Merge pull request #445 from nens/nici_sug_1dadvectie

minor changes in text

source/h_onedee_flow.rst

@@ -98,9 +98,10 @@ Some examples are shown in the figures below.
 1D momentum equation
 --------------------
 
-The flow in 1D networks is computed using the equations of conservation of mass and momentum, more specifically the 1D depth-averaged shallow water equations. The momentum equation for 1D flow is:
+The flow in a 1D network is computed using the equations of conservation of mass and momentum. For this type of flow, these are known as the 1D depth-averaged shallow water equations. The momentum equation for 1D flow in non-conservative form is:
 
 .. math::
+   :name: momentum_equation_1d
    :label: 1D momentum equation
 
    \frac{\partial u}{\partial t}+u \frac{\partial u}{\partial s}=-g\frac{\partial \zeta}{\partial s}-\frac{\tau_f}{R\rho}-\frac{\tau_w}{H \rho}
@@ -115,7 +116,39 @@ The flow in 1D networks is computed using the equations of conservation of mass
 | :math:`H` is the water depth
 | :math:`R` is the hydraulic radius
 
-In words; in 1D, 3Di takes inertia, advection, pressure gradients, bottom friction and wind shear stresses into account. This yields for all types of 1D network elements. However, there are some differences in the computation of advection and the effect of wind stress for specific 1D network This will be explained more elaborated, where these difference are relevant.
+In words; in 1D, 3Di takes inertia, advection, pressure gradients, bottom friction and wind shear stresses into account. This yields for almost all elements in a 1D network. However, there are, for example, some differences in the computation of advection and the effect of wind stress for specific 1D elements. This will be explained in detail in the following sections.
+
+.. _1d_advection:
+
+Advection in 1D domain
+----------------------
+
+The second term on the left-hand side of equation :ref:`momentum_equation_1d` :math:`u \frac{\partial u}{\partial s}` represents the advective term. Based on the spatial gradient of the velocity, it represents the transport of momentum. Advective terms can be numerically solved in various ways: implicit/explicit central difference method, first/second order upwind difference method, among others. Although all are mathematically correct and consistent, they all have their own advantages and disadvantages. They differ in their computational cost, accuracy, time step sensitivity, robustness and/or stability. Depending on the application, some of these characteristics are more pronounced than others. 
+
+All these methods are consistent under smooth conditions, but under certain conditions they can result in very different solutions, some even physically incorrect (:cite:t:`Stelling2003`). Exemplary are the results near sudden bed transitions or channel expansions or contractions. At those locations, there are large gradients in the velocity field. Under these circumstances the 1D momentum equation is not adequate. The vertical flow developed at the edge of these sudden transitions is at a scale too small to be resolved by large-scale models and more importantly, 3-dimensional approximations are required to model such complex flows. However, with applying correct conservation properties, accurate solutions can be achieved. Below, those that are used in 3Di are explained.
+
+3Di benefits from two main methods, both of which have been studied to be efficient and accurate. The first method is derived based on the momentum conservative form of equation :ref:`momentum_equation_1d`.
+
+.. math::
+   \frac{\partial (Hu)}{\partial t}+\frac{\partial (Hu^2 + \frac{1}{2}gH^2)}{\partial s}+c_{f}u=gH\frac{\partial d}{\partial s}
+
+For positive flow direction, the advection approximation of above equation yields a simple, first-order accurate, expression. In the discrete format, it is given by:
+
+.. math::
+   u\frac{\partial u}{\partial s} = \frac{q_{i-\frac{1}{2}}+q_{i+\frac{1}{2}}}{H_{i}+H_{i+1}} \frac{u_{i+\frac{1}{2}}-u_{i-\frac{1}{2}}}{ds}
+   
+The second method is derived based on the energy-head conservation form of equation :ref:`momentum_equation_1d`:
+
+.. math::
+   \frac{\partial u}{\partial t}+\frac{\partial (\frac{1}{2}u^2+g\zeta)}{\partial s}+c_{f}\frac{u|u|}{h}=0
+
+
+For positive flow direction, the advection approximation of above equation yields a first-order expression. In the discrete format, it is given by:
+
+.. math::
+   u\frac{\partial u}{\partial s} = \frac{u_{i-\frac{1}{2}}+u_{i+\frac{1}{2}}}{2} \frac{u_{i+\frac{1}{2}}-u_{i-\frac{1}{2}}}{ds}
+   
+All discretisation schemes produce errors. These errors can be observed in the results as extra or diminishing energy/momentum losses. In case of, for example, a channel flow, these losses would be translated in an increase in the backwater curve. This is in such case an artificial backwater curve. Therefore, it is important to examine the amount of energy losses due to numerical errors arising with different advection methods (artificial backwater). In case of an energy-conservative scheme, as the approach would suggest, the total energy head loss is zero. Then, the advection term has no contribution to the artificial backwater. This is a stable method without creating any numerical errors. However, it also generates no head loss in case it is expected, e.g., in sudden expansions. The momentum-conservative method, on the other hand, always produces a minimum amount of backwater. In case of sudden expansions, the head loss generated by this method is in line with expectations, however application of this principle at strong contractions would increase the energy head. This is wrong from the physical point of view and might affect the stability. 3Di supports a momentum-conservative method, a energy-conservative method and (as a default) a combined approach. In this default option, a momentum-conservative method is applied except at sudden contractions. This combination ensures stability and realistic results.
 
 .. _1d_friction:
 

source/literature.bib

@@ -157,3 +157,13 @@ @article{Baptist2007
    url = {https://doi.org/10.1080/00221686.2007.9521778},
    year = {2007},
 }
+@article{Stelling2003,
+   author = {G. S. Stelling and S. P. A. Duinmeijer},
+   doi = {10.1002/fld.537},
+   journal = {International Journal for Numerical Methods in Fluids},
+   pages = {1329-1354},
+   publisher = {Wiley},
+   title = {A staggered conservative scheme for every Froude number in rapidly varied shallow water flows},
+   volume = {43},
+   year = {2003},
+}