{\bar {S}}_{ij},} 其中 p ¯ {\displaystyle {\bar {p}}} S ¯ i j {\displaystyle {\bar {S}}_{ij}} 非线性 滤波平流项 u i u j ¯ {\displaystyle {\overline {u_{i}u_{j}}}}
过滤后的平流项可以按照莱昂纳德(1975)[ 11]
u i u j ¯ = τ i j + u ¯ i u ¯ j {\displaystyle {\overline {u_{i}u_{j}}}=\tau _{ij}+{\overline {u}}_{i}{\overline {u}}_{j}} 其中 τ i j {\displaystyle \tau _{ij}}
∂ u i ¯ ∂ t + ∂ ∂ x j ( u ¯ i u ¯ j ) = − 1 ρ ∂ p ¯ ∂ x i + 2 ν ∂ ∂ x j S ¯ i j − ∂ τ i j ∂ x j {\displaystyle {\frac {\partial {\bar {u_{i}}}}{\partial t}}+{\frac {\partial }{\partial x_{j}}}\left({\overline {u}}_{i}{\overline {u}}_{j}\right)=-{\frac {1}{\rho }}{\frac {\partial {\overline {p}}}{\partial x_{i}}}+2\nu {\frac {\partial }{\partial x_{j}}}{\bar {S}}_{ij}-{\frac {\partial \tau _{ij}}{\partial x_{j}}}} τ i j {\displaystyle \tau _{ij}} τ i j = L i j + C i j + R i j {\displaystyle \tau _{ij}=L_{ij}+C_{ij}+R_{ij}} L i j = u ¯ i u ¯ j ¯ − u ¯ i u ¯ j {\displaystyle L_{ij}={\overline {{\bar {u}}_{i}{\bar {u}}_{j}}}-{\bar {u}}_{i}{\bar {u}}_{j}} R i j = u i ′ u j ′ ¯ {\displaystyle R_{ij}={\overline {u_{i}^{\prime }u_{j}^{\prime }}}} C i j = u ¯ i u j ′ ¯ + u ¯ j u i ′ ¯ {\displaystyle C_{ij}={\overline {{\bar {u}}_{i}u_{j}^{\prime }}}+{\overline {{\bar {u}}_{j}u_{i}^{\prime }}}} [ 12] [ 11] τ i j {\displaystyle \tau _{ij}} τ i j {\displaystyle \tau _{ij}}
被动标量 ϕ {\displaystyle \phi }
∂ ϕ ¯ ∂ t + ∂ ∂ x j ( u ¯ j ϕ ¯ ) = ∂ J ϕ ¯ ∂ x j + ∂ q j ∂ x j {\displaystyle {\frac {\partial {\overline {\phi }}}{\partial t}}+{\frac {\partial }{\partial x_{j}}}\left({\overline {u}}_{j}{\overline {\phi }}\right)={\frac {\partial {\overline {J_{\phi }}}}{\partial x_{j}}}+{\frac {\partial q_{j}}{\partial x_{j}}}} 其中 J ϕ {\displaystyle J_{\phi }} ϕ {\displaystyle \phi } q j {\displaystyle q_{j}} ϕ {\displaystyle \phi } J ϕ ¯ {\displaystyle {\overline {J_{\phi }}}} J ϕ = D ϕ ∂ ϕ ∂ x i {\displaystyle J_{\phi }=D_{\phi }{\frac {\partial \phi }{\partial x_{i}}}} q j {\displaystyle q_{j}} τ i j {\displaystyle \tau _{ij}}
q j = ϕ ¯ u ¯ j − ϕ u j ¯ {\displaystyle q_{j}={\bar {\phi }}{\overline {u}}_{j}-{\overline {\phi u_{j}}}} 并且可以类似地分解为各种尺度之间相互作用的贡献。这种子过滤器通量也需要子过滤器模型。
使用爱因斯坦求和约定 ，笛卡尔坐标中不可压缩流体的 Navier-Stokes 方程为
