CN110457806A - The whole flow field analogy method of five rank WENO format of center based on staggered-mesh - Google Patents

The whole flow field analogy method of five rank WENO format of center based on staggered-mesh Download PDF

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CN110457806A
CN110457806A CN201910710350.XA CN201910710350A CN110457806A CN 110457806 A CN110457806 A CN 110457806A CN 201910710350 A CN201910710350 A CN 201910710350A CN 110457806 A CN110457806 A CN 110457806A
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崔生柱
朱君
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Nanjing University of Aeronautics and Astronautics
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Abstract

The whole flow field analogy method of five rank WENO format of center based on staggered-mesh.The invention discloses the whole flow field analogy methods that a kind of five ranks based under central frame weight non-oscillatory scheme substantially, comprising: using the variable cell mean value at five rank WENO format of the limited bulk reconstruct least bit of new model；It is discrete using the progress of Gauss quadrature formula to time integral term, obtain combined type of the flux in space integral point about the Gaussian node of time；Reconstruct the variable point value at integral point；Reconstruct value of the derivative of flux at integral point；Obtain Gaussian node of the flux in space integral point about the time；Gaussian node of the unit mean value variable and flux at the least bit in space integral point about the time is combined, the unit mean value at the least bit of next time horizon is obtained；Successively iteration obtains the numerical result in end time flow field in zoning.The present invention can reach better precision, have preferably capture interruption, while can analog portion traditional algorithm can not solve the problems, such as；Solve the problems, such as the double Mach reflection and step problem that traditional algorithm cannot be solved well under two-dimensional case.

Description

The whole flow field analogy method of five rank WENO format of center based on staggered-mesh
Technical field
The present invention relates to Fluid Mechanics Computation field of engineering technology, in particular to a kind of center based on staggered-mesh The whole flow field analogy method of five rank Jacobi weights.
Background technique
In Computational fluid mechanics numerical simulation, the construction of high accurate scheme and its application are always in the emphasis of research Hold.Because high accurate scheme can accurately simulate whole flow field, it can be good at simulating formal similarity and accurately Capturing shock position.Nineteen eighty-three, Harten have been put forward for the first time TVD (Total Variation Diminishing) format, and Proposed ENO (Essentially Non-Oscillatory) high accurate scheme, ENO lattice on the basis of this in 1987 with Osher The main thought of formula is to select most smooth template in the template gradually extended to find out at elementary boundary to construct multinomial Value, and then achieve the effect that high-precision, high-resolution and in the basic dead-beat of discontinuities.But during the realization of method, ENO format will cause the waste of calculated result, cause computational efficiency not high, and therefore, Liu, Osher and Chan etc. propose WENO (Weighted Essentially Non-oscillatory) format, improves the utilization rate of calculated result and makes r rank The ENO format of precision is increased to r+1 rank precision.1996, Jiang and Shu proposed a kind of new WENO format, made value Precision is increased to 2r-1 rank.During the realization of classics WENO format, linear power depends on caster, and its solution procedure phase Work as complexity, therefore, Zhu and Qiu improved the finite difference form of the WENO format in 2016, in the feelings for maintaining precision not subtract Under condition, randomly selects greater than zero and summation is one linear power, and proposed the limited bulk shape of the WENO format in 2017 Formula, the referred to as finite Volume Scheme of new model, the finite Volume Scheme of this new model is still in the feelings for maintaining precision not subtract Under condition, the linear power that greater than zero and summation is one is randomly selected.WENO reconstruct is used primarily for frame windward, then applies them Into Center Scheme, and then the advantages of acquisition Center Scheme.Such as without numerical flux, stability relatively preferably, do not need to answer With thinking position etc..In conjunction with Fig. 2, compared with frame windward, cell centered scheme is also a kind of effectively Hyperbolic Conservation scheme, and phase To simple.First second-order central format is to be proposed by Nessyahu and Tadmor in nineteen ninety.Later, Levy and Puppo Et al. CWENO format under two-dimensional case under the Center Scheme of three ranks was proposed in 2000, then, Qiu and Shu were in 2001 The CWENO format under one-dimensional case under the Center Scheme of five ranks and nine ranks, hereinafter referred to as CWENO-QS format are proposed, it Afterwards, Levy and Puppo et al. proposed the CWENO format under two-dimensional case under the Center Scheme of three ranks based on 2000.They The CWENO format under two-dimensional case under the Center Scheme of quadravalence, hereinafter referred to as CWENO-LP format were proposed in 2002 again. But under one-dimensional case, the effect is unsatisfactory for part example for the CWENO format of these traditional Center Schemes, and Calculation amount is larger；Under two-dimensional case, the CWENO formats of these traditional Center Schemes for complex flowfield simulation not yet It is ideal.
