CN101866392A - A Lattice Walking Method for Simulating One-dimensional Transport Process of Solute - Google Patents

Abstract

The invention discloses a grid walking method for simulating the one-dimensional migration process of solutes. Firstly, at the time t=0, a solute source is placed at the origin; the grid length Δx and the time interval Δt are set to satisfy

, then the concentration C _i (t+Δt)=K _{i, j} C _j (t) at the grid point i at the time t+Δt, the concentration at the next moment can be obtained from the concentration at the previous moment; iteratively , then the concentration at any time can be calculated and the result obtained. The present invention combines the advantages of the random walk method and the LBM method. Through the solute migration simulation of the one-dimensional instantaneous point source and the one-dimensional constant concentration point source, it can be found that the simulated results of the method are in good agreement with the analytical solution, especially It is suitable for dealing with the problem of convective dominance when the number of Pe is large, which is difficult for traditional methods.

Description

A kind of lattice walking method of simulating solute one-dimensional migration process

Technical field

The present invention relates to the hydraulics field, particularly a kind of analogy method of solute transfer process.

Background technology

Groundwater resource have important effect in people life and activity in production, northern area particularly, and the water consumption in some city 80% is all from underground water.Yet along with rapid economic development, the groundwater resource pollution problem of China is also serious day by day, and the whole nation has half city proper groundwater contamination more serious approximately, and is increasing by lack of water city and area that pollution causes.The many forms with solute of pollutant in the underground water exist, and the solute motion in the underground water is predicted accurately and control that very important realistic meaning is arranged.Solute transfer process in the underground water is described by convection current disperse equation usually, and its one dimension form is:

$\frac{\∂C}{\∂t}=D\frac{{\∂}^{2}C}{\∂x}+\frac{\∂\left(\mathrm{uC}\right)}{\∂x}---\left(1\right)$

Here C is a solute concentration, and D is a dispersion coefficient, and u is a flow velocity.This equation can be found the solution with the finite difference method and the finite elements method of routine, yet usually can run into two difficulties, i.e. numerical dispersion and numerical oscillation.The reason that produces this bad numerical value phenomenon is that convective-diffusive equation comprises hyperbolic item and parabolic item [Xue Yuqun, the red groundwater Numerical Simulation of Xie Chun Beijing: Science Press, 2007] simultaneously.In order to overcome this difficulty, people have done a large amount of effort, have invented a lot of methods such as windward method, characteristic line method, random walk method and lattice point Boltzmann method etc.The windward method can reduce numerical oscillation, but can increase numerical dispersion [N.Sun simultaneously, and W.Yeh, A proposed upstreamweight numerical method for simulating pollutant transport in ground water, Water Resour.Res.19 1489 (1983) .], and the application of windward method can become pretty troublesome in complex flowfield.Characteristic line method is effective ways relatively more commonly used, it handles convective term with Lagrangian method, handle diffusion term [S.P.Neumann with Euler method, Adaptive Eulerian Lagrangian finite-element method for advection dispersion, Int.J.Numer.Meth.Eng.20 321 (2003)], but shortcoming is the mass conservation that can not guarantee solute.The random walk method is typical Lagrangian method [A.Tompson and L.w.Gelhar, Numerical simulation of solute transport inthree-dimensional, randomly heterogeneous porous media, Water Resour.Res.26 2541 (1990), E.M.LaBolle, et al, Random-Walk Simulation of Transport in Heterogeneous PorousMedia:Local Mass-Conservation Problem and Implementation Methods, Water Resour.Res.32 583 (1996)], it has eliminated numerical dispersion, represented the convection current diffusion process very intuitively, but its applicable elements is that flow velocity and coefficient of diffusion must change slowly in the flow field, and needs abundant population represent mass distribution.When coefficient of diffusion was big, the efficient of random walk method can reduce.Grid Boltzmann (LBM) method on the kinetics basis of being based upon also is applied to [the Michael C.Sukop and DanielT.Thorne that transports of solute in the simulation fluid recently, Lattice Boltzmann methoding (Springer-Verlag, Berlin Heidelberg, 2006); D.Wolf-Gladrow, Lattice-gas cellular automata and lattice Boltzmann models:an Introduction (Springer-Verlag, Berlin Heidelberg, 2000], it can both provide analog result [X.Zhang preferably to be dominant problem and the convection current problem of being dominant of diffusion, et al, A novel three-dimensional lattice Boltzmann model forsolute transport in variably saturated porous media, Water Resour.Res.38 1167 (2002)].Yet in order to reduce the error of boundary, relaxation time τ needs delivery to intend used time interval Δ t, and this is diffusion coefficient D [Pore-scale simulation of dispersion, Phys.Fluids.12 2065 (2000) for R.Maier, et al] fixedly.The derivation of diffusion coefficient D is more loaded down with trivial details in addition, and changes with the variation of grid-like.

