3.3 Hydro-kinematics and Hydro-dynamics

3.3 Hydro-kinematics and Hydro-dynamics

Introduction to Flow Analysis

• Hydro-kinematics describes fluid motion without considering the forces that cause it, focusing on displacement, velocity, and acceleration.

• Hydro-dynamics incorporates the influence of forces to analyze the causes of fluid motion.

• Together, they form the core of analytical fluid mechanics, providing the fundamental equations to predict pressure, velocity, and force distributions in flowing systems.

• This section details the systematic classification of flows and the derivation and application of the core conservation laws: mass, momentum, and energy.

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1. Classification of Fluid Flow

• Understanding the nature of a flow is the first critical step in selecting the appropriate analytical method. Flows are categorized based on key characteristics.

1.1 Based on Variation with Time

• Steady Flow:

  • Definition: Flow parameters (velocity, pressure, density) at any point do not change with time.

    $\boldsymbol{\frac{\partial V}{\partial t} = 0, \quad \frac{\partial p}{\partial t} = 0, \quad \frac{\partial \rho}{\partial t} = 0}$

• Unsteady Flow:

  • Definition: Flow parameters at a point vary with time.

    $\boldsymbol{\frac{\partial V}{\partial t} \neq 0}$

• Example: Flow from a tank under a constant head is steady. Flow from a tank with a falling head is unsteady.

1.2 Based on Spatial Variation

• Uniform Flow:

  • Definition: The velocity vector (magnitude and direction) is identical at every point in the flow field at a given instant.

    $\boldsymbol{\frac{\partial V}{\partial s} = 0}$

• Non-uniform Flow:

  • Definition: Velocity changes from point to point.

    $\boldsymbol{\frac{\partial V}{\partial s} \neq 0}$

• Example: Flow in a constant-diameter pipe is uniform. Flow in a converging nozzle is non-uniform.

1.3 Based on Viscosity

• Ideal or Inviscid Flow:

  • Definition: Assumes zero viscosity $\boldsymbol{(\mu = 0)}$. No shear stresses exist.

  • Application: Useful for analyzing flow far from solid boundaries (e.g., potential flow theory).

• Real or Viscous Flow:

  • Definition: Accounts for fluid viscosity, leading to shear stresses, boundary layer formation, and energy dissipation.

  • Application: Required for accurate analysis of drag, pipe flow, and any flow near walls.

1.4 Based on Compressibility

• Incompressible Flow:

  • Definition: Density remains constant $\boldsymbol{(\rho = \mathrm{constant})}$.

  • Criteria: Mach number $\boldsymbol{M < 0.3}$.

  • Example: Most liquid flows and low-speed gas flows ($\boldsymbol{V < 100 \ \mathrm{m/s}}$ in air).

• Compressible Flow:

  • Definition: Density changes significantly during flow $\boldsymbol{(\rho \neq \mathrm{constant})}$.

  • Criteria: Mach number $\boldsymbol{M > 0.3}$.

  • Example: High-speed gas flows in nozzles, turbines, and around aircraft.

1.5 Based on Flow Pattern (Flow Regime)

• Laminar Flow:

  • Characteristics: Smooth, orderly motion in parallel layers or streamlines. No macroscopic mixing between layers.

  • Governing Parameter: Reynolds Number $\boldsymbol{Re = \frac{\rho V D}{\mu}}$.

  • Regime: Pipe flow: $\boldsymbol{Re < 2000}$.

• Turbulent Flow:

  • Characteristics: Chaotic, irregular motion with intense microscopic mixing (eddies). High momentum and heat transfer rates.

  • Regime: Pipe flow: $\boldsymbol{Re > 4000}$.

• Transitional Flow:

  • Characteristics: Intermittent, unstable flow switching between laminar and turbulent patterns.

  • Regime: Pipe flow: $\boldsymbol{2000 < Re < 4000}$.

1.6 Based on Dimensionality

• One-Dimensional (1D) Flow:

  • Definition: Flow parameters vary only along one spatial coordinate (e.g., along a pipe's centerline). Area-averaged values are used.

  • Example: Pipe flow analysis using average velocity.

• Two-Dimensional (2D) Flow:

  • Definition: Flow parameters vary in two directions (e.g., plane flow, axisymmetric flow).

  • Example: Flow over a long airfoil (plane), flow in a round jet (axisymmetric).

• Three-Dimensional (3D) Flow:

  • Definition: Flow parameters vary in all three spatial directions. Most general and complex.

  • Example: Flow around a bluff body like a car or building.

1.7 Based on Pathlines

• Rotational Flow:

  • Definition: Fluid elements rotate about their own axes as they move. Vorticity is non-zero $\boldsymbol{(\vec{\omega} \neq 0)}$.