Summary of the invention
It is an object of that present invention to provide the full streams that a kind of five ranks based under central frame weight non-oscillatory scheme substantially Field analogy method, can overcome above-mentioned difficulties well under one-dimensional case, and can be good at processing complexity under two-dimensional case The simulation in flow field.Novel five rank weights the whole flow field simulation side of non-oscillatory scheme substantially in the case of present invention one-dimensional first Method has been compared with the traditional method better precision, and can preferably capture interruption, while being capable of the calculation of analog portion tradition Method can not solve the problems, such as；Then, the whole flow field simulation that novel five rank under two-dimensional case weights non-oscillatory scheme substantially is provided Method has been compared with the traditional method better precision, and it is conspicuous to simulate the span that traditional algorithm cannot be solved well Reflection problems and step problem, hereinafter referred to as CWENO-ZQ format.
To reach above-mentioned purpose, in conjunction with Fig. 1, the present invention proposes a kind of five rank WENO format of center based on staggered-mesh Whole flow field analogy method, the five rank WENO format of limited bulk suitable for constructing new model under cartesian coordinate are compressible to calculate Flow field problems, the whole flow field analogy method include:
S1: integrating Hyperbolic Conservation equation space variable and time variable to obtain finite Volume Scheme, uses Variable cell mean value at five rank WENO format of the limited bulk reconstruct least bit of new model；
S2: it is discrete using the progress of Gauss quadrature formula to time integral term, flux is obtained in space integral point about the time The combined type of Gaussian node；
S3: based on unit mean value using the variable point value at five rank WENO format of the limited bulk reconstruct integral point of new model；
S4: the point value based on step S3 reconstruct reconstructs the derivative of flux using the five rank WENO format of limited bulk of new model Value at integral point；
S5: using the NCERK method of quadravalence, the integral point value of integral point value and flux derivative based on space variable, by repeatedly The process that five ranks weight essential dead-beat reconstruct is carried out for value of the derivative at integral point of flux, it is whole in space to obtain all flux Gaussian node of the point about the time；
The unit mean value variable and Gaussian node of all flux in space integral point about the time at the least bit are combined, is obtained down Unit mean value at the least bit of one time horizon；
Successively iteration obtains the numerical result in end time flow field in zoning.
Further, in step S1, using the variable list at five rank WENO format of the limited bulk reconstruct least bit of new model The process of first mean value the following steps are included:
S11: being based on unit mean value, selects caster and subtemplate, reconstructs the multinomial of several different accuracies；
S12: for the multinomial of several different accuracies of reconstruct, the linear power of satisfaction and any positive number for one is taken；
S13: calculating smooth indicator, for measuring smooth degree of the reconstruct multinomial on object element；
S14: nonlinear weight is calculated；
S15: portfolio restructuring multinomial and nonlinear weight obtain the reconstruct of the variable cell mean value at the corresponding least bit Value.
Further, if One-dimensional Hyperbolic Conservation Law Equations are as follows:
The whole flow field analogy method the following steps are included:
S1.1: it carries out the integral of time orientation and direction in space respectively to governing equation, obtains:
Wherein,It is the cell-average value at the least bit of time horizon n+1,It is that unit at the least bit of time horizon n is flat Mean value, h are the step-length of direction in space, f (u (xi+1, t), t) it is point value of the flux in variable at node, It is from time horizon tnTo time horizon tn+1Integral；
To the unit mean value variable at the least bitIt carries out five ranks and weights essential dead-beat reconstruct；
S1.2: time integral term is carried out using Gauss quadrature formula using following formula discrete:
Wherein, Δ t is time step, αlWith τlIt is gaussian coefficient, obtains Gauss section of the flux in space integral point about the time The combined type of point；
S1.3: the variable point at five rank WENO format of the limited bulk reconstruct integral point of new model is used based on unit mean value Value；
S1.4: the point value based on step S1.3 reconstruct reconstructs flux using the five rank WENO format of limited bulk of new model Value of the derivative at integral point；
S1.5: by the NCERK method of quadravalence, the integral point value of integral point value and flux derivative based on space variable passes through Value of the derivative of iteration flux at integral point carries out the process that five ranks weight essential dead-beat reconstruct, obtains all flux in space Integral point is arrived about the Gaussian node of time:
Wherein, bj, cj, bj(τl) it is resulting value be middle coefficient, τlFor Gaussian node coefficient, Δ t is time step, g(j) It is reconstruction value of the derivative based on flux at integral point and is obtained by NCERK method iteration；
The unit mean value variable and Gaussian node of all flux in space integral point about the time at the least bit are combined, is obtained down Unit mean value at the least bit of one time horizon；
Successively iteration obtains the numerical result in end time flow field in zoning.
