A large eddy simulation method based on quasi-dynamic subgrid kinetic equation model

Abstract

The invention discloses a large eddy simulation method based on a quasi-dynamic sub-grid kinetic energy equation model, comprising the following steps: step 1, deriving a transport equation of sub-grid kinetic energy, and using an infinite expansion relation to model an energy flow to obtain an energy flow relation; step 2, substituting an eddy viscosity model into the energy flow relation to obtain an energy flow calculated by the eddy viscosity model, making the energy flow calculated by the eddy viscosity model equal to the true value of the energy flow obtained by modeling, and dynamically obtaining the coefficient of the eddy viscosity model; and expanding the isotropic parts of sub-grid stresses, and obtaining the unknown coefficients in the energy flow relation based on the relationship between the isotropic parts of the sub-grid kinetic energy and the sub-grid stress; step 3, calculating and obtaining the coefficients of the dynamic sub-grid heat flow and the sub-grid component flux term; and step 4, modeling the unclosed terms in the sub-grid kinetic energy transport equation based on the infinite expansion to make the entire equation group closed.

Description

Technical Field

[0001] The invention relates to the technical field of fluid mechanics, and in particular to a large eddy simulation method of a quasi-dynamic sub-grid kinetic energy equation model. Background Art

[0002] Large eddy simulation has been widely used in complex engineering calculations and mechanism analysis. In large eddy simulation, the Navior-Stokes equations are filtered, and the filtered equations will produce some unclosed terms. These unclosed terms are modeled to complete the closure of the equations, and these constructed models are collectively called sub-grid models. At present, most of the sub-grid models of large eddy simulation are derived based on incompressible conditions and then extended to compressible forms. Most of the current large eddy simulation models are derived from the equations of incompressible flow, which leads to various problems in the application to compressible flow. Summary of the invention

[0003] In view of this, in order to solve the problems in the prior art, the present invention proposes a quasi-dynamic sub-grid one-equation model. Starting from the compressible equation, the present invention derives a large eddy simulation model suitable for compressible flow, namely a quasi-dynamic sub-grid kinetic energy one-equation model. The present invention directly derives the sub-grid model of large eddy simulation from the perspective of the compressible flow equation, which helps to solve problems in compressible flow, such as predicting transitions; compressible flow is widely present in complex internal flows such as aerospace engines. The present invention proposes a large eddy simulation model for compressible flow, which can help solve problems such as hypersonic transition prediction.

[0004] A large eddy simulation method based on a quasi-dynamic subgrid kinetic energy equation model comprises the following steps:

[0005] Step 1, derive the transport equation of sub-grid kinetic energy, and use the infinite expansion relation to model the energy flow to obtain the energy flow relation;

[0006] Step 2, after substituting the eddy viscosity model into the energy flow relationship, the energy flow calculated by the eddy viscosity model is obtained, the energy flow calculated by the eddy viscosity model is equal to the true value of the energy flow obtained by modeling, and the coefficient of the eddy viscosity model is dynamically obtained; and the isotropic parts of the sub-grid stress are expanded, and the unknown coefficients in the energy flow relationship are obtained based on the relationship between the sub-grid kinetic energy and the isotropic parts of the sub-grid stress;

[0007] Step 3, calculate and obtain the coefficients of the subgrid heat flux and subgrid component flux terms dynamically;

[0008] Step 4: Model the unclosed terms in the sub-grid kinetic energy transport equation based on infinite expansion to make the entire set of equations closed.

[0009] Beneficial effects:

[0010] The method of the present invention can accurately predict the sub-grid kinetic energy through the additional sub-grid kinetic energy equation. Then, the sub-grid eddy viscosity model can be accurately determined by utilizing the dual constraints of the sub-grid kinetic energy and the energy flow. Using a method similar to the sub-grid stress expansion, other unclosed quantities in the equation to be solved can be well modeled separately. The method was tested in different examples, and can accurately predict processes such as transition and turbulent mixing, achieving good simulation results, and also proving that the method has scale adaptability. The new model proposed in the present invention can dynamically obtain the model coefficients without secondary filtering, has good grid adaptability and time history effect, and is convenient for application in complex engineering calculations. BRIEF DESCRIPTION OF THE DRAWINGS

