Figure
1.
Basic characteristics of turbulence
| Citation: | QI Jin, WU Chuijie. LMS Method: a Spatiotemporal Optimal Low-Dimensional Dynamical Systems of Multi-Scale Numerical Simulation Method for Compressible Turbulence[J]. Applied Mathematics and Mechanics, 2024, 45(3): 318-336. doi: 10.21656/1000-0887.440294
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Professor P-Y CHOU, the father of computational modeling[1], published a groundbreaking paper[2] on turbulence modeling in Chinese Journal of Physics in 1940, in which he proposed that to study the numerical simulation of turbulence, it is necessary to analyze and solve the fluctuating velocity field. He first presented the second- and third-order velocity correlation equations. Then, he transformed the fourth-order correlation function into a closure equation by summing the product of two second-order correlation functions. This process is the foundation of turbulence modeling theory. However, it is not the best method for computationally modeling turbulence (see section 4).
In this report, for clarity, we present the term “small-scale” as “SGS-scale” in the LES method, and in the LMS method (see section 2), the SGS-scale includes middle-scale and small-scale.
Following Chou’s idea, based on first principles, with the box filtering and without the Favre filter and any artificial assumption, coupled with the LES equation, a set of spatiotemporal multiscale optimal low-dimensional dynamical systems equations is solved with a small-scale turbulent model to obtain the middle-scale flow field, together with the LES large-scale averaged flow field to obtain a new kind of approximate solution of turbulence.
In this report, several problems encountered in the research and their solutions are studied first, as shown in section 2 to section 5. Then, the LMS method is established systematically in section 6, and its application on the re-shock Richtmyer-Meshkov problem is discussed in section 6.3. Finally, concluding remarks are given in section 7.
Traditional numerical simulation methods of turbulence include RANS[3], LES[4], DES[5] and DNS[6]. We used the following categories to classify the numerical simulation methods of turbulence from the point view of scale decomposition.
1) Class L(Large): All scales of turbulence are solved by CFD methods to obtain a unified solution of turbulence $ f $, such as DNS.
2) Class LS(Large-Small): The turbulence flow field is decomposed into large-scale and SGS-scale fields. The large-scale equation is solved by the CFD method, and the SGS-scale equation can be solved by:
① Coupling the solution of the CFD method with the large-scale equation;
② Coupling the dynamic system or optimal low-dimensional dynamic system method with the large-scale equation;
③ Using the SGS model in the LES equation.
Class LS includes the RANS, LES and DES methods, and the numerical results are large-scale averaged solutions of turbulence $ \overline{f} $.
| $$ f = \overline{f}+f' .$$ | (1) |
Since the SGS-scale in class LS includes a wavenumber range with completely different turbulence characteristics, it is not easy to find accurate turbulence models in RANS, LES and DES.
3) Class LMS(Large-Middle-Small)
It can be seen from fig. 1 that for high Reynolds number turbulence, the Kolmogorov spectra can be divided into the energy-containing range and the universal equilibrium range, in which the inertial sub-range is dominated by convection and dissipation is very small. Moreover, the dissipation range in which most turbulent energy is dissipated. Therefore, according to the basic characteristics of turbulence, the whole range of the turbulence spectrum wavenumbers can be divided into three parts with completely different characteristics, i.e., the energy-containing range, the inertial subrange and the dissipation range.
According to the basic characteristics of turbulence, the approximate solution of the turbulence $ f $ is divided into three parts as in equation (2): large-scale averaged quantities $ \overline{f} $, middle-scale quantities $ f' $ and small-scale quantities $ f'' $.
| $$ f = \overline{f}+f'+f''. $$ | (2) |
Class LMS includes the LMS(Low-dimensional dynamical systems Multiscale Simulation) method (see section 6).
Large-scale quantities $ \overline{f} $ reflect the characteristics of the energy-containing range, which are related to specific turbulence problems and must be obtained by solving a large-scale equation using the CFD method. The SGS term in the large-scale equation includes the middle-scale SGS correlation term and the role of the small-scale SGS turbulence model. $ \overline{f} $ reflects the characteristics of the energy-containing range, which are related to specific turbulence problems.
The middle-scale quantities $ f' $ reflect the convection characteristics of the anisotropic inertial subrange, which is obtained with the spatiotemporal optimal low-dimensional dynamical system equation. This equation includes large- and middle-scale quantities, with the large-scale equation in the middle-scale grid $ {\Delta x}/{2^p},p\ge 1 $.
Since the dissipation range contains most of the dissipation effects of turbulence, the small-scale quantities are very small and isotropic, and the grid sizes in the dissipation range are very small, it is not suitable for direct solving. The effects of small-scale quantities $ f'' $ are found by a small-scale SGS turbulence model expressed by the middle-scale quantities together with the middle-scale deformation tensor and the large-scale stress tensor, which provides an appropriate dissipation effect.
In this section, we clarify the concepts of average and filtering operations and the correlation operation related to the box filter used in the LMS method.
The following are the definitions of RANS and LES, in which the key concepts are the averaging operations (RANS) and filtering operations (LES).
RANS: by using turbulence averaging operations (including the time average, spatial average, ensemble average, etc.), the flow variables are decomposed into mean and fluctuating quantities.
LES: by using turbulence filtering operations (including the box filter, Fourier filter, Gaussian filter, etc.), the flow variables are decomposed into large-scale quantities and SGS-scale quantities.
The given filtering function $ G(|{\boldsymbol{x}}-{\boldsymbol{y}}|) $ is used to decompose the flow variables into large-scale $ \overline{f({\boldsymbol{x}}, t)} $ and SGS-scale quantities $ f'({\boldsymbol{x}}, t) $ as follows:
| $$ \left\{\begin{aligned} & f({\boldsymbol{x}}, t) = \overline{f({\boldsymbol{x}}, t)}+f'({\boldsymbol{x}}, t),\\& \overline{f({\boldsymbol{x}}, t)} = \int_\varOmega G(|{\boldsymbol{x}}-{\boldsymbol{y}}|)f({\boldsymbol{y}}, t){\rm{d}}\varOmega,\\& f' = f-\overline{f} . \end{aligned}\right. $$ | (3) |
The filtering function of the box filter for the grid scale $ \Delta x_i $ is
| $$ G(|{\boldsymbol{x}}-{\boldsymbol{\xi}}|) = \left\{ \begin{aligned} & \dfrac{1}{\Delta x_1\Delta x_2\Delta x_3}, \quad&& \text{if }|x_i-\xi_i|\leq\dfrac{\Delta x_i}{2}\\& 0, \quad&&\text{others} , \end{aligned} \right. ,\;i = 1, 2, 3 .$$ | (4) |
As shown in fig. 2, the box filter is the volume average on the grid centered around $ {\boldsymbol{x}} $. In RANS, the grid space averaging is equivalent to the box filtering. Therefore, only the RANS and LES in DES use grid space averaging, then, the DES is determined to be conceptually correct. Since generally, the averaging (RANS) and the filter (LES) used in the DES method are not compatible with each other, therefore the DES method is conceptually wrong, even though sometimes good results of turbulence numerical simulation can be obtained.
Correlation function: the average or filter of the self-multiplication of the same or different fluctuating or SGS-scale quantities are generally not equal to zero, and their values are related to the degree of correlation between these two fluctuating or SGS-scale quantities.
| $$ \left\{\begin{aligned} & \overline{a'b'}\;>\;0, \qquad\text{positive correlation},\\& \overline{a'b'}\;<\;0,\qquad \text{negative correlation},\\& \overline{a'b'}\; = \;0, \qquad\text{non correlation} . \end{aligned}\right.$$ | (5) |
Correlation coefficient: using the correlation coefficient R to measure the degree of correlation between $ a',b' $ yields
| $$ R = \frac{\overline{a'b'}}{\sqrt{\overline{a'^2}\ \overline{b'^2}}} .$$ | (6) |
If $ R = \pm 1 $, then they are completely correlated. It can be seen that each fluctuating or SGS-scale quantity is completely related to itself.
Since we do not use time-averaged functions in this study, only spatial correlations are studied.
In spatial correlations, for a grid, when the distance is equal to 0, its spatial correlation coefficient is equal to 1; for an infinite distance, its spatial correlation coefficient is equal to 0.
Therefore, the spatial correlation coefficient of the box filtering is equal to 1 for this grid and is equal to 0 for other grids. Thus, the spatial correlation of the box filtering in the LMS method can be clearly calculated.
The above discussion shows that it is important to calculate the spatial correlations in coupling to solve LES+SGS equations; therefore, two questions arise:
1) How can we represent the effects of the correlation functions in the RANS and SGS models?
2) Furthermore, are the RANS/SGS models independent of the averaging (RANS) or filtering (LES) operations used in turbulence numerical simulations?
To establish the bases for the following discussion, we first provide two basic statements:
Consensus 1 Under a coarse grid, it is impossible to obtain turbulence with a numerical simulation of the Navier-Stokes equation.
Consensus 2 Only with the DNS grid can complete turbulence be obtained with the numerical simulation of the Navier-Stokes equation.
Second, the following abbreviation conventions are used in the discussion.
We let the arbitrary mathematical equation be $ A = 0 $, $ B = 0 $, and apply a function $\overline{(\;\;) }$ to $ A $ to obtain $ {\overline A} = 0 $ and $ A-{\overline A} = B = 0 $.
For the turbulence study, we let the Navier-Stokes equation be $ A = 0 $.
For RANS: the Reynolds average operation is defined as $\overline{(\;\;) } $, the Reynolds average equation is $ {\overline A} = {\rm{RANS}} = 0 $, and the fluctuation equation is $ A-{\overline A} = {\rm{FLUC}} = 0 $.