Further, in step S1.1, the unit mean value variable at the least bitFive ranks weighting essence is carried out without vibration Swing the process of reconstruct the following steps are included:
S1.1.1: by object element IijAnd totally 5 units form caster to surrounding, are denoted as R1={ Ii-2,Ii-1,Ii, Ii+1,Ii+2, and assume that mesh spacing is all h, and rememberU is represented in grid cell ImAverage value (m=i-2 ..., i+2)；
S1.1.2: 2 subtemplates are selected to be respectively as follows: R in caster2={ Ii-1,Ii, R3={ Ii,Ii+1}；
S1.1.3: reconstruct multinomial p is found out respectively on caster and subtemplaten(x, y), n=1,2,3, keep it full Foot:
N=1, k=i-2, i-1, i, i+1, i+2；N=2, k=i-1, i；N=3, k=i, i+1；
Then have:
Wherein,Indicate U in grid cell IkOn average value, xiIndicate the value at grid node.
The above technical solution of the present invention, compared with existing, significant beneficial effect is:
(1) central frame has used the technology of staggered-mesh, does not need to solve Riemannian problem for frame windward, And then it does not need using thinking position.
(2) any linear positive can be used to weigh for the novel WENO format of five rank of limited bulk under structured grid, compared to tradition Algorithm structure is easier for five rank WENO format of limited bulk, while possessing higher computational accuracy and more excellent number Value simulation effect.
(3) novel five ranks WENO format under one-dimensional case, has been compared with the traditional method more in conjunction with central frame Good numerical precision, and can preferably capture interruption, at the same can the low close problem of analog portion low pressure, this is traditional side What method cann't be solved；Then, the whole flow field analogy method that novel five rank under two-dimensional case weights non-oscillatory scheme substantially is provided, Better precision has been compared with the traditional method it, and being capable of preferably numerical simulation double Mach reflection problem and step problem
It should be appreciated that as long as aforementioned concepts and all combinations additionally conceived described in greater detail below are at this It can be viewed as a part of the subject matter of the disclosure in the case that the design of sample is not conflicting.In addition, required guarantor All combinations of the theme of shield are considered as a part of the subject matter of the disclosure.
Can be more fully appreciated from the following description in conjunction with attached drawing present invention teach that the foregoing and other aspects, reality Apply example and feature.The features and/or benefits of other additional aspects such as illustrative embodiments of the invention will be below Description in it is obvious, or learnt in practice by the specific embodiment instructed according to the present invention.
Detailed description of the invention
Attached drawing is not intended to drawn to scale.In the accompanying drawings, identical or nearly identical group each of is shown in each figure It can be indicated by the same numeral at part.For clarity, in each figure, not each component part is labeled. Now, example will be passed through and the embodiments of various aspects of the invention is described in reference to the drawings, in which:
Fig. 1 is the whole flow field analogy method of the center five rank Jacobi weights of the invention based on staggered-mesh Flow chart.
Fig. 2 is the comparison diagram of frame windward and central frame of the invention.
Fig. 3 is one-dimensional caster schematic diagram of the invention.
Fig. 4 is two-dimentional caster schematic diagram of the invention.
CWENO-ZQ algorithm and each leisure L of CWENO-QS algorithm when Fig. 5 is T=0.5/ π of the invention1With L∞Under precision Schematic diagram is showed with error.
CWENO-ZQ algorithm and each leisure L of CWENO-QS algorithm when Fig. 6 is t=2 of the invention1With L∞Under precision and mistake Difference performance schematic diagram.
CWENO-ZQ algorithm and each leisure L of CWENO-LP algorithm when Fig. 7 is T=0.5/ π of the invention1With L∞Under precision Schematic diagram is showed with error.
CWENO-ZQ algorithm and each leisure L of CWENO-LP algorithm when Fig. 8 is t=2 of the invention1With L∞Under precision and mistake Difference performance schematic diagram.
Fig. 9 is the test result schematic diagram of the high shock capturing problem in example five of the invention.
Figure 10 is the test result schematic diagram of the high shock capturing problem in example six of the invention.
Figure 11 is the test result schematic diagram of the high shock capturing problem in example seven of the invention.
Figure 12 is the test result schematic diagram of the high shock capturing problem in example eight of the invention.
Figure 13 is the test result schematic diagram of the double Mach reflection problem in example nine of the invention.
Figure 14 is the test result schematic diagram of the step problem in example ten of the invention.
Specific embodiment
In order to better understand the technical content of the present invention, special to lift specific embodiment and institute's accompanying drawings is cooperated to be described as follows.
