Fluid Kinematics

Fluid kinematics describes the motion of fluids without considering the forces causing that motion. This mathematical framework is essential for analyzing airflow patterns in duct systems, water flow in hydronic systems, and refrigerant flow in vapor compression cycles.

Lagrangian and Eulerian Descriptions

Two fundamental approaches exist for describing fluid motion.

Lagrangian Description

The Lagrangian approach tracks individual fluid particles through space and time. Each particle’s position is described as a function of time and its initial position:

Position Vector:

x = x(x₀, y₀, z₀, t)
y = y(x₀, y₀, z₀, t)
z = z(x₀, y₀, z₀, t)

Where (x₀, y₀, z₀) represents the particle’s initial position.

Velocity in Lagrangian Coordinates:

V = dx/dt = ∂x/∂t

Applications in HVAC:

• Tracer gas testing for infiltration measurement
• Smoke visualization in cleanrooms
• Particle transport in air filtration analysis
• Contaminant dispersion modeling

Eulerian Description

The Eulerian approach examines fluid properties at fixed points in space as a function of time. This is the standard framework for HVAC analysis.

Velocity Field:

V(x, y, z, t) = u(x, y, z, t)î + v(x, y, z, t)ĵ + w(x, y, z, t)k̂

Where:

• u = velocity component in x-direction
• v = velocity component in y-direction
• w = velocity component in z-direction

Steady vs. Unsteady Flow:

• Steady: ∂V/∂t = 0 (most HVAC design assumes steady flow)
• Unsteady: ∂V/∂t ≠ 0 (startup, shutdown, cycling equipment)

Velocity Field Analysis

The velocity field V(x, y, z, t) represents the fundamental description of fluid motion in HVAC systems.

Uniform Flow

Velocity is constant throughout the flow field:

V = constant (magnitude and direction)

Examples:

• Idealized flow in straight duct sections
• Freestream approach to building facades
• Far-field air velocities in occupied zones

Non-Uniform Flow

Velocity varies with position in the flow field.

Common HVAC Applications:

• Developing flow in duct entries (entrance length effects)
• Flow through diffusers and grilles
• Velocity profiles in pipes and ducts
• Flow around HVAC equipment in mechanical rooms

Velocity Measurement Techniques

Acceleration Field

Fluid acceleration is the rate of change of velocity for a fluid particle. In Eulerian coordinates, this involves both local and convective acceleration.

Material Derivative

The acceleration experienced by a fluid particle is given by the material derivative (substantial derivative):

a = DV/Dt = ∂V/∂t + (V·∇)V

Components:

• ∂V/∂t = local acceleration (unsteady effects)
• (V·∇)V = convective acceleration (spatial variation effects)

Acceleration Components

In Cartesian coordinates, the acceleration field becomes:

aₓ = ∂u/∂t + u(∂u/∂x) + v(∂u/∂y) + w(∂u/∂z)
aᵧ = ∂v/∂t + u(∂v/∂x) + v(∂v/∂y) + w(∂v/∂z)
aᵤ = ∂w/∂t + u(∂w/∂x) + v(∂w/∂y) + w(∂w/∂z)

HVAC Acceleration Examples

High Convective Acceleration:

• Flow through sudden contractions in ductwork
• Nozzle flow in balancing valves
• Velocity increase through VAV box dampers
• Flow acceleration through coil fins

Unsteady Acceleration:

• Compressor startup transients
• Pump start/stop water hammer potential
• Variable speed drive ramp-up/ramp-down
• Control valve modulation

Streamlines, Streaklines, and Pathlines

These visualization concepts describe different aspects of flow patterns.

Streamlines

Lines that are tangent to the velocity vector at every point in the flow field at a given instant. They represent instantaneous flow direction.

Mathematical Definition:

dx/u = dy/v = dz/w

Properties:

• No flow crosses a streamline
• Streamlines cannot intersect (except at stagnation points)
• For steady flow, streamlines are fixed in space
• Streamtube bounded by streamlines has constant mass flow

HVAC Applications:

• CFD visualization of airflow patterns in spaces
• Duct flow streamline patterns around obstructions
• Airflow around HVAC equipment
• Supply air jet trajectories from diffusers

Streaklines

Lines connecting all fluid particles that have passed through a common point. This is what smoke or dye visualization shows.

