2.1. Water properties and hydrodynamic processes

In hydrodynamics, the main property parameters of water are its density/specific weight and its viscosity, both of which have a significant impact on the solutions to the governing equations. The density of water can vary depending on the temperature, concentration of the suspended solids, and salinity. The influence of these factors on water density has been formulated as follows [16]:

$$
\rho = \rho\_T + \Delta \rho\_S + \Delta \rho\_\mathbb{C} \tag{1}
$$

where r<sup>T</sup> is the density of pure water as a function of temperature T [17, 18], Δr<sup>S</sup> is the change in density due to salinity S, and Δr<sup>C</sup> is the change in density due to total suspended sediments [19].

Viscosity represents the internal friction of water and is very important for hydrodynamic processes. The kinematic viscosity (m2 /s) of a river can be approximated as a function of temperature T (�C) [20].

$$w = \left(1.785 - 0.0584T + 0.00116T^2 - 0.0000102T^3\right) \times 10^{-6} \tag{2}$$

Here, hydrodynamic processes refer to water motion, circulation, mixing phenomena; the corresponding processes involving the materials suspended in the water include advection, dispersion and mixing. The results combine to form a hydrodynamic model that generally includes flow field, water depth and water surface elevation, salinity, temperature, and sediment concentration. Some of this information, for example, which related to temperature, salinity, and/or sediment, may also be utilized in water quality models.

#### 2.2. Cartesian coordinate-based governing equations

The governing equations for both water flow and the transport of contaminants are based on the conservation laws of mass, momentum, and energy. Hydrodynamic models include two main types of governing equations: continuity equation for the mass balance of water in the


where A is the cross sectional area, u is the flow velocity averaged over the cross section, a is the elevation of channel bottom, h is the water depth, εxx is the horizontal eddy coefficient, B is the channel width at the water surface, λ is the wetted perimeter of the cross section, τsx is the wind stress acting on water surface, and τbx is the frictional stress on bottom and bank surface [20].

Application of a Hydrodynamic and Water Quality Model for Inland Surface Water Systems

<sup>þ</sup> <sup>h</sup> <sup>∂</sup><sup>u</sup> ∂x þ ∂v ∂y � �

> εxxh ∂u ∂x � �

<sup>þ</sup> <sup>ζ</sup>W<sup>2</sup>

εyxh ∂v ∂x � �

<sup>þ</sup> <sup>ζ</sup>W<sup>2</sup>

þ ∂ ∂y

þ ∂ ∂y

� � � �

cosψ

sinψ

� � � �

εxyh ∂u ∂y

εyyh ∂v ∂y

2

<sup>∂</sup><sup>z</sup> <sup>¼</sup> <sup>0</sup> (9)

¼ 0 (5)

http://dx.doi.org/10.5772/intechopen.74914

(6)

91

(7)

(8)

2.2.2. Depth-averaged 2D equations

∂h ∂t þ u ∂ð Þh ∂x þ v ∂ð Þh ∂y

<sup>þ</sup> hv <sup>∂</sup><sup>u</sup>

<sup>þ</sup> hv <sup>∂</sup><sup>v</sup>

<sup>ε</sup> <sup>¼</sup> <sup>2</sup>Am <sup>¼</sup> <sup>α</sup><sup>A</sup> <sup>∂</sup><sup>u</sup>

<sup>∂</sup><sup>y</sup> � fvh <sup>¼</sup> <sup>1</sup>

� ugn<sup>2</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>u</sup><sup>2</sup> <sup>þ</sup> <sup>v</sup><sup>2</sup> ð Þ <sup>p</sup> h1=3

<sup>∂</sup><sup>y</sup> � fuh <sup>¼</sup> <sup>1</sup>

� vgn<sup>2</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>u</sup><sup>2</sup> <sup>þ</sup> <sup>v</sup><sup>2</sup> ð Þ <sup>p</sup> h1=3

