Fluid Properties

Understanding fluid properties is fundamental to the study of fluid mechanics. Fluids, unlike solids, deform continuously under the action of shear stress. This deformation behavior is governed by their intrinsic properties. We will explore these properties in detail, as they are crucial for analyzing fluid behavior in various engineering applications, from pipe flow to open channel hydraulics.

Density

Density is defined as the mass per unit volume of a substance. For fluids, it's a critical property that influences buoyancy, pressure, and flow dynamics. It is typically denoted by the Greek letter rho ($\rho$).

The formula for density is:

$\rho = \frac{m}{V}$

Where:

• $\rho$ is the density (e.g., kg/m³ in SI units)
• $m$ is the mass (e.g., kg)
• $V$ is the volume (e.g., m³)

For liquids, density generally decreases slightly with an increase in temperature. For gases, density is highly dependent on both temperature and pressure. For instance, water at 4°C has a density of approximately 1000 kg/m³. At higher temperatures, this value decreases. Air at standard atmospheric conditions has a density of about 1.225 kg/m³.

Specific Weight (or Weight Density)

Specific weight, denoted by gamma ($\gamma$), is the weight per unit volume of a substance. It is related to density by the acceleration due to gravity ($g$).

The formula is:

$\gamma = \rho \times g$

Where:

• $\gamma$ is the specific weight (e.g., N/m³ in SI units)
• $\rho$ is the density
• $g$ is the acceleration due to gravity (approximately 9.81 m/s²)

The specific weight of water at standard conditions is approximately 9810 N/m³.

Specific Gravity

Specific gravity (SG), also known as relative density, is the ratio of the specific weight of a fluid to the specific weight of a reference substance, usually water at a standard temperature (e.g., 4°C). It is a dimensionless quantity.

The formula is:

$SG = \frac{\gamma_{fluid}}{\gamma_{water}} = \frac{\rho_{fluid}}{\rho_{water}}$

For example, mercury has a specific gravity of about 13.6. This means mercury is 13.6 times heavier than water. A value less than 1 indicates the fluid is lighter than water and will float on it.

Compressibility and Bulk Modulus

Compressibility describes how much the volume of a fluid changes when subjected to a change in pressure. Most liquids are considered incompressible for practical engineering purposes, meaning their volume changes very little with pressure variations. Gases, however, are highly compressible.

The coefficient of compressibility ($\beta$) is defined as the ratio of the volumetric strain to the change in pressure. The bulk modulus ($K$) is its inverse.

$K = -\frac{dV}{VdP}$

Where:

• $K$ is the bulk modulus (e.g., Pa or N/m²)
• $V$ is the volume
• $P$ is the pressure
• $dV$ and $dP$ represent infinitesimal changes in volume and pressure.

A higher bulk modulus indicates a less compressible fluid. For water, $K$ is very large (around 2.2 GPa), confirming its low compressibility.

Viscosity

Viscosity is a measure of a fluid's resistance to deformation or flow. It represents the internal friction within the fluid. Viscosity is often described in two forms: dynamic viscosity and kinematic viscosity.

Dynamic Viscosity (Absolute Viscosity)

Dynamic viscosity ($\mu$) is the resistance to shearing stresses. Consider two parallel plates, one stationary and the other moving with a velocity $v$, separated by a distance $y$. The fluid between them experiences a shear stress ($\tau$) proportional to the velocity gradient ($dv/dy$).

Newton's law of viscosity states:

$\tau = \mu \frac{dv}{dy}$

Where:

• $\tau$ is the shear stress (e.g., Pa or N/m²)
• $\mu$ is the dynamic viscosity (e.g., Pa·s or N·s/m² in SI units, or Poise in CGS)
• $dv/dy$ is the velocity gradient or rate of shear strain.

For Newtonian fluids, $\mu$ is constant. For non-Newtonian fluids, $\mu$ varies with the shear rate. Dynamic viscosity of liquids decreases with increasing temperature, while for gases, it increases with increasing temperature.

Kinematic Viscosity

Kinematic viscosity ($\nu$) is the ratio of dynamic viscosity to density. It represents the ratio of viscous forces to inertial forces.

