CN105787162B - The method for numerical simulation of multimedium interface tracking is realized based on non-structural RKDG - Google Patents
The method for numerical simulation of multimedium interface tracking is realized based on non-structural RKDG Download PDFInfo
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Abstract
The present invention proposes a kind of method for numerical simulation that multimedium interface tracking is realized based on non-structural RKDG.It is discrete to the progress of multimedium interface using a series of index points on unstrctured grid, Riemannian problem is constructed in index point normal direction and is solved using two shock approximation method.The solution of Riemannian problem be applied not only to promote interface, and can directly update interface nearby real fluid side fluid state and extrapolation obtain corresponding Level Set method state.Meanwhile each single medium flow field is solved using the preferably non-structural RKDG method of space compactness.It is entire to solve scheme programming simply, it calculates efficient.By comparing with the result obtained based on finite Volume Scheme, the conservative errors that the present invention obtains are smaller, and numerical stability is more preferable.
Description
Technical field
The invention belongs to numerical simulation technology fields, and in particular to one kind realizes that multimedium interface chases after based on non-structural RKDG
The method for numerical simulation of track.
Background technique
The solution of compressible multimedium problem mainly includes two parts: first is that carrying out high-precision for each single medium flow field
Numerical simulation, second is that accurately handling multimedium interface and its near zone.In recent two decades, golden (DG) method of interruption gal the Liao Dynasty
Research hotspot is increasingly becoming on solving single medium flow field.In the numerous researchs carried out to DG method, it is worth mentioning at this point that
Golden (RKDG) method of Runge-Kutta interruption gal the Liao Dynasty that Cockburn et al. is proposed: reduced on time discrete using total variance
(TVD) high-order Runge-Kutta (RK) method obtains grid boundary flux, application by solving Riemannian problem on spatial spreading
Total variance bounded (TVB) limiter inhibits the false oscillation of numerical solution.The major advantage of RKDG method: the smooth domain in flow field
It is easy to construct high accurate scheme；The numerical solution of polynomial form can be obtained in flow field any position；Space compactness is good, net
The boundary flux of lattice is only related with the left and right grid cell on boundary.Based on these advantages, RKDG method is solving single medium flow field
Biggish success is achieved in problem.In multimedium interface processing, based on early γ model, mass fraction model, level
The method of set function often generates biggish numerical dissipation in multimedium interface.However, for interface-tracking method, boundary
Face be explicitly track and also interface location can remain very clear in calculating process.Level Set method method is mainly used
In definition interfaces boundary condition so that every kind of medium can be solved individually.But when pressure or speed are attached at interface
When closely there is biggish gradient, the expression effect of this method is poor.In fact, the definition of Level Set method state should be examined properly
Consider the interaction of interface wave system and the fluid properties of different medium.Subsequent correlation scholar proposes some modified versions
Level Set method method, such as actual and virtual fluid method (RGFM).This method updates interface by solving interface Riemannian problem
The fluid state of real fluid side and extrapolation obtains Level Set method state.Because the wave system of interface can upstream and downstream
It propagates simultaneously, so the interface boundary condition that actual and virtual fluid method defines has lesser conservative errors.Pass through these sides
The integrated application of method can carry out preferable numerical simulation to compressible multimedium problem.
Up to the present, the research for realizing multimedium interface tracking is coupled with Level Set method method for DG method both at home and abroad
Still less, and multimedium interface problem is mostly solved on homogeneous texture grid.In order to realize to single medium flow field
The accurate solution of field, the research of early stage often use the finite difference scheme or finite Volume Scheme of higher order accuracy.However, high
Rank format generally requires more template point since space compactness is poor, needs to define more Level Set methods near interface
State.This is often unfavorable in some multimedium problems with complicated interface.Because interface-tracking method is in very great Cheng
Dependent on the geological information on interface on degree, the geometric parameters such as normal vector are being solved there are biggish error far from interface, from
And it further results in the Level Set method state far from interface there are biggish inexactnesies.
Goal of the invention
It is above mostly individually carried out since the prior art is solved in multimedium interface tracking and Level Set method state, thus
Increase the complexity of programming and calculating.In addition, higher order accuracy numeric format needs more mould when solving single medium flow field
Plate point thus inevitably needs to define the Level Set method state far from interface.However it is remote in interface-tracking method
Often there is biggish error in the Level Set method state from interface, which results in the prior arts to realize multimedium interface tracking
It is upper that there is biggish inexactness.Therefore, the invention proposes a kind of simpler, more accurate multimedium interface-tracking methods.