For LES: the filtering operation is defined as $ \overline{(\;\;) } $, the large-eddy equation is $ {\overline A} = {\rm{LES}} = 0 $, and the SGS-scale equation is $ A-{\overline A} = {\rm{SGS}} = 0 $.
We take the LES as an example in the discussion. The results are the same as those of the RANS.
If the LES and SGS equations are coupled solving in the LES grid, i.e.
| $$ \text{Coupled solution of LES and SGS:} \left\{\begin{aligned} & { \cancel{\rm{LES}} = 0},\qquad&& { {{{\rm{with}}}} \;{{{\rm{LES}}}}\; {{{\rm{grid}}}}},\\& + \;\;\;\;\;\;\;\;\;\;\;\rightarrow {{\rm{NS}}} = 0,\qquad&& { {{{\rm{with}}}} \;{{{\rm{LES}}}}\; {{{\rm{grid}}}}},\\& { {\rm{SGS}} = {\rm{NS}}-{\cancel{\rm{LES}}} = 0},\qquad&& { {{{\rm{with}}}} \;{{{\rm{LES}}}}\; {{{\rm{grid}}}}}. \end{aligned} \right.$$ | (7) |
Then, since from Consensus 1, we know that turbulence cannot be calculated by the Navier-Stokes equation in the LES grid; therefore, the coupled solution of the LES+SGS equations in the LES grid or the single-scale SGS model that approximates the SGS equation has a logic problem.
That is, in the coupled solution of LES+SGS equations, the closer the result is to $ {\rm{SGS}} = 0 $, the less turbulent the solution is, i.e., there is a contradiction between the purpose and result, which means there is a logical error; or in the modeling, we cannot make $ {\rm{SGS}} = 0 $, then, the so-called turbulent results come from unphysical errors of modeling and/or numerical algorithms, which is absurd.
The basis of correct turbulence modeling theory is equation+grids.
To obtain complete turbulence, the DNS grid must be used with the Navier-Stokes equation.
| $$ \text{Coupled solution of LES+SGS:} \left\{ \begin{aligned} & { {\rm{LES}} = 0},\qquad &&{ {\rm{with}} \;{\rm{LES}}\; {\rm{grid}}},\\& + \;\;\;\;\;\;\;\;\;\;\;\rightarrow \text{similar to }{{\rm{NS}}} = 0,\qquad&& { {\rm{with}} \;{\rm{DNS}}\; {\rm{grid}}},\\& { {\rm{SGS}} = {\rm{NS}}-{\rm{LES}} = 0},\qquad&& { {\rm{with}} \;{\rm{small}}{\text{-}}{\rm{scale}}\; {\rm{grid}}}, \end{aligned}\right. $$ | (8) |
i.e., a multiscale turbulence model must be used in LES to obtain correct turbulent results.
Notably, since the SGS equation is different from the Navier-Stokes equation used in DNS, the solution of the SGS equation, i.e., the so-called SGS-scale flow field, is conceptually different from the small-scale flow structures obtained by DNS.
However, the amount of computation required to solve the SGS equation on the SGS-scale grid is enormous, even exceeding that of DNS. Therefore, we must develop a new method (see section 6).
Finally, we noticed that this type of “modeling” method also appears in other disciplines, so such logic errors widely exist.
The closure problem of turbulence is a long-standing unsolved problem in the theoretical bases of turbulence modeling. It is very important to understand the essence and key of the closure problem of turbulence. The closure problem of turbulence can be stated as follows: when constructing the equation of the SGS-scale correlation terms, there will be more higher-order SGS-scale correlation terms produced in each order of the equation of SGS-scale correlation terms.
First, the closure problem of turbulence is produced in the study of numerical simulation and modeling of turbulence; therefore, the essence of it is not a theoretical problem but a numerical problem.
Second, the key to the closure problem of turbulence is the requirement of the simultaneity of the SGS-scale and the SGS-scale correlation terms, which is similar to the following basic philosophy problem: did the chicken or the egg come at the same time?
Since the Reynolds stress equation of the RANS or LES comes from the fluctuation equation or SGS equation, it is not necessary to study the Reynolds stress equation but only to use a numerical method to couple the RANS+fluctuation equations or LES+SGS equations to overcome the closure problem of turbulence numerically.
The method of alternating coupling solutions of the LES and SGS equations is adopted, and the SGS-scale terms obtained at the last moment are used to calculate the SGS-scale correlation terms so that the SGS-scale and SGS-scale correlation terms are not simultaneous. Therefore, the closure problem of turbulence can be overcome with a numerical method.
In fact, in 2000 and 2009, we successfully applied the alternating coupling solution method of the LES equation and POD low-dimensional dynamical system equation in the numerical simulation of incompressible turbulence[7-8].
After overcoming the problems in section 2, section 3, section 4 and section 5, we can now develop the LMS method for the numerical simulation of compressible turbulence.
In the LMS method, the most important innovation is the application of a spatiotemporal multiscale optimal low-dimensional dynamical system of the middle-scale equation to ensure the accuracy of middle-scale turbulence modeling.
In a previous paper[9] on the spatiotemporal optimal low-dimensional dynamical system, there are important findings, such as those shown in fig. 3. In general, low-dimensional dynamical systems with a spatial bases, such as cases of the first eight legends in fig. 3 are not predictive, i.e., the solution errors increase with time, and the spatiotemporal optimal low-dimensional dynamical system with a spatiotemporal intrinsic bases, such as the case of last legend in fig. 3 is predictable, i.e., the solution errors can be neglected.
The key point is the spatiotemporal intrinsic bases $ {\boldsymbol{\xi}}_k({\boldsymbol{x}},t) $, which characterizes the spatiotemporal characteristics of the system. The approximated optimal solution space can be spened with the spatiotemporal intrinsic bases that advances with time and forms the spatiotemporal optimal low-dimensional dynamical system with the lowest $ N $, which will be used as the spatiotemporal optimal low-dimensional dynamical system of the middle-scale equation.
| $$ {\boldsymbol{u}}({\boldsymbol{x}},t) = {\boldsymbol{u}}_N+{\boldsymbol{u}}_R\approx\sum_{k = 1}^N a_k(t){\boldsymbol{\xi}}_k({\boldsymbol{x}},t). $$ | (9) |
The state equation is
| $$ p = \rho R_{\rm{g}} T ,$$ | (10) |
where $ R_{\rm{g}} $ is the universal gas constant ( $ R_{\rm{g}} = 287.0 $). The dimensional compressible Navier-Stokes equation is
| $$ \left\{ \begin{aligned} & {\rho}_{,t}+(\rho u_j)_{,j} = 0,\\& (\rho u_i)_{,t}+(\rho u_i u_j+p\delta_{ij})_{,j}-\sigma_{ij,j} = 0,\\& {E}_{,t}+[(E+p)u_j]_{,j}-\kappa T_{,jj}-(\sigma_{ij}u_i)_{,j} = 0 ,\end{aligned}\right. $$ | (11) |
where $ E $ is total energy,
| $$ E\equiv\frac{p}{(\gamma-1)\gamma {Ma}^2}+\frac 12 \rho(u_h u_h) ,$$ | (12) |
where the specific heat ratio is $ \gamma = 1.4 $ and $ {Ma} $ is the Mach number.
Since there are drawbacks with the Favre filter, we do not use the Favre filter in the LMS method. We applied the box filter to the dimensional compressible Navier-Stokes equation (11), to obtain the dimensional compressible turbulence large-scale equation.
| $$ \left\{ \begin{aligned} & \overline{\rho}_{,t}+(\overline{\rho}\ \overline{u}_j)_{,j}+r_{j,j}+{\cal{R}}_{j,j} = 0,\\& (\overline{\rho}\ \overline{u}_i+r_i)_{,t}+(\overline{\rho}\ \overline{u}_i \overline{u}_j)_{,j}+\overline{p}_{,i}-\overline{\sigma}_{ij,j}+\tau_{ij,j}+ {\cal{T}}_{ij,j} = 0,\\& \overline{E}_{,t}+(\overline{E}\ \overline{u}_j)_{,j}+(\overline{p}\ \overline{u}_j)_{,j}-(\overline{u}_i\ \overline{\sigma}_{ij})_{,j}-\kappa\overline{T}_{,jj}+q_{j,j}+C_v {\cal{Q}}_{j,j} = 0 .\end{aligned}\right. $$ | (13) |
$ r_{j} $, $ \tau_{ij} $ and $ q_{j} $ are middle-scale correlation terms of the LMS method,
| $$ \left\{ \begin{aligned} & r_j = \overline{\rho u_j}-\overline{\rho}\ \overline{u}_j = \overline{\rho' u'_j},\\& \tau_{ij} = \overline{\rho u_i u_j}-\overline{\rho}\ \overline{u}_i\ \overline{u}_j = \overline{\rho}\ \overline{u'_i\ u'_j}+\overline{u_i}\ \overline{\rho'\ u'_j}+\overline{u_j}\ \overline{\rho'\ u'_i}+\overline{\rho'\ u'_i\ u'_j},\\& q_j = \overline{(E+p)u}_j-\overline{(E+p)}\ \overline{u}_j-\overline{\sigma_{ij}u_i}+\overline{\sigma}_{ij}\ \overline{u}_i = \overline{(E'+p')u'_j}-\overline{\sigma'_{ij} u'_i}. \end{aligned}\right. $$ | (14) |
$ {\cal{T}}_{ij} $ is the small-scale SGS Reynolds stress tensor, $ {\cal{T}}_{ij} = \overline{\rho''}(\overline{u''_i u''_j}-\overline{u''}_i\ \overline{u''}_j) $, and the small-scale Smagorinsky SGS model based on middle-scale quantities is
| $$ {\cal{T}}_{ij}-\dfrac{\delta_{ij}}{3}{\cal{T}}_{kk} =-2\nu_t \bar S_{ij} =-2C^2_{\rm{s}}\Delta^2|\overline {S'}|\overline {S}_{ij}, $$ | (15) |
where $ |\overline {S'}| = \sqrt{2\overline {S'}_{ij} \overline {S'}_{ij}} $ is the modulus of middle-scale strain rate tensor, $\nu_t =C_{\rm{s}}^2 \Delta ^2|\overline{S'}| $ is the eddy viscosity, and $ \overline {S}_{ij} $ is the large-scale strain rate tensor.