The formula is:

$\nu = \frac{\mu}{\rho}$

Where:

• $\nu$ is the kinematic viscosity (e.g., m²/s in SI units, or Stokes in CGS)
• $\mu$ is the dynamic viscosity
• $\rho$ is the density

Kinematic viscosity is important in problems involving fluid motion where gravity is the only body force, such as the flow of a fluid under its own weight.

Surface Tension

Surface tension is a property of liquids that causes the free surface of a liquid to behave like a stretched elastic membrane. It arises from the cohesive forces between liquid molecules. Molecules at the surface have fewer neighbors than those in the bulk, resulting in a net inward force. This force tends to minimize the surface area.

It is defined as the force per unit length acting perpendicular to a line drawn on the surface, or as the energy per unit area required to create a new surface.

The formula is:

$\sigma = \frac{F}{L} = \frac{E}{A}$

Where:

• $\sigma$ is the surface tension (e.g., N/m)
• $F$ is the force
• $L$ is the length
• $E$ is the surface energy
• $A$ is the surface area

Surface tension is responsible for phenomena like the formation of droplets, capillary rise, and the ability of some insects to walk on water. It decreases with increasing temperature.

Vapor Pressure

Vapor pressure is the pressure exerted by the vapor of a liquid in thermodynamic equilibrium with its condensed phases (solid or liquid) at a given temperature in a closed system. When a liquid is placed in a closed container, some molecules escape from the liquid surface into the space above, forming vapor. Equilibrium is reached when the rate of evaporation equals the rate of condensation.

Vapor pressure increases with temperature. If the pressure above the liquid drops to the vapor pressure, the liquid will start to boil, even at low temperatures. This phenomenon is known as cavitation, which can cause significant damage in pumps and turbines.

Hydrostatics

Hydrostatics is the branch of fluid mechanics that deals with fluids at rest. In a fluid at rest, there are no relative motions between fluid layers, and therefore, no shear stresses exist. The only stress present is the normal pressure. This section covers the fundamental principles governing pressure in static fluids.

Pressure at a Point

Pressure is defined as force per unit area. In a fluid at rest, pressure at any point is the same in all directions (Pascal's Law). This is because if there were a pressure difference, it would cause motion, contradicting the condition of the fluid being at rest.

Mathematically, pressure ($P$) is given by:

$P = \frac{F}{A}$

Where $F$ is the force acting perpendicular to the area $A$. The SI unit for pressure is the Pascal (Pa), where 1 Pa = 1 N/m². Other common units include kPa, MPa, bar, and psi.

Variation of Pressure in a Static Fluid

Consider a small fluid element of weight $W$ and volume $dV$. In a static fluid, the pressure increases with depth. This is because the pressure at a certain level must support the weight of the fluid column above it.

Let's analyze the forces on a small vertical prism of fluid with cross-sectional area $dA$ and height $dh$. The pressure at the top surface is $P$, and at the bottom surface is $P + dP$. The weight of the fluid element is $dW = \rho g \, dV = \rho g \, dA \, dh$.

For equilibrium, the sum of vertical forces must be zero:

$(P + dP)dA - P dA - dW = 0$

$PdA + dPdA - P dA - \rho g \, dA \, dh = 0$

$dPdA = \rho g \, dA \, dh$

$dP = \rho g \, dh$

Integrating this equation from a reference point (e.g., surface, where $h=0$ and $P=P_0$) to a depth $h$ (where pressure is $P$):

$\int_{P_0}^{P} dP = \int_{0}^{h} \rho g \, dh$

$P - P_0 = \rho g h$

$P = P_0 + \rho g h$

This fundamental equation shows that pressure in a static fluid increases linearly with depth.

• $P$ is the absolute pressure at depth $h$.
• $P_0$ is the pressure at the surface (often atmospheric pressure).
• $\rho$ is the fluid density.
• $g$ is the acceleration due to gravity.
• $h$ is the depth from the surface.

The term $\rho g h$ is called the gauge pressure ($P_{gauge}$), which is the pressure relative to atmospheric pressure.

$P_{absolute} = P_{atmospheric} + P_{gauge}$

If the surface is exposed to vacuum ($P_0 = 0$), then $P = \rho g h$.

