CN102880797A - Mesoscopic method for processing dryness and wetness boundary problem - Google Patents

Abstract

The invention discloses a mesoscopic method for processing the dryness and wetness boundary problem. The method is characterized in that: on the basis of a grid Boltzmann model of a shallow water equation in computational hydraulics, information contents of dryness and wetness grid units are correlated by utilizing Chapman-Enskog expansion and Taylor formula, so that the change of a dryness and wetness boundary is numerically simulated. When the method disclosed by the invention is applied to processing the dryness and wetness boundary problem, virtualization of an artificial film or non-physical epitaxial difference of such variables as water depth and flow rate can be avoided; and meanwhile, exogenic action can be naturally and reasonably merged into calculation, and the grid local correlation of the grid Boltzmann model is kept, so that the model is suitable for large-scale parallel calculation.

Description

A kind of Jie who processes dried wet boundary problem sees method

Technical field

The innovation method belongs to the hydraulics field, relates to the processing of doing wet border in the computational hydraulics, and especially Jie based on shallow water equation sees numerical algorithm.

Background technology

In nature and hydraulic engineering, Shallow-water Flow is all very common, such as the river, and lake and harbour etc.Method for numerical simulation to Shallow-water Flow is a lot, and these methods often need to be processed the wet boundary problem of doing of current, and the boundary that namely solids such as water body and land is contacted carries out special processing.The research of doing wet boundary problem is subject to international extensive concern (Balzano, 1998) always.Closely during the last ten years, the research of doing wet boundary processing method mainly contains: the slit method that Kennedy et al. (2000) uses in the Finite Element Method, and namely supposing has a slit at the dried boundary bottom, makes the depth of water can be lower than the earth's surface; Lynett et al. (2002) uses linear extension that dried grid cell is carried out difference, thereby obtains the depth of water of dried grid cell and flow velocity etc.; Madsen et al. (2007) proposes film process to be applied to the dried grid of boundary, supposes that namely dried grid cell place has very thin one deck water, then it is calculated simulation as normal water body.Frandsen (2008) is applied to development in recent years sight method---the lattice Boltzmann method that is situated between rapidly with linear epitaxy and membrane process, simulates having the Shallow-water Flow of doing wet boundary problem; Shafiai (2011) also is dissolved into same thought in the turbulent flow lattice Boltzmann method and has carried out a large amount of measuring and calculations.In China open source literature and patent are arranged not yet and use Jie's sight method to process the correlative study of doing wet boundary problem.The linear extension of the non-physics of use that present these dried wet boundary processing methods have, the virtual hypothesis of some use films, all inevitably increased to a certain extent the numerical model error, and vital external force in the shallow water problems (such as wind-stress, bottom surface friction force etc.) is considered not enough, easily cause as a result accuracy decline, even obtain the non-results of Physical of distortion.

Summary of the invention

For overcoming now methodical deficiency, the present invention is based on is situated between sees the LATTICE BOLTZMANN theory, provide a kind of processing to do the new method of wet boundary problem, the method need not to adopt the linear extension to variablees such as the depth of water and flow velocitys, also based thin film is not supposed, and the physical distortions that External Force Acting is introduced model and avoided human factor to cause that can be convenient, flexible.

Description of drawings

Fig. 1. do the structure (d and w represent respectively the dry-wet grid unit) of wet border grid cell.

Fig. 2. the implementation step process flow diagram.

Fig. 3. three kinds of water surface curve and experimental data comparison diagrams that method calculates.