$ {\cal{R}}_{j} $ is the small-scale SGS turbulence mass flux, $ {\cal{R}}_j = \overline{\rho'' u''_j} $. Its small-scale SGS turbulence mass flux model is
| $$ {\cal{R}}_j = \overline{\rho'' u''_j} = -C^2_{\rm{s}} \Delta^2|\overline {S'}|\dfrac{\overline{\partial \rho}}{\partial x_j}. $$ | (16) |
$ {\cal{Q}}_j $ is the small-scale SGS heat flux term:
| $$ {\cal{Q}}_j = \nu_t C_p Pr T. $$ | (17) |
Navier-Stokes equation − large-scale equation = middle-scale equation:
| $$ \left\{ \begin{aligned} & \rho'_{,t} = (\overline{\rho' u'_j}-\overline{\rho} u'_j-\rho' u'_j-\rho'\overline u_j)_{,j},\\& (\overline{\rho} u'_i+\rho' \overline{u_i}+\rho' u'_i-\overline{\rho' u'_i})_{,t} = \Bigg[-\overline{\rho} \overline{u_i} u'_j-\overline{\rho} u'_i\overline{u_j}-\overline{\rho} u'_i u'_j-\rho' \overline{u_i} \overline{u_j}-\rho' \overline{u_i} u'_j-\rho' u'_i \overline{u_j}-\rho' u'_i u'_j+\\& \qquad \overline{\rho} \overline{u'_i u'_j}+\overline{u_i} \overline{\rho' u'_j}+\overline{u_j} \overline{\rho' u'_i}+\overline{\rho' u'_i u'_j}- R_{\rm{g}} \delta_{ij}(\overline{\rho}\ T'+\rho' \overline{T}+\rho' T'-\overline {\rho' T'}) +\mu\left(u'_{i,j}+u'_{j,i}-\frac 2 3\delta_{ij}u'_{h,h}\right)\Bigg]_{,j},\\& \Bigg[\frac{R_{\rm{g}}}{\gamma-1}(-\overline{\rho} T'-\rho' \overline{T}+\rho'\ T'-\overline{\rho' T'}) -\frac 1 2 (2\overline{\rho} \overline{u_{k}} u'_{k}+2\overline{\rho} u'_k u'_k+\rho' \overline{u_{k}} \overline{u_{k}}+\\& \qquad 2 \rho' \overline{u_{k}} u'_{k} +\rho' u'_k u'_k+\overline{\rho} \overline{u'_k u'_k}+2\overline{u_k} \overline{\rho' u'_k}+\overline{\rho' u'_k u'_k})\Bigg]_{,t}= \\& \qquad -\Bigg\{R_{\rm{g}} (u'_{j}\overline{\rho} \overline{T}-\overline{u_{j}} \overline{\rho}\ T'-u'_{j}\overline{\rho}\ T'-\overline{u_{j}} \rho' \overline{T}-u'_{j}\rho' \overline{T}+\overline{u_{j}} \rho'\ T'+u'_{j}\rho'\ T'-\\& \qquad \overline{\rho} \overline{T' u'_j}-\overline{T} \overline{\rho' u'_j} -\overline{u_j} \overline{\rho' T'}-\overline{\rho' T' u'_j} ) +\frac 1 2 \big( \overline{u_k} \overline{\rho} \overline{u_k} u'_{j}+\overline{u_k} \overline{\rho} u'_k \overline{u_{j}}+2\overline{u_k} \overline{\rho} u'_k u'_{j} +\\& \qquad u'_k \overline{\rho} \overline{u_k} \overline{u_{j}} +u'_k \overline{\rho} u'_k\overline{u_{j}} +u'_k \overline{\rho} u'_k u'_{j}-\overline{u_k} \rho' \overline{u_k} \overline{u_{j}}-\overline{u_k} \rho' \overline{u_k} u'_{j}-2u'_k \rho' \overline{u_k} \overline{u_{j}}-\\& \qquad u'_k \rho' u'_k \overline{u_{j}}-u'_k \rho' u'_k u'_{j} -\overline{\rho} \overline{u_j} \overline{u'_k u'_k}-2\overline{\rho} \overline{u_k} \overline{u'_j u'_k}-2\overline{u_j} \overline{u_k} \overline{\rho' u'_k}-\overline{u_k} \overline{u_k} \overline{\rho' u'_j}-\\& \qquad \overline{\rho} \overline{u'_j u'_k u'_k}-\overline{u_j} \overline{\rho' u'_k u'_k} -2\overline{u_k} \overline{\rho' u'_j u'_k} -\overline{\rho' u'_j u'_k u'_k}\big)+\kappa (T')_{,j}-\\& \qquad \mu[u'_{i,j} \overline{u_{i}}+u'_{j,i} \overline{u_{i}}+\overline{u_{i}}_{,j} u'_{i} +u'_{i,j} u'_{i}+\overline{u_{j}}_{,i} u'_{i} +u'_{j,i} u'_{i}-2\overline{u'_{j,i} u'_i}-\\& \qquad \frac 2 3 (u'_{k,k} \overline{u_{i}}+\overline{u_{k}}_{,k} u'_{i}+u'_{k,k} u'_{i}-\overline{u'_{i,j} u'_j})\delta_{ij}]\Bigg\}_{,j}. \end{aligned}\right. $$ | (18) |
We defined the function space $ {\cal{B}}^{u'_{N}} $, $ {\cal{B}}^{\rho'_{N}} $ and $ {\cal{B}}^{T'_{N}} $, where the research problem is located, to satisfy the conditions as follows:
| $$ \left\{ \begin{aligned} & {\cal{B}}^{u'_{N}} = \bigg\{\big[\xi^t_{ki}\big]_{k = 1}^{N}\Big| \; \xi^t_{ki}\in{\cal{H}}^{N}\left(\varOmega\right),\; \int_{\varOmega}\xi^t_{ki}\xi^t_{li}\; \mathrm{d}\varOmega = \delta_{kl}, \; \xi^t_{ki}\; \text{second order differentiable}\bigg\},\\& {\cal{B}}^{\rho'_{N}} = \bigg\{\big[\zeta^t_{k}\big]_{k = 1}^{N}\Big| \; \zeta^t_{k}\in{\cal{H}}^{N}\left(\varOmega\right),\; \int_{\varOmega}\zeta^t_{k}\zeta^t_{l}\; \mathrm{d}\varOmega = \delta_{kl}, \; \zeta^t_{k}\; \text{second order differentiable}\bigg\},\\& {\cal{B}}^{T'_{N}} = \bigg\{\big[\eta^t_{k}\big]_{k = 1}^{N}\Big| \; \eta^t_{k}\in{\cal{H}}^{N}\left(\varOmega\right),\; \int_{\varOmega}\eta^t_{k}\eta^t_{l}\; \mathrm{d}\varOmega = \delta_{kl}, \; \eta^t_{k}\; \text{second order differentiable}\bigg\}. \end{aligned}\right. $$ | (19) |
We expressed $ u'_{i} $, $ \rho' $ and $ T' $ as low-dimensional approximate forms in the function spaces $ {\cal{B}}^{u'_{N}} $, $ {\cal{B}}^{\rho'_{N}} $ and $ {\cal{B}}^{T'_{N}} $, respectively, as follows. For convenience, the spatiotemporal optimal bases $ \xi^t_{ki},\zeta^t_k,\eta^t_k $ are expressed as $ \xi_{ki},\zeta_k,\eta_k $:
| $$ \left\{ \begin{aligned} & u'_{i} = a_{k}\xi_{ki}+u'_{Ri}\approx a_{k}\xi_{ki},\\& \rho' = b_{k}\zeta_{k}+\rho'_{R}\approx b_{k}\zeta_{k},\\& T' = c_{k}\eta_{k}+T'_{R}\approx c_{k}\eta_{k}, \end{aligned}\right. $$ | (20) |
where the truncation dimension $ N $ is omitted.