> ∂x � �<sup>2</sup>

∂ð Þ Bu ∂x þ ∂ð Þ Bw

r ∂ ∂x

r ∂ ∂x

where u, v are the depth-averaged velocities in the x, y directions, respectively, W is the wind velocity, ψ is the wind direction, ζ is the empirical wind coefficient; f is the Coriolis parameter, g is the acceleration due to gravity, n is theManning's roughness coefficient, andε is the depth-averaged eddy viscosity. Horizontal mixing is described using the Smagorinsky eddy parameterization:

þ

where α is a constant in the range 0.01–0.5 [22] and Am is the area of the current element based

where u is the lateral averaged velocity in the x direction, w is the lateral averaged velocity in

∂v ∂y � �<sup>2</sup>

þ 1 2

� �<sup>2</sup> " #<sup>1</sup>

∂u ∂y þ ∂v ∂x

<sup>þ</sup> hu <sup>∂</sup><sup>u</sup> ∂x

<sup>þ</sup> hu <sup>∂</sup><sup>v</sup> ∂x

2.2.2.1. Continuity equation

2.2.2.2. Momentum equations

h ∂u ∂t

h ∂v ∂t

on the finite element scheme [21].

2.2.3. Lateral-averaged 2D equations

the z direction, and B is the water width.

2.2.3.1. Continuity equation

�gh <sup>∂</sup><sup>a</sup> ∂x þ ∂h ∂x � �

�gh <sup>∂</sup><sup>a</sup> ∂y þ ∂h ∂y � �

Table 1. Examples of different dimensional models and their applications.

flow and momentum equation that indicates the relationship between the driving forces and water acceleration of motion, which is Newton's second law. The forces acting on a water body include gravity, viscous force, and pressure as well as other external forces such as wind and the Coriolis force. The momentum equation in most hydrodynamic models is a simplification and/or modification of the Navier-Stokes equation. Most natural surface water systems are characterized as "shallow water," where the horizontal scale is far greater than the water depth. The hydrostatic assumption, which assumes that the hydrostatic balance in the vertical plane and the vertical acceleration can be neglected, also applies to shallow water. Technically, all surface water systems are three-dimensional scenarios, and a fully 3D model would thus provide the most complete description of their flow features, but the resulting model would be far too complex for the governing equations to be solved. Under most circumstances, it is reasonable to use a 1D, 2D, or quasi-3D model as this provides sufficient accuracy to solve nearly all practical problems. Based on the representation and spatial scale of water body and application purposes, a number of different dimensional models have been developed by researchers. Examples of these models and their applications are listed in Table 1. Their governing equations for continuity and momentum are described as follows [20] (2.2.1), [16, 20] (2.2.3), [21] (2.2.2), [23] (2.2.4).

#### 2.2.1. 1D equations

#### 2.2.1.1. Continuity equation

$$\frac{\partial A}{\partial t} + \frac{\partial (Au)}{\partial x} = 0 \tag{3}$$

#### 2.2.1.2. Momentum equation

$$\frac{\partial(Au)}{\partial t} + \frac{\partial(Au^2)}{\partial x} + gA \left[ \frac{\partial(a+h)}{\partial x} \right] - \frac{1}{\rho} \frac{\partial}{\partial x} \left( \varepsilon\_{\text{xx}} A \frac{\partial u}{\partial x} \right) - \frac{1}{\rho} (B\tau\_{\text{xx}} - \lambda \tau\_{bx}) = 0 \tag{4}$$

where A is the cross sectional area, u is the flow velocity averaged over the cross section, a is the elevation of channel bottom, h is the water depth, εxx is the horizontal eddy coefficient, B is the channel width at the water surface, λ is the wetted perimeter of the cross section, τsx is the wind stress acting on water surface, and τbx is the frictional stress on bottom and bank surface [20].