Hydrostatic Force on Surfaces

The force exerted by a static fluid on a submerged surface is perpendicular to the surface. The magnitude of this force depends on the pressure distribution, which varies with depth.

Horizontal Surfaces

For a horizontal surface submerged in a fluid, the pressure is constant across the entire surface because the depth is constant.

The hydrostatic force ($F_H$) on a horizontal surface is:

$F_H = P \times A$

Where $P$ is the pressure at the depth of the surface, and $A$ is the area of the surface.

Example: The force on the bottom of a rectangular tank filled with water. If the depth is 2m and the bottom area is 3 m², the pressure at the bottom is $P = \rho g h = 1000 \times 9.81 \times 2 = 19620$ Pa. The force is $F_H = 19620 \times 3 = 58860$ N.

Vertical Surfaces

For a vertical surface, the pressure varies with depth. The force calculation requires integrating the pressure over the area. The resultant hydrostatic force on a plane vertical surface submerged in a liquid is equal to the product of the pressure at the centroid of the area and the total area.

$F_V = P_c \times A$

Where:

• $F_V$ is the hydrostatic force on the vertical surface.
• $A$ is the area of the submerged surface.
• $P_c$ is the pressure at the centroid of the submerged area. $P_c = \rho g h_c$, where $h_c$ is the vertical distance from the liquid surface to the centroid of the area.

The point where this resultant force acts is called the center of pressure ($y_p$). For a plane vertical surface, the center of pressure is always below the centroid. The distance of the center of pressure from the free surface ($h_p$) is given by:

$h_p = h_c + \frac{I_c}{A h_c}$

Where $I_c$ is the moment of inertia of the submerged area about the horizontal axis passing through its centroid.

Example: A rectangular gate of width $b$ and height $h$, hinged at the top, submerged in water. The centroid is at $h/2$. The force is $F_V = (\rho g \frac{h}{2}) \times (b \times h) = \frac{1}{2} \rho g b h^2$. The moment of inertia of a rectangle about its centroid is $I_c = \frac{bh^3}{12}$. The center of pressure is $h_p = \frac{h}{2} + \frac{bh^3/12}{(bh)(h/2)} = \frac{h}{2} + \frac{h}{6} = \frac{2h}{3}$.

Curved Surfaces

For curved surfaces, the hydrostatic force is found by resolving the force into horizontal and vertical components.

• Horizontal Component ($F_h$): This is equal to the hydrostatic force on the vertical projection of the curved surface onto a vertical plane.
• Vertical Component ($F_v$): This is equal to the weight of the liquid column extending vertically from the curved surface up to the free surface (or to a horizontal plane if the surface is only partially submerged).

The resultant force is $F_R = \sqrt{F_h^2 + F_v^2}$, and its line of action passes through the intersection of the lines of action of $F_h$ and $F_v$.

Buoyancy and Flotation (Archimedes' Principle)

Archimedes' principle states that any body wholly or partially submerged in a fluid is buoyed up by a force equal to the weight of the fluid displaced by the body.

The buoyant force ($F_B$) is given by:

$F_B = \text{Weight of displaced fluid} = \rho_{fluid} \times g \times V_{submerged}$

Where $V_{submerged}$ is the volume of the submerged part of the body.

For a floating body, the buoyant force equals the weight of the body ($W_{body}$).

$W_{body} = F_B$

A body will float if its average density is less than the density of the fluid. It will sink if its average density is greater than the fluid density. It will remain in equilibrium at any level if its average density equals the fluid density.

The point through which the buoyant force acts is called the center of buoyancy (which is the centroid of the displaced volume).

Stability of Floating Bodies

The stability of a submerged or floating body depends on the relative positions of the center of gravity ($G$) and the center of buoyancy ($B$).

• Submerged Bodies: Stable if $G$ is below $B$. Unstable if $G$ is above $B$. Neutral if $G$ coincides with $B$.
• Floating Bodies: Stability depends on the metacenter ($M$). The metacenter is the point where the vertical line through the new center of buoyancy intersects the original vertical axis of the body when it is tilted slightly. The body is stable if the metacenter $M$ is above the center of gravity $G$. The distance $GM$ is called the metacentric height.