HVAC Applications:

• Smoke testing of kitchen exhaust hoods
• Tracer gas studies in laboratories
• Fume hood face velocity verification
• Cleanroom airflow pattern verification

Pathlines

The actual trajectory traced by an individual fluid particle over time.

Relationship Between Lines:

• For steady flow: streamlines = streaklines = pathlines
• For unsteady flow: these three concepts differ
• Most HVAC design assumes steady flow conditions

Flow Visualization Techniques

Experimental methods for observing flow patterns in HVAC systems.

Smoke Visualization

Best Practices:

• Avoid thermal effects from hot smoke sources
• Document lighting conditions for photography
• Record environmental conditions (temperature, pressure)
• Ensure adequate ventilation after testing

Particle Image Velocimetry (PIV)

Advanced optical technique using laser illumination and high-speed cameras to measure velocity fields.

Capabilities:

• Full-field velocity measurements
• Instantaneous flow field mapping
• Turbulence structure identification
• Validation of CFD models

HVAC Research Applications:

• Diffuser performance characterization
• VAV box airflow patterns
• Coil face velocity distribution
• Filter face velocity uniformity

Flow Tracers

Gas Tracers:

• Sulfur hexafluoride (SF₆): infiltration, building air exchange
• Carbon dioxide (CO₂): ventilation effectiveness
• Perfluorocarbon tracers (PFT): multi-zone airflow

Liquid Tracers:

• Fluorescent dyes: hydronic system flow patterns
• Thermal tracers: heat exchanger performance
• Salt solutions: conductivity-based flow tracking

Rotation and Vorticity

Vorticity quantifies the local rotation of fluid elements, critical for understanding mixing and energy dissipation.

Vorticity Vector

Vorticity is defined as the curl of the velocity field:

ω = ∇ × V

Cartesian Components:

ωₓ = ∂w/∂y - ∂v/∂z
ωᵧ = ∂u/∂z - ∂w/∂x
ωᵤ = ∂v/∂x - ∂u/∂y

Magnitude:

|ω| = √(ωₓ² + ωᵧ² + ωᵤ²)

Irrotational Flow

Flow with zero vorticity (∇ × V = 0). This simplifies analysis considerably.

Conditions for Irrotational Flow:

• Inviscid flow away from boundaries
• Flow regions outside boundary layers
• Potential flow theory applications

HVAC Examples:

• Freestream flow approaching buildings
• Core flow in large ducts (away from walls)
• Inlet flow to filters and coils

Rotational Flow

Flow with non-zero vorticity, typical in real HVAC systems due to viscosity.

High Vorticity Regions:

• Boundary layers along duct walls
• Flow separation zones (elbows, expansions)
• Turbulent mixing regions
• Fan blade tip vortices
• Coil wake regions

Vorticity Transport

Helmholtz’s theorems describe vorticity behavior:

1. Vortex filaments move with the fluid
2. Vortex strength is conserved along filaments
3. Vortex filaments cannot end in the fluid

HVAC Implications:

• Swirl generation in poor duct transitions
• Fan inlet vortex formation
• Mixing enhancement in air handling units
• Energy dissipation in flow restrictions

Circulation

Circulation Γ is the line integral of velocity around a closed curve.

Definition

Γ = ∮ V · ds

Where ds is a differential length element along the closed path.

Relationship to Vorticity

By Stokes’ theorem:

Γ = ∮ V · ds = ∫∫ (∇ × V) · n dA = ∫∫ ω · n dA

Circulation equals the flux of vorticity through any surface bounded by the closed curve.

HVAC Applications

High Circulation Flows:

• Centrifugal fan scroll design
• Swirl diffusers (high induction ratio)
• Cyclone separators in dust collection
• Vortex shedding from heat exchanger tubes

Low Circulation Design Goals:

• Straight duct runs before measurement stations
• Fan inlet approach conditions
• Flow straighteners upstream of flowmeters
• VAV box inlet conditions

Stream Function and Potential Function

These scalar functions simplify analysis of two-dimensional flows.

