ENV5056 Numerical Modeling ofFlow and Contaminant Transport in Rivers

Numerical Solution of Contaminant Transport Contaminant Transport

Equation

Asst. Prof. Dr. Orhan GÜNDÜZ

1-D Contaminant Transport Equation

( )( )r r

r x r

C A CVAC AD kC A

t x x x

∂ ∂ ∂ ∂ + = −

∂ ∂ ∂ ∂ Conservative form

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1r r rx r

C C CV AD kC

t t A x x

∂ ∂ ∂ ∂ + = −

∂ ∂ ∂ ∂ Non-conservative form

or;

The numerical solution of the advection-dispersion equation (ADE) is one of the most difficult numerical problems in the computational fluid dynamics area. Despite the advances achieved in the field of numerical modeling of partial differential equations, there is still not a globally accepted efficient algorithm to solve ADE.

Although the equation looks simple, it shows a dual behavior in terms of its characteristics depending on the relative significance of various terms of the equation.

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For advection dominated flows, the equation shows the characteristics of a hyperbolic equation, whereas it becomes a parabolic partial differential equation when the dispersion is the dominant term.

Considering the fact that this changing behavior of the transport phenomenon can occur in a time- and space-dependent fashion, the numerical solution becomes a challenging task. It is generally accepted that an entirely satisfactory and routinely accepted numerical method which possesses stability, accuracy, algorithmic simplicity, economy and boundedness simultaneously is yet to be discovered.

There are numerous numerical solution techniques for solving advection dispersion equation. These techniques can be classified as:

(i) Eularian methods(ii) Lagrangian methods(iii) Hybrid methods.

The conventional Eularian methods can further be classified as:

(i) finite difference methods(ii) finite element methods

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(ii) finite element methods(iii) finite volume methods

A common ground for all these methods is the fixed grid structure that these methods utilize. This is one of the reasons why Eularian methods are so popular in the computational fluid dynamics area. However, these methods are limited in the sense that they have a tendency to produce false smearing and false oscillations around steep front regions. Although these problems can be overcome by employing accurate upstream centered Eularian schemes on a dense computational grid but this necessitates complex numerical algorithms and substantial computational effort.

On the opposite extreme, the Lagrangian methods follow the natural motion of the water mass along a changing mesh. The necessity to keep track of moving coordinates can still be computationally cumbersome, particularly, if arbitrary grid node placement is allowed.

Hybrid techniques combine the advantageous aspects of each method and allow the solution of the advective transport with the powerful Lagrangianbased particle tracking scheme over a fixed Eularian grid.

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The applicability of hybrid schemes is still under development and limited to simple systems (i.e., single channel). They are not currently used in well-developed water quality models and the accurate algorithms are to be formulated for complex systems such as river networks before these models become an industry standard.

In this regard, it is clear that there is a wide-open field for future research in the area of water quality modeling with hybrid numerical techniques.

Considering the limitations of each technique, it is still tempting to use fixed grid methods since the flow models that supply vital input to transport models are almost always based on Eularian framework.

Therefore, information obtained from flow models at fixed points can best be used in the transport model with at the same grid points. Hence, the one-to-one correlation of flow and transport model discretizations greatly simplifies the implementation of the transport model and possibly increases its accuracy since it would not require the unnecessary interpolations that would otherwise be

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it would not require the unnecessary interpolations that would otherwise be inevitable. Furthermore, systematic modeling of complex channel networks is still only viable with fixed-grid methods

Within the context of Eularian methods, it is widely accepted by the numerical modelers that the dispersion component of the equation can generally be solved without many problems using a variety of schemes. The problem generally arises from the advection component of the equation, particularly for highly advectivetransport of contaminants with sharp fronts, where the numerical methods start to lose accuracy and computational efficiency.

While dispersion favors implicit solution algorithms with possible use of large time steps, advection modeling generally utilizes an explicit algorithm with time steps limited by the Courant number criteria. Hence, the two major transport processes essentially behave in a contradicting manner. Since dispersion modeling could also be done with an explicit algorithm, a fully explicit scheme for the entire ADE is possible. This method however has the limitation of smalltime steps that might make the simulations very computational costly.