$GM = BM - BG$

Where $BM = \frac{I_c}{V_{submerged}}$ ($I_c$ is the moment of inertia of the water-plane area about the longitudinal axis).

Bernoulli's Theorem

Bernoulli's theorem is a fundamental principle in fluid dynamics that describes the relationship between pressure, velocity, and elevation in a moving fluid. It is derived from the conservation of energy for an inviscid, incompressible, steady flow.

Assumptions for Bernoulli's Theorem

The theorem is based on several key assumptions:

• The fluid is ideal (inviscid, i.e., zero viscosity).
• The flow is steady (velocity, pressure, and density at any point do not change with time).
• The flow is incompressible (density is constant).
• The fluid is flowing along a streamline.
• There are no energy sources or sinks within the fluid.
• The flow is along a horizontal plane or the effect of gravity is accounted for.

The Equation

Bernoulli's equation states that the sum of the pressure energy, kinetic energy, and potential energy per unit volume of a fluid flowing along a streamline is constant.

The equation is typically written as:

$P + \frac{1}{2} \rho v^2 + \rho g h = \text{Constant}$

Where:

• $P$ is the static pressure (e.g., N/m² or Pa). This is the pressure that would be measured by a gauge moving with the fluid.
• $\frac{1}{2} \rho v^2$ is the dynamic pressure. It represents the kinetic energy per unit volume of the fluid.
• $\rho g h$ is the hydrostatic pressure or potential energy per unit volume. $h$ is the elevation above a datum.
• $\rho$ is the fluid density.
• $v$ is the fluid velocity.
• $g$ is the acceleration due to gravity.

Forms of Bernoulli's Equation

Bernoulli's equation can also be expressed in terms of head (energy per unit weight):

$\frac{P}{\rho g} + \frac{v^2}{2g} + h = H = \text{Constant}$

Where:

• $\frac{P}{\rho g}$ is the pressure head.
• $\frac{v^2}{2g}$ is the velocity head.
• $h$ is the elevation head.
• $H$ is the total head.

This form is often used in practical applications, especially in open channel flow and pipe flow analysis, as it relates different forms of energy to lengths (heads).

Applications of Bernoulli's Theorem

Bernoulli's theorem has numerous practical applications:

• Venturi Meter: Used to measure the flow rate of a fluid. As the fluid passes through a constriction (throat), its velocity increases, and the pressure decreases, allowing the flow rate to be calculated from the pressure difference.
• Orifice Meter: Similar to Venturi meter but uses a sharp-edged orifice.
• Pitot Tube: Used to measure fluid velocity. It measures the stagnation pressure (where velocity is zero) and the static pressure. The difference allows velocity calculation.
• Atomizers and Sprayers: The high velocity of air blown across the top of a tube causes a drop in pressure, allowing the liquid to be drawn up and atomized.
• Aerodynamic Lift: The shape of an airplane wing causes air to flow faster over the top surface than the bottom. This creates lower pressure on top, resulting in an upward lift force.
• Flow in pipes: Explains pressure changes as pipe diameter varies.

Limitations and Modifications

The ideal Bernoulli equation does not account for energy losses due to friction (viscosity) or turbulence. In real-world scenarios, these losses are significant. The extended Bernoulli equation, often called the energy equation, includes a term for head loss ($h_L$) due to friction and other factors:

$\frac{P_1}{\rho g} + \frac{v_1^2}{2g} + h_1 = \frac{P_2}{\rho g} + \frac{v_2^2}{2g} + h_2 + h_L$

Where subscript 1 refers to the inlet and subscript 2 to the outlet.

Flow Through Pipes

The study of fluid flow through pipes is crucial in many engineering applications, including water supply systems, oil pipelines, and HVAC systems. We will focus on the characteristics of pipe flow, energy losses, and methods to calculate flow rates and pressure drops.

Types of Pipe Flow

Pipe flow can be classified into two main types based on the flow regime:

• Laminar Flow: Occurs at low velocities. Fluid particles move in smooth, parallel layers (laminae) with no significant mixing between layers. It is characterized by a low Reynolds number.
• Turbulent Flow: Occurs at higher velocities. Fluid particles move in a chaotic, irregular manner with significant mixing and swirling. It is characterized by a high Reynolds number.

