CHME 330 Fluid Mechanics - Chapter 1 Notes

Introduction and Basic Concepts

  • Fluid mechanics: study of liquids and gases in motion or at rest.
  • Mechanics: oldest physical science; deals with bodies under forces.
    • Statics: bodies at rest.
    • Dynamics: bodies in motion.
  • Fluid mechanics encompasses [fluid statics] (fluids at rest) and [fluid dynamics] (fluids in motion) and interactions with boundaries.
  • Fluid dynamics is sometimes referred to as fluid mechanics, treating fluids at rest as a special case of motion with zero velocity.
  • Subfields often mentioned:
    • Hydrodynamics: motion of incompressible-like fluids (e.g., liquids, low-speed gases).
    • Hydraulics: liquid flows in pipes/open channels.
    • Gas dynamics: flows with significant density changes (e.g., high-speed gases).
    • Aerodynamics: flow of gases (air) over bodies (aircraft, rockets, cars).
    • Meteorology/oceanography/hydrology: naturally occurring flows.
  • What is a fluid?
    • Fluid: substance in the liquid or gas phase.
    • Fluids deform continuously under shear stress; solids resist shear by deforming up to a fixed strain angle.
    • In solids, stress ∝ strain; in fluids, stress ∝ strain rate.
    • Under constant shear, a solid deforms to a finite strain, while a fluid keeps deforming (reaches a constant rate of strain).
  • Stress concepts:
    • Stress: force per unit area.
    • Normal stress: normal component of force per unit area.
    • Shear stress: tangential component of force per unit area.
    • Pressure: normal stress in a fluid at rest.
    • A fluid at rest has zero shear stress; when motion occurs, shear stresses develop to re-establish flow.
  • Free surface behavior differs between liquids and gases (see further sections).

The No-Slip Condition

  • No-slip condition: as a fluid flows over a solid surface, the fluid at the surface has zero velocity relative to the surface.
    • The fluid “sticks” to the surface due to viscosity.
    • This condition leads to the development of boundary layers where viscous effects and velocity gradients are significant.
  • Boundary layer: region adjacent to the wall where viscous effects are significant and velocity gradients are large.
  • Flow over curved surfaces can lead to flow separation, where the flow detaches from the surface.
  • The no-slip condition is a key reason for boundary-layer formation and is essential for predicting viscous effects in real flows.

Classification of Fluid Flows

  • Viscous vs inviscid regions:
    • Viscous flows: frictional (viscous) effects are significant.
    • Inviscid flow regions: viscous effects are negligible compared to inertial/pressure forces (common away from walls).
  • Internal vs external flow:
    • Internal flow: bounded by solid surfaces (e.g., pipe flow).
    • External flow: unbounded fluid around a surface (e.g., flow over a ball).
    • Open-channel flow: duct flow with a free surface (partially filled duct).
  • Compressible vs incompressible flow:
    • Incompressible: fluid density remains nearly constant (e.g., liquids).
    • Compressible: density changes during flow (e.g., high-speed gas flows).
    • Mach number (Ma) is a common descriptor in compressible flows (Ma = velocity/speed of sound).
    • Examples: Sonic flow Ma = 1; Subsonic Ma < 1; Supersonic Ma > 1; Hypersonic Ma ≫ 1.
  • Laminar vs turbulent flow:
    • Laminar: highly ordered, layered flow; common for high-viscosity fluids at low speeds.
    • Turbulent: highly disordered with velocity fluctuations; common for low-viscosity fluids at high speeds.
    • Transitional flow: alternates between laminar and turbulent.
  • Natural (unforced) vs forced flow:
    • Forced flow: externally driven by pump/fan.
    • Natural flow: driven by buoyancy and density differences (thermal plumes, etc.).
  • Steady vs unsteady flow:
    • Steady: properties at a point do not change with time.
    • Unsteady: properties change with time; transient flows are a subset.
    • Uniform: properties do not vary with position over a region.
    • Periodic: unsteady flow that oscillates about a mean value.
    • Many engineering devices operate under steady flow conditions for long periods.
  • Dimensionality of flows:
    • One-, two-, and three-dimensional flows: velocity can vary in 1, 2, or 3 spatial directions.
    • Velocity field examples:
    • Entrance region in a circular pipe: V = V(r, z) (two-dimensional in the entrance region).
    • Downstream fully developed: V = V(r) (one-dimensional in the flow direction).
    • Some flows (e.g., car antenna) are effectively two-dimensional except near boundaries.
  • Axisymmetric flows (Example 1-1):
    • Axisymmetric body (e.g., a bullet) has rotational symmetry about an axis.
    • Upstream flow is parallel to the axis; time-averaged flow is axisymmetric and two-dimensional (depends on z and r, not on the angular coordinate).
    • Instantaneous flow is three-dimensional; bullets may also spin.