[0011] Figure 1 is a flow chart of the method of the present invention;

[0012] Figure 2 This is a schematic diagram of the slot calculation domain;

[0013] Figure 3 is the velocity profile obtained after Van Driest transformation;

[0014] Figure 4 is the dimensionless Reynolds deviatoric stress;

[0015] Figure 5 is the dimensionless turbulent heat flux;

[0016] Figure 6 It is a schematic diagram of the computational domain of a flat plate;

[0017] Figure 7 Velocity profiles and friction coefficients under different grid settings. A is the velocity profile of LES-gridA, B is the friction coefficient of LES-gridA, C is the velocity profile of LES-gridB, D is the friction coefficient of LES-gridB, E is the velocity profile of LES-gridC, and F is the friction coefficient of LES-gridC;

[0018] Figure 8 Schematic diagram of the computational domain for RM instability with spherical convergence;

[0019] Fig. 9 is the change of the inner and outer radius of the mixing layer with time (r1 is the inner radius and r2 is the outer radius). DETAILED DESCRIPTION

[0020] The technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, rather than all the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the protection scope of the present invention.

[0021] According to an embodiment of the present invention, a large eddy simulation method of a quasi-dynamic sub-grid one-equation model is proposed. Starting from the compressible equation, a large eddy simulation model suitable for compressible flow is derived, that is, a quasi-dynamic sub-grid kinetic energy one-equation model, and large eddy simulation is performed based on the model. The model uses the dual constraints of sub-grid kinetic energy and energy flow to construct sub-grid eddy viscosity, sub-grid heat flow and sub-grid diffusion terms, so that the model coefficients can be dynamically obtained without using secondary filtering, so that the gradient model and the eddy viscosity model are combined, which can obtain both high correlation and good stability.

[0022] like Figure 1 As shown, a large eddy simulation method based on a quasi-dynamic subgrid kinetic energy equation model of the present invention specifically comprises the following steps:

[0023] Step 1: First, the transport equation of sub-grid kinetic energy is derived, and the energy flow is modeled using the infinite expansion relation to obtain the energy flow relation;

[0024] Step 2, after substituting the traditional eddy viscosity model into the energy flow relationship, the energy flow calculated by the eddy viscosity model is obtained, the energy flow calculated by the eddy viscosity model is made equal to the true value of the energy flow obtained by modeling, and the coefficient of the eddy viscosity model is dynamically obtained; and the isotropic parts of the sub-grid stress are expanded, and the unknown coefficients in the energy flow relationship can be obtained based on the relationship between the sub-grid kinetic energy and the isotropic parts of the sub-grid stress;

[0025] Step 3: Based on the same method, the coefficients of the subgrid heat flow and subgrid component flux terms can be obtained dynamically;

[0026] Step 4: Model the unclosed terms in the sub-grid kinetic energy transport equation based on infinite expansion, so that the filtered Navier-Stokes and sub-grid kinetic energy transport equations are closed.

[0027] According to an embodiment of the present invention, the specific derivation process is as follows:

[0028] In compressible flow, the subgrid kinetic energy is defined as follows:

[0029]

[0030] Where, “-” indicates filtering, “~” indicates Favre filtering, ρ is the density, ksgs is the subgrid kinetic energy, and ui is the velocity.

[0031] The transport equation of sub-grid kinetic energy is derived, and the coefficients of the energy flow model are obtained based on the transport equation;

[0032] The "quasi-dynamic subgrid kinetic energy-equation model" of the present invention includes: the calculation expression of Csm, the subgrid Prandtl number Pr sgs The calculation expression formula of and the calculation expression of the sub-grid Schmidt number;

[0033]

[0034] in:

[0035]

[0036]

[0037] Among them, when i = j, δ ij =1, when i is not equal to j, δ ij =0;

[0038]

[0039]

[0040]

[0041]

[0042]

[0043] is the molecular viscosity, ε s is the spiral dissipation term, ε d is the expansion dissipation term, ∏ p is the pressure diffusion term, J j is the third-order correlation term. ij is the sub-grid stress, is the strain rate tensor, is the molecular viscosity coefficient, p is the pressure, Q j It is the subgrid heat flow.