Using a fairly recent development in the area that results in the formulation of the so-called ‘split operator’ approach, one can now separate the advection

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the so-called ‘split operator’ approach, one can now separate the advection operator from the dispersion and the rest of the operators and solve them using the most suitable scheme possible for each operator.

Although this approach appears to be a violation of the principle of “simultaneous presence” of these processes in nature, it provides a very powerful technique to handle the numerical difficulties associated with each particular operator. It provides a sound methodology that gives mathematically identical results to the traditional compact operator methods.

The numerical solution of the advection-dispersion equation on a fixed Eularian grid is characterized by two principle phenomena: (i) oscillation (i.e., overshoots and undershoots) and (ii) numerical dispersion. These two phenomena are closely related. When a numerical scheme is developed to minimize the numerical dispersion, oscillation is encountered but when the oscillation is controlled, it is generally at the expense of increased numerical dispersion, particularly in the case when advection term dominates the dispersive term.

Therefore, it is now widely accepted that whilst the first-order upwind difference scheme will not generate wiggles (or grid scale oscillations) in

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difference scheme will not generate wiggles (or grid scale oscillations) in regions of high solute gradients or discontinuities, the scheme will nonetheless give rise to excessive numerical diffusion and is therefore inadequate for practical applications.

On the other hand, although the conventional second-order-accurate central difference schemes (such as the Crank-Nicholson (C-N) scheme which has no numerical dissipation and the Lax-Wendroff (L-W) scheme which has no numerical diffusion but some higher order dissipation), give more accurate results with tolerable numerical diffusion for most studies, these schemes generally exhibit pronounced oscillations when applied to the ADE with relatively small diffusion coefficients.

These oscillations can spread throughout the model domain and swamp the true numerical results or even produce noise. These effects are particularly pronounced where steep gradients exist. Many explicit schemes can reduce these oscillations substantially using an explicit formulation.

To alleviate these problems, a convective modeling procedure based on three points upstream biased interpolation was developed:

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(i) QUICK (quadratic upwind interpolation for convection kinematics) scheme for quasi-steady flow

(ii) QUICKEST (quadratic upwind interpolation for convection kinematics with estimated streaming terms) scheme for unsteady flow situations.

Although both the QUICK and the QUICKEST schemes possess thedesirable properties of high accuracy, stability and algorithmic simplicity, certain fundamental problems remain.

When they are applied to purely convective flows of a scalar variable, unphysical overshoots and undershoots are generated in the vicinity of abrupt gradient changes. These problems are also encountered when the two schemes are used to model linear-advection diffusion. Otherwise, the performance of the QUICKEST scheme is very good for modeling the linear advection or advection-diffusion.

To get rid of the wiggles, the ULTIMATE strategy which is applicable to explicit conservative schemes of any order of accuracy was developed. Unphysical oscillations can be also removed by using TVD (total variation diminishing) schemes.

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schemes.

But all TVD schemes in common use conform to an overly restrictive limiter which tends to make them more diffusive than necessary. Generally, the ULTIMATE (universal limiter for transient interpolation modeling of advective transport equation) strategy is preferable to TVD schemes.

The application of ULTIMATE QUICKEST scheme successfully alleviates the problems of spurious oscillations for both linear advection and advection-diffusion. Consequently, the ULTIMATE QUICKEST scheme is a commonly used solution method in the advection operator.

The advection-dispersion equation describing the transport of contaminants in a river channel can be solved using a control-volume approach by evaluating the advection term explicitly in time and by evaluating the remaining terms implicitly in time. The discretized form of the equation becomes

( )( )r r

r x r

C A CVAC AD kC A

t x x x

∂ ∂ ∂ ∂ + = −

∂ ∂ ∂ ∂

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( ) ( ) ( )1

1

nnn n r

r r r x ri i

CC A C A t VAC t AD kC A

x x x

+

+ ∂ ∂ ∂ = + ∆ − + ∆ − ∂ ∂ ∂

Explicit advectionoperator

Implicit dispersionand reaction operators

The square brackets represent some form of spatial discretization. Since the advection term is treated explicitly, the equation may be rewritten in two substeps without compromising the algorithmic integrity

( ) ( ) ( )*

nn

r r ri iC A C A t VAC

x

∂ = + ∆ − ∂

( ) ( )1

1 *

n

n rr r x ri i

CC A C A t AD kC A

x x

+

+ ∂ ∂ = + ∆ − ∂ ∂

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In essence, this procedure of splitting the operators first allows the fluid to advect for a time step then lets it to disperse and decay in its new advectedlocation.

