Fluid Dynamics

name: fluid-dynamics

description: > Expert fluid dynamics assistant for engineers and physicists. Use this skill whenever the user needs: analysis of fluid flow, solving Navier-Stokes equations, calculating boundary layer behavior, understanding laminar and turbulent flow, analyzing aerodynamic forces, or designing systems involving fluid transport. Includes both theoretical foundations and practical engineering applications. trigger: Any engineering problem involving fluid flow - from aerospace to civil engineering to biomedical applications. license: MIT compatibility: opencode metadata: audience: engineers category: physics

Fluid Dynamics — Theory and Engineering Applications

Covers: Fluid Kinematics · Navier-Stokes · Laminar and Turbulent Flow · Boundary Layers · Compressible Flow · Aerodynamics

Fundamental Equations

Continuity Equation (Mass Conservation)

Differential form:

∂ρ/∂t + ∇·(ρv) = 0

For incompressible flow (ρ = constant):

∇·v = 0

Integral form (control volume):

d/dt ∫_CV ρ dV = -∫_CS ρ v·n̂ dA

Momentum Equation (Navier-Stokes)

For Newtonian fluid:

ρ(Dv/Dt) = -∇p + μ∇²v + ρg

Where Dv/Dt = ∂v/∂t + (v·∇)v is the substantial (material) derivative.

In component form (x-momentum):

ρ(∂u/∂t + u∂u/∂x + v∂u/∂y + w∂u/∂z) = -∂p/∂x + μ(∂²u/∂x² + ∂²u/∂y² + ∂²u/∂z²) + ρg_x

Energy Equation

ρ C_p DT/Dt = k ∇²T + Φ + Q

Where Φ is the viscous dissipation function:

Φ = μ[2(∂u/∂x)² + 2(∂v/∂y)² + 2(∂w/∂z)² + (∂v/∂x + ∂u/∂y)² + ...]

Equation of State

• Ideal gas: p = ρRT
• Incompressible: ρ = constant
• Real gas: Use tables or compressibility factor Z

Fluid Properties

Viscosity

Temperature dependence:

• Gases: μ ∝ T^0.7 (approx)
• Liquids: μ decreases with T (typically exponential)

Compressibility

Bulk modulus:

K = -V(dp/dV)

Speed of sound:

c = √(dp/dρ)|_s = √(γRT) for ideal gas

Surface Tension

σ = dγ/dT (Gibbs adsorption equation basis)

Capillary rise: h = 2σ cosθ/(ρgr)

Hydrostatics

Pressure Variation

In static fluid:

dp/dz = -ρg

For incompressible (ρ constant):

p = p₀ + ρg(h - z)

For compressible (ideal gas, isothermal):

p = p₀ exp(-Mgz/RT)

For compressible (adiabatic):

p = p₀[1 + (γ-1)gz/(c₀²)]^{γ/(γ-1)}

Buoyancy (Archimedes' Principle)

F_b = ρ_fluid V_displaced g

Stability: Center of buoyancy vs. center of gravity.

Pressure Measurement

• Manometer: p = p_ref + ρgΔh
• Bourdon gauge: Mechanical deformation
• Piezoelectric: Crystal deformation

Kinematics of Flow

Streamlines, Pathlines, and Streaklines

In steady flow: all three coincide.

Velocity Potential

For irrotational flow (∇×v = 0):

v = ∇φ

φ satisfies Laplace's equation: ∇²φ = 0

Stream Function

For 2D incompressible flow:

u = ∂ψ/∂y, v = -∂ψ/∂x

Constant ψ = streamlines. For 2D, ψ exists automatically if ∇·v = 0.

Circulation

Γ = ∮_C v·dl = ∬_S (∇×v)·n̂ dS

Kelvin's theorem: Circulation constant for barotropic flow with conservative body forces.

Inviscid Flow

Bernoulli's Equation

Along a streamline (steady, incompressible, inviscid):

p/ρ + v²/2 + gz = constant

With head loss (in real fluids):

p₁/ρ + v₁²/2 + gz₁ = p₂/ρ + v₂²/2 + gz₂ + h_L

Euler's Equations (Differential)

Dv/Dt = -∇p/ρ + g

Potential Flow Solutions

Superposition of elementary solutions:

Flow Around a Cylinder

Without circulation:

• Symmetric pressure distribution
• Zero lift

With circulation (Kutta-Joukowski):

L = ρU_∞Γ (per unit span)

Lift and Drag

Lift coefficient:

C_L = L/(½ρU²_∞ A)

Drag coefficient:

C_D = D/(½ρU²_∞ A)

Laminar Flow

Reynolds Number

Re = ρVL/μ = UL/ν

Poiseuille Flow (Pipe)

For laminar flow in circular pipe:

u(r) = (Δp/L)(R² - r²)/(4μ)

Average velocity: ū = (Δp/L)(R²)/(8μ)

Pressure drop:

Δp = 32μLū/D² = 128μLQ/(πD⁴)

Friction factor (Darcy):

f = 64/Re (laminar)

Couette Flow

Flow between parallel plates, one moving:

u(y) = U(y/h)

Laminar Boundary Layer (Flat Plate)

Blasius solution for laminar BL on flat plate:

δ(x) ≈ 5.0 √(νx/U_∞)

Boundary layer thickness grows as √x.

