In the SGS stress modeling, the constitutive relation of the unclosed SGS stress can be regarded as the function of the local filtered quantities, i.e., the filtered strain-rate tensor \(\bar {S}_{ij}\) and filtered rotation-rate tensor \(\bar {\Omega }_{ij}\), namely [34, 35]

$$ {\tau_{ij}} = f\left({{{\bar{S}}_{ij}},\;{\bar{\Omega}_{ij}};\;{\delta_{ij}},\bar{\Delta}} \right), $$

(19)

where the filtered rotation-rate tensor \(\bar {\Omega }_{ij} = \frac {1}{2}\left ({\partial {\bar {u}_{i}}}/{{\partial {x_{j}}}} - {\partial {\bar {u}_{j}}}/{{\partial {x_{i}}}} \right)\). For brevity and simplicity of the tensorial polynomials, the matrix multiplications for the tensor contractions are expressed as [34, 35, 40]

$$ {\bar{\mathbf{S}}^{2}} = {{\bar{S}}_{ik}}{{\bar{S}}_{kj}},\;\;\bar{\mathbf{S}}\bar{\mathbf{\Omega}} = {{\bar{S}}_{ik}}{\bar{\Omega}_{kj}},\;\;tr\left(\bar{\mathbf{S}}{\bar{\mathbf{\Omega}}^{2}} \right) = {{\bar{S}}_{ij}}{\bar{\Omega}_{jk}}{\bar{\Omega}_{ki}}. $$

(20)

A general expression of the modeled SGS stress [Eq. (19)] can be expanded to the sum of an infinite number of tensorial polynomials with the form \({\bar {\mathbf {S}}^{{m_{1}}}}{\bar {\mathbf {\Omega }}^{{n_{1}}}}{\bar {\mathbf { S}}^{{m_{2}}}}{\bar {\mathbf {\Omega }}^{{n_{2}}}} \cdots \), where *m*_{i} and *n*_{i} are positive integers. The infinite tensorial polynomials can be reduced to a finite number by the Cayley-Hamilton theorem [34, 35, 40], thus the modeled SGS stress can be expressed as the linear combination of the basis tensors formed by the product of \(\bar {\mathbf {S}}\) and \(\bar {\mathbf {\Omega }}\), namely [34]

$$ {\tau_{ij}} = \sum\limits_{n = 1}^{11} {{g_{n}}\left({{\lambda_{1}},{\lambda_{2}}, \cdots,{\lambda_{6}}} \right)T_{ij}^{\left(n \right)}}, $$

(21)

where \(T_{ij}^{\left (n \right)}\) is the *n*-th basis tensor and the model coefficients *g*_{n} are functions of the six integrity invariants *λ*_{m} (*m*=1,2,⋯,6). Here, the eleven basis tensors \(T_{ij}^{\left (n \right)}\) and six independent invariants *λ*_{m} are respectively expressed as [34]

$$ \begin{array}{l} {\mathbf{T}^{\left(1 \right)}} = \bar{\mathbf{S}},\;\;\;\;{\mathbf{T}^{\left(2 \right)}} = {\bar{\mathbf{ S}}^{2}},\\ {\mathbf{T}^{\left(3 \right)}} = {\bar{\mathbf{\Omega }}^{2}},\;\;{\mathbf{T}^{\left(4 \right)}} = \bar{\mathbf{S}}\bar{\mathbf{\Omega}} - \bar{\mathbf{\Omega}} \bar{\mathbf{S}},\\ {\mathbf{T}^{\left(5 \right)}} = {\bar{\mathbf{S}}^{2}}\bar{\mathbf{\Omega}} - \bar{\mathbf{\Omega}}{\bar{\mathbf{ S}}^{2}},\;\;{\mathbf{T}^{\left(6 \right)}} = \mathbf{I},\\ {\mathbf{T}^{\left(7 \right)}} = \bar{\mathbf{S}}{\bar{\mathbf{\Omega }}^{2}} + {\bar{\mathbf{\Omega }}^{2}}\bar{\mathbf{S}},\;\;{\mathbf{T}^{\left(8 \right)}} = \bar{\mathbf{\Omega}} \bar{\mathbf{S}}{\bar{\mathbf{\Omega }}^{2}} - {\bar{\mathbf{\Omega }}^{2}}\bar{\mathbf{S}}\bar{\mathbf{\Omega}},\\ {\mathbf{T}^{\left(9 \right)}} = \bar{\mathbf{S}}\bar{\mathbf{\Omega}}{\bar{\mathbf{S}}^{2}} - {\bar{\mathbf{ S}}^{2}}\bar{\mathbf{\Omega}} \bar{\mathbf{S}},\;{\mathbf{T}^{\left({10} \right)}} = {\bar{\mathbf{S}}^{2}}{\bar{\mathbf{\Omega }}^{2}} + {\bar{\mathbf{\Omega }}^{2}}{\bar{\mathbf{S}}^{2}},\\ {\mathbf{T}^{\left({11} \right)}} = \bar{\mathbf{\Omega}}{\bar{\mathbf{S}}^{2}}{\bar{\mathbf{\Omega }}^{2}} - {\bar{\mathbf{\Omega }}^{2}}{\bar{\mathbf{S}}^{2}}\bar{\mathbf{\Omega}}, \end{array} $$