With this approach, it is possible to use a suitable solution scheme for advection and other operators. QUICKEST algorithm can be used to model the advection operator. A time-averaged central difference scheme can then be used to simulate the dispersion process that does not create additional numerical problems. The remaining terms are just algebraic terms weighted between n and n+1 time lines.

In the control-volume finite difference principle, each node represents a control-volume within which the mass is conserved. A control-volume for node i is shown below. The left-backward and right-forward cell faces of the control volume are represented by the letters b and f. When similar control volumes are defined for all nodes of the domain, the domain is fully discretized in a control-volume sense guaranteeing mass balance. For volume i, the discretized equation could be written as:

( ) ( )( )

( ) ( )*

1

n n n

r r r ri i f b

tC A C A VAC VAC

x x

∆ = + −

∆ + ∆

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i 1i + 1i − 2i −

2ix −∆ 1ix −∆ ix∆ 1ix +∆

( )1

1

2i ix x−∆ + ∆

f b

2i +

FLOW

Numerical discretization of the domain for QUICKEST algorithm

( )1

1

2i ix x−

∆ + ∆

It is now essential to determine the parameter to advect along the channel. In the perfectly conservative form of ADE, the parameter CA is to be advected that also properly captures the unsteady behavior of the flow. However, it is important to note that the variation of the parameter CA may be totally different from C along the channel. The numerical algorithm must be suitable to handle such variations and possible non-monotonic behaviors

The solution of the contaminant concentrations from the above equation is based on the principle of properly representing the parameters along the cell faces. In the above equation, the flow variables V and A are provided from the solution of an hydrodynamics model and are known values for the transport solution. In general, flow model allows much larger time steps than the

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solution. In general, flow model allows much larger time steps than the explicit advection model shown here. Hence, for each flow model solution, numerous transport model solutions are to be made.

In all of these transport solution steps, two techniques can be used to extract the flow data. One first is to linearly interpolate the flow information between the two time level flow solution data according to the transport time step. The second way is to use constant values of all flow parameters from the current or future time level within the transport solution. When the variation of flow variables within a flow time step is significant, then the first method may be more suitable. Otherwise, using constant values of flow parameters would give sufficient results.

Once flow parameters are defined in all node points at each transport time line, then the next step is to find the cell wall values of these parameters. When CA is solved in the equation, a higher order interpolator is used to solve CA value and an extra interpolation (i.e., simple linear or higher order) is done to find the cell face values of the velocity. When C is solved in the equation, a higher order interpolator is used to solve for the concentration and separate extra interpolations (i.e., simple linear or higher order) are used to find the cell face values of the velocity and the area. Regardless of the dependent variable of the equation, it is generally accepted that the remaining flow parameters (i.e., V when CA is the dependent variable; and V and A when C is the dependent variable) can be linearly interpolated with sufficient accuracy.

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dependent variable) can be linearly interpolated with sufficient accuracy.

Since area (A) is a known parameter to the transport model and very small time steps are enforced due to the explicit nature of the numerical scheme where A varies very mildly, the dependent variable of the equation can be reduced to C without violating the conservative behavior of the algorithm. Then one can write the equation as:

( ) ( )( )

( ) ( )* 1

1

1

2

n n nn n n n n n

r i r i r f f r b bi i f b

i i

tC A C A C V A C V A

x x

+

−

∆ = + −

∆ + ∆

where the cell wall values of flow parameters are evaluated as

1 1

1 1

2 2

2 2

n n n nn ni i i if f

n n n nn ni i i i

b b

V V A AV A

V V A AV A

+ +

− −

+ += =

+ += =

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2 2b bV A= =

The cell face values of the concentration are evaluated using the higher order interpolator such as the QUICKEST scheme. For a non-uniform grid such as the one shown in Figure 1, the forward face concentration can be written using the QUICKEST algorithm as