Specific embodiment one
Below with reference to two specific examples, the five rank WENO lattice of limited bulk for the new model that the present invention refers to are elaborated Formula and whole flow field the analogy method disclosed in this invention detailed process in the case where solving one-dimensional case and two-dimensional case respectively.
One, the construction of one-dimensional case:
If One-Dimensional Euler equation are as follows:
Ut+f(U)x=0 (1)
Wherein, U=(ρ, ρ u, E)TIndicate conservation variable, f (U)=(ρ u, ρ u2+p,u(E+p))T, f (U) expression flux, Ut Indicate U to t derivation, f (U)xIndicate f (U) to x derivation, ρ, p, u, E respectively indicate fluid density, pressure, horizontal direction speed and Energy, T indicate transposition.Assuming that grid cell step-length is fixed, xi+1/2-xi-1/2=Δ x=h, grid element center areGrid cell is Ii=[xi-1/2,xi+1/2], wherein subscript i is coordinate serial number, xiThe value at place is known as Integral point value,The value at place is known as half point value.The integral for carrying out time orientation and direction in space respectively to governing equation, obtains:
Wherein,It is the cell-average value at the least bit of time horizon n+1,It is that unit at the least bit of time horizon n is flat Mean value, h are the step-length of direction in space, f (u (xi+1, t), t) it is point value of the flux in variable at node, It is from time horizon tnTo time horizon tn+1Integral.
Step 1.1. is to the unit mean value variable at the least bitIt carries out novel five rank and weights essential dead-beat reconstruct, specifically Process is as follows:
Step 1.1.1. is as shown in figure 3, by object element IijAnd totally 5 units form caster to surrounding, are denoted as R1= {Ii-2,Ii-1,Ii,Ii+1,Ii+2, and assume that mesh spacing is all h, and rememberU is represented in grid cell ImAverage value (m= i-2,…,i+2)。
Step 1.1.2. selects 2 subtemplates to be respectively as follows: R in caster2={ Ii-1,Ii, R3={ Ii,Ii+1}。
Step 1.1.3. finds out reconstruct multinomial p respectively on caster and subtemplaten(x, y), n=1,2,3, make it Meet:
N=1, k=i-2, i-1, i, i+1, i+2；N=2, k=i-1, i；N=3, k=i, i+1；
Then have:
Wherein,Indicate U in grid cell IkOn average value, xiIndicate the value at grid node.
Step 1.1.4. takes the linear power of satisfaction and any positive number for one, avoids complicated numerical procedure, we It takes:
γ1=10.0/12.0, γ2=1.0/12.0, γ3=1.0/12.0 (7)
As multinomial p1(x), p2(x), p3(x) linear power.
Step 1.1.5. calculates smooth indicator, for measuring smoothness of the reconstruct multinomial on object element.
Calculate smooth indicator βl, l=1,2,3, for measuring trigonometric function multinomial pl(x) in section [xi-1/2, xi+1/2] on smoothness, calculation formula are as follows:
Wherein, h is spatial mesh size,It is multinomial pj(x) l power derivative, r are multinomial pj(x's, y) Highest number.
Step 1.1.6. calculates nonlinear weight, calculation formula are as follows:
Wherein:
Wherein, ωnIt is nonlinear weight,It is median, ε=10 with τ-10。
Step 1.1.7. finds out the expression formula of unit mean value at the least bit are as follows:
Wherein, ω1,ω2,ω3It is nonlinear weight, γ1,γ2,γ3It is linearly to weigh.
Step 1.2. carries out time integral term using Gauss quadrature formula discrete, it may be assumed that
Wherein, Δ t is time step, αlWith τlIt is gaussian coefficient,
Step 1.3. carries out novel five rank in integral point value to space variable and weights essential dead-beat reconstruct, and detailed process is such as Under:
The interpolation polynomial and nonlinear weight that its obtained reconstruct multinomial and nonlinear weight and step 1.1 obtain are complete It is exactly the same.Find out the expression formula of unit mean value at the least bit are as follows:
Wherein, ω1,ω2,ω3It is nonlinear weight, pj(xi) it is multinomial pj(x) in integral point xiThe value at place, j=1,2,3,It is variable in xiThe reconstruction value at place, u (xi,tn) it is variable in xiThe exact value at place.
Step 1.4. carries out novel five rank at integral point to the derivative of flux and weights essential dead-beat reconstruct, integrative reconstruction Process is identical with step 1.1, in addition to the interpolation polynomial q of constructionn(x), n=1,2,3, meet:
qn(xk)=fk (13)
Wherein:
Then have:
Wherein, fkF (U) is respectively indicated in unit IkLocate the value at grid midpoint,It is multinomial qj(x) whole Point xiThe derivative value at place, j=1,2,3,It is flux derivative in xiThe reconstruction value at place.