(22)

$$ \begin{array}{l} {\lambda_{1}} = tr\left({{\bar{\mathbf{S}}^{2}}} \right),\;\;\;\;\;{\lambda_{2}} = tr\left({{\bar{\mathbf{\Omega }}^{2}}} \right),\\ {\lambda_{3}} = tr\left({{\bar{\mathbf{S}}^{3}}} \right),\;\;\;\;\;{\lambda_{4}} = tr\left(\bar{\mathbf{S}}{\bar{\mathbf{\Omega }}^{2}} \right),\\ {\lambda_{5}} = tr\left({{\bar{\mathbf{S}}^{2}}{\bar{\mathbf{\Omega }}^{2}}} \right),\;{\lambda_{6}} = tr\left({{\bar{\mathbf{S}}^{2}}{\bar{\mathbf{\Omega }}^{2}}\bar{\mathbf{S}}\bar{\mathbf{\Omega}}} \right). \end{array} $$

(23)

If the model coefficients *g*_{n} are relaxed as the ratios of polynomials of these integrity invariants, the number of the above basis tensors can be reduced from eleven to five. In accordance with the dimensional consistency, the anisotropic part of the modeled SGS stress can be given by [35]

$$ \begin{aligned} \tau_{ij}^{A} &= {{\bar{\Delta} }^{2}}\sum\limits_{n = 1}^{5} {{C_{n}}\mathbb{T}_{ij}^{\left(n \right),A}} \\ &= {{\bar{\Delta} }^{2}}\left[ {{C_{1}}\left| \bar{\mathbf{S}} \right|\bar{\mathbf{S}} + {C_{2}}{{\left({{\bar{\mathbf{ S}}^{2}}} \right)}^{A}} + {C_{3}}{{\left({{\bar{\mathbf{\Omega }}^{2}}} \right)}^{A}} + {C_{4}}\left(\bar{\mathbf{S}}\bar{\mathbf{\Omega}} - \bar{\mathbf{\Omega}} \bar{\mathbf{S}} \right) + \frac{{{C_{5}}}}{{\left| \bar{\mathbf{S}} \right|}}\left({{\bar{\mathbf{S}}^{2}}\bar{\mathbf{\Omega}} - \bar{\mathbf{\Omega}}{\bar{\mathbf{S}}^{2}}} \right)} \right], \end{aligned} $$

(24)

where the characteristic filtered strain rate \({\left | \bar {\mathbf {S}} \right |} = {(2{{\bar {S}}_{ij}}{{\bar {S}}_{ij}})^{1/2}}\), and *C*_{n} are five dimensionless model coefficients. The corresponding basis tensors \(\mathbb {T}_{ij}^{(n)}\) that satisfy the consistent dimension with the square of the velocity gradient are defined by

$$ {\mathbf{\mathbb{T}}^{\left(1 \right)}_{ij}} = \left| \bar{\mathbf{S}} \right|\bar{\mathbf{S}},\;{\mathbf{\mathbb{T}}^{\left(2 \right)}_{ij}} = {\bar{\mathbf{S}}^{2}},\;{\mathbf{\mathbb{T}}^{\left(3 \right)}_{ij}} = {\bar{\mathbf{\Omega }}^{2}},\; {\mathbf{\mathbb{T}}^{\left(4 \right)}_{ij}} = \bar{\mathbf{S}}\bar{\mathbf{\Omega}} - \bar{\mathbf{\Omega}} \bar{\mathbf{S}},\;\;{\mathbf{\mathbb{T}}^{\left(5 \right)}_{ij}} = \frac{1}{\left| \bar{\mathbf{S}} \right|}\left({{\bar{\mathbf{ S}}^{2}}\bar{\mathbf{\Omega}} - \bar{\mathbf{\Omega}}{\bar{\mathbf{S}}^{2}}} \right). $$

(25)

In the paper, two dynamic procedures are adopted to determine the model coefficients *C*_{n} of the dynamic nonlinear algebraic models (DNAM). One is the Germano identity dynamic (GID) procedure based on the scale-invariance assumption, and the other is the newly proposed scale-similarity dynamic (SSD) procedure in accordance with the scale-similarity relation. The rest of this section will be divided into two subsections to respectively introduce these two different modeling approaches.