The transition between laminar and turbulent flow is determined by the Reynolds number ($Re$).

$Re = \frac{\rho v D}{\mu} = \frac{v D}{\nu}$

Where:

• $\rho$ is the fluid density.
• $v$ is the average flow velocity.
• $D$ is the pipe diameter.
• $\mu$ is the dynamic viscosity.
• $\nu$ is the kinematic viscosity.

Generally:

• $Re < 2000$: Laminar flow
• $2000 < Re < 4000$: Transitional flow
• $Re > 4000$: Turbulent flow

For laminar flow in a circular pipe, the velocity profile is parabolic. For turbulent flow, the velocity profile is flatter and more uniform across the central part of the pipe, with a steep velocity gradient near the pipe walls.

Energy Losses in Pipes

When fluid flows through a pipe, energy is lost due to friction between the fluid and the pipe wall, and due to turbulence within the fluid. These losses manifest as a drop in pressure or head along the direction of flow.

Major Losses ($h_f$)

These are losses due to friction along the length of the pipe. They are the dominant losses in long, straight pipes. The Darcy-Weisbach equation is used to calculate major losses:

$h_f = f \frac{L}{D} \frac{v^2}{2g}$

Where:

• $h_f$ is the head loss due to friction.
• $f$ is the Darcy friction factor (dimensionless).
• $L$ is the length of the pipe.
• $D$ is the pipe diameter.
• $v$ is the average flow velocity.
• $g$ is the acceleration due to gravity.

The friction factor $f$ depends on the Reynolds number ($Re$) and the relative roughness of the pipe ($\epsilon/D$), where $\epsilon$ is the absolute roughness of the pipe material. For laminar flow ($Re < 2000$), $f = 64/Re$. For turbulent flow, $f$ is determined using the Moody chart or empirical equations like the Colebrook-White equation.

Minor Losses ($h_m$)

These are losses due to fittings, valves, bends, entrances, exits, and other changes in the pipe geometry. They are usually expressed in terms of the velocity head using a loss coefficient ($K_L$):

$h_m = K_L \frac{v^2}{2g}$

Common examples include:

• Entrance loss: $K_L \approx 0.5$ for a sharp-edged entrance.
• Exit loss: $K_L = 1.0$ (the kinetic energy is dissipated).
• Bend loss: $K_L$ depends on the radius of the bend and the pipe diameter.
• Valve loss: $K_L$ varies significantly with the type and opening of the valve.

Calculating Flow Rate and Pressure Drop

The energy equation (extended Bernoulli) is used to relate the pressures, velocities, and elevations between two points in a pipe system, accounting for all head losses:

$\frac{P_1}{\rho g} + \frac{v_1^2}{2g} + h_1 = \frac{P_2}{\rho g} + \frac{v_2^2}{2g} + h_2 + h_f + \sum h_m$

If the pipe diameter is constant ($D_1 = D_2$), then $v_1 = v_2$, and the equation simplifies to:

$\frac{P_1}{\rho g} + h_1 = \frac{P_2}{\rho g} + h_2 + h_{total\_loss}$

Where $h_{total\_loss} = h_f + \sum h_m$.

This equation can be rearranged to solve for pressure drop ($P_1 - P_2$) or flow rate ($v$, which is related to $Re$ and $f$).

Pipes in Series and Parallel

Pipes in Series: Connected end-to-end. The total head loss is the sum of the head losses in each pipe. The flow rate is the same through all pipes.

$h_{L, total} = h_{L,1} + h_{L,2} + ...$

Pipes in Parallel: Connected between the same two junctions. The head loss is the same for each pipe. The total flow rate is the sum of the flow rates through each parallel pipe.

$Q_{total} = Q_1 + Q_2 + ...$ $h_{L,1} = h_{L,2} = ...$

Open Channels

Open channel flow refers to the flow of liquid in a conduit that has a free surface exposed to the atmosphere. Examples include rivers, canals, sewers, and partially filled pipes. Unlike pipe flow, the flow depth and velocity can vary significantly, and the pressure at the free surface is atmospheric.