Embodiment

The technical scheme that technical solution problem of the present invention adopts comprises following a few step:

1. set up the lattice Boltzmann model of shallow water equation

The mathematical model of Shallow-water Flow is shallow water equation.Shallow water equation can derive out based on vertical hydrostatic force hypothesis from Neville-RANS, discusses for convenient, lists the one dimension form here, shown in (1) and (2),

$\frac{\∂h}{\∂t}+\frac{\∂\left(\mathrm{hu}\right)}{\∂x}=0---\left(1\right)$

$\frac{\&PartialD;\left(\mathrm{hu}\right)}{\&PartialD;t}+\frac{\&PartialD;\left({\mathrm{hu}}^2\right)}{\&PartialD;x}=-\frac{\mathrm{<figure-callout\; id="2"\; label="g"\; filenames="HSA00000779641100031.png"\; state="\{\{state\}\}">g</figure-callout>}}{2}\frac{{\&PartialD;\mathrm{<figure-callout\; id="2"\; label="h"\; filenames="HSA00000779641100031.png"\; state="\{\{state\}\}">h</figure-callout>}}^2}{\&PartialD;x}+v\frac{{\&PartialD;}^2\left(\mathrm{hu}\right)}{{\&PartialD;x}^2}+F,---\left(2\right)$

Use card Deere coordinate-system in the formula; V is kinematic coefficient of viscosity; H is the depth of water; X and u are respectively distance and flow velocity; F is external force.

A kind of Jie that lattice Boltzmann method is based on statistical physics sees method, with the microscopic behavior of very simple formal description particle, and in the motion of macroscopic view reflection fluid.Lattice Boltzmann model for shallow water equation is divided into three parts:

First is the migration collision equation of LATTICE BOLTZMANN

$f_{\mathrm{\&alpha;}}(x+e_{\mathrm{\&alpha;}}\mathrm{\&Delta;t},t+\mathrm{\&Delta;t})=f_{\mathrm{\&alpha;}}(x,t)-\frac{1}{\mathrm{\&tau;}}(f_{\mathrm{\&alpha;}}-f_{\mathrm{\&alpha;}}^{\mathrm{eq}})+\frac{\mathrm{\&Delta;t}}{2e^2}{e}_{\mathrm{\&alpha;}}F---\left(3\right)$

F wherein _αIt is particle distribution function; E=Δ x/ Δ t is Grid Velocity; Δ x is mesh scale; Δ t is time step; τ is slack time; e _αBe velocity vector; The second part is grid configuration, adopts D1Q3 type grid for one-dimensional model, so e _α=[0, e ,-e|0,1,2]; The 3rd part is the balanced distribution function, and the balanced distribution function of shallow water equation lattice Boltzmann model is

$f_{\mathrm{\&alpha;}}^{\mathrm{eq}}=\left\{\begin{array}{cc}h-\frac{{\mathrm{hu}}^2}{e^2}-\frac{{\mathrm{<figure-callout\; id="2"\; label="gh"\; filenames="HSA00000779641100031.png"\; state="\{\{state\}\}">gh</figure-callout>}}^2}{2{\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="HSA00000779641100031.png"\; state="\{\{state\}\}">e</figure-callout>}}^2},& \mathrm{\&alpha;}=0,\\ \frac{{\mathrm{<figure-callout\; id="2"\; label="gh"\; filenames="HSA00000779641100031.png"\; state="\{\{state\}\}">gh</figure-callout>}}^2}{{4\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="HSA00000779641100031.png"\; state="\{\{state\}\}">e</figure-callout>}}^2}+\frac{{\mathrm{<figure-callout\; id="2"\; label="hu"\; filenames="HSA00000779641100031.png"\; state="\{\{state\}\}">hu</figure-callout>}}^2}{{2\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="HSA00000779641100031.png"\; state="\{\{state\}\}">e</figure-callout>}}^2}+\frac{\mathrm{hu}}{2e},& \mathrm{\&alpha;}=1,\\ \frac{{\mathrm{<figure-callout\; id="2"\; label="gh"\; filenames="HSA00000779641100031.png"\; state="\{\{state\}\}">gh</figure-callout>}}^2}{4{\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="HSA00000779641100031.png"\; state="\{\{state\}\}">e</figure-callout>}}^2}+\frac{{\mathrm{hu}}^2}{2e^2}-\frac{\mathrm{hu}}{2e},& \mathrm{\&alpha;}=2,\end{array}\right.---\left(4\right)$