With spatiotemporal optimal bases $ \xi_{ri},\zeta_r,\eta_r $, the spectral expansion equation of the dimensional compressible turbulence middle-scale equation (18) is Galerkin formally projected onto unknown bases of $ \xi_{ri},\zeta_r,\eta_r $ respectively, to obtain the equation of the middle-scale dynamic system of compressible turbulence:
| $$ \left\{ \begin{aligned} & b_{r,t} = {\cal{A}}^{\rm{I}}_r,\\& a_{k,t}{\cal{B}}^{\rm{I}}_{kr} +b_{l,t}{\cal{B}}^{\rm{II}}_{lr} = {\cal{B}}^{\rm{III}}_{r},\\& c_{m,t}{\cal{C}}^{\rm{I}}_{mr} +b_{l,t}{\cal{C}}^{\rm{II}}_{lr} +a_{k,t}{\cal{C}}^{\rm{III}}_{kr} = {\cal{C}}^{\rm{IV}}_{r} , \end{aligned}\right. $$ | (21) |
where the coefficients are as follows:
| $$ \begin{split} & {\cal{A}}^{\rm{I}}_r = \int_\varOmega \overline{\rho' u'_{j,j}} \zeta_r\; \mathrm{d}\varOmega +\int_\varOmega \overline{\rho'_{,j} u'_{j}} \zeta_r\; \mathrm{d}\varOmega -\int_\varOmega \overline{\rho} u'_{j,j} \zeta_r\; \mathrm{d}\varOmega -\int_\varOmega \overline{\rho}_{,j} u'_{j} \zeta_r\; \mathrm{d}\varOmega-\\& \qquad \int_\varOmega \rho' u'_{j,j} \zeta_r\; \mathrm{d}\varOmega -\int_\varOmega \rho'_{,j} u'_{j} \zeta_r\; \mathrm{d}\varOmega -\int_\varOmega \rho' \overline{u_{j}}_{,j} \zeta_r\; \mathrm{d}\varOmega -\int_\varOmega \rho'_{,j} \overline{u_j} \zeta_r\; \mathrm{d}\varOmega , \end{split}$$ | (22) |
| $$ {\cal{B}}^{\rm{I}}_{kr} = \int_{\varOmega}\overline{\rho} \xi_{ki}\xi_{ri}\; \mathrm{d}\varOmega +\int_{\varOmega} \rho' \xi_{ki} \xi_{ri} \mathrm{d}\varOmega -\int_{\varOmega} \overline{\rho' \xi_{ki}} \xi_{ri} \mathrm{d}\varOmega, $$ | (23) |
| $$ {\cal{B}}^{\rm{II}}_{lr} = \int_{\varOmega} \overline{u_i} \zeta_{l} \xi_{ri}\mathrm{d}\varOmega +\int_{\varOmega} \zeta_{l} u'_i \xi_{ri} \mathrm{d}\varOmega -\int_{\varOmega} \overline{\zeta_{l} u'_i} \xi_{ri} \mathrm{d}\varOmega, $$ | (24) |
| $$ \begin{split} & {\cal{B}}^{\rm{III}}_{r} = -\int_{\varOmega}\overline{\rho}_{,t} u'_i \xi_{ri}\; \mathrm{d}\varOmega -\int_{\varOmega} \overline{u_{i}}_{,t} \rho' \xi_{ri}\mathrm{d}\varOmega -\int_{\varOmega} \overline{\rho}_{,j}\ \overline{u_i}\ u'_j \xi_{ri}\mathrm{d}\varOmega -\int_{\varOmega} \overline{\rho}\ \overline{u_{i}}_{,j}\ u'_j \xi_{ri}\mathrm{d}\varOmega -\int_{\varOmega} \overline{\rho}\ \overline{u_{i}}\ u'_{j,j} \xi_{ri}\mathrm{d}\varOmega-\\& \qquad \int_{\varOmega} \overline{\rho}_{,j}\ u'_i \overline{u_j} \xi_{ri}\mathrm{d}\varOmega -\int_{\varOmega} \overline{\rho}\ u'_{i,j} \overline{u_j} \xi_{ri}\mathrm{d}\varOmega -\int_{\varOmega} \overline{\rho}\ u'_i \overline{u_{j}}_{,j} \xi_{ri}\mathrm{d}\varOmega -\int_{\varOmega} \overline{\rho}_{,j}\ u'_i\ u'_j \xi_{ri}\mathrm{d}\varOmega -\int_{\varOmega} \overline{\rho}\ u'_{i,j}\ u'_{j} \xi_{ri}\mathrm{d}\varOmega-\\& \qquad \int_{\varOmega} \overline{\rho}\ u'_{i}\ u'_{j,j} \xi_{ri}\mathrm{d}\varOmega -\int_{\varOmega} \rho'_{,j} \overline{u_i}\ \overline{u_j} \xi_{ri}\mathrm{d}\varOmega -\int_{\varOmega} \rho' \overline{u_{i}}_{,j}\ \overline{u_j} \xi_{ri}\mathrm{d}\varOmega -\int_{\varOmega} \rho' \overline{u_{i}}\ \overline{u_{j}}_{,j} \xi_{ri}\mathrm{d}\varOmega -2\int_{\varOmega} \rho'_{,j} \overline{u_i}\ u'_{j} \xi_{ri}\mathrm{d}\varOmega-\\& \qquad\int_{\varOmega} \rho' \overline{u_{i}}_{,j}\ u'_{j} \xi_{ri}\mathrm{d}\varOmega -\int_{\varOmega} \rho' \overline{u_{i}}\ u'_{j,j} \xi_{ri}\mathrm{d}\varOmega -\int_{\varOmega} \rho' u'_{i,j} \overline{u_j} \xi_{ri}\mathrm{d}\varOmega -\int_{\varOmega} \rho' u'_{i} \overline{u_{j}}_{,j} \xi_{ri}\mathrm{d}\varOmega -\int_{\varOmega} \rho'_{,j} u'_{i} u'_{j} \xi_{ri}\mathrm{d}\varOmega-\\& \qquad\int_{\varOmega} \rho' u'_{i,j} u'_{j} \xi_{ri}\mathrm{d}\varOmega -\int_{\varOmega} \rho' u'_{i} u'_{j,j} \xi_{ri}\mathrm{d}\varOmega +\int_{\varOmega} \overline{\rho}_{,j}\ \overline{u'_{i} u'_{j}} \xi_{ri}\mathrm{d}\varOmega +\int_{\varOmega} \overline{\rho}\ \overline{u'_{i,j} u'_{j}} \xi_{ri}\mathrm{d}\varOmega +\int_{\varOmega} \overline{\rho}\ \overline{u'_{i} u'_{j,j}} \xi_{ri}\mathrm{d}\varOmega+\\& \qquad2\int_{\varOmega} \overline{u_{i}}_{,j}\ \overline{\rho' u'_{j}} \xi_{ri}\mathrm{d}\varOmega +2\int_{\varOmega} \overline{u_{i}}\ \overline{\rho'_{,j} u'_{j}} \xi_{ri}\mathrm{d}\varOmega +2\int_{\varOmega} \overline{u_{i}}\ \overline{\rho' u'_{j,j}} \xi_{ri}\mathrm{d}\varOmega +\int_{\varOmega} \overline{u_{j}}_{,j}\ \overline{\rho' u'_{i}} \xi_{ri}\mathrm{d}\varOmega+\\& \qquad\int_{\varOmega} \overline{u_{j}}\ \overline{\rho'_{,j} u'_{i}} \xi_{ri}\mathrm{d}\varOmega +\int_{\varOmega} \overline{u_{j}}\ \overline{\rho' u'_{i,j}} \xi_{ri}\mathrm{d}\varOmega +\int_{\varOmega} \overline{\rho'_{,j} u'_{i} u'_{j}} \xi_{ri}\mathrm{d}\varOmega +\int_{\varOmega} \overline{\rho' u'_{i,j} u'_{j}} \xi_{ri}\mathrm{d}\varOmega +\int_{\varOmega} \overline{\rho' u'_{i} u'_{j,j}} \xi_{ri}\mathrm{d}\varOmega-\\& \qquad R_{\rm{g}}\delta_{ij} \Bigg(\int_{\varOmega} \overline{\rho}_{,j}\ T' \xi_{ri}\mathrm{d}\varOmega +\int_{\varOmega} \overline{\rho}\ T'_{,j} \xi_{ri}\mathrm{d}\varOmega +\int_{\varOmega} \rho'_{,j} \overline{T} \xi_{ri}\mathrm{d}\varOmega +\int_{\varOmega} \rho' \overline{T}_{,j} \xi_{ri}\mathrm{d}\varOmega +\int_{\varOmega} \rho'_{,j} T' \xi_{ri}\mathrm{d}\varOmega+\\& \qquad\int_{\varOmega} \rho' T'_{,j} \xi_{ri}\mathrm{d}\varOmega -\int_{\varOmega} \overline {\rho'_{,j} T'} \xi_{ri}\mathrm{d}\varOmega -\int_{\varOmega} \overline {\rho' T'_{,j}} \xi_{ri}\mathrm{d}\varOmega\Bigg) +\mu\Bigg(\int_{\varOmega} u'_{i,jj} \xi_{ri} \mathrm{d}\varOmega +\frac 1 3\int_{\varOmega}u'_{j,ij} \xi_{ri} \mathrm{d}\varOmega\Bigg) , \end{split} $$ | (25) |
| $$ {\cal{C}}^{\rm{I}}_{mr} = C_v\Bigg(-\int_\varOmega \overline{\rho}\ \eta_{m} \eta_r\mathrm{d}\varOmega +\int_\varOmega \rho'\ \eta_{m} \eta_r\mathrm{d}\varOmega -\int_\varOmega \overline{\rho' \eta_{m}} \eta_r\mathrm{d}\varOmega\Bigg), $$ | (26) |
| $$ \begin{split} & {\cal{C}}^{\rm{II}}_{lr} = \Bigg[C_v\Bigg(-\int_\varOmega \zeta_{l}\ \overline{T} \eta_r\mathrm{d}\varOmega +\int_\varOmega \zeta_{l}\ T' \eta_r\mathrm{d}\varOmega -\int_\varOmega \overline{\zeta_{l} T'} \eta_r\mathrm{d}\varOmega\Bigg)-\\& \qquad\frac 1 2 \Bigg(\int_\varOmega \zeta_{l}\ \overline{u_{k}}\ \overline{u_{k}} \eta_r\mathrm{d}\varOmega +2\int_\varOmega \zeta_{l}\ \overline{u_{k}}\ u'_{k} \eta_r\mathrm{d}\varOmega +\int_\varOmega \zeta_{l}\ u'_{k} u'_{k} \eta_r\mathrm{d}\varOmega+\\& \qquad2\int_\varOmega \overline{u_k}\ \overline{\zeta_{l} u'_{k}} \eta_r\mathrm{d}\varOmega +\int_\varOmega \overline{\zeta_{l} u'_{k} u'_{k}} \eta_r\mathrm{d}\varOmega\Bigg)\Bigg], \end{split}$$ | (27) |