∂ u i ∂ x i = 0 {\displaystyle {\frac {\partial u_{i}}{\partial x_{i}}}=0} ∂ u i ∂ t + ∂ u i u j ∂ x j = − 1 ρ ∂ p ∂ x i + ν ∂ 2 u i ∂ x j ∂ x j . {\displaystyle {\frac {\partial u_{i}}{\partial t}}+{\frac {\partial u_{i}u_{j}}{\partial x_{j}}}=-{\frac {1}{\rho }}{\frac {\partial p}{\partial x_{i}}}+\nu {\frac {\partial ^{2}u_{i}}{\partial x_{j}\partial x_{j}}}.} 过滤动量方程导致
∂ u i ∂ t ¯ + ∂ u i u j ∂ x j ¯ = − 1 ρ ∂ p ∂ x i ¯ + ν ∂ 2 u i ∂ x j ∂ x j ¯ . {\displaystyle {\overline {\frac {\partial u_{i}}{\partial t}}}+{\overline {\frac {\partial u_{i}u_{j}}{\partial x_{j}}}}=-{\overline {{\frac {1}{\rho }}{\frac {\partial p}{\partial x_{i}}}}}+{\overline {\nu {\frac {\partial ^{2}u_{i}}{\partial x_{j}\partial x_{j}}}}}.} 如果我们假设过滤和微分对等，那么
∂ u i ¯ ∂ t + ∂ u i u j ∂ x j ¯ = − 1 ρ ∂ p ¯ ∂ x i + ν ∂ 2 u i ¯ ∂ x j ∂ x j . {\displaystyle {\frac {\partial {\bar {u_{i}}}}{\partial t}}+{\overline {\frac {\partial u_{i}u_{j}}{\partial x_{j}}}}=-{\frac {1}{\rho }}{\frac {\partial {\bar {p}}}{\partial x_{i}}}+\nu {\frac {\partial ^{2}{\bar {u_{i}}}}{\partial x_{j}\partial x_{j}}}.} 这个方程模拟了过滤变量的时间变化 u i ¯ {\displaystyle {\bar {u_{i}}}} u i {\displaystyle u_{i}} ∂ u i u j ∂ x j ¯ {\displaystyle {\overline {\frac {\partial u_{i}u_{j}}{\partial x_{j}}}}} ∂ u i ¯ u j ¯ ∂ x j {\displaystyle {\frac {\partial {\bar {u_{i}}}{\bar {u_{j}}}}{\partial x_{j}}}}
∂ u i ¯ ∂ t + ∂ u i ¯ u j ¯ ∂ x j = − 1 ρ ∂ p ¯ ∂ x i + ν ∂ 2 u i ¯ ∂ x j ∂ x j − ( ∂ u i u j ∂ x j ¯ − ∂ u i ¯ u j ¯ ∂ x j ) . {\displaystyle {\frac {\partial {\bar {u_{i}}}}{\partial t}}+{\frac {\partial {\bar {u_{i}}}{\bar {u_{j}}}}{\partial x_{j}}}=-{\frac {1}{\rho }}{\frac {\partial {\bar {p}}}{\partial x_{i}}}+\nu {\frac {\partial ^{2}{\bar {u_{i}}}}{\partial x_{j}\partial x_{j}}}-\left({\overline {\frac {\partial u_{i}u_{j}}{\partial x_{j}}}}-{\frac {\partial {\bar {u_{i}}}{\bar {u_{j}}}}{\partial x_{j}}}\right).} 令 τ i j = u i u j ¯ − u ¯ i u ¯ j {\displaystyle \tau _{ij}={\overline {u_{i}u_{j}}}-{\bar {u}}_{i}{\bar {u}}_{j}}
∂ u i ¯ ∂ t + u j ¯ ∂ u i ¯ ∂ x j = − 1 ρ ∂ p ¯ ∂ x i + ν ∂ 2 u i ¯ ∂ x j ∂ x j − ∂ τ i j ∂ x j . {\displaystyle {\frac {\partial {\bar {u_{i}}}}{\partial t}}+{\bar {u_{j}}}{\frac {\partial {\bar {u_{i}}}}{\partial x_{j}}}=-{\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 \tau _{ij}}{\partial x_{j}}}.} 对于可压缩流动的控制方程，每个方程都从质量守恒开始过滤。这给出了：