[0044] Energy flux is a key physical quantity reflecting the turbulent cascade process. In the quasi-dynamic one-equation method, the energy flux is first modeled, and the energy flux needs to be obtained from the sub-grid stress. After the energy flux is modeled, the model coefficients need to be solved, and the coefficients are obtained by ksgs, which is obtained by its transport equation, as follows:

[0045] Consider the following expansion:

[0046]

[0047] Among them, f and g are physical quantities, which can also be vectors, vectors, etc., and G(x, y) is a filter.

[0048] Using the above expansion for the subgrid stresses:

[0049]

[0050] Among them, C0 is the unknown coefficient, △ k is the filter width.

[0051] Then we can get the energy flow relationship:

[0052]

[0053] Using the isotropic part of the above expanded subgrid stresses:

[0054]

[0055] Based on the relationship between the isotropic parts of the subgrid kinetic energy and the subgrid stress, we can obtain:

[0056]

[0057] In this way, the unknown coefficient C0 can be obtained, thus forming the constraint of the sub-grid kinetic energy on the unclosed term. The unclosed term is for all sub-grid unclosed terms. The present invention adopts infinite series expansion to generate a coefficient. It is assumed that all coefficients are C0.

[0058] If the energy flow obtained above is equal to the energy flow obtained by the Smagorinsky model, then:

[0059]

[0060] in:

[0061]

[0062]

[0063]

[0064] ∏ Δ SMA is the anisotropic part of the Smagorinsky model, ∏ Δ SMI are homogeneous parts;

[0065] And △The anisotropic and homotropic parts of are as follows:

[0066]

[0067]

[0068] Equating the anisotropic parts of the energy flow in the quasi-dynamic one-equation method with the energy flow in the Smagorinsky model:

[0069]

[0070] The coefficients of the Smagorinsky model can be obtained as:

[0071]

[0072] Since the sub-grid kinetic energy is an unclosed term (here it is only the sub-grid kinetic energy itself), it is necessary to solve the transport equation of the sub-grid kinetic energy to obtain it. However, there are unclosed terms in its transport equation. The pressure diffusion term and the two dissipation terms are modeled using expansion as follows:

[0073]

[0074]

[0075]

[0076] For the third-order correlation term, the model is

[0077] For the sub-grid heat flow term in the compressible Navior-Stokes equation, C0 is also used to constrain it, and the modeling form is considered as follows:

[0078]

[0079] The present invention has obtained μ sgs Next, we need to obtain the subgrid Prandtl number Pr sgs .

[0080] Expanding the subgrid heat flow yields:

[0081]

[0082] Use the following relationship:

[0083]

[0084] You can get:

[0085]

[0086] In turbulent mixing problems, the component transport equations need to be solved:

[0087]

[0088] The subgrid component flux term is:

[0089]

[0090] This term can be modeled as follows:

[0091]

[0092] Sc sgs,k is the subgrid Schmidt number, which is an unknown constant. Using the same method as that for the subgrid Prandtl number, we can obtain:

[0093]

[0094] Next, we will verify the new model with an example. This model is called the quasi-dynamic subgrid kinetic energy equation model (QKM), which includes the solution of C sm ,Pr sgs and Sc sgs,k formula.

[0095] The process of applying this method to large eddy simulation is as follows:

[0096] For a certain calculation example, a suitable large eddy simulation grid is selected to solve the filtered Navior-Stokes equations. There are unclosed terms in the filtered NS equations. The above model is substituted for solving subgrid stress, subgrid heat flow and subgrid component flux terms, and the subgrid kinetic energy transport equation is added to solve the model coefficients. After time advancement, the flow field results are obtained. This quasi-dynamic method can obtain the model coefficients without the need for secondary filtering, and can be applied to complex boundary problems. This method has been proven to be able to well predict transition, turbulence and other processes, and has certain advantages over traditional models.

[0097] According to one embodiment of the present invention, the method of the present invention is described by taking a compressible channel as an example.