In multimedium Interface Construction Riemannian problem, the solution of Riemannian problem is applied not only to promote interface, and is used directly for updating true
Real fluid state and extrapolation obtain Level Set method state.In addition, being solved using the preferably non-structural RKDG method of space compactness
Single medium flow field greatly reduces the information of required Level Set method state.
Summary of the invention
It is a kind of in order to overcome the deficiencies in the prior art, the present invention provides that multimedium interface is realized based on non-structural RKDG
The numerical simulation solution of tracking, specific as follows:
Step 1, unstrctured grid is generated on computational domain；
Step 2, Level Set method region is determined according to the position at interface；
Step 3, using a series of discrete multimedium interface of index points, Riemannian problem is constructed in index point normal direction, using double
Shock wave approximation method solves.The movement velocity of index point is obtained according to the solution of Riemannian problem；
Step 4, the fluid state of real fluid side near interface is directly updated using the solution of Riemannian problem on index point,
And it extrapolates and obtains corresponding Level Set method state；
Step 5, every kind of medium is individually solved using non-structural RKDG method；
Step 6, multimedium interface is promoted by the speed of index point, reconstructs interface, obtains new interface location and index point
And the numerical solution in entire multimedium flow field is determined according to new interface；
Step 7, judge whether to meet and calculate termination condition, such as calculating also needs to continue, and return step 2 carries out lower a period of time
The calculating at quarter.
The present invention by adopting the above technical scheme, has the advantages that
(1) unstrctured grid has stronger adaptability for computational domain boundary；
(2) it is applied not only to promote interface in Interface Construction Riemannian problem, and can be directly used for defining Level Set method state,
Programming is simple, calculates efficient.
(3) it is calculated using the preferable RKDG method of space compactness so that required Level Set method status information greatly subtracts
It is few, improve the accuracy of Flow Field Calculation.
Detailed description of the invention
Fig. 1 is the construction of slope limiter.
Fig. 2 is the construction Riemannian problem at index point.
Fig. 3 is numerical result (example 5.1).
Fig. 4 is computational domain (example 5.2).
Fig. 5 is the flow field density (example 5.2) of different moments.
Fig. 6 is characteristic point motion diagram (example 5.2).
Fig. 7 is relative mass error (example 5.2).
Fig. 8 is computational domain (example 5.3).
Fig. 9 is the density cloud atlas (example 5.3) of different moments.
Figure 10 is characteristic point motion diagram (example 5.3).
Figure 11 is SF6Medium relative mass error (example 5.3).
Figure 12 is the density cloud atlas (example 5.4) of different moments.
Figure 13 is different moments density (left side) and pressure (right side) comparing result (example 5.4).
Figure 14 is air dielectric relative mass error (example 5.4).
Specific embodiment
The present invention combines RKDG method and interface-tracking method on unstrctured grid and passes through RGFM definition interfaces side
Boundary's condition solves multimedium problem.Other than unstrctured grid has stronger adaptability for zoning boundary, by
There is preferable space compactness in RKDG method, the Level Set method state far from interface is not engaged in calculating.By with base
It is compared in the result that finite Volume Scheme obtains, the present invention obtains that conservative errors are smaller, and numerical stability is more preferable.
Further explanation of the contents of the invention are made with example with reference to the accompanying drawing:
(1) non-structural RKDG method
Governing equation are as follows:
Wherein ρ is density, and u and v are speed, and p is pressure, and E is the total of unit volume
Energy, is defined as:
E=ρ e+ ρ (u2+v2)/2 (2)
Wherein e is the interior energy of unit mass.State equation are as follows:
P=(γ -1) ρ e- γ B (3)
γ and B is fluid constant.It is specific heat ratio for perfect gas γ, B is taken as 0.