∂ ρ ¯ ∂ t + ∂ u i ρ ¯ ∂ x i = 0 {\displaystyle {\frac {\partial {\overline {\rho }}}{\partial t}}+{\frac {\partial {\overline {u_{i}\rho }}}{\partial x_{i}}}=0} 这导致了一个额外的子过滤器项。然而，希望避免必须对质量守恒方程的子过滤器尺度进行建模。出于这个原因，Favre [ 13] ϕ {\displaystyle \phi }
ϕ ~ = ρ ϕ ¯ ρ ¯ {\displaystyle {\tilde {\phi }}={\frac {\overline {\rho \phi }}{\overline {\rho }}}} 在不可压缩的限制下，它变成了正常的过滤操作。这使得质量守恒方程化为：
∂ ρ ¯ ∂ t + ∂ ρ ¯ u i ~ ∂ x i = 0. {\displaystyle {\frac {\partial {\overline {\rho }}}{\partial t}}+{\frac {\partial {\overline {\rho }}{\tilde {u_{i}}}}{\partial x_{i}}}=0.} 然后可以将此概念扩展到编写可压缩流的 Favre 过滤动量方程。根据弗雷曼： [ 14]
∂ ρ ¯ u i ~ ∂ t + ∂ ρ ¯ u i ~ u j ~ ∂ x j + ∂ p ¯ ∂ x i − ∂ σ ~ i j ∂ x j = − ∂ ρ ¯ τ i j r ∂ x j + ∂ ∂ x j ( σ ¯ i j − σ ~ i j ) {\displaystyle {\frac {\partial {\overline {\rho }}{\tilde {u_{i}}}}{\partial t}}+{\frac {\partial {\overline {\rho }}{\tilde {u_{i}}}{\tilde {u_{j}}}}{\partial x_{j}}}+{\frac {\partial {\overline {p}}}{\partial x_{i}}}-{\frac {\partial {\tilde {\sigma }}_{ij}}{\partial x_{j}}}=-{\frac {\partial {\overline {\rho }}\tau _{ij}^{r}}{\partial x_{j}}}+{\frac {\partial }{\partial x_{j}}}\left({\overline {\sigma }}_{ij}-{\tilde {\sigma }}_{ij}\right)} 其中 σ i j {\displaystyle \sigma _{ij}}
σ i j = 2 μ ( T ) S i j − 2 3 μ ( T ) δ i j S k k {\displaystyle \sigma _{ij}=2\mu (T)S_{ij}-{\frac {2}{3}}\mu (T)\delta _{ij}S_{kk}} ∂ ∂ x j ( σ ¯ i j − σ ~ i j ) {\displaystyle {\frac {\partial }{\partial x_{j}}}\left({\overline {\sigma }}_{ij}-{\tilde {\sigma }}_{ij}\right)} μ ( T ) {\displaystyle \mu (T)} T ~ {\displaystyle {\tilde {T}}} τ i j r = u i ⋅ u j ~ − u i ~ u j ~ {\displaystyle \tau _{ij}^{r}={\widetilde {u_{i}\cdot u_{j}}}-{\tilde {u_{i}}}{\tilde {u_{j}}}}
以此类推，莱昂纳德分解也可以写成滤波三重积的残余应力张量 ρ ϕ ψ ¯ {\displaystyle {\overline {\rho \phi \psi }}} ρ ¯ ϕ ψ ~ {\displaystyle {\overline {\rho }}{\widetilde {\phi \psi }}} ϕ {\displaystyle \phi } ψ {\displaystyle \psi } ϕ ~ {\displaystyle {\tilde {\phi }}} ψ ~ {\displaystyle {\tilde {\psi }}} u i u j ¯ {\displaystyle {\overline {u_{i}u_{j}}}} ρ ¯ ( ϕ ψ ~ − ϕ ~ ψ ~ ) {\displaystyle {\overline {\rho }}\left({\widetilde {\phi \psi }}-{\tilde {\phi }}{\tilde {\psi }}\right)} L i j {\displaystyle L_{ij}} C i j {\displaystyle C_{ij}} R i j {\displaystyle R_{ij}} [ 15]