[0098] First, the test is carried out in a compressible channel. The schematic diagram of the channel calculation domain is shown in Figure 2 The computational grid settings and flow parameters are shown in Table 1. The calculated incoming flow Mach number is 1.5 and the Reynolds number is 3000. Direct numerical simulation (DNS), dynamic Smagorinsky model (DSM) and dynamic one-equation model (dk-equation) are used for comparison.

[0099] Table 1 Grid settings and main parameters

[0100] Grid <![CDATA[△x + ]]> <![CDATA[△y min + ]]> <![CDATA[△z + ]]> DNS 900×201×300 2.99 0.32 2.99 DSM 48×65×48 57.55 1.07 19.18 Dk-equation 48×65×48 57.88 1.06 19.20 QKM 48×65×48 57.45 1.07 19.18

[0101] Figure 3-Figure 5 The comparison results of key physical quantities in the slot are shown. It can be seen that the results of the new model are closer to DNS and significantly better than other comparison models.

[0102] According to yet another embodiment of the present invention, a compressible flat plate boundary layer is taken as an example to illustrate the method of the present invention.

[0103] This example is a spatially developed flat plate, including laminar flow, transitional flow, and turbulent flow. Figure 6 Table 2 shows the grid settings. There are three sets of settings for large eddy simulation: dense, medium, and sparse, to test the grid adaptability of the model. The calculated incoming flow Mach number is 2.25 and the Reynolds number is 635,000.

[0104] Table 2 Compressible plate grid settings

[0105] Grid <![CDATA[△x + ]]> <![CDATA[△y min + ]]> <![CDATA[△z + ]]> DNS 10000×90×320 6.0 0.58 5.47 LES-gridA 1500×90×100 40.1 0.58 17.5 LES-gridB 1000×90×80 60.2 0.58 21.9 LES-gridC 1000×60×50 60.2 1.00 35.1

[0106] Figure 7 Velocity profiles and friction coefficients for different grid settings. A and B are the velocity profiles and friction coefficients for LES-gridA, C and D are the velocity profiles and friction coefficients for LES-gridB, and E and F are the velocity profiles and friction coefficients for LES-gridC.

[0107] from Figure 7 From the displayed results, it can be seen that the new model can not only predict the turbulent part very well, but also predict the transition peak and transition position very well. Even when the grid is relatively sparse, the new model can still get good results, which shows that the model has good grid adaptability.

[0108] According to yet another embodiment of the present invention, the method of the present invention is described by taking the RM instability of spherical convergence as an example.

[0109] The next example is the RM instability of spherical convergence. This example is a time-evolving example, which is different from the above two examples. The grid settings are shown in Table 3, and the schematic diagram of the computational domain is shown in Figure 8 The Mach number is about 1.5 and the Atwood number is about 0.578. The traditional Smagorinsky model (SM) is used as a comparison model.

[0110] Table 3. Mesh settings for RM instability of spherical convergence

[0111] Calculation example Grid DNS <![CDATA[2048 3 ]]> QKM <![CDATA[384 3 ]]> SM <![CDATA[384 3 ]]>

[0112] from Fig. 9 It can be seen that the new model can well predict the changes in the inner and outer radii of the mixing layer, and the prediction results are better than those of the traditional Smagorinsky model, indicating that it has a good time history effect.

[0113] In summary, the model designed by the present invention can dynamically obtain the model coefficients without secondary filtering, and has good grid adaptability and time history effect. Based on the above model, the large eddy simulation method of the present invention can be conveniently applied to complex engineering calculations, such as hypersonic lifting bodies, hypersonic compression angles and other examples.

[0114] Although the above describes the illustrative specific embodiments of the present invention to facilitate those skilled in the art to understand the present invention, and it should be clear that the present invention is not limited to the scope of the specific embodiments, for those of ordinary skill in the art, as long as various changes are within the spirit and scope of the present invention as defined and determined by the attached claims, these changes are obvious, and all inventions and creations using the concept of the present invention are protected.