Summary of the invention

Technical matters to be solved by this invention provides the lattice walking method of solute one-dimensional migration process in a kind of Simulated Water mechanics, result and analytic solution that this method simulation obtains are very identical, no matter are to be dominant or to spread in the problem that is dominant in convection current to provide extraordinary analog result.

The lattice walking method of simulation solute one-dimensional migration process of the present invention, it mainly may further comprise the steps:

1), throws in the solute source at the initial point place in the moment of t=0;

2) set grid length Δ x and time interval Δ t, and it is satisfied

D is a dispersion coefficient;

3) concentration C at t+ Δ t moment lattice point i place _i(t+ Δ t)=K _{I, j}C _j(t), K _{I, j}Be transition matrix, some x _j+ u Δ t is the mean place that the motion of the particle process Δ t time of lattice point j reaches, and establishing lattice point m is the lattice point nearest apart from this mean place, and r is defined as x _j+ u Δ t-x _m, K then _{M-1, j}=(2D Δ t+r ²-r Δ x)/(2 Δ x ²), K _{M, j}=(2D Δ t+ Δ x ²-r ²)/Δ x ², K _{M+1, j}=(2D Δ t+r ²+ r Δ x)/(2 Δ x ²), the particle of lattice point j is to the transition matrix K of all the other lattice point place correspondences _{I, j}=0, just can obtain back one concentration constantly by the concentration of previous moment like this;

4) with this iteration, then can obtain the concentration of any time, obtain a result.

Have two kinds for the described solute of step 1) source, a kind of is the instantaneous point source of one dimension, and promptly the initial concentration at release position place is for throwing in concentration, afterwards in the time no longer the concentration to release position artificially interfere.

Another kind of solute source is that one dimension is decided the concentration continuous point source, and promptly the concentration at release position place equals the given concentration of boundary condition all the time, and the concentration at used release position place is the given concentration of boundary condition during iteration each time.

The present invention proposes a kind of method of brand-new simulation solute one-dimensional migration process, it combines the advantage of random walk method and LBM method.Similar with the LBM method, what the lattice walking method among the present invention was simulated is the motion of particle on lattice point.Different with it is that the distribution probability of particle is determined by the Gaussian distribution that random walk causes in lattice walking method, rather than is determined by Boltzmann equation.Square by two kinds of Probability Distribution equates that the present invention represents continuous Gaussian distribution with discrete grid point distribution.By to the simulation of the solute transfer of instantaneous point source of one dimension and one dimension constant density point source, can find that the result that this method simulation obtains coincide finely with analytic solution, be adapted to handle Pe number that classic method is difficult to the be competent at convection current greatly time problem that is dominant especially.

Analog result shows that finite difference method commonly used can bring serious numerical dispersion near the sharp side, and no matter the present invention is to be dominant or to spread in the problem that is dominant in convection current to provide extraordinary analog result.Another advantage of the present invention is insensitive to flow velocity, and grid interval delta x and time interval Δ t when needing only simulation satisfy the relation of formula (6), and this method can be simulated the solute transport process under any flow velocity.Therefore in the bigger flow field of change in flow, the present invention can improve the efficient that solute transports simulation greatly.