Characteristics of Open Channel Flow

• Free Surface: The upper boundary is open to the atmosphere, meaning the pressure at the free surface is constant (usually atmospheric).
• Varying Flow Depth: The depth of the liquid can change along the length of the channel.
• Velocity Distribution: Velocity is generally maximum near the free surface and at the center, and minimum near the boundaries (bed and sides) due to friction.

Key Parameters

• Wetted Perimeter ($P$): The length of the channel boundary in contact with the fluid.
• Cross-sectional Area ($A$): The area of the fluid flow perpendicular to the direction of flow.
• Hydraulic Radius ($R$): Defined as the ratio of the cross-sectional area to the wetted perimeter ($R = A/P$). It is a measure of the efficiency of the channel's cross-section for carrying flow. A larger hydraulic radius generally means less resistance to flow for a given area.
• Flow Rate ($Q$): $Q = A \times v$, where $v$ is the average velocity.
• Flow Depth ($y$): The vertical distance from the channel bed to the free surface.

Types of Open Channel Flow

• Steady vs. Unsteady: Steady flow has constant flow properties (depth, velocity) with time at any given point. Unsteady flow changes with time.
• Uniform vs. Non-uniform: Uniform flow has constant flow depth and velocity along the length of the channel. Non-uniform flow has varying depth and velocity. Non-uniform flow can be further classified as gradually varied flow (GVF) or rapidly varied flow (RVF).
• Laminar vs. Turbulent: Similar to pipe flow, determined by Reynolds number, but the definition uses hydraulic radius instead of diameter. $Re = \frac{v P}{\mu} = \frac{v (A/R)}{\mu}$. Generally, $Re > 500$ indicates turbulent flow. Most open channel flows are turbulent.

Uniform Flow - Manning's Equation

Uniform flow occurs when the water surface, the channel bed, and the velocity profile are all parallel. The depth and velocity remain constant along the channel length. Manning's equation is the most widely used formula for calculating the average velocity in uniform flow:

$v = C \frac{R^{2/3} S^{1/2}}{n}$ (SI units)

Where:

• $v$ is the average velocity.
• $C$ is a conversion factor (often omitted in SI units, or $C=1$).
• $R$ is the hydraulic radius ($A/P$).
• $S$ is the slope of the channel bed (and water surface for uniform flow).
• $n$ is Manning's roughness coefficient (dimensionless, depends on channel material and condition).

The flow rate $Q$ is then calculated as $Q = A \times v$.

Specific Energy and Critical Depth

Specific energy ($E$) is the energy per unit weight of fluid relative to the channel bed. For a given cross-sectional area $A$, it is defined as:

$E = \frac{v^2}{2g} + y$

Where $y$ is the flow depth.

For a given specific energy $E$, there can be two possible flow depths: a subcritical depth (where $y < E$) and a supercritical depth (where $y > E$).

Critical Depth ($y_c$): This is the depth at which the specific energy is minimum for a given flow rate ($Q$). At critical depth, the Froude number ($Fr$) is equal to 1.

$Fr = \frac{v}{\sqrt{gy}}$

For a rectangular channel, the minimum specific energy occurs when $y = y_c$, and $y_c = (\frac{q^2}{g})^{1/3}$, where $q = Q/b$ is the flow per unit width. Also, at critical depth, $v_c = \sqrt{g y_c}$.

• Subcritical Flow: $y > y_c$, $Fr < 1$, $v < \sqrt{gy}$. Flow is slow, deep, and tranquil. Disturbances can propagate upstream.
• Supercritical Flow: $y < y_c$, $Fr > 1$, $v > \sqrt{gy}$. Flow is fast, shallow, and shooting. Disturbances cannot propagate upstream.
• Critical Flow: $y = y_c$, $Fr = 1$. Represents the transition between subcritical and supercritical flow.

Hydraulic Jump

A hydraulic jump is a phenomenon where supercritical flow abruptly transitions to subcritical flow. This transition is characterized by a rapid rise in water depth, a significant decrease in velocity, and a substantial dissipation of energy (due to turbulence). Hydraulic jumps are often intentionally created in spillways and stilling basins to dissipate excess energy and prevent erosion.