除了过滤的质量和动量方程之外，过滤动能方程可以提供额外的见解。可以过滤动能场以产生总过滤动能：
E ¯ = 1 2 u i u i ¯ {\displaystyle {\overline {E}}={\frac {1}{2}}{\overline {u_{i}u_{i}}}} 过滤后的总动能可以分解为两项：过滤后的速度场的动能 E f {\displaystyle E_{f}}
E f = 1 2 u i ¯ u i ¯ {\displaystyle E_{f}={\frac {1}{2}}{\overline {u_{i}}}\,{\overline {u_{i}}}} 和剩余动能 k r {\displaystyle k_{r}}
k r = 1 2 u i u i ¯ − 1 2 u i ¯ u i ¯ = 1 2 τ i i r {\displaystyle k_{r}={\frac {1}{2}}{\overline {u_{i}u_{i}}}-{\frac {1}{2}}{\overline {u_{i}}}\,{\overline {u_{i}}}={\frac {1}{2}}\tau _{ii}^{r}} 这样 E ¯ = E f + k r {\displaystyle {\overline {E}}=E_{f}+k_{r}}
守恒方程为 E f {\displaystyle E_{f}} u i ¯ {\displaystyle {\overline {u_{i}}}}
∂ E f ∂ t + u j ¯ ∂ E f ∂ x j + 1 ρ ∂ u i ¯ p ¯ ∂ x i + ∂ u i ¯ τ i j r ∂ x j − 2 ν ∂ u i ¯ S i j ¯ ∂ x j = − ϵ f − Π {\displaystyle {\frac {\partial E_{f}}{\partial t}}+{\overline {u_{j}}}{\frac {\partial E_{f}}{\partial x_{j}}}+{\frac {1}{\rho }}{\frac {\partial {\overline {u_{i}}}{\bar {p}}}{\partial x_{i}}}+{\frac {\partial {\overline {u_{i}}}\tau _{ij}^{r}}{\partial x_{j}}}-2\nu {\frac {\partial {\overline {u_{i}}}{\bar {S_{ij}}}}{\partial x_{j}}}=-\epsilon _{f}-\Pi } 其中 ϵ f = 2 ν S i j ¯ S i j ¯ {\displaystyle \epsilon _{f}=2\nu {\bar {S_{ij}}}{\bar {S_{ij}}}} Π = − τ i j r S i j ¯ {\displaystyle \Pi =-\tau _{ij}^{r}{\bar {S_{ij}}}}
左边的项代表运输，右边的项是耗散动能的汇项。 [ 9]
Π {\displaystyle \Pi } Π {\displaystyle \Pi } Π {\displaystyle \Pi } 或 消极的，这意味着它也可以作为源项 E f {\displaystyle E_{f}} 反向散射 （同样，能量从已解析到未解析尺度的传递称为前向散射 ）。 [ 16]
大涡模拟涉及使用计算流体动力学 求解离散滤波控制方程。 LES 从域大小中解析尺度 L {\displaystyle L} Δ {\displaystyle \Delta } [ 9] Δ x {\displaystyle \Delta x} u ¯ ( x ) {\displaystyle {\overline {u}}({\boldsymbol {x}})} [ 17] Δ {\displaystyle \Delta } Δ x {\displaystyle \Delta x} [ 18]
大涡模拟中的滤波操作可以是隐式的，也可以是显式的。隐式过滤认识到子过滤器比例模型将以与许多数值方案相同的方式消散。通过这种方式，可以假设网格或数值离散化方案是 LES 低通滤波器。虽然这充分利用了网格分辨率，并消除了计算子滤波器比例模型项的计算成本，但很难确定与一些数值问题相关的 LES 滤波器的形状。此外，截断误差也可能成为问题。 [ 19]