Claims

1. A large eddy simulation method based on a quasi-dynamic subgrid kinetic energy equation model, characterized in that: The steps include: Step 1, derive the transport equation of sub-grid kinetic energy, and use the infinite expansion relation to model the energy flow to obtain the energy flow relation; Step 2: Based on the energy flow relationship, calculate and obtain the coefficients of the eddy viscosity model; based on the transport equation, obtain the unknown coefficients in the energy flow relationship; Step 3, calculate the coefficients of the dynamic subgrid heat flux and the subgrid component flux term; Step 4: Model the unclosed terms in the sub-grid kinetic energy transport equation based on infinite expansion, so that the entire system of equations is closed; The step 2 calculates the coefficients of the eddy viscosity model based on the energy flow relationship, and specifically comprises the following steps: After substituting the eddy viscosity model into the energy flow relationship, the energy flow calculated by the eddy viscosity model is obtained, so that the energy flow calculated by the eddy viscosity model is equal to the true value of the energy flow obtained by modeling, and the coefficient of the eddy viscosity model is dynamically obtained; The step 2 obtains the unknown coefficients in the energy flow relation based on the transport equation, specifically including: expanding the isotropic parts of the sub-grid stress, and obtaining the unknown coefficients in the energy flow relation based on the relationship between the sub-grid kinetic energy and the isotropic parts of the sub-grid stress; the specific calculation process is as follows: In compressible flow, the subgrid kinetic energy is defined as follows: In the formula, " " indicates filtering, " " represents Favre filtering, ρ is density, ksgs is subgrid kinetic energy, ui is velocity; The transport equation of sub-grid kinetic energy is derived, and the coefficients of the energy flow model are obtained based on the transport equation; The "quasi-dynamic subgrid kinetic energy-equation model" includes: the calculation expression of Csm, the subgrid Prandtl number The calculation expression formula of and the calculation expression of the sub-grid Schmidt number; in: ， Among them, when i=j, =1, when i is not equal to j, =0; is the molecular viscosity, is the spiral dissipation term, is the expansion dissipation term, is the pressure diffusion term, is the third-order correlation term, is the sub-grid stress, is the strain rate tensor, is the molecular viscosity coefficient, p is the pressure, Q j It is the subgrid heat flow; Energy flux is a key physical quantity reflecting the turbulent cascade process. In the quasi-dynamic one-equation method, the energy flux is first modeled, and the energy flux needs to be obtained from the sub-grid stress. After the energy flux is modeled, the model coefficients need to be solved, and the coefficients are obtained by ksgs, which is obtained by its transport equation, as follows: Consider the following expansion: Among them, f and g are vector and vector, is the filter; Using the above expansion for the subgrid stresses: Among them, C0 is the unknown coefficient, is the filter width; Then we can get the energy flow relationship: Using the isotropic part of the above expanded subgrid stresses: Based on the relationship between the isotropic parts of the subgrid kinetic energy and the subgrid stress, we obtain: In this way, the unknown coefficient C0 is obtained, which forms the constraint of the sub-grid kinetic energy on the unclosed term; the unclosed term is all the sub-grid unclosed terms, and all of them are expanded by infinite series, and all of them will produce a coefficient. It is assumed that all coefficients are C0; If the energy flow obtained above is equal to the energy flow obtained by the Smagorinsky model, then: in: is the anisotropic part of the Smagorinsky model, are homogeneous parts; and The anisotropic and homotropic parts of are as follows: Equating the anisotropic parts of the energy flow in the quasi-dynamic one-equation method with the energy flow in the Smagorinsky model: The coefficients of the Smagorinsky model are obtained as: The sub-grid kinetic energy is an open term, which needs to be obtained by solving the transport equation of the sub-grid kinetic energy. However, there are open terms in the transport equation. The pressure diffusion term and the two dissipation terms are modeled using expansion as follows: For the third-order correlation term, the model is ; For the sub-grid heat flow term in the compressible Navior-Stokes equation, C0 is also used to constrain it, and the modeling form is considered as follows: Already got , then we need to get the subgrid Prandtl number ; Expanding the subgrid heat flow yields: Use the following relationship: get: In turbulent mixing problems, the component transport equations need to be solved: The subgrid component flux term is: This term is modeled in the following way: is the subgrid Schmidt number, which is an unknown constant. The subgrid Schmidt number can be solved by the same method as that of the subgrid Prandtl number: 。

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