Can draw macroscopical depth of water thus and the flow velocity variable is

$h=\underset{\mathrm{\&alpha;}}{\mathrm{\&Sigma;}}{f}_{\mathrm{\&alpha;}}=\underset{\mathrm{\&alpha;}}{\mathrm{\&Sigma;}}{f}_{\mathrm{\&alpha;}}^{\mathrm{eq}},$ $u=\frac{1}{h}\underset{\mathrm{\&alpha;}}{\mathrm{\&Sigma;}}{e}_{\mathrm{\&alpha;}}{f}_{\mathrm{\&alpha;}}=\frac{1}{h}\underset{\mathrm{\&alpha;}}{\mathrm{\&Sigma;}}{e}_{\mathrm{\&alpha;}}{f}_{\mathrm{\&alpha;}}^{\mathrm{eq}}.---\left(5\right)$

2. do the information association of wet border grid cell

Do the structure of wet border grid cell as shown in the figure, the present invention is calculated the variable on the dried grid cell to set up communication between the dry-wet grid as basic thought.At first, suppose a micro-ε=Δ t, by the Chapman-Enskog formula, f α can be existed

(

Be f _αN rank amount of precision) near launch,

$f_{\mathrm{\&alpha;}}=f_{\mathrm{\&alpha;}}^{\left(0\right)}+\mathrm{\&epsiv;}{f}_{\mathrm{\&alpha;}}^{\left(1\right)}+o\left({\mathrm{\&epsiv;}}^2\right)---\left(6\right)$

Simultaneously, use the Taylor formula that formula (3) left term is launched at room and time yardstick (x, t), can get

$\mathrm{\&epsiv;}(\frac{\&PartialD;}{\&PartialD;t}+e_{\mathrm{\&alpha;}}\frac{\&PartialD;}{\&PartialD;x})f_{\mathrm{\&alpha;}}+\frac{{\mathrm{\&epsiv;}}^2}{2}{(\frac{\&PartialD;}{\&PartialD;t}+e_{\mathrm{\&alpha;}}\frac{\&PartialD;}{\&PartialD;x})}^2{f}_{\mathrm{\&alpha;}}+o\left({\mathrm{\&epsiv;}}^3\right)=-\frac{1}{\mathrm{\&tau;}}(f_{\mathrm{\&alpha;}}-f_{\mathrm{\&alpha;}}^{\left(0\right)})+\frac{\mathrm{\&Delta;t}}{2e^2}{e}_{\mathrm{\&alpha;}}F,---\left(7\right)$

In the formula

Ignore if contain the higher order term of ε, can obtain

$(\frac{\&PartialD;}{\&PartialD;t}+e_{\mathrm{\&alpha;}}\frac{\&PartialD;}{\&PartialD;x})f_{\mathrm{\&alpha;}}^{\left(0\right)}=-\frac{1}{\mathrm{\&tau;}}{f}_{\mathrm{\&alpha;}}^{\left(1\right)}+\frac{e_{\mathrm{\&alpha;}}F}{2{\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="HSA00000779641100031.png"\; state="\{\{state\}\}">e</figure-callout>}}^2}.---\left(8\right)$

For dried grid cell,

$f_{\mathrm{\&alpha;}}^{\left(0\right)}=0,$ $\frac{\&PartialD;f_{\mathrm{\&alpha;}}^{\left(0\right)}}{\&PartialD;t}=0---\left(9\right)$

Wherein adopt backward difference discrete particle balanced distribution function

To the derivative of time, by and uncomplicated derivation can draw

$f_{\mathrm{\&alpha;}}^{\left(1\right)}=\mathrm{\&tau;}(\frac{1}{{2e}^2}{e}_{\mathrm{\&alpha;}}F-e_{\mathrm{\&alpha;}}\frac{\&PartialD;f_{\mathrm{\&alpha;}}^{\left(0\right)}}{\&PartialD;x}).---\left(10\right)$