在显式滤波中， LES 滤波器应用于离散的 Navier-Stokes 方程，提供明确定义的滤波器形状并减少截断误差。然而，显式过滤比隐式过滤需要更精细的网格，并且计算量与 ( Δ x ) 4 {\displaystyle (\Delta x)^{4}} [ 10]
入口边界条件对LES的精度影响很大，对于LES入口条件的处理是一个复杂的问题。理论上，一个良好的 LES 边界条件应包含以下特征： [ 20]
(1) 提供准确的流动特性信息，即速度和湍流；
(2) 满足 Navier-Stokes 方程和其他物理；
(3)易于实施和适应不同情况。
目前，为 LES 生成入口条件的方法大致分为 Tabor 等人分类的两类： [ 21]
产生湍流入口的第一种方法是根据具体情况合成它们，例如傅里叶技术、原理正交分解（POD）和涡流方法。合成技术试图在入口处构建具有合适的类湍流特性的湍流场，并使其易于指定湍流参数，例如湍流动能和湍流耗散率。此外，使用随机数生成的入口条件在计算上并不昂贵。然而，该方法存在一个严重的缺陷。合成的湍流不满足由 Navier-Stokes 方程控制的流体流动的物理结构。 [ 20]
第二种方法涉及一个单独的前体计算，以生成一个湍流数据库，该数据库可以引入到入口处的主要计算中。数据库（有时称为“库”）可以通过多种方式生成，例如循环域、预先准备好的库和内部映射。然而，前驱体模拟产生湍流流入的方法需要很大的计算能力。
研究人员检查了各种合成和前体计算的应用，发现入口湍流越真实，LES 预测结果就越准确。 [ 20]
为了讨论未解析尺度的建模，首先必须对未解析尺度进行分类。它们分为两组：已解决的子过滤器尺度 (SFS) 和子网格尺度 (SGS)。
解析的子滤波器尺度表示波数大于截止波数的尺度 k c {\displaystyle k_{c}}
子网格比例是小于截止滤波器宽度 Δ {\displaystyle \Delta }
如果没有普遍有效的湍流描述，则在构建和应用 SGS 模型时必须利用经验信息，并辅以伽利略不变性[ 9] [ 22] 功能模型 ，第二类是结构模型 。一些模型可能被归类为两者。
功能模型比结构模型更简单，只关注以物理上正确的速率耗散能量。这些基于人工涡流粘度方法，其中湍流的影响集中在湍流粘度中。该方法将亚网格尺度上的动能耗散视为类似于分子扩散。在这种情况下，偏斜部分 τ i j {\displaystyle \tau _{ij}}
τ i j r − 1 3 τ k k δ i j = − 2 ν t S ¯ i j {\displaystyle \tau _{ij}^{r}-{\frac {1}{3}}\tau _{kk}\delta _{ij}=-2\nu _{\mathrm {t} }{\bar {S}}_{ij}} 其中 ν t {\displaystyle \nu _{\mathrm {t} }} S ¯ i j = 1 2 ( ∂ u ¯ i ∂ x j + ∂ u ¯ j ∂ x i ) {\displaystyle {\bar {S}}_{ij}={\frac {1}{2}}\left({\frac {\partial {\bar {u}}_{i}}{\partial x_{j}}}+{\frac {\partial {\bar {u}}_{j}}{\partial x_{i}}}\right)}
根据量纲分析，涡流粘度的单位必须为 [ ν t ] = m 2 s {\displaystyle \left[\nu _{\mathrm {t} }\right]={\frac {\mathrm {m^{2}} }{\mathrm {s} }}}
第一个成功开发的 SGS 模型是 Smagorinsky-Lilly SGS 模型，它由Smagorinsky [ 1] [ 2]
ν t = C Δ 2 2 S ¯ i j S ¯ i j = C Δ 2 | S ¯ | {\displaystyle \nu _{\mathrm {t} }=C\Delta ^{2}{\sqrt {2{\bar {S}}_{ij}{\bar {S}}_{ij}}}=C\Delta ^{2}\left|{\bar {S}}\right|} 其中 Δ {\displaystyle \Delta } C {\displaystyle C}
该方法假设小尺度的能量产生和耗散处于平衡状态——即， ϵ = Π {\displaystyle \epsilon =\Pi }
Germano 等 [ 23] C {\displaystyle C} f ¯ {\displaystyle {\overline {f}}} f ^ {\displaystyle {\hat {f}}} f {\displaystyle f} τ i j = u i u j ¯ − u ¯ i u ¯ j {\displaystyle \tau _{ij}={\overline {u_{i}u_{j}}}-{\bar {u}}_{i}{\bar {u}}_{j}} T i j = u i u j ¯ ^ − u ¯ ^ i u ¯ ^ j {\displaystyle T_{ij}={\widehat {\overline {u_{i}u_{j}}}}-{\hat {\bar {u}}}_{i}{\hat {\bar {u}}}_{j}}