System and Control Volume

  • System: a quantity of matter or a region in space chosen for study.
  • Surroundings: mass or region outside the system.
  • Boundary: surface that separates the system from its surroundings (can be fixed or movable).
  • Systems can be open or closed:
    • Closed system (control mass): fixed amount of mass; no mass crosses the boundary; boundary can move.
    • Open system (control volume): region in space enclosing devices with mass flow (e.g., compressor, turbine, nozzle); both mass and energy can cross the boundary.
  • Control surface: the boundary of a control volume (real or imaginary; fixed or moving).

Importance of Dimensions and Units

  • Physical quantities have dimensions; magnitudes are expressed in units.
  • Primary (fundamental) dimensions: mass (m), length (L), time (t), temperature (T), electric current (I), amount of substance (N or mol), luminous intensity (J or cd).
  • Derived or secondary dimensions: velocity, energy, volume, etc., expressed in terms of primary dimensions.
  • Metric SI system: simple, decimal-based.
  • English system: lacks a single systematic base; units relate arbitrarily.
  • Table: fundamental SI dimensions and their units (examples):
    • Length: meter (m)
    • Mass: kilogram (kg)
    • Time: second (s)
    • Temperature: kelvin (K)
    • Electric current: ampere (A)
    • Amount of substance: mole (mol)
    • Luminous intensity: candela (cd)
  • Standard prefixes in SI (examples):
    • kilo (k) = 10^3
    • mega (M) = 10^6
    • giga (G) = 10^9
    • tera (T) = 10^12
    • deci (d) = 10^-1
    • centi (c) = 10^-2
    • milli (m) = 10^-3
    • micro (μ) = 10^-6
    • nano (n) = 10^-9
  • SI and English units (selected):
    • Work: 1 J = 1 N·m
    • 1 cal = 4.1868 J
    • 1 Btu = 1.0551 kJ
    • Weight vs mass: distinct concepts; weight depends on gravity; mass is invariant.
  • Unity conversion ratios:
    • All nonprimary units can be formed by combinations of primary units.
    • Unity conversion ratios are dimensionless and equal to 1, useful for unit consistency in calculations.
  • Dimensional homogeneity:
    • All equations must be dimensionally homogeneous.
    • Always check units; unity conversion ratios are exact and equal to 1.

Modeling in Engineering

  • Engineering modeling approaches:
    • Experimental (testing and measurement) vs analytical (calculation-based).
    • Experimental advantages: captures true system behavior; accuracy limited by measurement errors.
    • Analytical advantages: fast and inexpensive; depends on assumptions and idealizations.
  • Mathematical modeling rationale:
    • Many problems require differential equations to relate rates of change; differential equations describe physical laws precisely.
    • Not all problems require differential equations; simplified models are common.
  • Simplified vs complex models:
    • Simplified models (e.g., rotor modeled as a disk, body as an ellipsoid) can capture essential features with less complexity.
    • The simplest model that yields satisfactory results is often preferred.

Problem-Solving Technique

  • Step-by-step approach (Example-driven):
    1. Problem Statement
    2. Schematic
    3. Assumptions and Approximations
    4. Physical Laws
    5. Properties
    6. Calculations
    7. Reasoning, Verification, and Discussion
  • The approach helps simplify and organize problem solving and ensures reasonableness of results.
  • Emphasis on reasonable assumptions and justification; neatness and organization are valued.

Engineering Software Packages

  • Software aids: not a substitute for understanding physics.
  • Excel: can solve systems of equations, enable parametric studies, plot results, handle what-if questions.
  • Engineering Equation Solver (EES): solves systems of linear or nonlinear algebraic or differential equations; large library of thermodynamic properties; users must formulate problems and provide equations.
  • Important caveat: equation solvers do not replace problem formulation; physics must be applied by the user.

Accuracy, Precision, and Significant Digits

  • Accuracy error: difference between reading and true value; often tied to systematic errors; accuracy relates to the closeness of the average reading to the true value.
  • Precision error: difference between a reading and the average of multiple readings; relates to random errors and instrument resolution.
  • Significant digits: meaningful digits in a number; reflect data precision.
  • Examples and table:
    • Example patterns of significant digits (e.g., 12.3 → 1.23 × 10^1 has 3 significant digits).
    • Table 1-3 illustrates various numbers and their significant digits in exponential notation.
  • Practical rule: avoid implying more precision than data provide; round results appropriately to the significant digits of the input data.

