Comprehensive Review of River Mechanics and Hydraulic Modeling

Fundamentals of River Motion and Topography

• Principle Driving Force: The primary force responsible for driving river motion is gravity.

• River Confluence: This term is jargon utilized to describe the junction where a tributary meets the main stem of a river.

• Defining River Banks: The left bank of a river is determined by the observer's left side when they are looking in the downstream direction.

• River Reach: A reach is defined as a relatively short section of a river where the flowrate can be approximated as being single/uniform throughout. Typically, this constitutes the stretch of a river located between two confluences.

• GIS "Blue Lines" and Water Presence: It is false to assume that a blue line in a Geographic Information System (GIS) always indicates flowing water.     - Explanation: GIS systems define blue lines as topographic connections between low points in the landscape. Water only flows through these lines if hydrology provides a supply via overland flow, groundwater inflow, or upward upstream flow.     - Headwater Streams: In many cases, these lines only experience actual water flow during periods of heavy rainfall.

• Groundwater vs. Surface Water Catchments: A groundwater catchment is not always aligned with the surface water catchment above it.     - Determinants: Surface water catchment boundaries are dictated by surface topography (all points from which water flows downhill). Groundwater catchment boundaries depend on subsurface geology. It is rare for these two boundaries to be exactly aligned.

• The Vadose Zone: This is the region of groundwater located near the surface that remains unsaturated. This zone is where rainfall leaches into the soil.

Hydraulic Flow and Geometry

• Steady Flow in a Reach: Steady flow implies that the flowrate ($Q$) does not change over time ($\frac{dQ}{dt} = 0$).     - Implications: If groundwater exchange and surface runoff are neglected, a reach between two confluences has a single steady flowrate across its length; thus, steady flow also implies uniform flow for short reaches.

• Gravity vs. Friction in Uniform Flow: A uniform flow is characterized by identical upstream and downstream flowrates, velocities ($V$), depths ($y$), and cross-sectional areas ($A$).     - Force Balance: This state only exists when the downstream component of gravity (along the slope) exactly balances the upstream force of friction.     - Non-Uniform Scenarios:         - If Gravity > Friction: Velocity increases downstream, causing depth or cross-sectional area to decrease for a steady $Q$.         - If Gravity < Friction: Velocity decreases downstream, causing depth or cross-sectional area to increase for a steady $Q$.

• Key Geometric Parameters Affecting Flow:     - Cross-sectional area ($A$).     - Average (hydraulic) depth ($D$).     - Hydraulic radius ($R_h$).     - Topwidth ($B$).     - Bankfull width.     - Wetted perimeter ($P$).

Flow Equations and Modeling

• The Chezy-Manning (C-M) Equation and Unsteady Flow: While the C-M equation is technically false for unsteady flow because it lacks a time-varying term, it is used in models of the Saint-Venant equations to approximate the friction slope ($S_f$) in the energy dissipation term. Thus, it becomes a component of unsteady Saint-Venant (S-V) calculations.

• Normal Flow Depth: The C-M equation is used with the bottom slope ($S_0$) to compute normal flow depth, not specifically the energy grade line ($S_f$).     - Condition: Normal flow depth occurs when the friction exactly balances the downslope gravitational acceleration, meaning $S_f$ is exactly parallel to $S_0$.

• Steady Flow Alternatives: Beyond the C-M equation, steady flow in a river can be represented by the energy equation or the Saint-Venant equation where $\frac{dQ}{dt} = 0$.

• Coupling S-V Momentum and Energy Equations: It is false that these can be coupled to provide a closed set for flowrate ($Q$) and area ($A$). Under steady-state conditions, the Saint-Venant equation collapses into the energy equation; they are not independent. Attempting to solve them together results in two copies of the same equation written differently.

Gradually Varying Flow (GVF)

• Definition: GVF curves provide water surface profiles for a steady flowrate that is gradually changing in depth over a reach. They are not used for unsteady flow.

• Channel Width Changes:     - Strict interpretation: Formally, GVF curves assume a constant channel cross-sectional shape.     - Practical application: If cross-sections change gradually, basic GVF shapes may be observed. However, sharp geometry changes imply that an upstream GVF curve may transition rapidly to a different downstream GVF curve.

• Critical GVF Curves and Hydraulic Jumps: A critical GVF curve implies that the normal depth ($y_n$) and critical depth ($y_c$) are equal. It is false that a critical curve applies whenever there is a jump. Jumps can occur on steep, mild, adverse, or horizontal slopes whenever the transition moves from upstream supercritical to downstream subcritical.

• Steep Slope Profiles: There is no GVF curve for a steep slope that transitions from normal upstream depth to critical downstream depth. Such a transition requires rapidly-varying flow, such as a hydraulic jump.

3D Momentum and Physics terms

• The Advective Term: Represented as $u_k rac{ ext{∂}u_i}{ ext{∂}x_k}$. It describes how the change of velocity over space affects the momentum of a particle in the flow.

• Einstein Summation (x-direction): For the x-direction ($i=1$), the expansion is:   $u_1 rac{ ext{∂}u_1}{ ext{∂}x_1} + u_2 rac{ ext{∂}u_1}{ ext{∂}x_2} + u_3 rac{ ext{∂}u_1}{ ext{∂}x_3}$

• Other Momentum Terms:     - Pressure gradient.     - Unsteady term ($\frac{ ext{∂}u_i}{ ext{∂}t}$, representing acceleration at a fixed point).     - Gravity term.     - Stress (viscous) terms.