For unstrctured grid K0, numerical solutionWith basic function space byGiven (its
Middle Pk(K0) it is grid K0The upper degree of polynomial is less than or equal to the multinomial set of k).Numerical solutionIt can indicate are as follows:
For P1Situation, freedom degreeIt is the value at each side of grid midpoint, basic function φi(x, y) is linear letter
Number, value is 1 on the midpoint on i-th side, and the midpoint value on other sides is 0.For P2Situation, freedom degree
It is the value of grid vertex and each side midpoint, basic function φi(x, y) is quadratic function, and value is 1 on above-mentioned i-th point,
It is 0 in other values.Wushu (4) substitutes into formula (1), to formula (1) multiplied by basic function φi(x, y) and in grid K0Upper branch's product
Point:
WhereinIt is mass matrix,φ (x, y)=[φ1(x,y),φ2(x,y),…,φN(x,y)]T, it is net
Lattice K0The unit normal vector of top e.It is approximate using Lax-Friedrichs flux.Right end integral
Item is solved using Gaussian integration method.Half discrete form of formula (5) indicates are as follows:
Time discrete is carried out using 3 rank TVD RK methods:
The false oscillation problem being likely to occur is overcome using slope limiter when flow field is interrupted.As shown in Figure 1,
K1, K2And K3It is and grid K0Three adjacent grids, m1, m2And m3It is grid K0The midpoint on upper three sides, b0, b1, b2And b3Point
Not Biao Shi grid centroid.It is obtained by geometry decomposition:
Wherein α1And α2For nonnegative real number.The average value of grid are as follows:
Note:
On other two sides midpointsWithIt similar can provide.For piecewise linear functionIt is available:
The specific configuration of limiter is as follows, note:
Wherein ν=1.5,It is the minmod function of amendment form:
Wherein M is limiter constant.
IfThen have:
Otherwise, it enables:
It takes:
Then have:
(2) interface-tracking method
As shown in Fig. 2, in tnMoment medium 1 and medium 2 are separated by interface.Mark point is the intersection point of interface and grid lines.WithIt is the normal vector and tangent vector of mark point respectively.It is mark point P (xP,yP) unit normal direction
Amount.Point A (xA,yA) and point B (xB,yB) in different media, it is apart that (h is that the maximum of grid circumscribed circle is straight to h with mark point P
Diameter).Here:
xA=xP+h·NPx,yA=yP+h·NPy
xB=xP-h·NPx,yB=yP-h·NPy
The state of point A and point B is directly obtained by formula (4):
Wherein KAAnd KBIt is the grid comprising point A and point B respectively.Density, normal velocity and the pressure of point A and point B are calculated,
It is expressed asWithMultitude is constructed along the normal direction of index point P
Graceful problem, primary condition are as follows:
Use two shock approximation method to solve above-mentioned Riemannian problem and remember its solution forWherein
Subscript I indicates interface, and subscript L and R are indicated at left and right sides of interface.The tangential velocity of index point P is defined as:
WhereinWithIt is the normal velocity and tangential velocity of index point P,WithIt is the tangential velocity of point A He point B.
After finding out the speed of all index points, the new position of index point is obtained using three rank TVD RK methods:
WhereinWithIt is index point in tnAnd tn+1The position at moment,It is the speed of index point, Δ t is time step
It is long.
(3) Level Set method region is determined
Since each medium is individually solved, it is therefore necessary to determine the Level Set method region of every kind of medium.Such as Fig. 2 institute
Show, it is specified that the normal vector of all index points is directed toward medium 2 by medium 1.By taking mesh point C as an example, itself and surrounding are calculated first
The distance of index point.It is assumed that index point P is minimum at a distance from mesh point C.Number of computations productWhereinIt is by point P
It is directed toward the position vector of point C.If scalar product is greater than 0, mesh point C is in the Level Set method region of medium 1, otherwise in medium 2
Level Set method region.Similar method can be used for other mesh points.Normal vector on Level Set method region uses
Area-weighted method acquires.Since the space compactness of RKDG method is preferable, the normal vector far from interface is not engaged in meter
It calculates from without being solved.
(4) actual and virtual fluid method
In actual and virtual fluid method, passes through construction Riemannian problem near interface and update real fluid state and extrapolate
Obtain Level Set method state.Due to the Riemannian problem at interface tracking part the structural sign point, the solution of Riemannian problem
It is used directly for defining Level Set method state.As shown in Fig. 2, point P, Q, R are three index points near lattice point C,It is
The normal vector of grid C.The state of grid C can be by being updated with the Riemann Solution on the smallest index point of its normal angle.Such as
Fruit index point P be with the smallest index point of grid C normal angle, then the Riemann Solution at mark point P is projected to feature space
The average value of grid C can be updated.Nearby real fluid state updating method is similar at other interfaces.
Level Set method state is obtained by solving following equation:
Wherein φ indicates density, normal velocity, tangential velocity and pressure.As shown in Fig. 2, in the perimeter strip for solving medium 1
"+" number is used when part, otherwise uses "-" number.Using alternative manner solve the equation, spatially using single order upstreame scheme from
It dissipates, it is discrete using the progress of three rank TVD RK methods on the time.Since RKDG method space compactness is fine, herein only with 2~3
A grid is as Level Set method region.By obtaining virtual grid state Level Set method state, as projected to basic function space
Average value can determine interface boundary condition in this way.
