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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Three-Dimensional Finite Difference Model for Transport of Conservative
Pollutants
Yuling Liu 1，X.D. Zhou 1，W. B. Fan 2
1
2
Institute of Water Conservancy and Hydraulic Engineering, Xi’an University of Technology, Xi’an, 710048 China
Economics and Management School, Northwest University, Xi’an, 710069 China
Email: [email protected]
Abstract A three-dimensional finite difference transport model appropriate for the coastal environment is developed
for the solution of the three-dimensional convection–diffusion equation. A higher order upwind scheme is used for
the convective terms of the convection–diffusion equation, to minimize the numerical diffusion. The validity of the
numerical model is verified through a test problem, whose exact solutions are known.
Keywords: convection–diffusion equation; a higher order upwind scheme; finite difference method
INTRODUCTION
Mathematical modeling of the transport of salinity, pollutants and suspended matter in shallow waters involves the
numerical solution of convection–diffusion equation. Numerical studies show that the use of central differencing for
the convective terms of the convection–diffusion equation results in negative species concentration. Lam (1975)
points out that the central difference approximation will overestimate the advective flux so much that it often causes
a negative concentration to appear in the neighbouring cell. To circumvent such a shortcoming of central
differencing, the upwind or donor cell method introduced by Gentry et al. (1966) is generally used. To overcome the
shortcomings of numerical dispersion, Leonard (1979) introduced an upstream interpolation method, namely
QUICK (Quadratic Upstream Interpolation Convective Kinematics) for one-dimensional unsteady flow. Later,
Leonard (1988) gave an improved version of the QUICK scheme, eliminating the wiggles completely by introducing
exponential integration into regions with sharp fronts. Chen and Falconer (1994) showed that the QUICK scheme is
only second-order accurate in space and presented different forms of the third-order convection, second-order
diffusion for the solution of the convection-diffusion equation. In the present study, super-upwinding difference
scheme as given in Barrett, K., S (1982) has been used for the convection terms of the convection-diffusion
equation.
MATHEMATICAL MODELLING OF CONVECTION DIFFUSION PROCESSES
1. Governing equations The mathematical model describing the transport processes in three dimensions is given by
the convection–diffusion equation
∂
∂
∂
∂
∂ 2C
∂ 2C
∂ 2C
( C ) + ( CU ) + ( CV ) + ( CW ) − K x 2 − K y 2 − K z 2 = G ( x, y, z )
∂t
∂x
∂y
∂z
∂x
∂y
∂z
(1)
subject to the initial conditions, C(x, y, z, 0), and the boundary conditions, ∂ C ( x, y, t ) ∂ z at z = 0 on the top surface,
∂ C ( x, y, t ) ∂ z at z = - h on the bottom surface, and ∂ C ( x, y, t ) ∂ n along the lateral boundaries, where C is the
unknown pollutant concentration at location (x, y, z), U, V and W are the flow velocities in the x, y and z directions,
respectively, Kx, Ky and Kz are the diffusion coefficients in the x, y and z directions, respectively, G represents any
source or sink. x, y and z are the Cartesian coordinates with the x y plane horizontal and occupying the
undisturbed position of the water surface, and the z-axis pointing vertically upwards, and n is the outward drawn
normal in the x-y plane. The position of the free surface is denoted by z = z(x, y, t) as shown in Fig. 1 and that of the
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bottom by z = −h(x, y). The values of z, U, V and W are to be obtained from a hydrodynamic model such as that of
Shankar et al (1996).
¦Ζ
Figure 1: Definition sketch
Since ∂ U ∂ x + ∂ V ∂ y + ∂ W ∂ z = 0 , Eq. (1) reduces to
∂C
∂C
∂C
∂C
∂ 2C
∂ 2C
∂ 2C
K
K
+U
+V
+W
− Kx
−
−
= G ( x, y , z )
y
z
∂t
∂x
∂y
∂z
∂ x2
∂ y2
∂ z2
(2)
2. Finite difference formulation of the convection term using super-unwinding scheme In the present study, a
supper-upwind differencing has been used as given in Barrett, K., S (1982). A typical convective term ∂C ∂x is
discretised using the supper-upwind differencing as
Ci +1, j ,k − Ci −1, j , k Ci +1, j , k − 3Ci , j , k + 3Ci −1, j , k − Ci − 2, j , k
11Ci , j , k − 18Ci −1, j , k + 9Ci − 2, j , k − 2Ci −3, j , k
∂C
= λ(
−
+ (1 − λ )(
)
∂x
2Δx
6 Δx
6 Δx
(3a)
for U i , j ,k f 0 and i = 4, nx-3.
Ci +1, j ,k − Ci −1, j , k Ci + 2, j , k − 3Ci +1, j , k + 3Ci , j , k − Ci −1, j , k
2Ci + 3, j , k − 9Ci + 2, j , k + 18Ci +1, j , k − 11Ci , j , k
∂C
= λ(
−
+ (1 − λ )(
)
∂x
2Δx
6Δx
6Δx
(3b)
for U i , j ,k p 0 and i = 4, nx-3.
Where λ is computed using the formula in Barrett, K., S (1982).
However, for the boundary points, a four-point upstream formula can be written such that either point to the left or to
the right are considered in the finite difference approximation.
Near the left boundary, i.e. for i = 2
∂C −11Ci , j ,k + 18Ci +1, j , k − 9Ci + 2, j , k + 2Ci + 3, j , k
=
∂x
6Δx
(4a)
Near the right boundary, i.e. for i = nx−1
∂C −2Ci −3, j , k + 9Ci − 2, j , k − 18Ci −1, j , k + 11Ci , j , k
=
∂x
6Δx
(4b)
where nx is the number of grid points in the x direction.