| $$ \begin{aligned} & {\cal{C}}^{\rm{III}}_{kr} = -\Bigg(\int_\varOmega \overline{\rho} \ \overline{u_{l}}\ \xi_{kl} \eta_r\mathrm{d}\varOmega +2\int_\varOmega \overline{\rho}\ \xi_{kl} u'_{l} \eta_r\mathrm{d}\varOmega +\int_\varOmega \rho'\ \overline{u_{l}}\ \xi_{kl} \eta_r\mathrm{d}\varOmega+\\& \qquad \int_\varOmega \rho'\ \xi_{kl} u'_{l} \eta_r\mathrm{d}\varOmega +\int_\varOmega \overline{\rho}\ \overline{\xi_{kl} u'_{l}} \eta_r\mathrm{d}\varOmega +\int_\varOmega \overline{u_l}\ \overline{\rho' \xi_{kl}} \eta_r\mathrm{d}\varOmega +\int_\varOmega \overline{\rho' \xi_{kl} u'_{l}} \eta_r\mathrm{d}\varOmega\Bigg) , \end{aligned} $$ | (28) |
| $$ \begin{split} & {\cal{C}}^{\rm{IV}}_{r} = -C_v\gamma\Bigg( -\int_\varOmega \overline{\rho}_{,t}\ T' \eta_r\mathrm{d}\varOmega -\int_\varOmega \rho'\ \overline{T}_{,t} \eta_r\mathrm{d}\varOmega +\int_\varOmega u'_{j,j} \overline{\rho}\ \overline{T} \eta_r\mathrm{d}\varOmega +\int_\varOmega u'_{j} \overline{\rho}_{,j}\ \overline{T} \eta_r\mathrm{d}\varOmega+\\& \qquad \int_\varOmega u'_{j} \overline{\rho}\ \overline{T}_{,j} \eta_r\mathrm{d}\varOmega -\int_\varOmega \overline{u_{j}}_{,j}\ \overline{\rho}\ T' \eta_r\mathrm{d}\varOmega -\int_\varOmega \overline{u_{j}}\ \overline{\rho}_{,j}\ T' \eta_r\mathrm{d}\varOmega -\int_\varOmega \overline{u_{j}}\ \overline{\rho}\ T'_{,j} \eta_r\mathrm{d}\varOmega-\\& \qquad \int_\varOmega u'_{j,j} \overline{\rho}\ T' \eta_r\mathrm{d}\varOmega -\int_\varOmega u'_{j} \overline{\rho}_{,j}\ T' \eta_r\mathrm{d}\varOmega -\int_\varOmega u'_{j} \overline{\rho}\ T'_{,j} \eta_r\mathrm{d}\varOmega -\int_\varOmega \overline{u_{j}}_{,j}\ \rho'\ \overline{T} \eta_r\mathrm{d}\varOmega-\\& \qquad \int_\varOmega \overline{u_{j}}\ \rho'_{,j}\ \overline{T} \eta_r\mathrm{d}\varOmega -\int_\varOmega \overline{u_{j}}\ \rho'\ \overline{T}_{,j} \eta_r\mathrm{d}\varOmega -\int_\varOmega u'_{j,j} \rho'\ \overline{T} \eta_r\mathrm{d}\varOmega -\int_\varOmega u'_{j} T'_{,j}\ \overline{T} \eta_r\mathrm{d}\varOmega-\\& \qquad \int_\varOmega u'_{j} T'\ \overline{T}_{,j} \eta_r\mathrm{d}\varOmega +\int_\varOmega \overline{u_{j}}_{,j}\ \rho'\ T' \eta_r\mathrm{d}\varOmega +\int_\varOmega \overline{u_{j}}\ \rho'_{,j}\ T' \eta_r\mathrm{d}\varOmega +\int_\varOmega \overline{u_{j}}\ \rho'\ T'_{,j} \eta_r\mathrm{d}\varOmega+\\& \qquad \int_\varOmega u'_{j,j} \rho'\ T' \eta_r\mathrm{d}\varOmega +\int_\varOmega u'_{j} \rho'_{,j}\ T' \eta_r\mathrm{d}\varOmega +\int_\varOmega u'_{j} \rho'\ T'_{,j} \eta_r\mathrm{d}\varOmega -\int_\varOmega \overline{\rho}_{,j}\ \overline{T' u'_{j}} \eta_r\mathrm{d}\varOmega-\\& \qquad \int_\varOmega \overline{\rho}\ \overline{T'_{,j} u'_{j}} \eta_r\mathrm{d}\varOmega -\int_\varOmega \overline{\rho}\ \overline{T' u'_{j,j}} \eta_r\mathrm{d}\varOmega -\int_\varOmega \overline{T}_{,j}\ \overline{\rho' u'_{j}} \eta_r\mathrm{d}\varOmega -\int_\varOmega \overline{T}\ \overline{\rho'_{,j} u'_{j}} \eta_r\mathrm{d}\varOmega-\\& \qquad \int_\varOmega \overline{T}\ \overline{\rho' u'_{j,j}} \eta_r\mathrm{d}\varOmega -\int_\varOmega \overline{u_j}_{,j}\ \overline{\rho' T'} \eta_r\mathrm{d}\varOmega -\int_\varOmega \overline{u_j}\ \overline{\rho'_{,j} T'} \eta_r\mathrm{d}\varOmega -\int_\varOmega \overline{u_j}\ \overline{\rho' T'_{,j}} \eta_r\mathrm{d}\varOmega-\\& \qquad \int_\varOmega \overline{\rho'_{,j} T' u'_{j}} \eta_r\mathrm{d}\varOmega -\int_\varOmega \overline{\rho' T'_{,j} u'_{j}} \eta_r\mathrm{d}\varOmega -\int_\varOmega \overline{\rho' T' u'_{j,j}} \eta_r\mathrm{d}\varOmega\Bigg) -\\& \qquad\frac 1 2 \Bigg( -2 \int_\varOmega \overline{\rho}_{,t}\ \overline{u_{k}}\ u'_{k} \eta_r\mathrm{d}\varOmega -2 \int_\varOmega \overline{\rho}\ \overline{u_{k}}_{,t}\ u'_{k} \eta_r\mathrm{d}\varOmega -2 \int_\varOmega \overline{\rho}_{,t}\ u'_{k} u'_{k} \eta_r\mathrm{d}\varOmega -2\int_\varOmega \rho'\ \overline{u_{k}}_{,t}\ \overline{u_{k}} \eta_r\mathrm{d}\varOmega - \end{split} $$ |
| $$ \begin{split} & \qquad 2 \int_\varOmega \rho'\ \overline{u_{k}}_{,t}\ u'_{k} \eta_r\mathrm{d}\varOmega -\int_\varOmega \overline{\rho}_{,t}\ \overline{u'_{k} u'_{k}} \eta_r\mathrm{d}\varOmega -2 \int_\varOmega \overline{u_k}_{,t}\ \overline{\rho' u'_{k}} \eta_r\mathrm{d}\varOmega +2 \int_\varOmega \overline{u_k}_{,j}\ \overline{\rho}\ \overline{u_k}\ u'_{j} \eta_r\mathrm{d}\varOmega+\\& \qquad\int_\varOmega \overline{u_k}\ \overline{\rho}_{,j}\ \overline{u_k}\ u'_{j} \eta_r\mathrm{d}\varOmega +\int_\varOmega \overline{u_k}_{,j}\ \overline{\rho}\ u'_{k}\ \overline{u_{j}} \eta_r\mathrm{d}\varOmega +\int_\varOmega \overline{u_k}\ \overline{\rho}_{,j}\ u'_{k}\ \overline{u_{j}} \eta_r\mathrm{d}\varOmega +\int_\varOmega \overline{u_k}\ \overline{\rho}\ u'_{k}\ \overline{u_j}_{,j} \eta_r\mathrm{d}\varOmega+\\& \qquad2\int_\varOmega \overline{u_k}_{,j}\ \overline{\rho}\ u'_{k} u'_{j} \eta_r\mathrm{d}\varOmega +2\int_\varOmega \overline{u_k}\ \overline{\rho}_{,j}\ u'_{k} u'_{j} \eta_r\mathrm{d}\varOmega +2\int_\varOmega \overline{u_k}\ \overline{\rho}\ u'_{k,j} u'_{j} \eta_r\mathrm{d}\varOmega +2\int_\varOmega \overline{u_k}\ \overline{\rho}\ u'_{k} u'_{j,j} \eta_r\mathrm{d}\varOmega+\\& \qquad\int_\varOmega u'_{k}\ \overline{\rho}_{,j}\ \overline{u_k}\ \overline{u_{j}} \eta_r\mathrm{d}\varOmega +\int_\varOmega u'_{k}\ \overline{\rho}\ \overline{u_k}_{,j}\ \overline{u_{j}} \eta_r\mathrm{d}\varOmega +\int_\varOmega u'_{k}\ \overline{\rho}\ \overline{u_k}\ \overline{u_j}_{,j} \eta_r\mathrm{d}\varOmega +\int_\varOmega u'_{k}\ \overline{\rho}_{,j}\ u'_{k} \overline{u_{j}} \eta_r\mathrm{d}\varOmega+\\& \qquad 2\int_\varOmega u'_{k}\ \overline{\rho}\ u'_{k,j} \overline{u_{j}} \eta_r\mathrm{d}\varOmega +\int_\varOmega u'_{k}\ \overline{\rho}\ u'_{k} \overline{u_j}_{,j} \eta_r\mathrm{d}\varOmega +2\int_\varOmega u'_{k,j}\ \overline{\rho}\ u'_{k}\ u'_{j} \eta_r\mathrm{d}\varOmega+\\& \qquad\int_\varOmega u'_{k}\ \overline{\rho}_{,j}\ u'_{k}\ u'_{j} \eta_r\mathrm{d}\varOmega +\int_\varOmega u'_{k}\ \overline{\rho}\ u'_{k}\ u'_{j,j} \eta_r\mathrm{d}\varOmega -2\int_\varOmega \overline{u_k}_{,j}\ \rho'\ \overline{u_k}\ \overline{u_{j}} \eta_r\mathrm{d}\varOmega -\int_\varOmega \overline{u_k}\ \rho'_{,j}\ \overline{u_k}\ \overline{u_{j}} \eta_r\mathrm{d}\varOmega - \end{split} $$ |