#### 2.2.2. Depth-averaged 2D equations

#### 2.2.2.1. Continuity equation

$$
\rho \frac{\partial h}{\partial t} + \mu \frac{\partial (h)}{\partial x} + \upsilon \frac{\partial (h)}{\partial y} + h \left( \frac{\partial \mu}{\partial x} + \frac{\partial \upsilon}{\partial y} \right) = 0 \tag{5}
$$

#### 2.2.2.2. Momentum equations

flow and momentum equation that indicates the relationship between the driving forces and water acceleration of motion, which is Newton's second law. The forces acting on a water body include gravity, viscous force, and pressure as well as other external forces such as wind and the Coriolis force. The momentum equation in most hydrodynamic models is a simplification and/or modification of the Navier-Stokes equation. Most natural surface water systems are characterized as "shallow water," where the horizontal scale is far greater than the water depth. The hydrostatic assumption, which assumes that the hydrostatic balance in the vertical plane and the vertical acceleration can be neglected, also applies to shallow water. Technically, all surface water systems are three-dimensional scenarios, and a fully 3D model would thus provide the most complete description of their flow features, but the resulting model would be far too complex for the governing equations to be solved. Under most circumstances, it is reasonable to use a 1D, 2D, or quasi-3D model as this provides sufficient accuracy to solve nearly all practical problems. Based on the representation and spatial scale of water body and application purposes, a number of different dimensional models have been developed by researchers. Examples of these models and their applications are listed in Table 1. Their governing equations for continuity and momentum are described as follows [20] (2.2.1), [16,

Zero Completely mixed, no flow, spatially uniform Estimation of ponds, lakes, tanks with no or low

flow

(strongly stratified)

estuaries

Long creeks, rivers, streams, narrow channels

Narrow and deep lakes, reservoirs, river reaches

Deep large lakes, reservoirs, coastal regions,

Wide open lakes and estuaries (weakly stratified), ponds, wide shallow river reaches

Characteristics Applications

One Shallow narrow flow, well mixed, spatial variation in transverse and vertical directions neglected

Three Deep, stratified flow, significant variation in vertical direction, all three directions considered

vertical direction neglected

90 Applications in Water Systems Management and Modeling

neglected

Shallow wide flow, well mixed, spatial variations in

Deep narrow flow, spatial variation in lateral direction

Table 1. Examples of different dimensional models and their applications.

∂A ∂t þ

<sup>þ</sup> gA <sup>∂</sup>ð Þ <sup>a</sup> <sup>þ</sup> <sup>h</sup> ∂x  ∂ð Þ Au

� 1 r ∂ ∂x

<sup>ε</sup>xxA <sup>∂</sup><sup>u</sup> ∂x  � 1 r

<sup>∂</sup><sup>x</sup> <sup>¼</sup> <sup>0</sup> (3)

ð Þ¼ Bτsx � λτbx 0 (4)

20] (2.2.3), [21] (2.2.2), [23] (2.2.4).

2.2.1. 1D equations

Model dimensions

Two (horizontal)

Two (vertical)

2.2.1.1. Continuity equation

2.2.1.2. Momentum equation

∂ð Þ Au ∂t þ ∂ Au<sup>2</sup> ∂x

$$\begin{aligned} \hbar \frac{\partial u}{\partial t} + hu \frac{\partial u}{\partial x} + hv \frac{\partial u}{\partial y} - fvh &= \frac{1}{\rho} \left[ \frac{\partial}{\partial x} \left( \overline{\varepsilon}\_{xx} h \frac{\partial u}{\partial x} \right) + \frac{\partial}{\partial y} \left( \overline{\varepsilon}\_{xy} h \frac{\partial u}{\partial y} \right) \right] \\ -gh \left( \frac{\partial u}{\partial x} + \frac{\partial h}{\partial x} \right) &- \frac{u g n^2 \sqrt{(u^2 + v^2)}}{h^\circ} + \zeta \mathcal{W}^2 \cos \psi \\ h \frac{\partial v}{\partial t} + hu \frac{\partial v}{\partial x} + hv \frac{\partial v}{\partial y} - fvh &= \frac{1}{\rho} \left[ \frac{\partial}{\partial x} \left( \overline{\varepsilon}\_{yx} h \frac{\partial v}{\partial x} \right) + \frac{\partial}{\partial y} \left( \overline{\varepsilon}\_{yy} h \frac{\partial v}{\partial y} \right) \right] \\ -gh \left( \frac{\partial a}{\partial y} + \frac{\partial h}{\partial y} \right) &- \frac{v g n^2 \sqrt{(u^2 + v^2)}}{h^\circ} + \zeta \mathcal{W}^2 \sin \psi \end{aligned} \tag{7}$$

