Numerical Calculation of Flow Field by Application of Superposition Theory of Partial Differential Equation

Hong-gang Wang, Jian-rang Zhang,Feng-liang Wu, Hai-fei Lin, Xin-tan Chang School of Energy, Xi’an University of Science and Technology

Xi’an, CHINA maxwell79@126.com, hnyzwhg@yahoo.com.cn

Abstract—Based on the superposition theory of partial differential equation (PDE), a new method is introduced here to solve complicated problems of flow field. In this method, a problem of flow field with complex boundary conditions and source items is divided into several problems of flow fields with simple boundary conditions and source items, and then the solution of a complicated problem of flow field can be determined by the solutions’ superposition of several simple problems of flow fields. For an instance, by using a software in computational fluid dynamics (CFD), a linear seeping problem in porous media is solved by numerical simulation as well as solutions’ superposition of two simpler problems deriving from the proto-problem, and thus the application feasibility of superposition theory in numerical simulation of flow field is verified. More importantly, the application of the new method in CFD can decrease the number of necessary numerical examination greatly.

Keywords-superposition theory; boundary condition; linear seepage; computational fluid dynamics

I. INTRODUCTION Traditional methods of studying flow problems of fluid

are pure experiment and pure theory. In view of the complicacy of flow problems, traditional method of pure theory can only obtain solutions of some simple flow problems. To non-linear flow problems with multi-variables and irregular geometry shape as well as complex boundary conditions, pure theory method is now incompetent. At the same time, limited by the model size、safety factors and precision of measure, sometimes it is difficult to solve some flow problems by experimental method. The method of CFD, however, overcomes rightly the defects of the two traditional methods [1]. Since 1960’s, with the rapid development of computer technology and the advent of many steady、 precise and smart algorithmic, CFD has witnessed a fast development and gradually become a new subject, which has been employed in many areas of science and technology and displayed its marvelous potential in solving problems of scientific theories and engineering projects [2]. Even so, it will spend considerable computer resource to solve a difficult flow problem with complicated boundary conditions and sources by using CFD method.

In fact, lots of physical phenomena in the nature word have obvious characteristic of superposition. The superposition theory in physics points out that, total effect of

several co-existing physical factors equals the sum of several effects, each of which comes from a single existing physical factor [3]. The reason that some different physical phenomena have a same characteristic of superposition lies in the same governing equations obeyed by these different physical phenomena, and solutions of these governing equations have a characteristic of superposition in nature, which means a complicated flow problem can divide into several simple flow problems. On the other hand, several solutions of simple flow problems can yield uncountable solutions of complicated problems, which illuminates us that, in the process of numerical simulation using CFD method we can plan reasonably some simple flow problems spending little computer resource to solve them, and then use the arbitrary superposition of these solutions to simulate solutions of many big flow problems spending more computer resource.

II. SUPERPOSITION THEORY OF PDE A definite problem with linear universal equation and

linear definite conditions is called linear definite problem, which can be written as below:

1

1 2

1 1

[ ] , ( ... ) , ( ) |

....... ( ) |

k

T

s

k s k

L u f x x universal equationl u g

definite conditionsl u g

⎧ = ∈Ω⎪

= ⎫⎪⎨ ⎪

⎬⎪⎪⎪ = ⎭⎩

（1）

Where [ ]L u is a linear differential operator with Ω as its definition domain , ( )il u is a linear differential operator defined on is , is ⊂ Ω , and the order of ( )il u is usually lower than [ ]L u ,even zero order when ( )il u u= ; f is a known function defined in area Ω , and ig is a known function defined on is .

For a linear problem, linear superposition theory takes effect and dominates its research. For simplicity, each linear differential equation of the linear problem can be written as

[ ]L u f= which can denote both the universal equation and the definite conditions, thus linear superposition theory can be depicted as below:

1) If iu satisfies [ ] ,i iL u f=

1 1[ ]

n n

i i i ii i

L c u c f= =

=∑ ∑ （2）

2009 International Conference on Engineering Computation

978-0-7695-3655-2/09 $25.00 © 2009 IEEE

DOI 10.1109/ICEC.2009.27

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2) If iu satisfies [ ] ,i iL u f=

1 1[ ]i i i i

i iL c u c f

∞ ∞

= ==∑ ∑ （3）

3) If ( , )u x ε satisfies [ ( , )] ( , )L u x f xε ε= , where ε is a parametric variable, and Vε is the varying range of ε

which may be a region or a surface or a curve of nR , we get [ ( ) ( , ) ] ( ) ( , )V V

L a u x dV a f x dVε ε

ε εε ε ε ε=∫ ∫ （3）

Where integral aiming at ε can be expressed as volume integral or surface integral or line integral for different Vε ofε ; ( )a ε is an arbitrary function.