| $$ \begin{split} & \qquad \int_\varOmega \overline{u_k}\ \rho'\ \overline{u_k}\ \overline{u_j}_{,j} \eta_r\mathrm{d}\varOmega -2\int_\varOmega \overline{u_k}_{,j}\ \rho'\ \overline{u_k}\ u'_{j} \eta_r\mathrm{d}\varOmega -\int_\varOmega \overline{u_k}\ \rho'_{,j}\ \overline{u_k}\ u'_{j} \eta_r\mathrm{d}\varOmega-\\& \qquad \int_\varOmega \overline{u_k}\ \rho'\ \overline{u_k}\ u'_{j,j} \eta_r\mathrm{d}\varOmega -2\int_\varOmega u'_{k,j}\ \rho'\ \overline{u_k}\ \overline{u_{j}} \eta_r\mathrm{d}\varOmega -2\int_\varOmega u'_{k}\ \rho'_{,j}\ \overline{u_k}\ \overline{u_{j}} \eta_r\mathrm{d}\varOmega-\\& \qquad 2\int_\varOmega u'_{k}\ \rho'\ \overline{u_k}_{,j}\ \overline{u_{j}} \eta_r\mathrm{d}\varOmega -2\int_\varOmega u'_{k}\ \rho'\ \overline{u_k}\ \overline{u_j}_{,j} \eta_r\mathrm{d}\varOmega -\int_\varOmega u'_{k,j}\ \rho'\ u'_{k}\ \overline{u_{j}} \eta_r\mathrm{d}\varOmega -\int_\varOmega u'_{k}\ \rho'_{,j}\ u'_{k}\ \overline{u_{j}} \eta_r\mathrm{d}\varOmega-\\& \qquad\int_\varOmega u'_{k}\ \rho'\ u'_{k,j}\ \overline{u_{j}} \eta_r\mathrm{d}\varOmega -\int_\varOmega u'_{k}\ \rho'\ u'_{k}\ \overline{u_j}_{,j} \eta_r\mathrm{d}\varOmega -2\int_\varOmega u'_{k,j}\ \rho'\ u'_{k}\ u'_{j} \eta_r\mathrm{d}\varOmega -\int_\varOmega u'_{k}\ \rho'_{,j}\ u'_{k}\ u'_{j} \eta_r\mathrm{d}\varOmega-\\& \qquad\int_\varOmega u'_{k}\ \rho'\ u'_{k}\ u'_{j,j} \eta_r\mathrm{d}\varOmega -\int_\varOmega \overline{\rho}_{,j}\ \overline{u_j}\ \overline{u'_{k}\ u'_{k}} \eta_r\mathrm{d}\varOmega -\int_\varOmega \overline{\rho}\ \overline{u_j}_{,j}\ \overline{u'_{k}\ u'_{k}} \eta_r\mathrm{d}\varOmega -2\int_\varOmega \overline{\rho}\ \overline{u_j}\ \overline{u'_{k,j}\ u'_{k}} \eta_r\mathrm{d}\varOmega- \end{split} $$ |
| $$ \begin{split} & \qquad2\int_\varOmega \overline{\rho}_{,j}\ \overline{u_k}\ \overline{u'_{j}\ u'_{k}} \eta_r\mathrm{d}\varOmega -2\int_\varOmega \overline{\rho}\ \overline{u_k}_{,j}\ \overline{u'_{j}\ u'_{k}} \eta_r\mathrm{d}\varOmega -2\int_\varOmega \overline{\rho}\ \overline{u_k}\ \overline{u'_{j,j}\ u'_{k}} \eta_r\mathrm{d}\varOmega -2\int_\varOmega \overline{\rho}\ \overline{u_k}\ \overline{u'_{j}\ u'_{k,j}} \eta_r\mathrm{d}\varOmega-\\& \qquad 2\int_\varOmega \overline{u_j}_{,j}\ \overline{u_k}\ \overline{\rho'\ u'_{k}} \eta_r\mathrm{d}\varOmega -2\int_\varOmega \overline{u_j}\ \overline{u_k}_{,j}\ \overline{\rho'\ u'_{k}} \eta_r\mathrm{d}\varOmega -2\int_\varOmega \overline{u_j}\ \overline{u_k}\ \overline{\rho'_{,j}\ u'_{k}} \eta_r\mathrm{d}\varOmega -2\int_\varOmega \overline{u_j}\ \overline{u_k}\ \overline{\rho'\ u'_{k,j}} \eta_r\mathrm{d}\varOmega-\\& \qquad 2\int_\varOmega \overline{u_k}_{,j}\ \overline{u_k}\ \overline{\rho'\ u'_{j}} \eta_r\mathrm{d}\varOmega -\int_\varOmega \overline{u_k}\ \overline{u_k}\ \overline{\rho'_{,j}\ u'_{j}} \eta_r\mathrm{d}\varOmega -\int_\varOmega \overline{u_k}\ \overline{u_k}\ \overline{\rho'\ u'_{j,j}} \eta_r\mathrm{d}\varOmega -\int_\varOmega \overline{\rho}_{,j}\ \overline{u'_{j}\ u'_{k}\ u'_{k}} \eta_r\mathrm{d}\varOmega-\\& \qquad \int_\varOmega \overline{\rho}\ \overline{u'_{j,j}\ u'_{k}\ u'_{k}} \eta_r\mathrm{d}\varOmega -2\int_\varOmega \overline{\rho}\ \overline{u'_{j}\ u'_{k,j}\ u'_{k}} \eta_r\mathrm{d}\varOmega -\int_\varOmega \overline{u_j}_{,j}\ \overline{\rho'\ u'_{k}\ u'_{k}} \eta_r\mathrm{d}\varOmega -\int_\varOmega \overline{u_j}\ \overline{\rho'_{,j}\ u'_{k}\ u'_{k}} \eta_r\mathrm{d}\varOmega-\\& \qquad 2\int_\varOmega \overline{u_j}\ \overline{\rho'\ u'_{k,j}\ u'_{k}} \eta_r\mathrm{d}\varOmega -2\int_\varOmega \overline{u_k}_{,j}\ \overline{\rho'\ u'_{j}\ u'_{k}} \eta_r\mathrm{d}\varOmega -2\int_\varOmega \overline{u_k}\ \overline{\rho'_{,j}\ u'_{j}\ u'_{k}} \eta_r\mathrm{d}\varOmega -2\int_\varOmega \overline{u_k}\ \overline{\rho'\ u'_{j,j}\ u'_{k}} \eta_r\mathrm{d}\varOmega-\\& \qquad 2\int_\varOmega \overline{u_k}\ \overline{\rho'\ u'_{j}\ u'_{k,j}} \eta_r\mathrm{d}\varOmega -\int_\varOmega \overline{\rho'_{,j}\ u'_{j}\ u'_{k}\ u'_{k}} \eta_r\mathrm{d}\varOmega -\int_\varOmega \overline{\rho'\ u'_{j,j}\ u'_{k}\ u'_{k}} \eta_r\mathrm{d}\varOmega -2\int_\varOmega \overline{\rho'\ u'_{j}\ u'_{k,j}\ u'_{k}} \eta_r\mathrm{d}\varOmega\Bigg)-\\& \qquad \kappa \int_\varOmega T'_{,jj} \eta_r\mathrm{d}\varOmega+\mu\Bigg[ \int_\varOmega u'_{i,jj}\ \overline{u_{i}} \eta_r\mathrm{d}\varOmega +\int_\varOmega u'_{i,j}\ \overline{u_i}_{,j} \eta_r\mathrm{d}\varOmega +\int_\varOmega u'_{j,ij}\ \overline{u_{i}} \eta_r\mathrm{d}\varOmega +\int_\varOmega u'_{j,i}\ \overline{u_i}_{,j} \eta_r\mathrm{d}\varOmega+\\& \qquad \int_\varOmega \overline{u_{i}}_{,jj}\ u'_{i} \eta_r\mathrm{d}\varOmega +\int_\varOmega \overline{u_{i}}_{,j}\ u'_{i,j} \eta_r\mathrm{d}\varOmega +\int_\varOmega u'_{i,jj}\ u'_{i} \eta_r\mathrm{d}\varOmega +\int_\varOmega u'_{i,j}\ u'_{i,j} \eta_r\mathrm{d}\varOmega +\int_\varOmega \overline{u_{j}}_{,ij}\ u'_{i} \eta_r\mathrm{d}\varOmega+\\& \qquad \int_\varOmega \overline{u_{j}}_{,i}\ u'_{i,j} \eta_r\mathrm{d}\varOmega +\int_\varOmega u'_{j,ij}\ u'_{i} \eta_r\mathrm{d}\varOmega +\int_\varOmega u'_{j,i}\ u'_{i,j} \eta_r\mathrm{d}\varOmega -2\int_\varOmega \overline{u'_{j,ij} u'_{i}} \eta_r\mathrm{d}\varOmega-\\& \qquad 2\int_\varOmega \overline{u'_{j,i} u'_{i,j}} \eta_r\mathrm{d}\varOmega -\frac 2 3 \Bigg( \int_\varOmega u'_{k,kj}\ \overline{u_{i}} \delta_{ij} \eta_r\mathrm{d}\varOmega +2\int_\varOmega u'_{k,k}\ \overline{u_i}_{,j} \delta_{ij} \eta_r\mathrm{d}\varOmega +\int_\varOmega \overline{u_k}_{,kj}\ u'_{i} \delta_{ij} \eta_r\mathrm{d}\varOmega+\\& \qquad\int_\varOmega u'_{k,kj}\ u'_{i} \delta_{ij} \eta_r\mathrm{d}\varOmega +\int_\varOmega u'_{k,k}\ u'_{i,j} \delta_{ij} \eta_r\mathrm{d}\varOmega -\int_\varOmega \overline{u'_{i,jj} u'_{j}} \delta_{ij} \eta_r\mathrm{d}\varOmega -\int_\varOmega \overline{u'_{i,j} u'_{j,j}} \delta_{ij} \eta_r\mathrm{d}\varOmega\Bigg)\Bigg] . \end{split}$$ | (29) |
The optimal functional for approximating the initial conditions of a dimensional compressible turbulence middle-scale dynamical system $ J $ can be found below.