### 4.1 DNAM models with Germano identity dynamic procedure (DNAM-GID)

Similar to the conventional dynamic SGS models (e.g. DSM and DMM models), the DNAM model with Germano identity dynamic procedure, abbreviated as DNAM-GID, introduces the test-filter level SGS stress \(\mathcal {T}_{ij}\) at the double-filtering scale \(\tilde {\bar {\Delta }} = 2\bar {\Delta } \), modeled by

$$ \begin{aligned} \mathcal{T}_{ij}^{A} &= {\tilde{\bar{\Delta}}}^{2}\sum\limits_{n = 1}^{5} {{C_{n}} \mathbb{H}_{ij}^{\left(n \right),A}} \\ &= {{\tilde{\bar{\Delta}}}^{2}}\left[ {{C_{1}}\left| {\tilde{\bar{\mathbf{S}}}} \right|\tilde{\bar{\mathbf{S}}} + {C_{2}}{{\left({{{\tilde{\bar{\mathbf{S}}}}^{2}}} \right)}^{A}} + {C_{3}}{{\left({{{\tilde{\bar{\mathbf{\Omega}} }}^{2}}} \right)}^{A}} + {C_{4}}\left({\tilde{\bar{\mathbf{S}}} \tilde{\bar{\mathbf{\Omega}}} - \tilde{\bar{\mathbf{\Omega}}} \tilde{\bar{\mathbf{S}}}} \right) + \frac{{{C_{5}}}}{{\left| {\tilde{\bar{\mathbf{S}}}} \right|}}\left({{{\tilde{\bar{\mathbf{S}}}}^{2}}\tilde{\bar{\mathbf{\Omega}}} - \tilde{\bar{\mathbf{\Omega}}}{{\tilde{\bar{\mathbf{S}}}}^{2}}} \right)} \right]. \end{aligned} $$

(26)

Here, \(\mathbb {H}^{(n)}_{ij}\) is the *n*-th basis tensor at the test filter scale \(\tilde {\bar {\Delta }} = 2\bar {\Delta } \), expressed as

$$ {\mathbf{\mathbb{H}}^{\left(1 \right)}_{ij}} = \left| {\tilde{\bar{\mathbf{S}}}} \right|\tilde{\bar{\mathbf{S}}},\;{\mathbf{\mathbb{H}}^{\left(2 \right)}_{ij}} = {{\tilde{\bar{\mathbf{S}}}}^{2}},\;{\mathbf{\mathbb{H}}^{\left(3 \right)}_{ij}} = {\tilde{\bar{\mathbf{\Omega}}}^{2}},\; {\mathbf{\mathbb{H}}^{\left(4 \right)}_{ij}} = \tilde{\bar{\mathbf{S}}} \tilde{\bar{\mathbf{\Omega}}} - \tilde{\bar{\mathbf{\Omega}}} \tilde{\bar{\mathbf{S}}},\;\;{\mathbf{\mathbb{H}}^{\left(5 \right)}_{ij}} = \frac{1}{\left| {\tilde{\bar{\mathbf{S}}}} \right|}\left({{{\tilde{\bar{\mathbf{S}}}}^{2}}\tilde{\bar{\mathbf{\Omega}}} - \tilde{\bar{\mathbf{\Omega}}}{{\tilde{\bar{\mathbf{S}}}}^{2}}} \right). $$

(27)

Consistent with Eq. (12), the modeled SGS stresses *τ*_{ij} and \(\mathcal {T}_{ij}\) at different filter scales satisfy the Germano identity, namely

$$ \mathcal L_{ij}^{A}=\mathcal{T}_{ij}^{A}-\tilde{\tau}_{ij}^{A}=\sum\limits_{n = 1}^{5} {{C_{n}}\mathbb{M}_{ij}^{\left(n \right)}}, $$