Bringing formula (10) into formula (6) draws

$f_{\mathrm{\&alpha;}}=f_{\mathrm{\&alpha;}}^{\mathrm{eq}}+\mathrm{\&Delta;t\&tau;}(\frac{1}{2e^2}{e}_{\mathrm{\&alpha;}}F-e_{\mathrm{\&alpha;}}\frac{\&PartialD;f_{\mathrm{\&alpha;}}^{\left(0\right)}}{\&PartialD;x}),---\left(11\right)$

In formula (11), only need to use simple difference discrete balanced distribution

Namely can be used for calculating the variable of dried grid cell.In addition for the f of dried grid cell the unknown ₀Can adopt that the mean value of grid cell respective amount calculates around it.It can also be seen that from formula (11), what the effect of external force F can be very natural is dissolved in the middle of the calculating of doing wet border, need not special processing.At last, through type (5) can calculate the correlated variables of dried grid cell.

By the processing of the innovation method, can the quantity of information of doing on the wet boundary element is interrelated, thus processing non-physical property extension difference or the virtual hypothesis of film of avoiding when doing wet boundary problem variablees such as the depth of water and flow velocitys.Simultaneously, External Force Acting can reasonably be dissolved in the calculating naturally.In addition, the method is simple, has the local correlativity of grid, is fit to the large-scale parallel computing.

3. implementation step

Only need in the specific implementation process to adopt explicit calculating to finish, implementation step is decomposed as follows:

(1). set depth of water h and flow velocity u initial value;

(2). calculate f according to formula (4) ^Eq

(3) if. f ₂(d ₂)＞0, through type (11) calculates f ₁(d ₂), and utilize mean value calculation f ₀(d ₂); Otherwise utilization standard bounce-back form calculates;

(4). calculate f by the formula (3) that comprises collision and migration step _α

(5). again used for the 3rd step calculated at the unknown f of next circulation (being t=t+ Δ t) _α

(6). utilization formula (5) is upgraded flow velocity u and depth of water h;

(7). returned for the 2nd step, and repeated for the 2nd step to the 6th step, finish until calculate to find the solution.

Fig. 2 is the implementation step process flow diagram.

4. embodiment

One isolated wave is propagated to the seashore direction.Wave height H and hydrostatic depth of water h ₀Ratio be H/h ₀=0.0185, the bank slope gradient is 1/19.85, and the nondimensionalization time is

(t is the time, and g is acceleration of gravity) disregards the bottom surface friction.Its wave climbing is showed with the water surface over time with dropping process.Result of calculation such as table 1 and shown in Figure 3 are compared with film subjunctive and linear epitaxial process method, use the innovation method acquired results and experimental data to coincide better.

Three kinds of computing method of table 1. are in the error of calculation of different time to the depth of water

Time t *	The film subjunctive	Linear epitaxy	The innovation method
				35	1.73％	1.73％	1.72％
50	6.75％	3.68％	1.84％
				70	7.67％	2.65％	2.43％

Claims (7)

1. Jie who processes dried wet boundary problem sees method.This method adopts the lattice Boltzmann model of shallow water equation, it is characterized in that: use Chapman-Enskog to launch, the Taylor formula is interrelated with dry-wet grid unit information amount, adopts the particle equilibrium distribution function of the discrete dried grid cell of backward difference

Derivative to time t.

2. a kind of Jie who processes dried wet boundary problem according to claim 1 sees method, it is characterized in that: use shallow water equation to be control equation, shown in (1) and formula (2).

$\frac{\&PartialD;h}{\&PartialD;t}+\frac{\&PartialD;\left(\mathrm{hu}\right)}{\&PartialD;x}=0---\left(1\right)$

$\frac{\&PartialD;\left(\mathrm{hu}\right)}{\&PartialD;t}+\frac{\&PartialD;\left({\mathrm{hu}}^2\right)}{\&PartialD;x}=-\frac{g}{2}\frac{{\&PartialD;h}^2}{\&PartialD;x}+v\frac{{\&PartialD;}^2\left(\mathrm{hu}\right)}{{\&PartialD;x}^2}+F,---\left(2\right)$

3. a kind of Jie who processes dried wet boundary problem according to claim 1 sees method, it is characterized in that: adopt the migration collision model of LATTICE BOLTZMANN, shown in (3).