  • Example: Flow inside a rotating tank, viscous flows near boundaries.

• Irrotational Flow:

  • Definition: Fluid elements translate and deform but do not rotate. Vorticity is zero $\boldsymbol{(\vec{\omega} = 0)}$.

  • Example: Ideal flow outside the boundary layer, potential flow approximations.

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2. Conservation of Mass: Continuity Equation and Applications

• The principle of conservation of mass states that mass can neither be created nor destroyed. For a fluid system, this leads to the Continuity Equation.

2.1 General Form (Integral Form for a Control Volume)

• Statement: The net mass flow rate out of a fixed control volume equals the time rate of decrease of mass inside it.

  $\boldsymbol{\frac{\partial}{\partial t} \int_{CV} \rho \ d\forall + \int_{CS} \rho \vec{V} \cdot d\vec{A} = 0}$

  where, $\boldsymbol{CV}$ is the control volume, $\boldsymbol{CS}$ is the control surface, and $\boldsymbol{d\vec{A}}$ is the area vector pointing outward.

2.2 Simplified Forms for Common Applications

• Steady Flow: Accumulation term is zero.

  $\boldsymbol{\int_{CS} \rho \vec{V} \cdot d\vec{A} = 0}$

• For a Streamtube with Inlet (1) and Outlet (2):

  $\boldsymbol{\dot{m}_1 = \dot{m}_2}$

  $\boldsymbol{\rho_1 A_1 V_1 = \rho_2 A_2 V_2}$

  where, $\boldsymbol{\dot{m}}$ is the mass flow rate.

• For Incompressible Flow ($\boldsymbol{\rho = \mathrm{constant}}$): Density cancels.

  $\boldsymbol{Q_1 = Q_2}$

  $\boldsymbol{A_1 V_1 = A_2 V_2}$

  where, $\boldsymbol{Q}$ is the volumetric flow rate.

2.3 Differential Form (For a Fluid Element)

• Represents conservation of mass at a point in the flow field.

• Cartesian Coordinates (x, y, z):

  $\boldsymbol{\frac{\partial \rho}{\partial t} + \frac{\partial (\rho u)}{\partial x} + \frac{\partial (\rho v)}{\partial y} + \frac{\partial (\rho w)}{\partial z} = 0}$

  where, $\boldsymbol{u, v, w}$ are velocity components.

• For Steady, Incompressible Flow: Simplifies to the divergence-free condition.

  $\boldsymbol{\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0}$

  or $\boldsymbol{\nabla \cdot \vec{V} = 0}$

2.4 Practical Applications

• Pipe Flow: Relating velocity changes to cross-sectional area changes. $\boldsymbol{V_2 = V_1 (A_1 / A_2)}$.

• Branching Networks: Sum of mass flows into a junction equals sum of mass flows out.

• Tank Filling/Emptying: Unsteady flow application: $\boldsymbol{A_{tank} \frac{dh}{dt} = Q_{in} - Q_{out}}$.

• Venturi Meter & Nozzles: Basis for flow measurement devices.

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3. Conservation of Momentum: Equations and Applications

• Newton's second law applied to a fluid system: The time rate of change of momentum of a fluid system equals the sum of external forces acting on it.

3.1 Linear Momentum Equation (Integral Form)

• General Form for a Control Volume:

  $\boldsymbol{\sum \vec{F} = \frac{\partial}{\partial t} \int_{CV} \vec{V} \rho \ d\forall + \int_{CS} \vec{V} \rho (\vec{V} \cdot d\vec{A})}$

  where, $\boldsymbol{\sum \vec{F}}$ is the sum of all external forces (body + surface).

• Forces Include:

  • Body Forces: $\boldsymbol{\vec{F}_b = \int_{CV} \vec{g} \rho \ d\forall}$ (e.g., gravity).

  • Surface Forces: Pressure and shear forces acting on the control surface.

• Steady Flow Simplification:

  $\boldsymbol{\sum \vec{F} = \int_{CS} \vec{V} \rho (\vec{V} \cdot d\vec{A})}$

3.2 Applications of the Linear Momentum Equation

• Force on a Pipe Bend or Nozzle:

  • Used to calculate anchoring forces required to hold a bend or reducer in place.

  • Procedure: Define CV enclosing the bend. Calculate pressure forces, then solve momentum equation for reaction force components $\boldsymbol{R_x, R_y}$.

• Jet Propulsion & Rocket Thrust:

  • Thrust force is generated due to the change in momentum of the exhaust fluid.

    $\boldsymbol{T = \dot{m} V_{jet}}$ (for a stationary rocket with atmospheric pressure at exit).