where u, v are the depth-averaged velocities in the x, y directions, respectively, W is the wind velocity, ψ is the wind direction, ζ is the empirical wind coefficient; f is the Coriolis parameter, g is the acceleration due to gravity, n is theManning's roughness coefficient, andε is the depth-averaged eddy viscosity. Horizontal mixing is described using the Smagorinsky eddy parameterization:

$$\overline{\varepsilon} = 2A\_m = \alpha A \left[ \left( \frac{\partial u}{\partial x} \right)^2 + \left( \frac{\partial v}{\partial y} \right)^2 + \frac{1}{2} \left( \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} \right)^2 \right]^{\frac{1}{2}} \tag{8}$$

where α is a constant in the range 0.01–0.5 [22] and Am is the area of the current element based on the finite element scheme [21].

#### 2.2.3. Lateral-averaged 2D equations

#### 2.2.3.1. Continuity equation

$$\frac{\partial(Bu)}{\partial x} + \frac{\partial(Bw)}{\partial z} = 0 \tag{9}$$

where u is the lateral averaged velocity in the x direction, w is the lateral averaged velocity in the z direction, and B is the water width.

#### 2.2.3.2. Momentum equation

$$\frac{\partial \partial (Bu)}{\partial t} + \frac{\partial (Bu^2)}{\partial x} + \frac{\partial (uvB)}{\partial z} + gB \frac{\partial (z\_s)}{\partial x} - \frac{\partial}{\partial x} \left( BA\_H \frac{\partial u}{\partial x} \right) - \frac{\partial}{\partial x} \left( BA\_v \frac{\partial u}{\partial z} \right) - \tau\_x = 0 \tag{10}$$

where AH is the horizontal eddy viscosity, Av is the vertical turbulent mixing coefficient, and zs is the water surface elevation. For gradually varying flows, the z-direction momentum equation can be neglected due to the insignificant effects of inertia, diffusion, and dispersion in the vertical direction [16, 20].

#### 2.2.4. 3D stratified flow equations

#### 2.2.4.1. Continuity equation

$$
\frac{\partial u}{\partial x} + \frac{\partial y}{\partial y} + \frac{\partial w}{\partial z} = 0 \tag{11}
$$

rh ∂u ∂t

�ð Þ b � a

�ð Þ b � a

�ð Þ b � a

�ð Þ b � a

2.3. Boundary and initial conditions

2.3.1. The free water surface

2.3.2. The bottom surface

rh ∂v ∂t

<sup>þ</sup> <sup>r</sup>hu <sup>∂</sup><sup>u</sup> ∂x

> ∂ ∂x εxx

∂ ∂z εxz ∂u ∂z 

<sup>þ</sup> <sup>r</sup>hu <sup>∂</sup><sup>v</sup> ∂x

> ∂ ∂x εyx

∂ ∂z εyz ∂v ∂z 

<sup>þ</sup> <sup>r</sup>hv <sup>∂</sup><sup>u</sup> ∂y þ r ∂u

> h ð Þ b � a

<sup>þ</sup> <sup>r</sup>hv <sup>∂</sup><sup>v</sup> ∂y þ r ∂v

> h ð Þ b � a

∂u ∂x

<sup>þ</sup> <sup>r</sup>sgh <sup>∂</sup><sup>a</sup> ∂x

> ∂v ∂x

<sup>þ</sup> <sup>r</sup>sgh <sup>∂</sup><sup>a</sup> ∂y

the water surface. Subscripts s and b denote surface and bottom, respectively.

sediment and salinity are to be considered) at all boundaries must be specified.

traction, such as wind stress, acting on the water surface.