L i j = T i j − τ ^ i j . {\displaystyle L_{ij}=T_{ij}-{\hat {\tau }}_{ij}.}
这里 L i j = u ¯ i u ¯ j ^ − u ¯ i ^ u ¯ j ^ {\displaystyle L_{ij}={\widehat {{\bar {u}}_{i}{\bar {u}}_{j}}}-{\widehat {{\bar {u}}_{i}}}{\widehat {{\bar {u}}_{j}}}} τ i j − ( τ k k / 3 ) δ i j = − 2 C Δ 2 | S ¯ i j | S ¯ i j {\displaystyle \tau _{ij}-(\tau _{kk}/3)\delta _{ij}=-2C\Delta ^{2}|{\bar {S}}_{ij}|{\bar {S}}_{ij}} T i j − ( T k k / 3 ) δ i j = − 2 C Δ ^ 2 | S ¯ ^ i j | S ¯ ^ i j {\displaystyle T_{ij}-(T_{kk}/3)\delta _{ij}=-2C{\hat {\Delta }}^{2}|{\hat {\bar {S}}}_{ij}|{\hat {\bar {S}}}_{ij}} C {\displaystyle C}
为了做到这一点，需要在原始推导中引入x两个额外的步骤。首先，假设即使 C {\displaystyle C} C ( . ) ^ = C ( . ) ^ {\displaystyle {\widehat {C(.)}}=C{\widehat {(.)}}} C {\displaystyle C} C {\displaystyle C} [ 24] C {\displaystyle C} C {\displaystyle C} C {\displaystyle C} C {\displaystyle C} C = L i j m i j m k l m k l . {\displaystyle C={\frac {L_{ij}m_{ij}}{m_{kl}m_{kl}}}.}
其中 m i j = α i j − β ^ i j {\displaystyle m_{ij}=\alpha _{ij}-{\widehat {\beta }}_{ij}} α i j = − 2 Δ ^ 2 | S ¯ ^ | S ¯ ^ i j {\displaystyle \alpha _{ij}=-2{\hat {\Delta }}^{2}|{\hat {\bar {S}}}|{\hat {\bar {S}}}_{ij}} β i j = − 2 Δ 2 | S ¯ | S ¯ i j {\displaystyle \beta _{ij}=-2\Delta ^{2}|{\bar {S}}|{\bar {S}}_{ij}} C {\displaystyle C} − τ i j S ¯ i j {\displaystyle -\tau _{ij}{\bar {S}}_{ij}}
C = ⟨ L i j m i j ⟩ ⟨ m k l m k l ⟩ . {\displaystyle C={\frac {\left\langle L_{ij}m_{ij}\right\rangle }{\left\langle m_{kl}m_{kl}\right\rangle }}.} 当平均涉及足够大的统计样本时，计算 C {\displaystyle C} [ 25]
Lilly 对 Germano 方法的修改，然后是统计平均或综合去除负粘度区域似乎是临时的，即使它可以“工作”。 Ghosal 等人提出了一种称为“动态定位模型”（DLM）的最小二乘最小化过程的替代公式。 [ 26]
E i j = L i j − T i j + τ ^ i j {\displaystyle E_{ij}=L_{ij}-T_{ij}+{\hat {\tau }}_{ij}} 张量 τ i j {\displaystyle \tau _{ij}} T i j {\displaystyle T_{ij}} C {\displaystyle C}
C ( . ) ^ = C ( . ) ^ {\displaystyle {\widehat {C(.)}}=C{\widehat {(.)}}} 使得 E i j {\displaystyle E_{ij}} C {\displaystyle C} E i j E i j {\displaystyle E_{ij}E_{ij}} C {\displaystyle C} C {\displaystyle C} C {\displaystyle C} C {\displaystyle C}