• Force on a Stationary or Moving Vane:

  • Determines the force exerted by a fluid jet impinging on a flat or curved plate.

• Hydraulic Jump Analysis:

  • Applies momentum conservation across a sudden transition from supercritical to subcritical flow in an open channel.

3.3 Angular Momentum Equation

• Principle: The time rate of change of angular momentum equals the sum of external torques.

• Integral Form for a CV:

  $\boldsymbol{\sum \vec{T} = \frac{\partial}{\partial t} \int_{CV} (\vec{r} \times \vec{V}) \rho \ d\forall + \int_{CS} (\vec{r} \times \vec{V}) \rho (\vec{V} \cdot d\vec{A})}$

• Primary Application: Analysis of turbo-machinery (pumps, turbines, fans). The torque on the rotor is related to the change in the tangential component of fluid velocity.

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4. Bernoulli’s Equation and Applications

• A specialized form of the energy equation derived from Euler's equation (momentum equation for inviscid flow) along a streamline.

4.1 Derivation and Assumptions

• Derived from: Newton's second law along a streamline for steady, inviscid, incompressible flow.

• Core Assumptions:

  1. Steady flow $\boldsymbol{(\partial / \partial t = 0)}$.

  2. Incompressible flow $\boldsymbol{(\rho = \mathrm{constant})}$.

  3. Inviscid flow $\boldsymbol{(\mu = 0)}$, no shear stresses.

  4. Flow along a single streamline.

  5. No shaft work or heat transfer between points 1 and 2.

• Caution: Bernoulli's equation is NOT a universal energy equation. It represents the mechanical energy balance under restrictive conditions.

4.2 The Bernoulli Equation

• Form 1 (Energy per Unit Mass):

  $\boldsymbol{\frac{p}{\rho} + \frac{1}{2} V^2 + g z = \mathrm{constant}}$

• Form 2 (Energy per Unit Weight - "Head" Form): Most common in engineering.

  $\boldsymbol{\frac{p}{\gamma} + \frac{V^2}{2g} + z = H = \mathrm{constant}}$

  where,

  • $\boldsymbol{\frac{p}{\gamma}}$ is the Pressure Head $\boldsymbol{[\mathrm{L}]}$.

  • $\boldsymbol{\frac{V^2}{2g}}$ is the Velocity Head $\boldsymbol{[\mathrm{L}]}$.

  • $\boldsymbol{z}$ is the Elevation Head (datum head) $\boldsymbol{[\mathrm{L}]}$.

  • $\boldsymbol{H}$ is the Total Head $\boldsymbol{[\mathrm{L}]}$.

• Physical Interpretation: Along a streamline in an ideal flow, the sum of pressure energy, kinetic energy, and potential energy per unit weight remains constant. Energy can be converted between these forms, but the total is conserved.

4.3 Extended Bernoulli Equation (For Real Viscous Flows)

• Accounts for energy losses due to friction and minor losses, and energy addition (e.g., by a pump) or extraction (e.g., by a turbine).

• Between two points (1) and (2) along a streamline:

  $\boldsymbol{\frac{p_1}{\gamma} + \frac{V_1^2}{2g} + z_1 + h_p = \frac{p_2}{\gamma} + \frac{V_2^2}{2g} + z_2 + h_t + h_L}$

  where,

  • $\boldsymbol{h_p}$ is the pump head (added energy) $\boldsymbol{[\mathrm{L}]}$.

  • $\boldsymbol{h_t}$ is the turbine head (extracted energy) $\boldsymbol{[\mathrm{L}]}$.

  • $\boldsymbol{h_L}$ is the head loss due to friction and fittings $\boldsymbol{[\mathrm{L}]}$.

4.4 Key Applications of Bernoulli's Equation

• Venturi Meter: Measures flow rate by relating pressure drop to velocity increase in a constriction.

• Pitot Tube: Measures flow velocity by converting kinetic energy to pressure (stagnation pressure).

• Siphon: Determines flow rate and pressure in a tube that carries fluid over an obstacle.

• Orifice & Nozzle Flow: Calculates jet velocity and discharge from an opening.

• Flow Over a Hump/Constriction in an Open Channel: Predicts water surface profile changes.

• Aircraft Lift (Qualitative): Explains higher velocity over the wing upper surface leading to lower pressure (lift generation).

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5. Flow Measurement

• Devices that utilize the principles of continuity, momentum, and Bernoulli's equation to quantify flow rate or velocity.

5.1 Pressure-Based Measurement (Obstruction Meters)

• Measure differential pressure created by a constriction in the flow.