There is zero pressure on the water surface and no leakage across the water surface.

methods for the horizontal and vertical eddy viscosity, respectively.

� ð Þ b � a

� ð Þ b � a

<sup>þ</sup> <sup>r</sup>sgh <sup>∂</sup><sup>h</sup>

where u, v, w are the velocities in the x, y, z direction, respectively, εij is the turbulent eddy coefficient (i, j ¼ x, y, z), Γ<sup>i</sup> is the external tractions on the water body, and r<sup>s</sup> is the density at

Eddy viscosity can be determined using a number of different methods, but the Mellor Yamada formulation [25] and the Smagorinsky [22] method are the among most widely used

To numerically solve these governing equations, boundary conditions (and initial conditions for transition problems) must be specified for all external surfaces of the water body. Boundaries of a surface water system consist of the top water surface and the bottom and side surfaces. Accordingly, the water surface elevation, the velocities at the bottom and side surfaces, and the flux (if

ws <sup>¼</sup> dh

where ws is the vertical velocity at the water surface. There may also be some external surface

Two types of boundary conditions are applied to the bottom surface [23], which will be either a

no leakage boundary or a no slip boundary (ub ¼ vb ¼ wb ¼ 0) condition.

<sup>þ</sup> <sup>r</sup>sgh <sup>∂</sup><sup>h</sup> ∂x

<sup>∂</sup><sup>z</sup> ð Þ <sup>b</sup> � <sup>a</sup> <sup>w</sup> � uTx � vTy

<sup>∂</sup><sup>z</sup> ð Þ <sup>b</sup> � <sup>a</sup> <sup>w</sup> � uTx � vTy

∂ ∂y εyy

∂ ∂y εxy

� ð Þ <sup>z</sup> � <sup>a</sup>

� ð Þ <sup>z</sup> � <sup>a</sup>

∂v ∂y

dt (17)

h ð Þ b � a

<sup>∂</sup><sup>h</sup> <sup>þ</sup> gyh � <sup>Γ</sup><sup>y</sup> <sup>¼</sup> <sup>0</sup>

∂u ∂y

h ð Þ b � a

Application of a Hydrodynamic and Water Quality Model for Inland Surface Water Systems

þ gxh � Γ<sup>x</sup> ¼ 0

∂h ∂t

http://dx.doi.org/10.5772/intechopen.74914

∂h ∂t (15)

93

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Based on the hydrostatic assumption Eq. (12), vertical momentum equations can be eliminated from 3D governing equations.

$$\frac{\partial p}{\partial z} + \rho \mathbf{g} = \mathbf{0} \tag{12}$$

For the purposes of this analysis, a water system is divided into a series of horizontal layers that interact with each other. The topography of a real 3D environment is generally nonuniform, and the bathymetry may spatially differ over a wide range. To avoid any problems due to the nonuniform water depth, it is necessary to transform the system to a constant geometric structure with uniform resolution. Based on the Sigma method [16, 23, 24], a transformation scheme was developed as follows:

$$\begin{aligned} x' &= x \\ y' &= y \\ z' &= a + \frac{(z-a)(b-a)}{h} \end{aligned} \tag{13}$$

where x<sup>0</sup> ,y0 ,z<sup>0</sup> are the transformed coordinates, p is the water pressure, g is the acceleration due to gravity, r is the water density, z is the vertical coordinate, a is the bottom elevation, and b is the fixed vertical location to which the water surface will be transformed. Thus, the continuity equation and momentum equations along the horizontal direction are as follows [23]:

$$\begin{split} \frac{\partial h}{\partial t} + \int\_{a}^{b} \left[ \frac{\partial u}{\partial x} - \frac{(b-a)}{h} \frac{\partial u}{\partial z} T\_x + \frac{\partial v}{\partial y} - \frac{(b-a)}{h} \frac{\partial v}{\partial z} T\_y \right] dz + u\_s \frac{\partial (a+h)}{\partial x} - u\_b \frac{\partial a}{\partial x} \\ + v\_s \frac{\partial (a+h)}{\partial y} - v\_b \frac{\partial a}{\partial y} = 0 \end{split} \tag{14}$$