• Gravity Alignment ($g_i$): If x and y axes are perfectly horizontal, $g_1$ and $g_2$ are zero as gravity aligns with the z-axis. $g_1$ is only non-zero if the axis is tilted, such as in reach analysis where the x-axis follows the bottom slope.

• Viscous Force and Dissipation: Viscous force terms redistribute momentum by smoothing velocity gradients; they do not reduce (dissipate) total momentum ($u$). However, they do dissipate kinetic energy ($u^2$) because smoothing the velocity field reduces its variance.

• Particle Acceleration: The total acceleration of a particle is the sum of the unsteady term (acceleration at a fixed point) and the advective term (acceleration due to spatial velocity gradients).

1D Saint-Venant (S-V) Equations

• Alternative Name: The Dynamic Wave equation (distinguished from Kinematic and Diffusive waves).

• Derivation Forms: The S-V equations can be written in an integrated form (integrated over the cross-section) or an averaged form (where terms are divided by cross-sectional area $A$).

• Bottom-Slope vs. Piezometric Head Forms:     - Bottom-Slope Form: Includes $gAS_0$ and the gradient $gA rac{ ext{∂}D}{ ext{∂}x}$. Used when the x-axis is tilted with the bottom.     - Head Form: Uses the gradient $gA rac{ ext{∂}H}{ ext{∂}x}$, where $H$ is Piezometric head.     - Calculation of $S_0$: $S_0 = - rac{ ext{∂}z_b}{ ext{∂}x}$, where $z_b$ is bottom elevation.

• Model Selection for Large Networks: For a 100 km network with sharp slope changes, the Piezometric head form is preferred because the bottom-slope form is only applicable if the slope changes slowly.

• Mass Conservation Discrepancy:     - 3D Case: $\frac{ ext{∂}u}{ ext{∂}x} + rac{ ext{∂}v}{ ext{∂}y} + rac{ ext{∂}w}{ ext{∂}z} = 0$.     - 1D Case: $\frac{ ext{∂}A}{ ext{∂}t} + rac{ ext{∂}Q}{ ext{∂}x} = 0$.     - Explanation: The transition from no time derivative (3D) to a time derivative (1D) occurs because the river's free surface boundaries move. Integrating over the cross-section requires the "Kinematic Boundary Condition" (KBC) to describe surface motion over time.

• Turbulent Energy Dissipation: Represented by $gAS_f$. The most common model is the Chezy-Manning equation, which in SI units is $Q = rac{1}{n} A R_h^{2/3} S_f^{1/2}$. Inverted for friction slope, it is $S_f = rac{Q^2 n^2}{A^2 R_h^{4/3}}$.

Simplified Routing Models

• River Routing Definition: Models that estimate flows and stage heights without full river mechanics (momentum), often using approximations and calibration. Frequently used with hydrological runoff models.

• Kinematic Wave Equation (KWE):     - Key Limitation: Cannot represent backwater effects (where subcritical flow is constrained, e.g., by a bridge, and backs up).     - Usage Warning: Should not be used for floodwave amplitude or duration in most rivers.

• Diffusive Wave Equation (DWE): Adds a diffusion term with an adjustable coefficient to the KWE, allowing for calibration to data sets that include backwater effects.

• Muskingum Method: A key limitation is that it does not guarantee mass conservation.

• Mechanics vs. Approximation: Only the S-V equations include basic momentum and mass mechanics. KWE, DWE, and Muskingum are calibrated approximations. A good result from an approximation does not mean the underlying mechanics (like backwater) are unimportant; it may just be a result of the specific calibration data used.

Advanced Modeling Considerations

• Energy Dissipation and Roughness ($n$): Formally, 1D S-V equations allow only one roughness value at a cross-section. However, a "composite $n$" can be calculated to represent different roughnesses in the floodplain and the main channel.

• River Bends in 1D Models: Standard 1D S-V models fail to include bend effects because they use the hydrostatic approximation and cross-sectional averaging, losing the 3D helical flow physics.

• The Hydrostatic Approximation:     - Definition: Represents water pressure at any point solely as the weight of the water column above it.     - Neglected Factors: Neglects non-hydrostatic pressure (created by fluid acceleration/deceleration) and vertical acceleration. This requires neglecting the vertical momentum equation.     - Vertical Velocity: Hydrostatic models still include vertical velocity ($w$) to satisfy mass conservation. If horizontal momentum causes more water to enter a point than leave it, the water level must rise, creating upward vertical velocity.

• Helical Flow in Bends:     - Mechanics: Fast-flowing surface water moves toward the outside of the bend, piles up (creating a hydrostatic pressure gradient), and increases non-hydrostatic pressure at the wall. This forces surface water downward. The pressure gradient pushes slower bottom water toward the channel center. Particles follow a helical path downstream.     - Morphological Impacts: Shear on the outer bank causes erosion and bank collapse, leading to outward meandering. On the inside of the bend, slower flow leads to aggradation (sediment dropping out) and the formation of point bars, such as gravel collections.

• 1D vs. 2D River Equations:     - 1D superiority: Better for within-channel bankfull flows as they involve less numerical dissipation than 2D models which rely only on hydrostatic gradients for redirection.     - 2D superiority: Preferred for flood modeling (overbank flow), braided rivers, and ecological habitat analysis where local velocities and depths are critical.