(28)

where \(\mathbb {M}_{ij}^{\left (n \right)} = {{\tilde {\bar {\Delta } }}^{2}}\mathbb {H}_{ij}^{\left (n \right),A} - {{\bar {\Delta } }^{2}}\tilde {\mathbb {T}}_{ij}^{\left (n \right),A}\). The model coefficients *C*_{n} can be further calculated by the least-squares algorithm, derived by

$$ \sum\limits_{m = 1}^{5} {{C_{m}}\left\langle {\mathbb{M}_{ij}^{\left(m \right)}\mathbb{M}_{ij}^{\left(n \right)}} \right\rangle} = \left\langle {\mathcal {L}_{ij}^{A} \mathbb{M}_{ij}^{\left(n \right)}} \right\rangle,\;\;\left({n = 1,2, \cdots,5} \right). $$

(29)

For the DNAM-GID model, the optimal model coefficients *C*_{n} can be obtained by solving the system of five linear equations in Eq. (29).

### 4.2 DNAM models with scale-similarity dynamic procedure (DNAM-SSD)

In this paper, we propose a novel scale-similarity dynamic procedure for the DNAM model to determine the optimal model coefficients dynamically. The real SGS stress can be regarded as the nonlinear function of the velocity *u*_{i} and the filter kernel at scale \(\bar {\Delta }\), whereas the SGS stress modeled by the DNAM model has the nonlinear constitutive relation with the local filtered physical quantities (e.g. the filtered strain-rate and rotation-rate tensors \(\bar {\mathbf {S}}\) and \(\bar {\mathbf {\Omega }}\)),

$$ \begin{aligned} \tau_{ij}^{A} &= \tau_{ij}^{A}(u_{i};\bar{\Delta}) = {\left({\overline {{u_{i}}{u_{j}}} - {\bar{u}_{i}}{\bar{u}_{j}}} \right)^{A}} = \sum\limits_{n = 1}^{5} {{C_{n}}\mathbb{T}_{ij}^{(n),A}} \left(\bar{\mathbf{S}},\bar{\mathbf{\Omega}} \right)\\ &= {\bar{\Delta}^{2}}\left[ {{C_{1}}\left| \bar{\mathbf{S}} \right|\bar{\mathbf{S}} + {C_{2}}{{\left({{\bar{\mathbf{ S}}^{2}}} \right)}^{A}} + {C_{3}}{{\left({{\bar{\mathbf{\Omega }}^{2}}} \right)}^{A}} + {C_{4}}\left(\bar{\mathbf{S}}\bar{\mathbf{\Omega}} - \bar{\mathbf{\Omega}} \bar{\mathbf{S}} \right) + \frac{{{C_{5}}}}{{\left| \bar{\mathbf{S}} \right|}}\left({{\bar{\mathbf{S}}^{2}}\bar{\mathbf{\Omega}} - \bar{\mathbf{\Omega}}{\bar{\mathbf{S}}^{2}}} \right)} \right]. \end{aligned} $$

(30)

Based on the scale-similarity hypothesis, the modeled SGS stress at the filter scale \(\tilde {\Delta }\) shares the consistent model coefficients *C*_{n} with that at the filter scale \(\bar {\Delta }\), namely

$$ \begin{aligned} \tau_{ij}^{A}(u_{i};\tilde{\Delta}) &= {\left({\widetilde {{u_{i}}{u_{j}}} - {\tilde{u}_{i}}{\tilde{u}_{j}}} \right)^{A}} = \sum\limits_{n = 1}^{5} {{C_{n}}\mathbb{T}_{ij}^{(n),A}} \left({\tilde{\mathbf{S}},\tilde{\mathbf{\Omega}}} \right)\\ &= \tilde{\Delta}^{2}\left[ {{C_{1}}\left| \tilde{\mathbf{S}} \right|\tilde{\mathbf{S}} + {C_{2}}{{\left({{\tilde{\mathbf{S}}^{2}}} \right)}^{A}} + {C_{3}}{{\left({{\tilde{\mathbf{\Omega}}^{2}}} \right)}^{A}} + {C_{4}}\left(\tilde{\mathbf{S}}\tilde{\mathbf{\Omega}} - \tilde{\mathbf{\Omega}} \tilde{\mathbf{S}} \right) + \frac{{{C_{5}}}}{{\left| \tilde{\mathbf{S}} \right|}}\left({{\tilde{\mathbf{S}}^{2}}\tilde{\mathbf{\Omega}} - \tilde{\mathbf{\Omega}}{\tilde{\mathbf{S}}^{2}}} \right)} \right]. \end{aligned} $$