Application of a Hydrodynamic and Water Quality Model for Inland Surface Water Systems http://dx.doi.org/10.5772/intechopen.74914 93

$$\begin{aligned} &\rho h \frac{\partial u}{\partial t} + \rho h u \frac{\partial u}{\partial x} + \rho h v \frac{\partial u}{\partial y} + \rho \frac{\partial u}{\partial z} \left[ (b-a) \left( w - u T\_x - v T\_y \right) - (z-a) \frac{\partial h}{\partial t} \right] \\ &- (b-a) \frac{\partial}{\partial x} \left( \varepsilon\_{xx} \frac{h}{(b-a)} \frac{\partial u}{\partial x} \right) - (b-a) \frac{\partial}{\partial y} \left( \varepsilon\_{xy} \frac{h}{(b-a)} \frac{\partial u}{\partial y} \right) \\ &- (b-a) \frac{\partial}{\partial z} \left( \varepsilon\_{xz} \frac{\partial u}{\partial z} \right) + \rho\_s gh \frac{\partial a}{\partial x} + \rho\_s gh \frac{\partial h}{\partial x} + g\_x h - \Gamma\_x = 0 \\ &\rho h \frac{\partial v}{\partial t} + \rho h u \frac{\partial v}{\partial x} + \rho h v \frac{\partial v}{\partial y} + \rho \frac{\partial v}{\partial z} \left[ (b-a) \left( w - u T\_x - v T\_y \right) - (z-a) \frac{\partial h}{\partial t} \right] \\ &- (b-a) \frac{\partial}{\partial x} \left( \varepsilon\_{yx} \frac{h}{(b-a)} \frac{\partial v}{\partial x} \right) - (b-a) \frac{\partial}{\partial y} \left( \varepsilon\_{yy} \frac{h}{(b-a)} \frac{\partial w}{\partial y} \right) \\ &- (b-a) \frac{\partial}{\partial z} \left( \varepsilon\_{yz} \frac{h}{\partial z} \right) + \rho\_s gh \frac{\partial a}{\partial y} + \rho\_s gh \frac{\partial h}{\partial t} + g\_y h - \Gamma\_y = 0 \end{aligned} \tag{16}$$

where u, v, w are the velocities in the x, y, z direction, respectively, εij is the turbulent eddy coefficient (i, j ¼ x, y, z), Γ<sup>i</sup> is the external tractions on the water body, and r<sup>s</sup> is the density at the water surface. Subscripts s and b denote surface and bottom, respectively.

Eddy viscosity can be determined using a number of different methods, but the Mellor Yamada formulation [25] and the Smagorinsky [22] method are the among most widely used methods for the horizontal and vertical eddy viscosity, respectively.

#### 2.3. Boundary and initial conditions

To numerically solve these governing equations, boundary conditions (and initial conditions for transition problems) must be specified for all external surfaces of the water body. Boundaries of a surface water system consist of the top water surface and the bottom and side surfaces. Accordingly, the water surface elevation, the velocities at the bottom and side surfaces, and the flux (if sediment and salinity are to be considered) at all boundaries must be specified.