The integral point value of step 1.5. the integral point value based on space variable and flux derivative, by the derivative of iteration flux in whole Value at point carries out the process that novel five rank weights essential dead-beat reconstruct, obtains all flux in space integral point about the time Gaussian node, that is:
Wherein,
And g(2), g(3), g(4)It is to be based respectively onCarry out step 1.4 weight Structure obtains.
Wherein:
b1(τl(the 1-4b of)=21)τl 3+3(3b1-1)τl 2+τl, (18)
bj(τl(the 3c of)=4j-2)bjτl 3+3(3-4cj)bjτl 2, j=2,3,4, (19)
Step 1.6. combines unit mean value variable and Gauss section of all flux in space integral point about the time at the least bit Point obtains the unit mean value at the least bit of next time horizon, thus whole flow field numerical simulation when successively obtaining stablizing.
Two, the construction of two-dimensional case
Consider Two-dimensional Euler Equations:
Ut+f(U)x+g(U)y=0 (21)
Wherein, t expression time variable, x, y representation space variable, U=(ρ, ρ u, ρ v, E) expression conservation variable, f (U)= (ρu,ρu2+p,ρuv,u(E+p))T, g (U)=(ρ v, ρ uv, ρ v2+p,v(E+p))T, f (U), g (U) indicate flux, f (U)xIt indicates F (U) is to x derivation, g (U)yIndicate g (U) to y derivation, ρ, p, u, v, E respectively indicate fluid density, pressure, horizontal direction speed, Vertical direction speed and energy, T indicate transposition.Assuming that grid cell step-length is fixed, xi+1/2-xi-1/2=Δ x, yj+1/2- yj-1/2=Δ y, grid element center areGrid cell is Iij= [xi-1/2,xi+1/2]×[yj-1/2,yj+1/2], wherein subscript i, j is coordinate serial number, (xi,yj) at value be known as integral point value,The value at place is known as half point value.The integral for carrying out time orientation and direction in space respectively to governing equation, obtains:
Wherein,It is the cell-average value at the least bit of time horizon n+1,It is at the least bit of time horizon n Cell-average value.
Step 2.1. is to the unit mean value variable at the least bitIt carries out novel five rank and weights essential dead-beat reconstruct, tool Body process is as follows:
Step 2.1.1. is as shown in figure 4, by object element IijAnd totally 25 units form caster to surrounding, are denoted as R1, And assume that mesh spacing is all h, and rememberU is represented in grid cell Im,nAverage value (m=i-2 ..., i+2；N=j- 2,…,j+2)。
Step 2.1.2. selects 5 subtemplates to be respectively as follows: in caster
R2={ Ii-1,j+1,Ii,j+1,Ii+1,j+1,Ii-1,j,Ii,j,Ii+1,j,Ii-1,j-1,Ii,j-1,Ii+1,j-1}；
R2-1={ Ii-1,j+1,Ii,j+1,Ii-1,j,Ii,j}；
R2-2={ Ii,j+1,Ii+1,j+1,Ii,j,Ii+1,j}；
R2-3={ Ii-1,j,Ii,j,Ii,j-1,Ii-1,j-1}；
R2-4={ Ii,j,Ii+1,j,Ii,j-1,Ii+1,j-1}。
Step 2.1.3. finds out interpolation polynomial p respectively on caster and subtemplaten(x, y), n=5,3, (2_1), (2_2), (2_3), (2_4) make its satisfaction:
Wherein:
Then have:
Wherein,Indicate U in Ik,lGrid cell average value, xi,yjIndicate the value at grid node, Δ x, Δ y divide Not Biao Shi the direction x and the direction y step-length.
Step 2.1.4. takes the linear power of satisfaction and any positive number for one, avoids complicated numerical procedure, we It takes:
Wherein γ3 (3),γ2-1 (3),γ2-2 (3),γ2-3 (3),γ2-4 (3)It is three rank multinomials when institute in conjunction with second order polynomial The linear power needed, γ5 (5),γ3 (5),γ2-1 (5),γ2-2 (5),γ2-3 (5),γ2-4 (5)Five rank multinomials, three rank multinomials and Second order polynomial required linear power when combining.
Step 2.1.5. calculates smooth indicator, for measuring smoothness of the reconstruct multinomial on object element.
Calculate smooth indicator βl, l=5,3, (2_1), (2_2), (2_3), (2_4) is multinomial for measuring trigonometric function Formula pl(x, y) is in section [xi-1/2,xi+1/2]×[yj-1/2,yj+1/2] on smoothness, calculation formula are as follows:
Wherein, α is multiple indexes, and D is derivation operator, and r is multinomial plThe highest number of (x, y).