With middle-scale optimal bases for approximating the initial conditions of large-scale fields $ \xi_{ki},\;\zeta_k,\;\eta_k $, the following optimal functional can be obtained:
| $$ \left\{ \begin{aligned} & J(\xi_{ki}, \zeta_{k}, \eta_{k}) = \int_{\varOmega} (\overline{{u_R}_i(0)}, \overline{{u_R}_i(0)}) \; \mathrm{d} \varOmega + \int_{\varOmega} (\overline{\rho_R(0)}, \overline{\rho_R(0)}) \; \mathrm{d} \varOmega + \int_{\varOmega} (\overline{T_R(0)}, \overline{T_R(0)}) \; \mathrm{d} \varOmega = \\& \qquad \int_{\varOmega} \Bigg( \overline{{u}_i(0)} - a_k(0) \xi_{ki} \Bigg) \Bigg( \overline{{u}_i(0)} - a_k(0) \xi_{ki} \Bigg) \; \mathrm{d} \varOmega+\\& \qquad \int_{\varOmega} \Bigg( \overline{{\rho}(0)} - b_k(0) \zeta_{k} \Bigg) \Bigg( \overline{{\rho}(0)} - b_k(0) \zeta_{k} \Bigg) \; \mathrm{d} \varOmega+\\& \qquad \int_{\varOmega} \Bigg( \overline{{T}(0)} - c_k(0) \eta_{k} \Bigg) \Bigg( \overline{{T}(0)} - c_k(0) \eta_{k} \Bigg) \; \mathrm{d} \varOmega=\\& \qquad \int_{\varOmega} \Bigg[\overline{{u}_i(0)}\ \overline{{u}_i(0)} - 2 a_k(0) \xi_{ki}\overline{{u}_i(0)} + a_k(0) \xi_{ki} a_k(0) \xi_{ki}\Bigg] \; \mathrm{d} \varOmega +\\& \qquad \int_{\varOmega} \Bigg[\overline{{\rho}(0)}\ \overline{{\rho}(0)} - 2 b_k(0) \zeta_{k}\overline{{\rho}(0)} + b_k(0) \zeta_{k} b_k(0) \zeta_{k}\Bigg] \; \mathrm{d} \varOmega +\\& \qquad \int_{\varOmega} \Bigg[\overline{{T}(0)}\ \overline{{T}(0)} - 2 c_k(0) \eta_{k}\overline{{T}(0)} + c_k(0) \eta_{k} c_k(0) \eta_{k}\Bigg] \; \mathrm{d} \varOmega , \end{aligned}\right. $$ | (30) |
where $ \overline{{u}_i(0)},\;\overline{\rho(0)},\;\overline{{T}(0)} $ are the initial values of the known large-scale field.
At the initial moment, the concept of the turbulence intensity[10-11] is adopted:
| $$ I\equiv\frac{u'}{U} = 0.16 {{Re}}^{-\tfrac 1 8} .$$ | (31) |
The initial values of the middle-scale flow field variables $ u_i(0),\;\rho(0),\;T(0) $ are obtained, where $ U = \sqrt{\overline{u_i}\ \overline{u_i}} $ is the large-scale velocity and $ {{Re}} $ is the Reynolds number,
With the scale similarity concept, the large-scale POD bases of $ \xi^0_{ki},\;\zeta^0_k,\;\eta^0_k $ and the large-scale flow field in a large-scale grid are interpolated into a middle-scale grid ( $ {\Delta {\boldsymbol{x}}}/{2^p},\; p\geq 1 $), and the large-scale POD bases are used as the initial middle-scale bases to obtain the initial condition of their coefficients:
| $$ \left\{ \begin{aligned} & b_k(0) = \left(\rho(0),\zeta_{k}\right),\\& a_k(0) = \left(u_i(0),\xi_{ki}\right),\\& c_k(0) = \left(T(0),\eta_{k}\right) . \end{aligned}\right. $$ | (32) |
The fourth-order Runge-Kutta method and ill-conditioned AFD algorithm are used to solve the middle-scale optimal low-dimensional dynamical system equation (21) to obtain $ a^i_k,b^i_k,c^i_k $;
♣ We used equation (20) to obtain the new solution of the middle-scale flow field and the correlation functions (14), and the small-scale SGS terms equations (15), (16) and (17);
Using the middle-scale flow field variables and the correlation functions, which are on the overlapped large- and middle-scale grids and the small-scale SGS terms equations (15), (16) and (17), to obtain the terms on the right-hand side of equation (13), the new solution of the large-scale flow field can be obtained by substituting these into the large-scale equation (13), and the new solution of the large-scale flow field is applied to the middle-scale dynamical system equation (21).
We used equation (30) together with the new values of the large-scale flow field interpolated into the middle-scale grid to obtain the new spatiotemporal optimal bases $ \xi_{ki},\zeta_k,\eta_k $.
Goto ♣ or the LMS simulation is completed.
In this way, the approximate solution of the turbulence flow field (= large-scale mean field+middle-scale flow field) can also be obtained.
The initial state of the re-shock RM instability problem is shown in fig. 4.
The re-shock RM instability of light/heavy fluid interfaces with multimodal initial disturbances under plane shock acceleration in the shock tube is studied.
| $$ D = \left \{{\boldsymbol{x}}| 0\le y, z\le S, 0\le x \le 2L\right\} ,$$ | (33) |
where $ S = 0.01 $ m and $ L = 0.014 $ m. The numbers of equal spacing grids are flow direction grids × spanwise grids × normal grid = $ 147\times 61\times 61 = 546\;987 $. The time step is $ 0.000\;000\;1\;{\rm{s}} $. The shock wave initially located at $ x = 0.01 $ m moves in the $ x $ direction, and the shock Mach number is $ Ma_{\rm{S}} = 1.5 $. The initial interface is located at $ x = L $, and the initial interface function is
| $$ \varphi({\boldsymbol{x}}) = x-L-\eta(y,z) = 0. $$ | (34) |
We adopted the multimodal perturbation form provided by Tritschler et al.[12]
| $$ \eta(y,z) = a_0\sum^{13}\limits_{n = 1}\sum^{15}\limits_{m = 3}a_{n,m}\sin(k_n y+\phi_n)\sin(k_m z+\chi_m), $$ | (35) |
where the main amplitude is $ a_0 = 4\times10^{-5} $ m, the secondary amplitudes of different modes are $ a_{n,m} = \sin(nm)/2 $, the wave numbers are $ k_n = 2{{\text{π}}} n/L $ and $ k_m = 2{{\text{π}}} m/L $, and the phase shifts are $ \phi_n = \tan(n) $ and $ \chi_n = \tan(m) $. Although equation (35) is similar to the form of random perturbation, it has a definite expression and can be repeatedly constructed in different examples.