Description of drawings

Fig. 1 is that particle is from x _jPoint is to x _MeanThe movement locus at contiguous lattice point place.

Fig. 2 is the solute Distribution curve map of the given time that transports of the instantaneous point source solute of one dimension.

Fig. 3 is the solute Distribution curve map of the given time that transports of one dimension constant density point source solute.

Embodiment

As shown in Figure 1, be that particle is from x _jPoint is to x _MeanThe motion at contiguous lattice point place, wherein r=x _Mean-x _mSuppose that t moment particle is at position x _jThe place, the spacing of lattice point is made as Δ x.After the elapsed time interval of delta t, the mean place of these particles is x _Mean, i.e. x _j+ u Δ t is designated as x from its nearest lattice point _mBecause diffusion, particle may be to the contiguous lattice point motion of mean place.We suppose that particle is at x at t+ Δ t constantly _M-1The probability at place is P _M-1, at x _mThe probability at place is P _m, at x _M+1The probability at place is P _M+1, the probability at other lattice point place is 0.

If particle moves in continuous space, so at x _jBehind the particle process Δ t of point, its space distribution is

$P\left(x\right)=\frac{1}{\sqrt{4\mathrm{\πD\Δt}}}\exp [-\frac{{(x-x_j-\mathrm{u\Δt})}^2}{4\mathrm{D\Δt}}].---\left(2\right)$

We make discrete distribution P _i0～2 rank square equate that with the P (x) of continuous distribution respectively corresponding quality equates that mean place equates and diffusion equates, can obtain the equation of three simultaneous like this

$\left\{\begin{array}{c}\underset{i=m-1}{\overset{m+1}{\mathrm{\Σ}}}{P}_i=\underset{-\∞}{\overset{+\∞}{\∫}}P\left(x\right)\mathrm{dx}=1\\ \underset{i=m-1}{\overset{m+1}{\mathrm{\Σ}}}{P}_i(x_i-x_{\mathrm{mean}})=\underset{-\∞}{\overset{+\∞}{\∫}}P\left(x\right)(x-x_{\mathrm{mean}})\mathrm{dx}=0\\ \underset{i}{\mathrm{\Σ}}{P}_i{(x_i-x_{\mathrm{mean}})}^2=\underset{-\∞}{\overset{+\∞}{\∫}}P\left(x\right){(x-x_{\mathrm{mean}})}^2\mathrm{dx}=2\mathrm{D\Δt}\end{array}\right.,---\left(3\right)$

Obtain thus

$\left\{\begin{array}{c}{P}_{m-1}=\frac{2\mathrm{D\Δt}+r^2-\mathrm{r\Δx}}{2\mathrm{\Δ}{x}^2}\\ P_m=\frac{-2\mathrm{D\Δt}+\mathrm{\Δ}{x}^2-r^2}{\mathrm{\Δ}{x}^2}\\ P_{m+1}=\frac{2\mathrm{D\Δt}+r^2+\mathrm{r\Δx}}{2\mathrm{\Δ}{x}^2}\end{array}\right..---\left(4\right)$

At t+ Δ t constantly, the solute concentration of each node can be calculated by following formula

C _i(t+Δt)＝K _i，jC _j(t)， (5)

K wherein _{I, j}=P _i, be transition matrix (transition matrix). obviously, the stability condition of lattice walking method is P _iMust be non-negative.

Verify effect of the present invention below in conjunction with specific embodiment.