E [ C ] = ∫ E i j E i j d V {\displaystyle E[C]=\int E_{ij}E_{ij}dV} 其中积分范围在整个流体体积上。这个全局误差 E [ C ( x , y , z , t ) ] {\displaystyle E[C(x,y,z,t)]} C ( x , y , z , t ) {\displaystyle C(x,y,z,t)} t {\displaystyle t} C {\displaystyle C} C {\displaystyle C}
C ( x ) = f ( x ) + ∫ K ( x , y ) C ( y ) d y {\displaystyle C({\boldsymbol {x}})=f({\boldsymbol {x}})+\int K({\boldsymbol {x}},{\boldsymbol {y}})C({\boldsymbol {y}})d{\boldsymbol {y}}} 其中函数 K ( x , y ) {\displaystyle K({\boldsymbol {x}},{\boldsymbol {y}})} f ( x ) {\displaystyle f({\boldsymbol {x}})} L i j , α i j , β i j {\displaystyle L_{ij},\alpha _{ij},\beta _{ij}} C ( x , y , z , t ) {\displaystyle C(x,y,z,t)} E [ C ] {\displaystyle E[C]} C ( x , y , z , t ) ≥ 0 {\displaystyle C(x,y,z,t)\geq 0} C {\displaystyle C}
C ( x ) = [ f ( x ) + ∫ K ( x , y ) C ( y ) d y ] + {\displaystyle C({\boldsymbol {x}})=\left[f({\boldsymbol {x}})+\int K({\boldsymbol {x}},{\boldsymbol {y}})C({\boldsymbol {y}})d{\boldsymbol {y}}\right]_{+}} 这里后缀 + 表示“正数部分”，即， x + = ( x + | x | ) / 2 {\displaystyle x_{+}=(x+|x|)/2} C = C ( y , t ) {\displaystyle C=C(y,t)}
DLM(+) 模型的一个缺点是它没有描述反向散射，而反向散射在分析 DNS 数据时被认为是真实的“事物”。开发了两种方法来解决这个问题。由于 Carati 等人的一种方法。 [ 27] Δ | S ¯ | {\displaystyle \Delta |{\bar {S}}|} k {\displaystyle {\sqrt {k}}} C {\displaystyle C} α i j = − 2 Δ ^ K S ¯ ^ i j {\displaystyle \alpha _{ij}=-2{\hat {\Delta }}{\sqrt {K}}{\hat {\bar {S}}}_{ij}} β i j = − 2 Δ ^ k S ¯ i j {\displaystyle \beta _{ij}=-2{\hat {\Delta }}{\sqrt {k}}{\bar {S}}_{ij}} K = k + L i i / 2 {\displaystyle K=k+L_{ii}/2}
∂ k ∂ t + u j ∂ k ∂ x j = − τ i j S ¯ i j − C ∗ Δ k 3 / 2 + ∂ ∂ x j ( D Δ k ∂ k ∂ x j ) + ν ∂ 2 k ∂ x j ∂ x j {\displaystyle {\frac {\partial k}{\partial t}}+u_{j}{\frac {\partial k}{\partial x_{j}}}=-\tau _{ij}{\bar {S}}_{ij}-{\frac {C_{*}}{\Delta }}k^{3/2}+{\frac {\partial }{\partial x_{j}}}\left(D\Delta {\sqrt {k}}{\frac {\partial k}{\partial x_{j}}}\right)+\nu {\frac {\partial ^{2}k}{\partial x_{j}\partial x_{j}}}} 其中 ν {\displaystyle \nu } C ∗ , D {\displaystyle C_{*},D}
动态模型起源于 1990 年斯坦福大学 湍流研究中心(CTR) 的暑期项目 （页面存档备份 ，存于互联网档案馆 ）。一系列“CTR-Tea”研讨会庆祝了湍流建模这一重要里程碑的30 周年 （页面存档备份 ，存于互联网档案馆 ）。
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