Explained from physical meaning, for a linear definite problem, function ( )f x in right of the universal equation and other functions ig can be treated as some additional sources. Superposition theory shows that, the result of co-operation of many sources equals the superposition of results, each of which comes from an independent source, in this way the theory of superposition is also called independent action principle. When analyzing the total effect induced by several boundary conditions or sources, we can firstly obtain the solution of each problem induced by only one boundary condition or source, and then add these solutions. Some kinds of nonlinear problems also have some kinds of nonlinear superposition theory, the discovery of which is significant for the research of nonlinear problems [4]. Limited by the length of this paper, only the linear theory of superposition is discussed here.

III. APPLICATION OF LINEAR SUPERPOSITION THEORY IN NUMERICAL SOLUTION OF SEEPING FLOW FIELD

A. Ideal model of porous media Given a square area of porous media as shown in figure

1, the fluid in the porous media is set mixture of methane and air with a viscosity coefficient 1.72e-05; the porous media is made of crushed coal and rock with a permeability 6 210 m− ; the left and bottom edges of the square area are set pressure inlets, the right and top edges are set pressure outlets. The length of each edge in the square is 160 m , and the square area is divided into 160 160× meshes.

For convenience, line-1, which lies in the middle of the square area as shown in figure 1, is selected as an observing line to verify the superposition theory in the next section.

To the seeping flow problem presented here, the following supposes are adopted:

1) The function relation between seeping velocity and pressure satisfies Darcy’s law;

2) The shape of porous media is fixed, which means an unchanging permeability when time changes

3) The fluid in porous media is incompressible; 4) Heat transfer is beyond our problem.

Figure 1. porous media and its mesh division.

With the above supposes, the seeping flow in porous media satisfies the following governing equations [5~7]:

1) Mass conservation equation

0yx vvx y

∂∂+ =

∂ ∂ （5）

Where xv denotes the velocity component in x direction,

yv denotes the velocity component in y direction. 2) Darcy’s Law

x xpv kx

∂= −∂

（6）

y ypv ky

∂= −∂

（7）

Where ,x yk k denote the permeability of ,x y directions respectively, p denotes the static pressure of fluid in porous media.

The following equation can be derived by combination of equation (5)、（6）and (7):

0x yp pk k

x x y y⎛ ⎞∂ ∂ ∂ ∂⎛ ⎞ + =⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠

（8）

The expanding form of (8) can be written as: 2 2

2 2 0yxx y

kkp p p pk kx x y yx y

∂∂∂ ∂ ∂ ∂+ + + =∂ ∂ ∂ ∂∂ ∂

（9）

In view of the suppose of unchanging permeability when time changes, ,x yk k are functions of only variables ,x y , thus to unknown function p , equation (9) belongs to a second-order PDE with variable coefficients; when ,x yk k are kept constants, the second and third items of equation (9) equal to zero, which means a second-order PDE with constant coefficients as shown below:

2 2

2 2 0x yp pk k

x y∂ ∂+ =∂ ∂

（10）

Furthermore, when x yk k k= = , equation (10) becomes a Laplace equation:

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2 2

2 2 0p px y

∂ ∂+ =∂ ∂

（11）

B. Using superposition theory to solve the pressure field and velocity field As linear PDE, equation (8) to (11) can be solved by

employing the linear superposition theory. There are uncountable solutions fitting one partial differential equation, so to obtain a definite solution, some boundary conditions must be set. As shown in figure 1, for a porous media with both a homogeneous isotropic permeability and a boundary condition given pressure values, the definite problem of pressure field in the porous media can be described as following:

2 2

2 2 0

10, On left and down boundaries0, On right and up boundaries

p px y

pp

⎧∂ ∂+ =⎪∂ ∂⎪

⎪ =⎨⎪ =⎪⎪⎩

（12）

From the superposition theory of linear PDE, the solution of “complicated” problem of equation (14) can be expressed by solutions of the following two “simple” problems:

2 2

2 2 0

10, On the down boundary0, On the other three boundaries

p px y

pp

⎧∂ ∂+ =⎪∂ ∂⎪

⎪ =⎨⎪ =⎪⎪⎩

（13）

2 2

2 2 0

10, On the left boundary0, On the other three boundaries

p px y

pp

⎧∂ ∂+ =⎪∂ ∂⎪

⎪ =⎨⎪ =⎪⎪⎩

（14）

The pressure field solutions of problem (12)、（13）and (14) by adopting FLUENT are shown as figure 2、figure 3 and figure 4：

Figure 2. pressure contours of problem 14

Figure 3. pressure contours of problem 15

Figure 4. pressure contours of problem 16

The precision of numerical calculation applying superposition theory can be verified by comparing the superposition solutions of problem (13) and (14) to the solution of problem (12). As far as line-1 is concerned, the superposition results of pressures and a velocity component are shown as figure 5 and figure 6:

Figure 5. pressure superposition on line-1

Figure 6. superposition of x -velocity on line-1

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In both figure 5 and figure 6, white line denotes the solution of problem (12), red line denotes the solution of problem (13), and green line denotes the solution of problem of (14). The two figures express a high precision with the relative errors below 1%, which means a good feasibility of the PDE superposition theory when being applied in numerical solution of flow fields.

IV. EXTEND APPLICATION OF SUPERPOSITION THEORY In the above sections, a small example is used to describe

the application of superposition theory of PDE in numerical solution of flow fields. In this example, solutions of two “simple” problems are added to obtain the solution of a relative complicated problem. In fact, from the superposition theory, the varying rate of pressure and velocity in flow area are proportional to the varying rate of that of boundaries. Thus with solutions of the two simple problems, uncountable solutions of complicated problems with boundary conditions set arbitrary pressure values can be determined.

Finally the following conclusion can be deduced: a complicated problem with n boundary conditions and m sources can be partitioned into n m+ simple problems, and the solution of complicated problem can be obtained by superposition of solutions of simple problems; more importantly, solutions of n m+ simple problems can work out lots of solutions of complicated problems, the number of which can be expressed by the following equation:

1 1( , ) 2

n m

i ji j

x yf n m = =

+∑ ∑= （17）

Where ( , )f n m denotes the number of solutions that can be generated from n m+ simple solutions; ix denotes the number of possible value of the unknown field variable on the i th boundary condition, and jy denotes the number of that on the j th source.

In addition, the method to solve a complicated flow field introduced in this paper can be used more than seeping problem in porous media. All flow problems with their governing equations satisfying the linear PDE can employ this method. For example, the pressure field of steady Stokes flow, to which the viscosity of fluid is much more important

than the inertia of fluid, satisfies Laplace equation [8], thus can also be solved by using superposition theory.

V. CONCLUSIONS 1) A new method to solve complicated flow field

problems, which employs the superposition theory of PDE, is detailed in this paper

2) By using a CFD software, an example of seeping flow problem in porous media is solved to verify the feasibility of the superposition theory of PDE in numerical solutions of complicated flow fields.

3) The application of this new method in solving complicated flow problems can decrease greatly the necessary number of numerical simulations and thus save plenty of computing time.

4) The new method can be used to solve any complicated flow field with a linear governing PDE.

ACKNOWLEDGMENT This research was funded by National Natural Science

Foundation of China (&nbsp: 50574072).

REFERENCES [1] F.J. Wang, Analysis of Computational Fluid Dynamics:the Theory of

CFD Software and its Application, QingHua University Publishing Houses, Beijing, 2004.

[2] J.D.Anderson, Computational Fluid Dynamics:The Basics with Applications, McGrawHill,1995, QingHua University Publishing Houses，Beijing, 2002

[3] X.Y. Ma, G.S. Han, “The Mathematical Basis of the Superpose Principle and Its Usage in Physics”, Journal of Anyang Teachers College,2006，no..5,pp.24

[4] X.H. Xue，Mathematical Physics and Partial Differential Equation, Publishing House of University of Science and Technology of China, Hefei, 1995

[5] M.T. Zhang, Mine-rock Hydrodynamics, Publishing House Of Science, Beijing, 1995

[6] W.C. Gu, The Calculation Principle of Seepage and its Application, China Building Materials Industry Publish House,Beijing, 2000

[7] CH.X. Mao, Analysis and Control of Seeping Calculation,China Hydraulic and Hydroelectric Publishing House, Beijing, 2003

[8] G.C. Dai, M.H. Chen , Chemical Fluid Dynamics, Chemical Industry Publishing House, Beijing, 2005

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