#### 2.3.1. The free water surface

2.2.3.2. Momentum equation

∂ Bu<sup>2</sup> � � ∂x

92 Applications in Water Systems Management and Modeling

þ

∂ð Þ uwB ∂z

<sup>þ</sup> gB <sup>∂</sup>ð Þ zs

∂u ∂x þ ∂y ∂y þ ∂w

<sup>∂</sup><sup>x</sup> � <sup>∂</sup> ∂x

where AH is the horizontal eddy viscosity, Av is the vertical turbulent mixing coefficient, and zs is the water surface elevation. For gradually varying flows, the z-direction momentum equation can be neglected due to the insignificant effects of inertia, diffusion, and dispersion in the

Based on the hydrostatic assumption Eq. (12), vertical momentum equations can be eliminated

For the purposes of this analysis, a water system is divided into a series of horizontal layers that interact with each other. The topography of a real 3D environment is generally nonuniform, and the bathymetry may spatially differ over a wide range. To avoid any problems due to the nonuniform water depth, it is necessary to transform the system to a constant geometric structure with uniform resolution. Based on the Sigma method [16, 23, 24], a

> <sup>z</sup><sup>0</sup> <sup>¼</sup> <sup>a</sup> <sup>þ</sup> ð Þ <sup>z</sup> � <sup>a</sup> ð Þ <sup>b</sup> � <sup>a</sup> h

to gravity, r is the water density, z is the vertical coordinate, a is the bottom elevation, and b is the fixed vertical location to which the water surface will be transformed. Thus, the continuity

> <sup>∂</sup><sup>y</sup> � ð Þ <sup>b</sup> � <sup>a</sup> h

equation and momentum equations along the horizontal direction are as follows [23]:

� �

,z<sup>0</sup> are the transformed coordinates, p is the water pressure, g is the acceleration due

∂v ∂z Ty

dz þ us

∂ð Þ a þ h <sup>∂</sup><sup>x</sup> � ub ∂a ∂x

∂p ∂z

x<sup>0</sup> ¼ x y<sup>0</sup> ¼ y

BAH ∂u ∂x � �

� ∂ ∂x

BAv ∂u ∂z � �

<sup>∂</sup><sup>z</sup> <sup>¼</sup> <sup>0</sup> (11)

þ rg ¼ 0 (12)

� τ<sup>x</sup> ¼ 0 (10)

(13)

(14)

∂ð Þ Bu ∂t þ

vertical direction [16, 20].

2.2.4.1. Continuity equation

2.2.4. 3D stratified flow equations

from 3D governing equations.

where x<sup>0</sup>

,y0

∂h ∂t þ ð b

þvs

a

∂u

∂ð Þ a þ h <sup>∂</sup><sup>y</sup> � vb

<sup>∂</sup><sup>x</sup> � ð Þ <sup>b</sup> � <sup>a</sup> h

∂u ∂z Tx þ ∂v

∂a <sup>∂</sup><sup>y</sup> <sup>¼</sup> <sup>0</sup>

transformation scheme was developed as follows:

There is zero pressure on the water surface and no leakage across the water surface.

$$w\_s = \frac{dh}{dt} \tag{17}$$

where ws is the vertical velocity at the water surface. There may also be some external surface traction, such as wind stress, acting on the water surface.

#### 2.3.2. The bottom surface

Two types of boundary conditions are applied to the bottom surface [23], which will be either a no leakage boundary or a no slip boundary (ub ¼ vb ¼ wb ¼ 0) condition.

### 2.3.3. The side surface

The side surface refers to the shoreline of the water body or a defined boundary of the model such as the entrance or exit of a river reach. The shoreline conditions are zero normal velocity, that is, no leakage across the boundary surface. Information on the water surface elevation, velocities, or flow rate need to be specified for the defined boundaries, but combinations of different conditions should be avoided on the same boundary.

are widely used to reflect these processes. Some examples of water quality model applications include the simulation and prediction of water temperature, dissolved oxygen (DO), biochemical oxygen demand (BOD), the nitrogen cycle (including levels of organic nitrogen, ammonia, nitrite and nitrate), the phosphorous cycle (including levels of organic phosphorous and phosphates), algae growth and decay, cohesive/noncohesive suspended sediment, salinity, heavy metals, and pathogens, among others [27]. Depending on the results for advection, dispersion, and turbulent mixing obtained from the hydrodynamic model, water quality models may also incorporate the