Stream Function (ψ)

For 2D incompressible flow, the stream function satisfies:

u = ∂ψ/∂y
v = -∂ψ/∂x

Properties:

• Automatically satisfies continuity equation
• Lines of constant ψ are streamlines
• Difference in ψ between streamlines equals volumetric flow rate per unit depth

HVAC Applications:

• Analytical solutions for duct flow development
• Mixing analysis in plenums
• Flow around obstacles in ductwork
• Simplified modeling of room airflow patterns

Velocity Potential (φ)

For irrotational flow, a velocity potential exists:

V = ∇φ

Components:

u = ∂φ/∂x
v = ∂φ/∂y
w = ∂φ/∂z

Governing Equation: For incompressible, irrotational flow:

∇²φ = 0 (Laplace's equation)

HVAC Applications:

• Idealized flow around buildings (wind loading)
• Core flow in large supply plenums
• Academic study of fundamental flow patterns
• Simplified CFD validation cases

Combined Use

For 2D, incompressible, irrotational flow, both functions exist and satisfy:

∂φ/∂x = ∂ψ/∂y
∂φ/∂y = -∂ψ/∂x

These are the Cauchy-Riemann equations, allowing powerful complex variable techniques.

Design Considerations

Duct System Design

Minimize Acceleration/Deceleration:

• Gradual transitions (15° maximum taper angle)
• Avoid sudden expansions and contractions
• Use curved elbows instead of sharp bends
• Maintain uniform velocity distribution

Streamline Duct Layouts:

• Align ductwork with anticipated streamlines
• Minimize flow separation regions
• Use turning vanes in elbows >90°
• Provide flow straightening upstream of critical components

Measurement Station Requirements

ASHRAE 111 Traverse Locations:

• Minimum 7.5 duct diameters downstream of disturbances
• Minimum 3 duct diameters upstream of disturbances
• Longer straight runs improve accuracy
• Flow straighteners may reduce required lengths

Velocity Profile Considerations:

• Fully developed flow requires 50-100 diameters
• Turbulent flow develops faster than laminar
• Square/rectangular ducts develop more slowly
• Non-uniform profiles cause measurement errors

Flow Visualization in Design

CFD Model Validation:

• Compare streamline patterns with physical tests
• Verify separation zones match expectations
• Check vorticity magnitudes in critical regions
• Validate velocity field measurements

Commissioning Applications:

• Document baseline airflow patterns
• Identify short-circuiting in spaces
• Verify stratification in large volumes
• Confirm exhaust capture effectiveness

Code and Standard References

ASHRAE Standards

• ASHRAE 111: Measurement, Testing, Adjusting, and Balancing of Building HVAC Systems

  • Pitot tube traverse procedures
  • Velocity profile measurement requirements
  • Flow visualization acceptance criteria

• ASHRAE Fundamentals Handbook: Chapter 21, Duct Design

  • Fluid mechanics principles
  • Velocity profile development
  • Pressure loss calculations

Test Standards

• AMCA 210: Laboratory Methods of Testing Fans for Certified Aerodynamic Performance

  • Inlet flow conditions
  • Velocity measurement plane requirements
  • Flow uniformity criteria

• ASHRAE 70: Method of Testing for Rating the Performance of Air Outlets and Inlets

  • Discharge velocity profiles
  • Throw and drop characteristics
  • Entrainment ratio determination

Practical Measurement Guidelines

Duct Traverse Procedures

Velocity Measurement Points:

Pitot Tube Alignment:

• Orient parallel to flow within ±5°
• Insert to traverse point center
• Allow stabilization (5-10 seconds)
• Record static and total pressure

Data Reduction

Average Velocity Calculation:

V_avg = √[(V₁² + V₂² + ... + Vₙ²) / n]

Root-mean-square averaging accounts for velocity squared relationship with kinetic energy.

Volumetric Flow Rate:

Q = V_avg × A

Where A is the duct cross-sectional area.

Velocity Pressure:

VP = (V/4005)² (in water, standard air)

Where V is in feet per minute.

Summary

Fluid kinematics provides the mathematical framework for describing and analyzing fluid motion in HVAC systems:

• Eulerian description is standard for fixed measurement points
• Velocity fields characterize flow patterns throughout spaces and ductwork
• Acceleration fields identify regions of pressure change and energy transformation
• Streamlines, streaklines, and pathlines visualize flow patterns
• Vorticity quantifies rotation and mixing in flows
• Stream and potential functions simplify 2D flow analysis
• Proper measurement techniques require attention to flow development and uniformity

Understanding these kinematic concepts enables HVAC engineers to design efficient systems, interpret test data correctly, validate computational models, and troubleshoot performance issues related to airflow and water flow distribution.