Do contrast earlier with one dimension instantaneous point source model checking lattice walking method, and with analytic solution and method of finite difference numerical solution result.In the moment of t=0, throw in concentration at the initial point place and be 1 instantaneous source.The Pe number is defined as Pe=u Δ x/D, and its is described is convective term and the diffusion term Relative Contribution to solute transfer.Fig. 2 is that one dimension instantaneous source solute transports, and gets Δ x=1, Δ t=1.The Pe number of two process correspondences among Fig. 2 is about 30 and 0.5, represents convection current problem and the diffusion problem that is dominant that is dominant respectively.Occupy problem for convection current, it is fine that lattice walking method and exact solution meet, and serious numerical dispersion has appearred in finite difference method.For the diffusion problem that is dominant, lattice walking method, finite difference method are all quite identical with exact solution.

What Fig. 3 showed is the analog result that one dimension is decided concentration continuous point source model, and all processes are all got D=0.2, Δ x=1, Δ t=1.The numerical solution of lattice walking method and analytic solution and method of finite difference numerical solution result are done contrast.From t=0 constantly, dropping into concentration continuously at the initial point place is 1 constant source.Three process Pe numbers (u Δ x/D) among Fig. 3 are respectively 0.5,1.5 and 6.Similar with the analog result of instantaneous point source among Fig. 2, no matter be that diffusion is dominant or the convection current problem that is dominant, analog result that lattice walking method obtains and analytic solution are very approaching.And method of finite difference numerical solution result except when diffusion be dominant (u=0.1 can provide comparatively approaching separating in the time of Pe=0.5), other two kinds of situations to separate near the sharp side numerical dispersion very severe.The above results shows that lattice walking method can overcome the numerical dispersion problem in the convection current disperse equation numerical solution well.

We have also done the simulation contrast of other situations.Repeatedly simulating comparing result and show, is not the precondition of negative value (this condition nature in real process should satisfy) as long as satisfy the probability P that particle drops on each lattice point, and lattice walking method simulation gained result is exactly very reliable.Probability P and particle are irrelevant from which lattice point, and only and r, D is relevant.As long as satisfy

$\sqrt{2\mathrm{D\Δt}}\≤\mathrm{\Δx}\≤2\sqrt{2\mathrm{D\Δt}}---\left(6\right)$

Just can simulate the solute transport process under any flow velocity, the lattice spacing and the time interval chose when this was very easy to simulation.

Claims (3)

1. A lattice walking method for simulating the one-dimensional migration process of solute, is characterized in that comprising the following steps: 1) At the moment of t=0, put in the solute source at the origin; 2) Set the grid length Δx and the time interval Δt, and make it satisfy

D is the diffusion coefficient; 3) Concentration C _i (t+Δt) at grid point i at time t+Δt=K _i,j C _j (t), K _i,j is the transition matrix, point x _j +uΔt is the particle passing through grid point j The average position reached by the movement of Δt time, the grid point m is the grid point closest to the average position, r is defined as x _j +uΔt-x _m , then K _{m-1, j} = (2DΔt+r ² -rΔx) /(2Δx ² ), K _m,j ＝(-2DΔt+Δx ² -r ² )/Δx ² , K _m+1,j ＝(2DΔt+r ² +rΔx)/(2Δx ² ), grid point j The transfer matrix K _{i, j} = 0 corresponding to the particle to the rest of the lattice points, so that the concentration at the next moment can be obtained from the concentration at the previous moment; 4) With this iteration, the concentration at any time can be obtained, and the result can be obtained. 2. The lattice walking method for simulating the one-dimensional migration process of solute according to claim 1, wherein the solute source described in step 1) is a one-dimensional instantaneous point source, that is, the initial concentration at the delivery point is the delivery concentration, In the future, there will be no human intervention on the concentration of the delivery point. 3. The lattice walking method for simulating the one-dimensional migration process of solute according to claim 1, characterized in that the solute source described in step 1) is a one-dimensional constant concentration continuous point source, that is, the concentration at the delivery point is always equal to the boundary The concentration given by the conditions, the concentration at the delivery point used in each iteration is the concentration given by the boundary conditions.

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