Examples

  • Example 1-2: Electric Power Generation by a Wind Turbine
    • Given: turbine rated power = 30 kW; operates 2200 h/year; electricity cost = $0.09/kWh.
    • Determine annual energy generated:
    • Rate = 30 kW = 30 kJ/s
    • Total energy per year:
      extEnergy=(30extkW)(2200exth)=66,000extkWhext{Energy} = (30 ext{ kW})(2200 ext{ h}) = 66{,}000 ext{ kWh}
    • Money saved per year:
      ext{Money saved} = (66{,}000 ext{ kWh})(0.09 ext{ ext{$/kWh$}}) = 5940 ext{ dollars}
    • Alternate unit manipulation path converts 66,000 kWh to energy in kJ for verification:
      66,000extkWh=2.38imes108extkJ66{,}000 ext{ kWh} = 2.38 imes 10^8 ext{ kJ}
  • Example 1-3: Obtaining Formulas from Unit Considerations
    • Given density $p = 850 ext{ kg/m}^3$, volume $V = 2 ext{ m}^3$.
    • Mass $m$ should have unit kilograms; density and volume combine to give $m$:
      m=pV=(850extkg/m3)(2extm3)=1700extkgm = pV = (850 ext{ kg/m}^3)(2 ext{ m}^3) = 1700 ext{ kg}
    • Note: not all formulas can be derived purely from units; nondimensional constants may appear.
  • Example 1-4: The Weight of One Pound-Mass
    • Show that 1.00 lbm weighs 1.00 lbf on Earth.
    • Using $W = m g$ with standard gravity $g = 32.174 ext{ ft/s}^2$ and unity conversion:
      W=(1.00extlbm)(32.174extft/s2)=1extlbfW = (1.00 ext{ lbm})(32.174 ext{ ft/s}^2) = 1 ext{ lbf}
    • Note: This relies on the customary conversion between lbm and lbf units under Earth gravity; different gravity yields different weights.
  • Example 1-5: Solving a System of Equations with EES
    • Problem: difference of two numbers is 4; sum of squares equals sum plus 20.
    • Equations to encode:
      x−y=4x - y = 4
      x2+y2=x+y+20x^2 + y^2 = x + y + 20
    • Solution (via EES): $x = 5$, $y = 1$.
    • Concept: equation solvers allow focusing on the physics; treat equations as written and let the solver handle math.
  • Example 1-6: Significant Digits and Volume Flow Rate
    • Given: volume $V = 1.1 ext{ gal}$, time $t = 45.62 ext{ s}$.
    • Volume flow rate $Q$ (in $ ext{m}^3/ ext{min}$):
    • Convert volume: $1.1 ext{ gal} = 3.7854 imes 10^{-3} ext{ m}^3$.

    • Q=Vt=1.1extgalimes3.7854imes10−3extm3/extgal45.62extsimes60exts/min =5.5imes10−3extm3/extminQ = \frac{V}{t} = \frac{1.1 ext{ gal} imes 3.7854 imes 10^{-3} ext{ m}^3/ ext{gal}}{45.62 ext{ s}} imes 60 ext{ s/min} \ = 5.5 imes 10^{-3} ext{ m}^3/ ext{min}
    • Reporting: two significant digits due to input precision; if used as intermediate step, maintain more digits to avoid round-off.
    • Discuss limitations: precision of measurements vs accuracy; systematic errors not known here.

CFD Software

  • CFD (Computational Fluid Dynamics) is widely used in engineering and research; discussed in Chap. 15 and illustrated with CFD visuals.
  • Example: unsteady vortex rope in a model Francis turbine draft tube; shown via isocontours of swirling strength using ANSYS-FLUENT.

Summary: Key Concepts to Remember

  • The No-Slip Condition leads to boundary layers and viscous effects near walls.
  • Flows are classified by viscous effects, compressibility, dimensionality, steadiness, and forcing.
  • System and Control Volume concepts are essential for formulating conservation laws.
  • Dimensions and Units: primary vs derived, SI prefixes, and dimensional homogeneity are critical for correct modeling and unit consistency.
  • Modeling choices balance simplicity and accuracy; both experimental and analytical approaches have roles.
  • Problem-solving is a disciplined, seven-step process; consider assumptions carefully.
  • Software tools are aids, not replacements for fundamental understanding.
  • Accuracy, precision, and significant digits must guide data interpretation and reporting.

Appendix: Notable Formulas and Concepts (quick reference)

  • Mass-Volume relation:
    m=pVm = pV
  • Weight relation:
    W=mgW = mg
  • Wind turbine energy example:
    • Power: $P = 30$ kW, time: $t = 2200$ h, energy: E=(30extkW)(2200exth)=66,000extkWhE = (30 ext{ kW})(2200 ext{ h}) = 66{,}000 ext{ kWh}
    • Cost saved: extMoney=Eimes0.09=5940extdollarsext{Money} = E imes 0.09 = 5940 ext{ dollars}
  • Dimensional analysis reminder:
    • Units must be consistent; unity conversion ratios are exact (equal to 1).
  • Mach number definitions (compressible flow):
    • Ma < 1: Subsonic
    • Ma = 1: Sonic
    • Ma > 1: Supersonic
    • Ma ≫ 1: Hypersonic
  • Typical SI prefixes (selected):
    • kilo (k) = $10^3$, mega (M) = $10^6$, giga (G) = $10^9$; milli (m) = $10^{-3}$, micro (μ) = $10^{-6}$, nano (n) = $10^{-9}$, etc.
  • Dimension naming:
    • Fundamental: mass (m), length (L), time (t), temperature (T), electric current (I), amount of substance (mol), luminous intensity (cd).
    • Derived: velocity (L/t), energy (ML^2/t^2), volume (L^3), etc.