Step 2.1.6. calculates nonlinear weight:
Wherein, ωnIt is nonlinear weight,It is median, ε=10 with τ-10。
Step 2.1.7. finds out the expression formula of unit mean value at the least bit are as follows:
Step 2.2. is discrete using the progress of Gauss quadrature formula to time integral term, carries out extrapolation process to space integral, That is:
Wherein, Δ t is time step, and Δ x, Δ y are direction x spatial mesh size and y director space step-length, α respectivelylWith τlIt is Gauss system Number,It is extrapolation coefficient,
Step 2.3. carries out novel five rank at integral point to space variable and weights essential dead-beat reconstruct, and detailed process is such as Under:
The reconstruct multinomial and nonlinear weight that its obtained reconstruct multinomial and nonlinear weight and step 2.1 obtain are complete It is exactly the same.Find out the expression formula of unit point value at integral point are as follows:
Wherein, ω5,ω3,ω2-1,ω2-2,ω2-3,ω2-4It is nonlinear weight, γl (5), γll (3)It is linearly to weigh, l=5,3, 2_1,2_2,2_3,2_4, ll=3,2_1,2_2,2_3,2_4.
Step 2.4. is respectively to the derivative of flux in flux integral point value f (u (xi,yj)),g(u(xi,yj)) carry out novel five Rank weights essential dead-beat reconstruct, and integrative reconstruction process is identical with step 2.1, in addition to the interpolation polynomial q of constructionn (x, y), n=5,3, (2-1), (2-2), (2-3), (2-4) meets:
Wherein:
Then there is the expression formula of flux derivative at integral point:
Wherein, ω5,ω3,ω2-1,ω2-2,ω2-3,ω2-4It is nonlinear weight, γl (5), γll (3)It is linearly to weigh, l=5,3, 2_1,2_2,2_3,2_4, ll=3,2_1,2_2,2_3,2_4.
The integral point value of step 2.5. the integral point value based on space variable and flux derivative, by the derivative of iteration flux in whole Value at point carries out the process that novel five rank weights essential dead-beat reconstruct, obtains all flux in space integral point about the time Gaussian node:
Wherein,g(2), g(3), g(4) It is to be based respectively onIt carries out step 2.4 reconstruct to obtain, gaussian coefficient and step 1.5 identical.
Step 2.6. combines unit mean value variable and Gauss section of all flux in space integral point about the time at the least bit Point obtains the unit mean value at the least bit of next time horizon, thus whole flow field numerical simulation when successively obtaining stablizing.
Specific embodiment two
The method disclosed in the present is described further below by ten examples.
Example one
Test about precision problem.It is contemplated that one-dimensional Bugers equation:
Initial value takes u (x, 0)=0.5+sin (π x), service life boundary condition, when T=0.5/ π is set forth in Fig. 5 CWENO-ZQ algorithm and each leisure L of CWENO-QS algorithm1With L∞Under precision and error show.
Example two
Test about precision problem.It is contemplated that One-Dimensional Euler equation
Ut+f(U)x=0, (43)
Wherein, U=(ρ, ρ u, E)TIndicate conservation variable, f (U)=(ρ u, ρ u2+p,u(E+p))T, f (U) expression flux, Ut Indicate U to t derivation, f (U)xIndicate f (U) to x derivation, ρ, p, u, E respectively indicate fluid density, pressure, horizontal direction speed and Energy, T indicate transposition.Initial value takes ρ (x, 0)=1+0.2sin (π x), v (x, 0)=1, p (x, 0)=1, service life perimeter strip Part, CWENO-ZQ algorithm and each leisure L of CWENO-QS algorithm when t=2 is set forth in Fig. 61With L∞Under precision and errors table It is existing.
Example three
Test about precision problem.It is contemplated that two dimension Bugers equation
Initial value takes u (x, y, 0)=0.5+sin (π (x+y)/2), and T=is set forth in service life boundary condition, Fig. 7 CWENO-ZQ algorithm and each leisure L of CWENO-LP algorithm when 0.5/ π1With L∞Under precision and error show.
Example four
Test about precision problem.It is contemplated that Two-dimensional Euler Equations
Ut+f(U)x+G(U)y=0, (45)
Wherein, t indicates time variable, x, y representation space variable, U=(ρ, ρ u, ρ v, E)TExpression conservation variable, f (U)= (ρu,ρu2+p,ρuv,u(E+p))T, g (U)=(ρ v, ρ uv, ρ v2+p,v(E+p))T, f (U), g (U) indicate flux, f (U)xIt indicates F (U) is to x derivation, g (U)yIndicate g (U) to y derivation, ρ, p, u, v, E respectively indicate fluid density, pressure, horizontal direction speed, Vertical direction speed and energy, T indicate transposition.Initial value takes ρ (x, y, 0)=1+0.2sin (x+y), u (x, y, 0)=1, v (x, Y, 0)=1, p (x, y, 0)=1, service life boundary condition, CWENO-ZQ algorithm and CWENO- when t=2 is set forth in Fig. 8 Each leisure L of LP algorithm1With L∞Under precision and error show.