The second derivatives occurring in diffusion terms are evaluated using a central difference.
A typical second derivative ∂ 2C ∂x 2 is evaluated using
∂ 2C Ci +1, j ,k − 2Ci , j , k + Ci −1, j , k
=
∂x 2
( Δx) 2
(5)
3. Stability criterion Hindmarsh et al (1984) established the stability criterion for the multi-dimensional advection
diffusion equation for an explicit scheme as
Δt ≤
1
K
Kz
U
V W
2[ x 2 +
]+
+
+
+
2
2
( Δx )
( Δy )
(Δz )
Δx Δy Δz
(6)
Ky
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VALIDATION OF THE NUMERICAL MODEL
It is necessary to compare the model results with analytical solutions, to validate the developed numerical model. A
test case used by Lardner and Song (1991), three-dimensional transport of a point source pollutant is used to validate
the model developed in the present study. The test case involving the transport of point source pollutant
concentration by convection and diffusion are taken, since they represent an important practical case of a sudden
influx of a pollutant, such as due to oil spill. The test case used in the present study has a analytical solution, which
makes it possible to compare the result obtained in the numerical solution. The test case involving three-dimensional
convection and diffusion is in uniform flow.
For an infinite region, the exact solution C(x, y, z, t) at a given time t for a point source placed at the centre of the
domain is given by C ( x, y, z , t ) = C x ( x, t ,U )C y ( y, t ,V )C z ( z , t ,W ) , where
C x ( x, t ,U ) = exp[ − ( x − x0 − Ut ) 2 (4 K x t )]
4π K x t , C y ( x, t ,V ) = exp[− ( y − y0 − Vt ) 2 (4 K y t )]
Cz ( z , t ,W ) = exp[− ( z − z0 − Wt ) 2 (4 K z t )]
4π K z t , where (x0, y0) is the location of the point source.
(a) analytical solution
4π K y t
(b) numerical solution
Figure 2: Concentration distributions for three-dimensional convection and diffusion by analytical and
numerical solutions, at z = 32.5 m from the surface, after 18,000 s, Dx = Dy = 2000 m, Dz = 6 m, Dt
= 50 s,
Kx = Ky = 2000 m2/s, Kz = 0.01 m2/s, U = V = 0.2 m/s, W = 0.0 m/s, t0 = 5000 s.
For applying the boundary conditions, the derivative of C(x, y, z, t) is evaluated as
[ z + w(t + t0 ) − z ]
∂[C ( x, y, z , t )]
= C x ( x, t ,U )C y ( y, t ,V )C z ( z , t ,W ) 0
∂z
2[ K z (t + t0 )]
(7)
Substituting the values of ∂ C ( x, y , t ) ∂ z at z =0 and z =-h, the boundary conditions at the top and bottom surfaces are
satisfied in the numerical solution.
The initial condition is given by C (x, y, z, t0), where if t0 f 0 , then the problem has a smooth initial condition.
The transport of an instantaneous point source pollutant initially placed at the center of a domain of size
40,000 × 40,000 × 65m3, by convection and diffusion is simulated using the model. Uniform horizontal velocities of
U = V = 0.2 m/s and W = 0 m/s, grid spacing Dx = Dy = 2000 m, time step Dt = 30 s and Kx = Ky = 2000 m2/s, Kz =
0.01 m2/s are used. For a smooth initial condition, the value of t0 is taken to be 5000 s. The solution for the bounded
region is taken large enough so that the solution remains essentially zero at the boundaries for the time interval of the
computation (Lardner and Song, 1991). The concentration distribution obtained after t = 18,000s using the model
developed in the present study compares well with that given by the analytical solution, and shows good
comparisons as in Fig. 2(a) and Fig. 2(b) respectively. The maximum relative error between the analytical and
numerical solutions is about 10%.
CONCLUSIONS
A three-dimensional finite difference transport model appropriate for the coastal environment is developed for the
numerical solution of the three-dimensional convection–diffusion equation. A higher order upwind scheme is used
for the convective terms of the transport equation, to minimize the numerical diffusion. The validity of the numerical
model is verified through the test problem, for the transport of point source pollutants in uniform flow, by comparing
the result obtained in the present study with that of analytical solution. The validity of the numerical solution can
also be tested against analytical solutions for the case of transport of a Gaussian pulse in unsteady and nonuniform
flows in the future work. Work is currently in progress to couple the transport model developed in the present study
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with the multi-level three-dimensional hydrodynamic model.
REFERENCES
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Advances in Water Resources, 1994; 17: 147-170.
2. Gentry RA, Martin RE, Daly BJ. An Eulerian differencing method for unsteady compressible flow problems.
Journal of Computational Physics, 1966; 8: 55-76.
3. Lam DCL. Computer modeling of pollutant transport in Lake Erie. Water Pollution, 1975; 25: 75-86.
4. Leonard BP. A stable and accurate convective modeling procedure based on upstream formulation. Computer
Methods in Applied Mechanics and Engineering, 1979; 19: 59-98.
5. Leonard BP. Simple high accuracy resolution program for convective modeling of discontinuities. International
Journal for Numerical Methods in Fluids, 1988; 8: 1291-1318.
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horizontal and vertical scales. International Journal for Numerical Methods in Engineering, 1991; 32:
1303-1319.
9. Barrett KS. Super-upwinding-element of doubt and discrete difference of opinion on the numerical modeling
of the incomprehensible defective confusion equation. in Coldwell J, Moscadini AO eds. Numerical Modeling
in Diffusion Convection, Pentech Press, London, Plymouth, 1982.
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