(31)

The constitutive equation of the SGS stress is assumed to be invariant to the physical field, therefore we can replace the unfiltered velocity *u*_{i} with the filtered velocity \(\bar {u}_{i}\) in Eq. (31) and obtain

$$ \begin{aligned} \mathcal{L}_{ij}^{A} &= \tau_{ij}^{A}(\bar{u}_{i};\tilde{\Delta}) = {\left({\widetilde {{\bar{u}_{i}}{\bar{u}_{j}}} - {\tilde{\bar{u}}_{i}}{\tilde{\bar{u}}_{j}}} \right)^{A}} = \sum\limits_{n = 1}^{5} {{C_{n}}\mathbb{T}_{ij}^{(n),A}} \left({\tilde{\bar{\mathbf{S}}},\tilde{\bar{\mathbf{\Omega}}}} \right)\\ &= {\tilde{\Delta}^{2}}\left[ {{C_{1}}\left| {\tilde{\bar{\mathbf{S}}}} \right|\tilde{\bar{\mathbf{S}}} + {C_{2}}{{\left({{{\tilde{\bar{\mathbf{S}}}}^{2}}} \right)}^{A}} + {C_{3}}{{\left({{{\tilde {\bar{\mathbf{\Omega}}}}^{2}}} \right)}^{A}} + {C_{4}}\left({\tilde{\bar{\mathbf{S}}} \tilde{\bar{\mathbf{\Omega}}} - \tilde{\bar{\mathbf{\Omega}}} \tilde{\bar{\mathbf{S}}}} \right) + \frac{{{C_{5}}}}{{\left| {\tilde{\bar{\mathbf{S}}}} \right|}}\left({{{\tilde{\bar{\mathbf{S}}}}^{2}}\tilde{\bar{\mathbf{\Omega}}} - \tilde{\bar{\mathbf{\Omega}}}{{\tilde{\bar{\mathbf{S}}}}^{2}}} \right)} \right]. \end{aligned} $$

(32)

\(\mathcal {L}_{ij}^{A}\) is the anisotropic part of the resolved Leonard stress. For simplicity, we let \(\mathbb {N}_{ij}^{(n)} =\mathbb {T}_{ij}^{(n),A} \left ({\tilde {\bar {\mathbf {S}}},\tilde {\bar {\mathbf {\Omega }}}} \right)\) and Eq. (32) is abbreviated as \(\mathcal {L}_{ij}^{A} = \sum \limits _{n = 1}^{5} {C_{n}} \mathbb {N}_{ij}^{(n)}\). Since \(\mathcal {L}_{ij}^{A}\) and \(\mathbb {N}_{ij}^{(n)}\) are both resolved in the filtered field, the model coefficients *C*_{n} can be determined by the least-squares method,

$$ \sum\limits_{m = 1}^{5} {{C_{m}}\left\langle {\mathbb{N}_{ij}^{\left(m \right)}\mathbb{N}_{ij}^{\left(n \right)}} \right\rangle} = \left\langle {\mathcal {L}_{ij}^{A} \mathbb{N}_{ij}^{\left(n \right)}} \right\rangle,\;\;\left({n = 1,2, \cdots,5} \right). $$

(33)

It is worth noting that the DNAM-SSD model only calculates \(\mathbb {T}_{ij}^{(n),A} \left ({\bar {\mathbf {S}},{{\bar {\mathbf {\Omega }} }}} \right)\) and \(\mathbb {T}_{ij}^{(n),A} \left ({\tilde {\bar {\mathbf {S}}},\tilde {\bar {\mathbf {\Omega }}}} \right)\) rather than \(\mathbb {T}_{ij}^{(n),A} \left ({{{\bar {\mathbf {S}}}},{{\bar {\mathbf {\Omega }} }}} \right), \mathbb {T}_{ij}^{(n),A} \left ({\tilde {\bar {\mathbf {S}}},\tilde {\bar {\mathbf {\Omega }}}} \right)\) and \(\tilde {\mathbb {T}}_{ij}^{(n),A} \left ({{{\bar {\mathbf {S}}}},{{\bar {\mathbf {\Omega }} }}} \right)\) in the DNAM-GID model, therefore the scale-similarity dynamic procedure simplifies the conventional dynamic procedure based on the Germano identity. Besides, in the following sections, we can show that the DNAM model with the proposed scale-similarity dynamic procedure performs better than that with the conventional GID procedure both in the *a priori* and the *a posteriori* testings of incompressible turbulence.