For a rectangular channel, the depths before ($y_1$, supercritical) and after ($y_2$, subcritical) the jump are related by the Bélanger's equation:

$\frac{y_2}{y_1} = \frac{1}{2} \left( \sqrt{1 + 8 Fr_1^2} - 1 \right)$

Where $Fr_1$ is the Froude number in the supercritical flow just before the jump.

Weirs

Weirs are overflow structures, typically built across open channels or rivers, used to measure or control the flow rate. They are essentially barriers over which the liquid flows. The flow characteristics over a weir depend on its shape and the head of liquid above its crest.

Types of Weirs

Weirs are broadly classified based on their crest shape and whether they are sharp-crested or broad-crested.

• Sharp-Crested Weirs: Have a thin, sharp edge over which the water flows. They are commonly used for accurate flow measurement in laboratories and smaller streams. Examples include:
  • Rectangular Weir (V-notch is a type of sharp-crested weir): Water flows over a horizontal crest.
  • Triangular Weir (V-Notch Weir): The notch is triangular. This type is particularly useful for measuring low flow rates as the head increases significantly for small changes in flow.
  • Trapezoidal Weir: Combines features of rectangular and V-notch weirs.
• Broad-Crested Weirs: Have a wide, flat crest. The flow over these weirs is often controlled by a critical depth condition.

Flow Over Sharp-Crested Weirs

The flow over a sharp-crested weir is generally considered to be supercritical, with the nappe (the sheet of water flowing over the weir) springing clear of the crest. The flow rate ($Q$) is primarily a function of the head ($H$) of liquid above the weir crest.

Rectangular Weir (Suppressed)

For a rectangular weir of length $L$ and head $H$:

$Q = C_d \frac{2}{3} \sqrt{2g} L H^{3/2}$

Where:

• $Q$ is the flow rate.
• $C_d$ is the coefficient of discharge, typically between 0.6 and 0.65 for sharp-crested rectangular weirs.
• $g$ is the acceleration due to gravity.
• $L$ is the length of the weir crest.
• $H$ is the head of water above the weir crest.

Note: $H$ should be measured upstream of the weir where the velocity of approach is negligible. If the velocity of approach is significant, a correction term is added.

Triangular Weir (V-Notch Weir)

For a V-notch weir with an angle $\theta$:

$Q = C_d \frac{8}{15} \sqrt{2g} \tan(\frac{\theta}{2}) H^{5/2}$

Commonly used V-notch angles are 90° ($\theta = 90^\circ$, $\tan(45^\circ)=1$) and 45° ($\theta = 45^\circ$, $\tan(22.5^\circ) \approx 0.414$).

For a 90° V-notch weir, the equation simplifies to:

$Q \approx 1.4 H^{5/2}$ (using typical $C_d$ and $g$ values)

Flow Over Broad-Crested Weirs

On a broad-crested weir, the flow often becomes critical over the crest. The discharge is primarily dependent on the weir width ($L$) and the head ($H$) upstream. A common empirical formula is:

$Q = C L H^{3/2}$

Where $C$ is an empirical coefficient that depends on the geometry of the weir and the fluid properties.

Flumes

Flumes are constructed channels, often trapezoidal or rectangular, designed to measure flow rates in open channels. Unlike weirs which obstruct flow significantly, flumes are designed to minimize flow obstruction while still allowing for accurate measurement based on the relationship between flow depth and flow rate within the flume.

Types of Flumes

Flumes are typically designed such that the flow within a specific section (the throat) is critical, meaning the flow depth is uniquely related to the flow rate.

• Parshall Flume: Perhaps the most common type. It has a converging section, a throat of constant width, and a diverging section. The flow is critical in the throat section. It is relatively insensitive to upstream conditions and can handle a range of flow rates and suspended solids.
• Trapezoidal Flumes: Simple trapezoidal channels, often with a modified entrance or throat section to establish critical flow.
• Rectangular Flumes: Used in situations where the channel is already rectangular or where specific flow conditions need to be met.

Operation Principle

The key principle behind flume flow measurement is establishing critical flow conditions within a constricted section (the throat). In critical flow, the Froude number is 1, and for a given geometry and flow rate, there is a unique flow depth. By measuring the depth of the flow ($H$) at a specific point upstream of or within the throat, the flow rate ($Q$) can be determined using established empirical formulas.