• General Discharge Equation (for incompressible flow):

  $\boldsymbol{Q = C_d A_t \sqrt{\frac{2 (p_1 - p_2) / \rho}{1 - (A_t / A_1)^2}}}$

  where, $\boldsymbol{C_d}$ is the discharge coefficient accounting for real-fluid effects.

5.1.1 Venturi Meter

• Design: A smooth, gradual contraction to a throat, followed by a gradual expansion.

• Characteristics: High accuracy, low permanent head loss $\boldsymbol{(h_L \sim 10\%)}$. $\boldsymbol{C_d \approx 0.98}$.

• Applications: Clean fluid flow in pipes (water, oil, gas).

5.1.2 Orifice Meter

• Design: A thin, sharp-edged plate with a hole, placed perpendicular to the flow.

• Characteristics: Inexpensive, simple, but high permanent head loss $\boldsymbol{(h_L \sim 60-70\%)}$. $\boldsymbol{C_d \approx 0.61}$ (varies with Reynolds number).

• Applications: Widely used for temporary or permanent flow measurement where efficiency is not critical.

5.1.3 Flow Nozzle

• Design: A smooth contraction profile but no downstream expansion.

• Characteristics: Performance and cost between Venturi and orifice. $\boldsymbol{C_d \approx 0.95-0.99}$.

• Applications: High-velocity steam and gas flow.

5.2 Velocity-Based Measurement

5.2.1 Pitot-Static Tube

• Principle: Stagnation pressure (Pitot) minus static pressure gives velocity head.

• Formula (Ideal):

  $\boldsymbol{V = \sqrt{\frac{2 (p_0 - p_s)}{\rho}}}$

  where, $\boldsymbol{p_0}$ is stagnation pressure and $\boldsymbol{p_s}$ is static pressure.

• Applications: Point velocity measurement in pipes, ducts, wind tunnels, and aircraft airspeed indicators.

5.2.2 Current Meter & Anemometer

• Types: Cup-type, propeller-type, hot-wire, laser Doppler.

• Applications: Measure local velocity in rivers (current meter) or air flows (anemometer).

5.3 Volumetric & Mass Flow Measurement

5.3.1 Rotameter (Variable Area Meter)

• Principle: A float rises in a tapered tube until the pressure drop balances its weight. Height indicates flow rate.

• Characteristics: Direct reading, low pressure drop, requires vertical installation.

• Applications: Laboratory and industrial applications for clean fluids.

5.3.2 Positive Displacement Meters

• Principle: Traps and counts discrete volumes of fluid (e.g., nutating disk, oval gear, piston).

• Characteristics: High accuracy, independent of viscosity over a range.

• Applications: Custody transfer of oil, gas, and water (billing).

5.3.3 Turbine Flow Meter

• Principle: Flow spins a turbine; rotational speed is proportional to flow velocity.

• Characteristics: Good accuracy, wide range, requires clean fluid.

• Applications: Aerospace fuel flow, industrial process control.

5.3.4 Electromagnetic Flow Meter

• Principle: Faraday's Law. Voltage induced across a conductive fluid moving through a magnetic field is proportional to velocity.

• Characteristics: No obstruction, works for slurries and corrosive liquids.

• Applications: Wastewater, chemical, and food industries.

5.4 Open Channel Flow Measurement

5.4.1 Weirs

• Rectangular Weir:

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

  where, $\boldsymbol{H}$ is the head over the weir crest, and $\boldsymbol{L}$ is the crest length.

• V-Notch (Triangular) Weir:

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

  where, $\boldsymbol{\theta}$ is the notch angle.

• Applications: Measurement in rivers, canals, and wastewater treatment plants.

5.4.2 Flumes (Parshall Flume, Venturi Flume)

• Principle: A constriction in the channel causes a change in water depth related to flow rate.

• Characteristics: Low head loss, self-cleaning.

• Applications: Irrigation channels, sewage flows.

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Key Interrelationships and Problem-Solving Strategy

• Integrated Use: Real-world problems often require sequential application of continuity, Bernoulli, and momentum equations.

• Typical Strategy:

  1. Classify the flow (steady? incompressible?).

  2. Define a suitable control volume or streamline.

  3. Apply Continuity to relate velocities and areas.

  4. Apply Bernoulli (if assumptions hold) to relate pressures and velocities.

  5. Apply Momentum to solve for unknown forces or relate pressures between two sections.

• Limitations Awareness: Recognize when viscosity (friction), compressibility, or unsteady effects invalidate the simple Bernoulli equation, necessitating the use of the extended form or other methods.

This structured approach to flow classification and the application of fundamental conservation laws provides the essential toolkit for analyzing a vast array of fluid mechanics problems in engineering practice.

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