Fluid Dynamics Cheat Sheet

This cheat sheet covers fundamental concepts in fluid statics, kinematics, and dynamics, including pressure, fluid motion, conservation laws, dimensional analysis, and compressible flow phenomena. It provides key equations and definitions for understanding fluid behavior.

Core Principles

Pressure is force per unit area, acting equally in all directions in a static fluid.
Pressure in a static fluid increases with depth due to gravity (hydrostatic equation).
Fluid kinematics describes fluid motion using Eulerian or Lagrangian perspectives.
Conservation laws (mass, momentum, energy) govern fluid behavior.
Dimensional analysis simplifies complex problems using dimensionless groups (e.g., Reynolds number).
Laminar flow is smooth and orderly; turbulent flow is chaotic.
Bernoulli's equation relates pressure, velocity, and elevation in inviscid, steady flow.
Compressible flow involves significant density changes, with Mach number indicating flow speed relative to sound speed.

Action Steps

Identify the flow regime (laminar vs. turbulent) using the Reynolds number.
Determine if the flow is compressible or incompressible based on Mach number.
Apply appropriate conservation laws (mass, momentum, energy) for control volume analysis.
Use Bernoulli's equation for steady, inviscid flow along a streamline.
Employ dimensional analysis (Buckingham Pi theorem) to simplify problems and scale results.
Calculate hydrostatic forces on submerged surfaces.
Analyze buoyancy and stability for submerged or floating bodies.
Use Moody chart or Colebrook equation for turbulent pipe flow friction factor.
Consider minor losses in pipe flow calculations.
Apply isentropic or shock relations for compressible flow problems.

Formulas

$P_{abs} = P_{gage} + P_{atm}$
$dP/dz = -ρg$
$P_2 = P_1 + ρgh$
$F_R = P_C A$
$F_B = ρ_f g V_{displaced}$
$d\vec{V}/dt = ∂\vec{V}/∂t + (\vec{V}⋅∇)\vec{V}$
$Re = ρVL/μ$
$P/ρ + V^2/2 + gz = constant$
$ṁ = ρV_{avg}A_c$
$∑F = d/dt ∫_{CV} ρV dV + ∫_{CS} ρV(V_r⋅n)dA$
$c = \sqrt{kRT}$
$Ma = V/c$

Key Terms

Pressure: Normal force exerted by a fluid per unit area.
Hydrostatic Equation: Relates pressure change to fluid density, gravity, and vertical height.
Eulerian Description: Observes fluid flow from fixed points in space.
Material Derivative: Describes the rate of change of a fluid property following a fluid particle.
Streamline: A curve tangent to the instantaneous velocity vector at every point.
Vorticity: Measure of the local spinning motion of a fluid element (∇ × V).
Bernoulli's Equation: Relates pressure, velocity, and elevation in steady, inviscid flow.
Reynolds Number: Ratio of inertial forces to viscous forces, indicating flow regime.
Mach Number: Ratio of flow velocity to the speed of sound, indicating compressibility effects.
Stagnation Properties: Properties of a fluid brought to rest isentropically.
Normal Shock: An abrupt, irreversible wave where supersonic flow becomes subsonic.

Timeline

Ancient Greece: Archimedes' principle established the concept of buoyancy.
17th Century: Torricelli invented the barometer and studied fluid efflux (Torricelli's Law).
18th Century: Daniel Bernoulli published Hydrodynamica, laying groundwork for the Bernoulli equation.
19th Century: Hagen and Poiseuille independently studied laminar flow in pipes (Hagen-Poiseuille equation).
Late 19th Century: Osborne Reynolds conducted experiments differentiating laminar and turbulent flow (Reynolds number).
Early 20th Century: Prandtl introduced boundary layer theory, bridging inviscid and viscous flow concepts.
Mid 20th Century: Development of compressible flow theory, including shock waves and supersonic aerodynamics.