The light/heavy fluids on both sides of the interface are composed of air/SF6 components, and their molar masses are $ M_1 = 28.964 \;{\rm{g}}\cdot {\rm{mol}}^{-1} $ and $ M_2 = 146.057 \;{\rm{g}}\cdot {\rm{mol}}^{-1} $, respectively. We set the initial pressure and temperature to $ p_0 = 23\;000 \;{\rm{Pa}} $ and $ T_0 = 298\;{\rm{K}} $, respectively. Then, the density is obtained from the equation of state (10):
| $$ \rho_i = \dfrac{p_0 M_i}{R T_0}, $$ | (36) |
i.e.,
| $$ \rho_1 = \dfrac{p_0 M_1}{R T_0} = \dfrac{23\;000 \;{\rm{Pa}}\times 28.964 \;{\rm{g}}\cdot{\rm{mol}}^{-1}}{8.314\; {\rm{Pa}}\cdot{\rm{m}}^3\cdot{\rm{K}}^{-1}\times 298\;{\rm{K}}} = 0.269\;{{\rm{kg}}}/{{\rm{m}}^3}, $$ | (37) |
| $$ \rho_2 = \dfrac{p_0 M_2}{R T_0} = \dfrac{23\;000 \;{\rm{Pa}}\times 146.057\; {\rm{g}}\cdot{\rm{mol}}^{-1}}{8.314\; {\rm{Pa}}\cdot{\rm{m}}^3\cdot{\rm{K}}^{-1}\times 298\;{\rm{K}}} = 1.356\;{{\rm{kg}}}/{{\rm{m}}^3}. $$ | (38) |
Atwood number is
| $$ A = \dfrac{\rho_2 -\rho_1}{\rho_1+\rho_2} = \dfrac{1.356-0.269}{0.269+1.356} = 0.669 .$$ | (39) |
1) Initial condition
In summary, it can be seen that in the study of RM instability, the fluid inside the shock tube at the initial moment is divided into three regions by the shock wave and interface: light fluid behind the wave (zone Ⅰ), light fluid before the wave (zone Ⅱ), and heavy fluid before the wave (zone Ⅲ). The light fluid after the shock wave needs to be determined on the shock wave, before and after, by the Rankine Hugoniot condition[13], and the specific heat ratio is $ \gamma_1 = 1.4 $. The specific expression of the density is as follows:
| $$ \rho_{\rm{S}} = \dfrac{(\gamma_1 + 1) Ma_{\rm{S}}^2}{2+(\gamma_1-1)Ma_{\rm{S}}^2}\rho_1 = \dfrac{(1.4+1)\times 1.5^2}{2+(1.4-1)\times 1.5^2}\times 0.269 = 0.501\;{{\rm{kg}}}/{{\rm{m}}^3}. $$ | (40) |
The pressure can be expressed as follows:
| $$ p_{\rm{S}} = \dfrac{2\gamma_1 Ma_{\rm{S}}^2-\gamma_1 + 1}{\gamma_1+1}p_0 = \dfrac{2\times 1.4\times 1.5^2-1.4+1}{1.4+1}\times 23\;000 = 56\;541.667\; {\rm{Pa}} .$$ | (41) |
Finally, the velocity can be represented as follows:
| $$ {\boldsymbol{U}}_{\rm{S}} = U_{\rm{S}} {\boldsymbol{i}},\; U_{\rm{S}} = \dfrac{2\left (Ma_{\rm{S}}^2-1\right )}{(\gamma_1+1)Ma_{\rm{S}}}\sqrt{\dfrac{\gamma_1 p_0}{\rho_1}} = \dfrac{2\times (1.5^2-1)}{(1.4+1)\times 1.5}\sqrt{\dfrac{1.4\times 23\;000}{0.269}} = 240.26\;{\rm{m}}/{\rm{s}} . $$ | (42) |
The temperature is also determined by the equation of state (10) as follows:
| $$ T_{\rm{S}} = \dfrac{p_{\rm{S}} M_1}{\rho_{\rm{S}} R} = \dfrac{56\;541.667\times 28.964}{0.501\times 8.314} = 393\;169.121\;{\rm{K}}. $$ | (43) |
Therefore, the initial conditions for the RM instability problems can be summarized as follows:
| $$ \begin{split} & \rho({\boldsymbol{x}}, t = 0) = (\rho_2 - \rho_1) {\rm{H}}(\varphi) - (\rho_{\rm{S}} - \rho_1) {\rm{H}}\left (\dfrac{x-L}{L} + \dfrac{1}{2}\right ) + \rho_{\rm{S}}=\\& \qquad (1.356-0.269){\rm{H}}(x-0.01-\eta(y,z))-(0.501-0.269){\rm{H}}\left (\dfrac{x-0.01}{0.01} + 0.5\right )+0.501 =\\&\qquad 0.5435\left [{\rm{erf}}\left (\dfrac{x-0.01-\eta(y,z)}{2\delta_0}\right )+1\right ]-0.116\left [{\rm{erf}}\left (\dfrac{100 x-0.5}{2\delta_0}\right )+1\right ]+0.501\;{{\rm{kg}}}/{{\rm{m}}^3} , \end{split} $$ | (44) |
| $$ \begin{split} & {\boldsymbol{u}}({\boldsymbol{x}}, t = 0) = \left [1 -{\rm{H}}\left (\dfrac{x-L}{L} + \dfrac{1}{2}\right )\right ] U_{\rm{S}}{\boldsymbol{i}}= \left [1 -{\rm{H}}\left (\dfrac{x-0.01}{0.01} + 0.5\right )\right ] 240.26{\boldsymbol{i}} = \\&\qquad \left \{1 -0.5\left [{\rm{erf}}\left (\dfrac{100x-0.5}{2\delta_0}\right )+1\right ]\right \} 240.26{\boldsymbol{i}}\;{{\rm{m}}}/{{\rm{s}}}, \end{split} $$ | (45) |
| $$ \begin{split} & p({\boldsymbol{x}}, t = 0) = - (p_{\rm{S}} - p_0) {\rm{H}}\left (\dfrac{x-L}{L} + \dfrac{1}{2}\right ) + p_{\rm{S}}=\\& \qquad - (56\;541.667 - 23\;000) {\rm{H}}\left (\dfrac{x-0.01}{0.01} + 0.5\right ) + 56\;541.667 \\& \qquad -16\;770.833\;5\left [{\rm{erf}}\left (\dfrac{100x-0.5}{2\delta_0}\right )+1\right ] + 56\;541.667 \;{\rm{Pa}}, \end{split} $$ | (46) |
| $$ \begin{split} & T({\boldsymbol{x}}, t = 0) = - (T_{\rm{S}} - T_0) {\rm{H}}\left (\dfrac{x-L}{L} + \dfrac{1}{2}\right ) + T_{\rm{S}}=\\&\qquad - (393\;169.121 - 298) {\rm{H}}\left (\dfrac{x-0.01}{0.01} + 0.5\right ) + 393\;169.121=\\&\qquad -196\;435.560\;5\left [{\rm{erf}}\left (\dfrac{100x-0.5}{2\delta_0}\right )+1\right ] + 393\;169.121\;{\rm{K}}, \end{split}$$ | (47) |
where $ {\rm{H}} $ is the Heaviside function
| $$ {\rm{H}} = \left \{ \begin{aligned} & 0,&& t<0,\\ & 1,&& t\le 0. \end{aligned}\right . $$ | (48) |
We used the error function to perform the following smoothing operations:
| $$ {\rm{H}} = \dfrac 1 2\left [{\rm{erf}}\left (\dfrac{\alpha}{2\delta_0}\right )+1 \right ] .$$ | (49) |
Its thickness can be set to a grid scale of $ \delta_0 = \Delta x $. Meanwhile, due to the acceleration of the shock wave at the interface, it will generate an $ x $ directional translation velocity of $ U_0 ^{+} = 158.1 \;{\rm{m\cdot s}}^{-1} $. Therefore, to maintain the interface in the initial position during the calculation process, the initial velocity in the direction of (45) should be subtracted from $ U_0^{+} $.
2) Boundary condition
The RM instability satisfies the periodic boundary conditions in the $ y $ and $ z $ directions, and the boundaries in the $ x $ direction are non-slip solid walls.
The parameters used in the following numerical results of the re-shock Richtmyer-Meshkov instability are the dimensions of the spatiotemporal multiscale optimal low-dimensional dynamical system of the middle-scale equation $ N = 3 $, Reynolds number $ {Re} = 3\;608\;549 $, Mach number $ {{Ma}} = 0.6 $, Prandtl number $ { {Pr}} = 0.72 $ and simulation time $ t = 0\sim 0.002\;4 $ s. In following figures, the upper, middle and lower figures show the iso-surface of middle-scale fields ( $\rho ',\;T',\;u',\,v',\,w' $), the large-scale mean fields ( $\bar \rho ,\;\bar T,\;\bar u,\,\bar v,\,\bar w $), and the approximate solutions of turbulence ( $\rho ,\;T,\;u,\,v,\,w $), respectively.
From figs. 5 to 9, it can be seen that the RM instability interface is repeatedly rubbed by the re-shock wave to produce a large amount of smaller flow structures close to the middle part in the middle-scale and large-scale flow fields and to obtain another kind of very interesting approximate solution of turbulence other than that of DNS for the first time.
Because the purpose of this paper is to point out that there are a few big problems in the theoretical basis of turbulence modeling, and then setup a new and multi-scale numerical simulation method for compressible turbulence (LMS), therefore there is no need to conduct a careful physical analysis of the results.
Since the LMS method is a new independent numerical simulation method of turbulence based on first principles and without any artificial assumptions, its theoretical basis is consistent; therefore, it is not necessary to verify its correctness through quantitative comparison with DNS coarse-grained results.
It must be noted that it is wrong to directly compare the numerical results of the RANS, LES, DES or LMS with those of the DNS. The correct method is to first coarse-grain the DNS numerical results on the RANS, LES, DES or LMS grid and then compare their results quantitatively.
According to Professor P-Y CHOU’s idea[2] in which “the turbulence should be studied by analyzing and solving the fluctuating velocity field”, we revisit the theoretical bases of turbulence modeling before setting up the LMS method and examined the following aspects.
Based on the physical characteristics of turbulence, a new concept of large-, middle- and small-scale decompositions of turbulence is proposed. The relationship between the correlation function and filtering (averaging) function is analyzed, and the algorithm for the correlation function is clarified. A long-standing logical error in the theory of turbulence models is pointed out, and the correct theoretical basis of the numerical simulation of turbulence and the concept of a multiscale turbulence model are given. The philosophical essence is indicated, and an iterative multiscale method is successfully employed to overcome the closure problem of turbulence. The errors in the quantitative comparison of turbulence numerical simulation results are studied, and a coarse-grained quantitative comparison method is provided.
Then, the LMS method is systematically established, and the turbulent middle-scale flow field and the spatiotemporally accurate solution of turbulence are obtained with the LES grid, which deepens the understanding of the complexity of turbulence. The multiscale characteristics of turbulence are naturally included in the middle- and small-scale modeling, there is no logic error in the LMS method, and the approximate solution of turbulence $ f $ can be obtained. Due to the use of first principles in the LMS method, the LMS method is suitable for the numerical simulation of various complex turbulent flows, and fewer grid points can be used in the LMS method to obtain a more accurate solution of turbulence. With the box filtering, the space grid average, and in the sense of a large-scale grid, the essence of the LMS method is a turbulence numerical simulation method that integrates the RANS, LES, DES and DNS.
Finally, it is necessary to indicate that the LMS method can also be served as an auxiliary tool for turbulence model research to examine whether the turbulence model corresponding to each term in the SGS-scale (fluctuations) equation is correct.
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Figures(9)
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