Application of a Hydrodynamic and Water Quality Model for Inland Surface Water Systems

The governing equations for contaminant fate and transport are as follows [16, 21, 23]:

DxA <sup>∂</sup><sup>C</sup> ∂x 

> Dxh ∂C ∂x

� khC � G ¼ 0

BDx ∂C ∂x 

<sup>∂</sup><sup>z</sup> � ð Þ <sup>b</sup> � <sup>a</sup> Tx

<sup>þ</sup> <sup>h</sup> <sup>∂</sup> ∂x

> ∂ ∂z

<sup>h</sup> DxyTx <sup>þ</sup> DyTy <sup>∂</sup><sup>C</sup>

þ Dxyh

� ∂ ∂x

∂ð Þ uC ∂z

ð Þ b � a

ð Þ b � a

∂ð Þ vsC <sup>∂</sup><sup>z</sup> <sup>¼</sup> <sup>0</sup>

∂C ∂y

BDz ∂C ∂z 

<sup>h</sup> DxTx <sup>þ</sup> DxyTy <sup>∂</sup><sup>C</sup>

<sup>h</sup> DxTx <sup>þ</sup> DxyTy <sup>∂</sup><sup>C</sup>

∂z

∂z

∂z

� kAsC � G ¼ 0 (18)

http://dx.doi.org/10.5772/intechopen.74914

(19)

95

(20)

(21)

sources/sinks, chemical and biological reactions of contaminants.

3.1. Governing equations for contaminant fate and transport

∂ð Þ AuC <sup>∂</sup><sup>x</sup> � <sup>∂</sup> ∂x

∂ð Þ hC ∂x

Dxyh ∂C ∂y

∂ð Þ uBC ∂x

∂ð Þ vC ∂y

þ v

þ Dyh

þ v

þ ð Þ b � a

∂C ∂y

<sup>þ</sup> <sup>h</sup> <sup>∂</sup> ∂y

<sup>h</sup> DxyTx <sup>þ</sup> DyTy <sup>∂</sup><sup>C</sup>

> ∂C ∂z

∂ð Þ hC <sup>∂</sup><sup>y</sup> � <sup>∂</sup> ∂x

> ∂C ∂y

∂ð Þ wBC <sup>∂</sup><sup>z</sup> � <sup>∂</sup> ∂x

∂ð Þ wC

∂C ∂y

� ð Þ b � a Tx

∂z

ð Þ b � a

� khC � G � ð Þ b � a

∂ð Þ AsC ∂t þ

∂ð Þ hC <sup>∂</sup><sup>t</sup> <sup>þ</sup> <sup>u</sup>

� ∂ ∂y

�khC � G ¼ 0

þ h

∂ð Þ vC <sup>∂</sup><sup>z</sup> � <sup>h</sup> <sup>∂</sup> ∂x Dx ∂C ∂x þ Dxy

ð Þ b � a

ð Þ b � a h

∂ ∂z Dx ∂C ∂x þ Dxy

∂ ∂z

Dxy ∂C ∂x þ Dy ∂C ∂y

∂ ∂z Dz

3.1.1. 1D equation

3.1.2. 2D depth-averaged equation

3.1.3. 2D laterally averaged equation

3.1.4. 3D stratified equation

�ð Þ b � a Ty

þð Þ b � a Tx

þð Þ b � a Ty

�ð Þ b � a

�<sup>h</sup> <sup>∂</sup> ∂y

h ∂C <sup>∂</sup><sup>t</sup> <sup>þ</sup> <sup>h</sup> ∂ð Þ BC <sup>∂</sup><sup>t</sup> <sup>þ</sup> <sup>u</sup>

∂ð Þ uC ∂x