(5) new method is verified
Example 5.1. shock tube problem.This example is for studying the expansion issues of pressure-air in water.Computational domain design
For circle, the center of circle is located at (1,1), diameter 2.Initial time, the circular bubble that diameter is 0.8 are placed in computational domain
The heart.Primary condition inside and outside bubble are as follows:
Outer boundary is all made of nonreflecting boundary condition.Grid dividing is carried out to computational domain, there are 4746 as shown in Fig. 3 (a), inside region
Grid, lattice point number 2443.Using P2RKDG method, M are taken as 0.1.Fig. 3 (b) gives in the close of the moment flow field t=0.002
Spend cloud atlas.Wherein it will be clear that interface, shock wave and dilatational wave.Dilatational wave is mobile towards computational domain center and shock wave to
The Boundary Moving of computational domain.In order to be compared with the Finite Volume Method with the discrete precision of same space, here RKDG
Method replaces with three rank limited bulk WENO methods and keeps the other parts of program constant.RKDG method, WENO method and boundary
The combination of face method for tracing is abbreviated as RKDG-FT method and WENO-FT method respectively.Fig. 3 (c) is compared based on the side RKDG-FT
Method and WENO-FT method density along y=1 distribution situation.First and last, Density Distribution is almost the same.However, in shock wave and
In the capture of rarefaction wave, RKDG-FT method has more advantage.It is apparent that the shock region obtained based on RKDG-FT method
Domain is narrower.In order to further make comparisons to two methods, Fig. 3 (d) illustrates the relative mass error Me of air.It is based on
The error that RKDG-FT method obtains is significantly smaller, this sufficiently illustrates advantage of the RKDG-FT method in interface processing.
Example 5.2. shock wave hits helium bubbles problem.Flow Field Calculation domain is as shown in figure 4, wherein parameter a=1, b=0.5, c
=2, d=6.5, e=0.89.The shock wave of Mach 2 ship 1.22 hits helium bubbles in air.Since flow field is about central axis pair
Claim, only studies the flow field problem in half of flow field above herein.Symmetrical boundary condition is taken on central axes, right boundary is taken as
Nonreflecting boundary condition, coboundary are wall boundary condition.Primary condition are as follows: wavefront air ρ=1, u=0, v=0, p=1/
1.4, γ=1.4, B=0, air ρ=1.3764, u=-0.3336, v=0, p=1.5698/1.4, γ=1.4, B=after wave
0, helium bubbles ρ=0.1819, u=0, v=0, p=1/1.4, γ=1.648, B=0.Grid dividing, region are carried out to computational domain
Inside generates 58364 grids, lattice point number 29604.Using P2RKDG method, M=0.1.Fig. 5 gives different moments flow field
The situation of change of density.It carves at the beginning, after interface is hit by shock wave, part interface is moved.In the compression of helium bubbles
In the process, the length at interface becomes smaller and some index points fade away.Helium bubbles roll in a counterclockwise direction, lead to bubble
Form biggish pressure below to push bubble to be moved to the left.Shock wave is divided into two parts after hitting helium bubbles, wherein helium
Shock wave in bubble there is higher intensity and also movement velocity also faster than incident shock.In order to the result of study of forefathers into
Row comparison, Fig. 6 illustrate motion diagram and Terashima of three characteristic points (Jet, Downstream, Upstream) et al.
Calculated result.It can be seen that these results are almost the same.In order to further make comparison with WENO-FT method, distinguish herein
The relative mass error of helium bubbles is calculated, as shown in Figure 7.As can be seen that helium bubbles rupture before two methods it is opposite
Quality error is respectively less than 7%.Error generally based on RKDG-FT method is smaller.
