Derivations used in rivr

Introduction

This vignette documents the mathematical derivations for gradually-varied and unsteady flow analysis with rivr. All derivations are based on Chaudry (2007). In Section 1, important definitions related to channel geometry are discussed. In Section 2, basic concepts of open-channel flow are discussed and one-dimensional shallow-water equations are derived. In Section 3, gradually-varied flow is derived and the standard-step method is discussed. In Section 4, the kinematic and dynamic wave models for simulating unsteady flow are derived along with the three discretization methods available in rivr. Finally, Section 5 discusses the Method of Characteristics and its application to boundary conditions of unsteady flow simulations.

Section 1: Channel geometry relations

A channel is prismatic if it has the same slope and cross-section throughout its entire length. The rivr package currently supports prismatic trapezoidal channels of arbitrary bottom width \(B\) and side slope \(SS = H:V\). Given a flow depth \(y\), the flow area is then \[ A = By + SSy^2 \] Another important definition is the wetted perimeter \(P\), which for a trapezoidal cross-section is \[ P = B + 2y\sqrt{1 + SS^2} \] The hydraulic radius is the ratio of the cross-sectional flow area to the wetted perimeter, i.e. \[ R = \frac{A}{P} \] The hydraulic depth \(D\) is defined as the ratio of the cross-sectional flow area to the top width \(T\), where \[ T = \frac{dA}{dy} \] which for a trapezoidal cross-section yields \[ T = B + 2ySS \] and \[ D = \frac{A}{T} = \frac{By + SSy^2}{B + 2ySS} \] Finally, the term \(\bar{y}\) is related to hydrostatic pressure and is defined as the distance from the water surface to the centroid of the cross-sectional flow area, i.e. \[ \bar{y} = \frac{2B + T}{3(B + T)} y \]

The definitions described above are fundamental properties related to one-dimensional open-channel flow. These relations (and their derivatives) are used extensively in the following sections to derive important flow relations and appear repeatedly in numerical solution schemes.

Section 2: Basic concepts of one-dimensional open-channel flow

The flow depth of a channel in an equilibrium state is called the normal depth, i.e. the flow depth at which gravitational forces (bed slope) are balanced by friction forces (bed roughness). The rivr package expresses the normal depth via the semi-empirical Manning's equation \[ Q = \frac{C_m}{n} AR^{2/3}S_0^{1 / 2} \] where \(Q\) is the channel flow, \(S_0\) is the channel slope, \(A\) is the cross-sectional flow area, \(R\) is the hydraulic depth and \(C_m\) is a conversion factor based on the unit system used (1.49 in US customary units and 1.0 in SI units). The expression is often rewritten as \[ Q = KS_0^{1 / 2} \] Where \(K\) is the channel conveyance. The normal depth \(y_n\) factors into the expressions of both \(A\) and \(R\) (i.e. \(K\)) and the above equation cannot be rearranged to solved explicitly for the normal depth; an implicit (iterative) solution is needed. The univariate Netwon-Raphson method is often used to provide efficient and precise solutions for \(y_n\). Generally, the Newton-Raphson method is defined as \[ x = x^* - \frac{f(x^*)}{\frac{df}{dx}\bigg|_{x^*}} \] where \(x\) is the updated guess for the parameter for which a solution is sought, \(x^*\) is the prior guess for the parameter value, \(f(x)\) is a zero-valued function of the parameter and \(\frac{df}{dx}\) is the derivative of said function. To apply the Newton-Raphson method here, Manning's equation is rewritten as \[ f(y_n) = AR^{2/3} - \frac{nQ}{C_m S_0^{1 / 2}} = 0 \] and its derivative is \[ \frac{df}{dy_n} = \frac{5}{3}T R^{2/3} - \frac{2}{3}R^{5/3}\frac{dP}{dy_n} \] where \(\frac{dP}{dy_n} = 2\sqrt{1 + SS^2}\). A related concept is the critical depth, the flow depth which minimizes the specific energy of the flow. The specific energy is the sum of flow depth and velocity head, i.e. \[ E = y + \frac{u^2}{2g} = \frac{1}{2g}\left(\frac{Q}{A}\right)^2 \] where \(u\) is the uniform flow velocity. The critical depth is then the flow depth that satisfies \[ \frac{dE}{dy} = 0 = 1 - \frac{Q^2}{gA^3}\frac{dA}{dy} = \frac{Q^2 T}{gA^3} \] For a trapezoidal channel it can be mathematically proved that only one critical depth exists for a given flow rate. The critical depth can be solved using the Newton-Raphson method with \(f(y_c) = \frac{dE}{dy} = 0\) and \[ \frac{df}{dy_c} = \frac{3}{2}\left(AT\right)^{1 / 2} - \frac{1}{2}\left(\frac{A}{T}\right)^{3/2}\frac{dT}{dy_c} \] where \(\frac{dT}{dy_c} = 2SS\) for a trapezoidal channel. The critical depth is also the depth at which the Froude number of the flow is unity; the Froude number is a dimensionless measure of bulk flow characteristics that represents the relative importance of inertial forces and gravitational forces and is defined as \[ Fr = \frac{Q}{A\sqrt{gD}} \]

Flows are referred to as subcritical if the flow depth is greater than the critical depth (\(Fr < 1\)) and supercritical if the flow depth is less than the critical depth (\(Fr > 1\)). Flows can transition gradually from subcritical to supercritical conditions; when the rate of variation of flow depth is small with respect to the longitudinal distance over which the change occurs, the river state is referred to as gradually-varied flow. In contrast, the transition from supercritical to subcritical conditions occur abruptly in the form of a hydraulic jump and is an example of rapidly-varied flow. Both gradually-varied and rapidly-varied flows can be either steady (flow is constant through time) or unsteady (flow rate varies with respect to time). The rivr package provides solutions for steady gradually-varied and unsteady flow problems.