$f_{\mathrm{\&alpha;}}(x+e_{\mathrm{\&alpha;}}\mathrm{\&Delta;t},t+\mathrm{\&Delta;t})=f_{\mathrm{\&alpha;}}(x,t)-\frac{1}{\mathrm{\&tau;}}(f_{\mathrm{\&alpha;}}-f_{\mathrm{\&alpha;}}^{\mathrm{eq}})+\frac{\mathrm{\&Delta;t}}{2e^2}{e}_{\mathrm{\&alpha;}}F---\left(3\right)$

4. a kind of Jie who processes dried wet boundary problem according to claim 1 sees method, it is characterized in that: use the Chapman-Enskog formula with particle distribution function f _αAt its balanced distribution function

Near the expansion is shown in (4).

$f_{\mathrm{\&alpha;}}=f_{\mathrm{\&alpha;}}^{\left(0\right)}+\mathrm{\&epsiv;}{f}_{\mathrm{\&alpha;}}^{\left(1\right)}+o\left({\mathrm{\&epsiv;}}^2\right)---\left(4\right)$

5. according to claim 1ly a kind ofly process Jie who does wet boundary problem and see method, it is characterized in that: use the Taylor formula that formula will collide particle distribution function f after moving _α, i.e. f _α(x+e _αΔ t, t+ Δ t) launch at room and time yardstick (x, t), then formula (3) becomes

$\mathrm{\&epsiv;}(\frac{\&PartialD;}{\&PartialD;t}+e_{\mathrm{\&alpha;}}\frac{\&PartialD;}{\&PartialD;x})f_{\mathrm{\&alpha;}}+\frac{{\mathrm{\&epsiv;}}^2}{2}{(\frac{\&PartialD;}{\&PartialD;t}+e_{\mathrm{\&alpha;}}\frac{\&PartialD;}{\&PartialD;x})}^2{f}_{\mathrm{\&alpha;}}+o\left({\mathrm{\&epsiv;}}^3\right)=-\frac{1}{\mathrm{\&tau;}}(f_{\mathrm{\&alpha;}}-f_{\mathrm{\&alpha;}}^{\left(0\right)})+\frac{\mathrm{\&Delta;t}}{2e^2}{e}_{\mathrm{\&alpha;}}F.---\left(5\right)$

6. a kind of Jie who processes dried wet boundary problem according to claim 1 sees method, it is characterized in that: the particle equilibrium distribution function that adopts the discrete dried grid cell of backward difference

To the derivative of time t, and the particle distribution function f of derivation boundary _αFor

$f_{\mathrm{\&alpha;}}=f_{\mathrm{\&alpha;}}^{\mathrm{eq}}+\mathrm{\&Delta;t\&tau;}(\frac{1}{2e^2}{e}_{\mathrm{\&alpha;}}F-e_{\mathrm{\&alpha;}}\frac{{\&PartialD;f}_{\mathrm{\&alpha;}}^{\left(0\right)}}{\&PartialD;x}).---\left(6\right)$

7. a kind of Jie who processes dried wet boundary problem according to claim 1 sees method, it is characterized in that: the calculating of final depth of water h and flow velocity u is based on the microscopic particle distribution function and draws, shown in (7).

$h=\underset{\mathrm{\&alpha;}}{\mathrm{\&Sigma;}}{f}_{\mathrm{\&alpha;}}=\underset{\mathrm{\&alpha;}}{\mathrm{\&Sigma;}}{f}_{\mathrm{\&alpha;}}^{\mathrm{eq}},$ $u=\frac{1}{h}\underset{\mathrm{\&alpha;}}{\mathrm{\&Sigma;}}{e}_{\mathrm{\&alpha;}}{f}_{\mathrm{\&alpha;}}=\frac{1}{h}\underset{\mathrm{\&alpha;}}{\mathrm{\&Sigma;}}{e}_{\mathrm{\&alpha;}}{f}_{\mathrm{\&alpha;}}^{\mathrm{eq}}.---\left(7\right)$

Images