Example five
Test about high shock capturing problem.The Lax problem of our One-Dimensional Euler equations (44), wherein initial value takes:
Fig. 9 is the partial enlarged view of density map and density respectively from top to bottom, is from left to right CWENO-ZQ method respectively With CWENO-QS.Solid line in method figure is accurately to solve, and grid is numerical solution, takes 200 mesh points.
Example six
Test about high shock capturing problem.It is contemplated that the shock density wave of One-Dimensional Euler equation (44) Interaction problem, wherein initial value takes:
Figure 10 is the partial enlarged view of density map and density respectively from top to bottom, is from left to right CWENO-ZQ method respectively With CWENO-QS.Solid line in method figure is accurately to solve, and grid is numerical solution, takes 400 mesh points.
Example seven
Test about high shock capturing problem.It is contemplated that the two blast wave problem of One-Dimensional Euler equation (44), Wherein initial value takes:
Using reflective boundary condition, for this example there is no good performance, Figure 11 from left to right distinguishes CWENO-QS The density of CWENO-ZQ when giving t=0.001, the distribution map of speed and pressure, the solid line in figure is accurately to solve, and grid is Numerical solution takes 800 mesh points.
Example eight
Test about high shock capturing problem.It is contemplated that Sedov blast wave problem, wherein initial value is taken:
Using extrapolated boundary condition, for this example there is no good performance, Figure 12 from left to right distinguishes CWENO-QS The density, speed of CWENO-ZQ and the distribution map of pressure when giving t=0.001, the solid line in figure is accurately to solve, and grid is Numerical solution takes 800 mesh points.
Example nine
Double Mach reflection problem.The problem describes one and x-axis is mapped at the intense shock wave at 60 ° of angles and occurs on reflecting wall Variation, incoming flow is the intense shock wave of Mach 2 ship 10.Zoning is [0,4] × [0,1].Sections bottom fromIt is opened at y=0 Begin to be reflective boundary condition, other bottom boundaries (from x=0 toPart) it is wavefront condition.CWENO-LP algorithm pair In this example there is no good performance, when t=0.2 is set forth in Figure 13 from top to bottom CWENO-ZQ algorithm [0,3] × The density isogram and its partial enlarged view in [0,1] region, take 900 × 300 mesh points.
Example ten
Step problem.The problem is propose in nineteen sixty-eight one of Emery for examining non-linear hyperbolic conservation law lattice The classical example of formula.Primary data is that horizontal free stream Mach number is 3, density 1.4, horizontal velocity 3, and vertical speed is 0, pressure Strong is 1, and conduit region is [0,3] × [0,1], and having a height at boundary 0.6 From Left is 0.2 step, and step extends To the end of pipeline.Up-and-down boundary is reflecting boundary, and left margin is incoming flow boundary, and right margin is Outlet boundary.CWENO-LP is calculated Method is for this example there is no good performance, and the density isogram of CWENO-ZQ algorithm, takes 600 when Figure 14 gives t=4 × 200 mesh points.
Various aspects with reference to the accompanying drawings to describe the present invention in the disclosure, shown in the drawings of the embodiment of many explanations. Embodiment of the disclosure need not be defined on including all aspects of the invention.It should be appreciated that a variety of designs and reality presented hereinbefore Those of apply example, and describe in more detail below design and embodiment can in many ways in any one come it is real It applies, this is because conception and embodiment disclosed in this invention are not limited to any embodiment.In addition, disclosed by the invention one A little aspects can be used alone, or otherwise any appropriately combined use with disclosed by the invention.
Although the present invention has been disclosed as a preferred embodiment, however, it is not to limit the invention.Skill belonging to the present invention Has usually intellectual in art field, without departing from the spirit and scope of the present invention, when can be used for a variety of modifications and variations.Cause This, the scope of protection of the present invention is defined by those of the claims.