Section 3: Solutions to steady gradually-varied flow problems

The standard-step method can be used to solve the steady gradually-varied flow profile when the channel flow and geometry are known. Additionally, the flow depth \(y\) must be known at a specified channel cross-section; this cross-section is referred to as the control section and the flow depth associated with the channel flow rate \(Q\) at the control section is \(y_1\). The total head \(H_1\) at the control section is the sum of the elevation head, flow depth head, and velocity head, i.e. \[ H_1 = y_1 + z_1 + \frac{1}{2g}\left(\frac{Q}{A_1}\right)^2 = z_1 + E_1 \] where \(z_1\) is the elevation of the control section bottom relative to some datum and \(A_1\) is the cross-sectional flow area at the control section. From conservation of energy, it follows that the total head \(H_2\) at some downstream cross-section, referred to as the target section, is \[ H_2 = H_1 - h_f \] where \(h_f\) is the head loss. While the head loss term generally combines both friction loss and form drag, the latter component is neglected by rivr. The friction component is expressed as the average friction slope between the control and target sections: \[ h_f = \frac{{S_f}_1 + {S_f}_2}{2} \left(x_2 - x_1\right) \] where \(x_2 - x_1\) is the longitudinal distance between the control and target section. Note that the sign of \(h_f\) therefore depends on whether the control section is upstream (\(x_1 < x_2\)) or downstream (\(x_1 > x_2\)) of the target section. Substituting these terms into the governing equation and rearranging yields \[ y_2 + z_2 + \frac{1}{2g}\left(\frac{Q}{A_2}\right)^2 + \frac{{S_f}_1 + {S_f}_2}{2} \left(x_2 - x_1\right) = y_1 + z_1 + \frac{1}{2g}\left(\frac{Q}{A_1}\right)^2 \] Note that all terms on the right-hand side of the equation are known, while \(A_2\) and \({S_f}_2\) on the left-hand side of the equation are functions of \(y_2\). Transposing all terms to the left-hand side yields a zero-value function of \(y_2\): \[ f(y_2) = 0 = y_2 + z_2 + \frac{1}{2g}\left(\frac{Q}{A_2}\right)^2 + \frac{{S_f}_1 + {S_f}_2}{2} \left(x_2 - x_1\right) - y_1 - z_1 - \frac{1}{2g}\left(\frac{Q}{A_1}\right)^2 \] This function is suitable for solving using the Newton-Raphson method discussed previously, i.e. \[ y_2 = y_2^* - \frac{f(y_2^*)}{\frac{df}{dy}\bigg|_{y_2^*}} \] where \[ \frac{df}{dy_2} = 1 - \frac{Q^2}{gA_2^3}\frac{dA_2}{dy_2} + \frac{1}{2}\left(x_2 - x_1\right) \frac{d{S_f}_2}{dy_2} \] and \[ \frac{d{S_f}_2}{dy} = -2\left(\frac{{S_f}_2}{A} \frac{dA}{dy} + \frac{2}{3} \frac{{S_f}_2}{R} \frac{dR}{dy} \right) \] Once the flow depth at the target section is found, the target section becomes the new control section and the flow depth at the next target section is computed, with the algorithm "stepping" up or down the channel to a specified distance from the initial control section. The standard-step method is accessed via the function compute_profile.

Section 4: Solutions to unsteady-flow problems

Unsteady flow problems are generally characterized using the Shallow Water Equations, with the one-dimensional form expressed as \[ \frac{\partial A}{\partial t} + \frac{\partial Q}{\partial x} = 0 \] \[ \frac{\partial Q}{\partial t} + \frac{\partial}{\partial x} \left(Qu + g\bar{y}A\right) - gA\left(S_0 - S_f\right) = 0 \] where the first equation expresses mass conservation and the second expresses momentum conservation. Without further simplification, these equations are often referred to as the Dynamic Wave Model (DWM). The Kinematic Wave Model (KWM) refers to a simplification of the momentum equation by assuming \(S_0 = S_f\), i.e. the momentum equation is instead expressed through the relation \[ A = \left( \frac{nQP^{2/3}}{C_m S_0^{1 / 2}} \right)^{3/5} \] Both the KWM and DWM can be solved using numerical discretization methods such as finite-difference schemes. Finite-difference schemes discretize a continuous model domain into a series of nodes separated by an incremental distance \(\Delta x\). The model time domain is similarly discretized into a series of time steps separated by an incremental time \(\Delta t\). A finite-difference scheme is called explicit if the value of the variable being solved for on time step \(k + 1\) depends explicitly on the value of the variable at the previous time step \(k\). Explicit methods are advantageous because they are easier to program and implement, but are disadvantageous because they are subject to stability constraints. The stability constraint is defined by the Courant number \[ C = \frac{\Delta t}{\Delta x} u \] which represents the ratio of the flow velocity to the rate of propagation of information through the model domain. The numerical solution is unstable if \(C > 1\).

The rivr package provides an interface to one finite-difference numerical scheme for the KWM and two finite-difference schemes for the DWM. In addition, the DWM interface supports boundary condition solutions using the Method of Characteristics (MOC). These schemes are accessed via the function route_wave and their derivations are discussed below.