( ) ( ) ( ) ( )2

2 21 2

1 2

1

2 6r cf

V t dP d PC P x x V t

dx dx

∆ = − − ∆ − ∆

where Δxc is the control-volume length given by ½( Δxi-1+Δxi), P1 and P2 are the first and second order Lagrangian interpolating polynomials written such that

( ) ( ) ( )1

1 1

1 1

i ir ri i

i i i i

x x x xP x C C

x x x x

+

++ +

− −= +

− −

( )( ) ( )

( ) ( )( )

( )( )( )( )

( )( ) ( )

( ) ( )( )1 1 1 1

2 1 1

1 1 1 1 1 1 1 1

i i i i i i

r r ri i i

i i i i i i i i i i i i

x x x x x x x x x x x xP x C C C

x x x x x x x x x x x x

− + − +

+ −+ + − + − − + −

− − − − − −= + +

− − − − − −

where x denotes the position of the forward cell face. The first and the

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where x denotes the position of the forward cell face. The first and the second derivatives of these functions are taken to evaluate the expressions required for the QUICKEST algorithm

( )( ) ( )1

1

1 1

1 1r ri i

i i i i

dP xC C

dx x x x x++ +

= −− −

( )( ) ( )

( )( )( )

( )( ) ( )

( )2

2

2 1 1

1 1 1 1 1 1 1 1

2 2 2r r ri i i

i i i i i i i i i i i i

d P xC C C

dx x x x x x x x x x x x x+ −+ + − + − − + −

= + +− − − − − −

When substituted back in the interpolation equation, one would obtain the expression for the front cell face concentration value. Note that for a uniform grid, the forward cell face concentration value becomes:

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( )( )2

1 1 1 1

1 11 2

2 2 6r r r r r r r rf i i i i i i i

CNC C C C C CN C C C

+ + + − = + − − − − − +

where CN is the Courant number given by

V t∆

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V tCN

x

∆=

∆

A similar procedure is implemented for the backward face concentration can be written using the QUICKEST algorithm as

( ) ( ) ( ) ( )2

2 21 2

1 2

1

2 6r cb

V t dP d PC P x x V t

dx dx

∆ = − − ∆ − ∆

where P1 and P2 are the first and second order Lagrangian interpolating polynomials written such that

( ) ( ) ( )1i ix x x xP x C C−− −

= +

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( ) ( ) ( )1

1 1

1 1

i ir ri i

i i i i

x x x xP x C C

x x x x

−

−− −

− −= +

− −

( )( ) ( )( ) ( )

( )( ) ( )

( ) ( )( )

( ) ( )( ) ( )

( )1 2 2 1

2 1 2

1 2 1 1 2 2 2 1

i i i i i i

r r ri i i

i i i i i i i i i i i i

x x x x x x x x x x x xP x C C C

x x x x x x x x x x x x

− − − −

− −− − − − − − − −

− − − − − −= + +

− − − − − −

where x now denotes the position of the backward cell face. The first and the second derivatives of these functions are taken to evaluate the expressions required for the QUICKEST algorithm

( )( ) ( )1

1

1 1

1 1r ri i

i i i i

dP xC C

dx x x x x −− −

= −− −

( )( )( )

( )( )( )

( )( ) ( )

( )2

2

2 1 2

1 2 1 1 2 2 2 1

2 2 2r r ri i i

i i i i i i i i i i i i

d P xC C C

dx x x x x x x x x x x x x− −− − − − − − − −

= + +− − − − − −

When substituted back in the interpolation equation, one would obtain the expression for the back cell face concentration value. Note that for a uniform grid, the backward cell face concentration value becomes

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( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( )( )2

1 1 1 2

1 11 2

2 2 6r r r r r r r rb i i i i i i i

CNC C C C C CN C C C

− − − − = + − − − − − +

Hence, for a control-volume i, the QUICKEST algorithm uses 4 nodal points to interpolate the forward and backward cell face concentration values giving a third order accurate scheme.