Flow Rate Equations (Example: Parshall Flume)

The flow rate through a Parshall flume is calculated using empirical formulas that depend on the width of the throat ($W$) and the measured head ($H$). For free flow conditions (where the downstream water level does not affect the flow in the throat), the formulas are generally of the form:

$Q = C W^a H^b$

Where $C$, $a$, and $b$ are empirical constants that vary depending on the throat width $W$.

For example, for a throat width $W$ between 1 and 8 feet, the formula is approximately:

$Q = 4.0 W H^{1.62}$ (for $W$ in feet, $Q$ in cubic feet per second)

For submerged flow conditions (where the downstream water level backs up into the throat), a correction factor based on the ratio of downstream head to upstream head is applied.

Advantages of Flumes

• Minimal head loss compared to weirs.
• Can handle larger flow rates and suspended solids.
• Less prone to silting or erosion compared to weirs.
• Relatively easy to install and operate.

Spillways

Spillways are structures built on dams or levees to provide a safe passage for excess water to flow from the reservoir or upstream side to the downstream side. Their primary purpose is to prevent the dam from being overtopped, which could lead to catastrophic failure. Spillways are designed to discharge large volumes of water safely and efficiently, dissipating the energy of the flowing water to prevent erosion of the downstream channel.

Types of Spillways

Spillways are classified based on their design and function:

• Gated Spillways: These have controllable gates (e.g., radial gates, tainter gates) on the crest. Gates allow for regulation of the discharge, enabling the reservoir level to be controlled precisely. They can discharge water only when needed.
• Ungated Spillways (e.g., Ogee Spillways): These have a fixed crest shape, typically an "ogee" curve, designed to match the profile of a free-flowing nappe under design head conditions. Water flows over them continuously when the reservoir level exceeds the crest elevation. They are simpler and more reliable as they have no moving parts.
• Chute Spillways: A common type where water flows down a sloped channel (chute) from the crest to the downstream river.
• Side Channel Spillways: The channel carrying water away from the crest is located parallel to the upstream side of the dam.
• Siphon Spillways: Utilize the principle of siphoning to discharge water. They can discharge large amounts of water at low head and are self-regulating.
• Shaft Spillways: Water drops vertically down a shaft.

Ogee Spillway Profile

The Ogee (or overflow) spillway crest is designed to approximate the shape of the underside of a nappe flowing freely over a sharp-crested weir. This shape ensures that the pressure along the spillway surface is close to atmospheric, minimizing suction or excessive pressure. The design aims to match the ideal trajectory of the water flow.

The discharge ($Q$) over an Ogee spillway is typically calculated using a weir-like formula:

$Q = C L H^{3/2}$

Where:

• $C$ is the coefficient of discharge, which depends on the design head, spillway geometry, and the presence of piers or gates. It is usually determined from model tests or empirical data.
• $L$ is the effective length of the spillway crest.
• $H$ is the design head (the difference between the upstream water level and the crest elevation).

The coefficient $C$ is typically higher for Ogee spillways (around 2.2 to 2.6 in SI units) than for simple sharp-crested weirs because the curved profile reduces the contraction of the nappe and minimizes energy losses.

Energy Dissipation

The high velocity and kinetic energy of water flowing down a spillway can cause significant erosion and damage to the downstream channel. Therefore, energy dissipation structures are essential. Common methods include:

• Stilling Basins: Concrete-lined areas downstream of the spillway designed to create turbulence and hydraulic jumps, converting kinetic energy into heat and sound.
• Sills and Baffles: Structures placed within the stilling basin to enhance turbulence and energy dissipation.
• Bucket Spillways (e.g., Flip Bucket): The water trajectory is "flipped" into the downstream river or a plunge pool, using the impact with the water body to dissipate energy.

Stilling Basins

Stilling basins are crucial components of spillway design. They are designed to ensure that the hydraulic jump, which dissipates energy, occurs within the basin itself, rather than eroding the downstream riverbed. The length and design of the stilling basin depend on the Froude number of the flow entering the basin and the desired downstream conditions.