Claims (4)

1. a kind of whole flow field analogy method of the five rank WENO format of center based on staggered-mesh is suitable for the structure under cartesian coordinate The five rank WENO format of limited bulk of new model is made to calculate compressible flow field problem, which is characterized in that the whole flow field simulation Method includes:
S1: being integrated to obtain finite Volume Scheme to Hyperbolic Conservation equation space variable and time variable, using new shape Variable cell mean value at five rank WENO format of the limited bulk reconstruct least bit of formula；
S2: it is discrete using the progress of Gauss quadrature formula to time integral term, obtain Gauss of the flux in space integral point about the time The combined type of node；
S3: based on unit mean value using the variable point value at five rank WENO format of the limited bulk reconstruct integral point of new model；
S4: the point value based on step S3 reconstruct is using the derivative of five rank WENO format of the limited bulk reconstruct flux of new model whole Value at point；
S5: using the NCERK method of quadravalence, the integral point value of integral point value and flux derivative based on space variable is logical by iteration Value of the derivative of amount at integral point carries out the process that five ranks weight essential dead-beat reconstruct, obtains all flux and closes in space integral point In the Gaussian node of time；
The unit mean value variable and Gaussian node of all flux in space integral point about the time at the least bit are combined, is obtained next Unit mean value at the least bit of time horizon；
Successively iteration obtains the numerical result in end time flow field in zoning.
2. the whole flow field analogy method of the center five rank WENO format according to claim 1 based on staggered-mesh, special Sign is, in step S1, using the mistake of the variable cell mean value at five rank WENO format of the limited bulk reconstruct least bit of new model Journey the following steps are included:
S11: being based on unit mean value, selects caster and subtemplate, reconstructs the multinomial of several different accuracies；
S12: for the multinomial of several different accuracies of reconstruct, the linear power of satisfaction and any positive number for one is taken；
S13: calculating smooth indicator, for measuring smooth degree of the reconstruct multinomial on object element；
S14: nonlinear weight is calculated；
S15: portfolio restructuring multinomial and nonlinear weight obtain the reconstruction value of the variable cell mean value at the corresponding least bit.
3. the whole flow field analogy method of the center five rank WENO format according to claim 1 based on staggered-mesh, special Sign is, if One-dimensional Hyperbolic Conservation Law Equations are as follows:
The whole flow field analogy method the following steps are included:
S1.1: it carries out the integral of time orientation and direction in space respectively to governing equation, obtains:
Wherein,It is the cell-average value at the least bit of time horizon n+1,It is the cell-average value at the least bit of time horizon n, h It is the step-length of direction in space, f (u (xi+1, t), t) it is point value of the flux in variable at node,Be from when Interbed tnTo time horizon tn+1Integral；
To the unit mean value variable at the least bitIt carries out five ranks and weights essential dead-beat reconstruct；
S1.2: time integral term is carried out using Gauss quadrature formula using following formula discrete:
Wherein, Δ t is time step, αlWith τlIt is gaussian coefficient, obtains Gaussian node of the flux in space integral point about the time Combined type；
S1.3: the variable point value at five rank WENO format of the limited bulk reconstruct integral point of new model is used based on unit mean value；
S1.4: the point value based on step S1.3 reconstruct reconstructs the derivative of flux using the five rank WENO format of limited bulk of new model Value at integral point；
S1.5: by the NCERK method of quadravalence, the integral point value of integral point value and flux derivative based on space variable passes through iteration Value of the derivative of flux at integral point carries out the process that five ranks weight essential dead-beat reconstruct, obtains all flux in space integral point Gaussian node about the time is to get arriving:
Wherein, bj, cj, bj(τl) it is resulting value be middle coefficient, τlFor Gaussian node coefficient, Δ t is time step, g(j)It is base In flux reconstruction value of the derivative at integral point and obtained by NCERK method iteration；
The unit mean value variable and Gaussian node of all flux in space integral point about the time at the least bit are combined, is obtained next Unit mean value at the least bit of time horizon；
Successively iteration obtains the numerical result in end time flow field in zoning.
4. the whole flow field analogy method of the center five rank WENO format according to claim 3 based on staggered-mesh, special Sign is, in step S1.1, the unit mean value variable at the least bitCarry out the mistake that five ranks weight essential dead-beat reconstruct Journey the following steps are included:
S1.1.1: by object element IijAnd totally 5 units form caster to surrounding, are denoted as R1={ Ii-2,Ii-1,Ii,Ii+1, Ii+2, and assume that mesh spacing is all h, and rememberU is represented in grid cell ImAverage value (m=i-2 ..., i+2)；
S1.1.2: 2 subtemplates are selected to be respectively as follows: R in caster2={ Ii-1,Ii, R3={ Ii,Ii+1}；
S1.1.3: reconstruct multinomial p is found out respectively on caster and subtemplaten(x, y), n=1,2,3, make its satisfaction:
N=1, k=i-2, i-1, i, i+1, i+2；N=2, k=i-1, i；N=3, k=i, i+1；
Then have
Wherein,Indicate U in grid cell IkOn average value, xiIndicate the value at grid node.
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Application publication date: 20191115