The dispersion operator does not pose any numerical problems and hence can be accurately discretized by using a central difference scheme as shown below

i 1i + 1i −

1ix −∆ ix∆

( )1

1

2i ix x−∆ + ∆

f b

Numerical discretizationof the domain for dispersion

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The dispersion part of the second equation shown above can be discretizedsuch that the inner part of the derivative is written between the forward and the backward face

( )1

1

1

2

r r rx x x

f bi i

C C CAD AD AD

x x x xx x−

∂ ∂ ∂ ∂ = −

∂ ∂ ∂ ∂ ∆ + ∆

Then, the derivative at the cell faces can simply be evaluated using the central differencing principle

( )( )

( ) ( )( )

( ) ( )1 1

11

1

1

2

r r r ri i i irx f x b xf b

i ii i

C C C CCAD A D A D

x x x xx x

+ −

−−

− − ∂ ∂ = −

∂ ∂ ∆ ∆ ∆ + ∆

The remaining parts of the equation can directly be evaluated for the control volume. Combining all terms and implementing a time-averaging idea would give:

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( ) ( )

( )( )

( ) ( )( )

( ) ( )( )

( )( )

( )( ) ( )

( )( ) ( )

1 *

1 1 1 1

11 1 11 1

11

1 1

11

1

1

2

11

1

2

n

r ri i

n n n n

nr r r rn n ni i i if x b x r if b i

i ii i

n n n n

r r r rn ni i i if x b xf b

i ii i

C A C A

C C C Ct A D A D k C A

x xx x

C C C Ct A D A D

x xx x

θ

θ

+

+ + + +

++ + ++ −

−−

+ −

−−

=

− − +∆ − − ∆ ∆ ∆ + ∆

− − +∆ − − ∆ ∆

∆ + ∆ ( )

n n

r iik C A

−

Initial and Boundary Conditions

The initial conditions state the contaminant concentration at all points in the system at time t=0

( ) ( )xCtxC or == 0,

where Co is the initial concentration distribution along the channel. Two boundary conditions are required: (i) at the upstream boundary and (ii) at

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boundary conditions are required: (i) at the upstream boundary and (ii) at the downstream boundary. In general, a specified concentration time series is used as the upstream boundary condition. The concentration time series is either available from continuous measurement (i.e., specified concentration) or from simple contaminant load computation (i.e., specified mass flux):

( ) ( )

( ))(

)(,0

,0

tQ

tWtC

tCtC

r

ubcr

=

=

where W(t) is the mass loading rate from some upstream source (MT-1) and Q(t)is the river flow at the upstream boundary (L3T-1) and Cubc is the corresponding upstream boundary concentration time series (ML-3). At the downstream boundary, a zero concentration gradient is used as the boundary condition when the boundary is far away from the contaminant zone

0=∂

∂

=Lx

rx

x

CDA

which states that advection dominates at the outflow and the contaminant

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which states that advection dominates at the outflow and the contaminant propagates out of the domain unhindered. When the boundary is not far and the outflow concentration is measured, a specified total mass flux is used as the downstream boundary condition

*

*

x

x L

CQC A D f

x=

∂− =

∂

where C* is the measured concentration at the downstream boundary and function f represents a known total flux out of the domain

Evaluation of Dispersion Coefficient

Although the accurate way to compute the longitudinal dispersion coefficient in a channel is to perform experiments along the channel, this is generally not practical for most of the cases. Numerous researchers have developed empirical expressions to approximate the longitudinal dispersion coefficients in rivers. Of these, the expression of Fischer et al (1979) has been widely applied in numerical modeling applications. This expression strictly uses the flow and channel characteristics and hence is very suitable for use in a numerical modeling code. The expression is written as:

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numerical modeling code. The expression is written as:

2 2

*0.011

V BK

V d=

where V is the flow velocity, V* is the shear velocity, B is the top width and d is the water depth

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