Example 5.3. gas-gas Richtmyer-Meshkov instability problem.As shown in figure 8, Flow Field Calculation domain be [0,
4] × [0,0.5] moves downward strike gas vapor interface positioned at the shock wave of x=3.2, Mach 2 ship 1.24.Interface original shape are as follows:
X=2.9-0.1sin (2 π (y+0.25)), 0 < y < 0.5.Flow field primary condition are as follows: SF6ρ=5.04, u=0, v=0, p=1,
γ=1.093, B=0, wavefront air ρ=1, u=0, v=0, p=1, γ=1.4, B=0, air ρ=1.411, u=- after wave
0.39, v=0, p=1.628, γ=1.4, B=0.Right boundary takes nonreflecting boundary condition, and up-and-down boundary is symmetrical border item
Part.Grid dividing is carried out to computational domain, generates 105542 grids, lattice point number 53357 inside region.Using P2The side RKDG
Method, M=0.1.The density cloud atlas of different moments is as shown in Figure 9, it can be clearly seen that shock wave is in SF6Motion process in medium
And the variation at interface.As verifying, Figure 10 illustrates the motion diagram of characteristic point (Spike, Bubble), also gives simultaneously
The calculated result of Terashima et al. is as a comparison.As can be seen that these curve movements are almost the same.It is similar with example 5.2,
Figure 11 shows the SF based on RKDG-FT method and WENO-FT method6The relative mass error of medium.As can be seen that at the beginning
It carves, error is not much different.However as shock wave in SF6It is constantly moved in medium, interface shape becomes complicated, is based on WENO-FT
The error of method increases comparatively fast, and the error change based on RKDG-FT method is gentle.
Example 5.4. solution-air Richtmyer-Meshkov instability problem.It is counted using with the identical flow field of example 5.3
Calculate domain, interface original shape and grid dividing.It is located at x=3.025 in liquid, the shock wave of Mach 2 ship 1.95 moves downward strike
Gas-liquid interface.Flow field primary condition are as follows: air ρ=1, u=0, v=0, p=1, γ=1.4, B=0, wavefront liquid ρ=5, u=
0, v=0, p=1, γ=4, B=1, liquid ρ=7.093, u=-0.7288, v=0, p=10, γ=4, B=1 after wave.Using
P2RKDG method, M=0.1.Figure 12 gives the situation of change of density in flow field.It can be clearly seen that shock wave is hit from figure
Wave system structure behind interface.For the correctness of proving program, Figure 13 give based on γ model, along y=0.5 density,
Pressure distribution compares.It can be seen that result is almost the same, it was demonstrated that the accuracy of RKDG-FT method.In order to further open up
Show that the superiority of RKDG-FT method, the relative mass error that Figure 14 gives air dielectric change over time situation.It can be seen that
In the initial stage, error is almost the same.After shock wave enters air dielectric, with the increase of time, based on WENO-FT method
Error rapid development, and it is smaller based on the error that RKDG-FT method obtains.
Claims (1)
1. realizing the method for numerical simulation of multimedium interface tracking based on non-structural RKDG, steps are as follows:
Step 1, unstrctured grid is generated on computational domain；
Step 2, Level Set method region is determined according to the position at interface；
Step 3, using a series of discrete multimedium interface of index points, Riemannian problem is constructed in index point normal direction, using bidifly wave
Approximation method solves, and obtains the movement velocity of index point according to the solution of Riemannian problem；
Step 4, the fluid state of real fluid side near interface is directly updated using the solution of Riemannian problem on index point, and outer
It pushes away to obtain corresponding Level Set method state；
Step 5, each single medium flow field is solved respectively using non-structural RKDG method；
Step 6, multimedium interface is promoted by the speed of index point, reconstructs interface, obtain new interface location and index point and root
The numerical solution in entire multimedium flow field is determined according to new interface；
Step 7, judge whether to meet and calculate termination condition, such as calculating also needs to continue, and return step 2 carries out subsequent time
It calculates；
Wherein, interface processing method uses interface-tracking method, in tnMoment medium 1 and medium 2 are separated by interface, and index point is
The intersection point at interface and grid lines,WithIt is the normal vector and tangent vector of index point respectively,It is index point P
(xP,yP) unit normal vector, point A (xA,yA) and point B (xB,yB) in different media, it is apart h with index point P, h is grid
The maximum gauge of circumscribed circle, here:
xA=xP+h·NPx,yA=yP+h·NPy
xB=xP-h·NPx,yB=yP-h·NPy
The state of point A and point B:
Wherein KAAnd KBIt is the grid comprising point A and point B respectively, calculates density, normal velocity and the pressure of point A and point B, respectively
It is expressed asWithIt is asked along the normal direction construction Riemann of index point P
Topic, primary condition are as follows:
Use two shock approximation method to solve above-mentioned Riemannian problem and remember its solution forWherein subscript
I indicates interface, and subscript L and R are indicated at left and right sides of interface, the tangential velocity of index point P is defined as:
WhereinWithIt is the normal velocity and tangential velocity of index point P,WithIt is the tangential velocity of point A He point B.
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