Shear stress:

τ_w = 0.664 μ U_∞ √(U_∞/νx)

Entrance Length

Laminar: L_e ≈ 0.05 Re D Turbulent: L_e ≈ 50 D

Turbulent Flow

Characteristics

• Random, chaotic, 3D
• Vortical structures at many scales
• Enhanced momentum/heat/mass transfer
• Statistically described

Turbulent Viscosity

Reynolds decomposition: u = ū + u' Boussinesq hypothesis:

-ρ<u'u'> = μ_t ∂ū/∂y

where μ_t >> μ

Velocity Profile

Log-law (inner region):

u⁺ = (1/κ) ln y⁺ + B

Where:

• u⁺ = u/u_τ
• y⁺ = yu_τ/ν
• u_τ = √(τ_w/ρ)
• κ ≈ 0.41 (von Kármán constant)
• B ≈ 5.0

Friction Factor (Moody Diagram)

Smooth pipe (Blasius):

f = 0.316/Re^{0.25} for Re < 10⁵

Rough pipe:

1/√f = -2 log₁₀(ε/(3.7D) + 2.51/(Re√f))

Turbulent Boundary Layer

Thickness:

δ ≈ 0.37 x/Re_x^{1/5}

99% velocity at approximately δ.

Turbulence Models

Boundary Layers

Boundary Layer Equations

For steady 2D laminar BL (∂p/∂x known from inviscid flow):

u∂u/∂x + v∂u/∂y = U dU/dx + ν ∂²u/∂y²

Momentum Integral Equation (von Kármán)

d/dx(∫_0^δ ρ u(u - U) dy) = τ_w + (dU/dx)∫_0^δ ρ(u - U) dy

Thermal Boundary Layer

For constant properties, same structure:

δ_T ≈ δ/Pr^{1/3} (laminar)
δ_T ≈ δ/Re_x^{1/5} Pr^{-1/3} (turbulent)

Prandtl number Pr = ν/α

Heat Transfer (Laminar Flat Plate)

Nusselt number:

Nu_x = 0.332 Re_x^{1/2} Pr^{1/3}

Average over length L:

Nu_L = 0.664 Re_L^{1/2} Pr^{1/3}

Separation

Adverse pressure gradient (dU/dx < 0) can cause flow separation:

• Boundary layer thickens
• Velocity gradient at wall becomes zero
• Flow reverses near wall

Compressible Flow

Mach Number

M = V/a

• M < 1: Subsonic
• M = 1: Sonic
• M > 1: Supersonic
• M > 5: Hypersonic

Isentropic Flow Relations

For ideal gas, γ = C_p/C_v:

Area-Mach relation:

(A/A*) = (1/M)[(2/(γ+1))(1 + (γ-1)/2 M²)]^{(γ+1)/(2(γ-1))}

Normal Shock Waves

Across shock (Rankine-Hugoniot):

p₂/p₁ = 1 + (2γ/(γ+1))(M₁² - 1)
T₂/T₁ = [1 + (2γ/(γ+1))(M₁² - 1)][2 + (γ-1)M₁²]/[(γ+1)M₁²]
M₂² = [1 + (γ-1)/2 M₁²]/[γ M₁² - (γ-1)/2]

Entropy increases across shock.

Oblique Shocks

Shock angle β related to deflection θ:

tan θ = 2 cot β (M₁² sin²β - 1)/[M₁²(γ + cos 2β) + 2]

Nozzle Flow

• Subsonic: Area decreases → M increases, p decreases
• Supersonic: Area increases → M increases, p decreases
• Sonic at throat (A = A*)
• Underexpanded/overexpanded nozzles

Fanno Flow (Friction)

Adiabatic with friction in constant area duct:

• Choking at M = 1
• Maximum entropy at M = 1

Rayleigh Flow (Heat Transfer)

Constant area with heat addition:

• Maximum temperature at M = 1/√((γ+1)/(γ-1))
• Can lead to thermal choking

Aerodynamics

Lift Generation

L = ½ ρ V² A C_L

For infinite wing:

C_L = 2πα (thin airfoil theory)

With stall: C_L peaks then drops.

Drag

D = ½ ρ V² A C_D

Components:

• Pressure drag: Due to pressure differences (form drag)
• Friction drag: Due to viscous shear
• Induced drag: Due to downwash from finite span
• Wave drag: Due to shock waves (supersonic)

Aspect Ratio Effects

C_D = C_D₀ + C_L²/(π e AR)

Where e = Oswald efficiency (≈ 0.9 for typical wings).

Airfoil Nomenclature

• Chord: c (leading to trailing edge)
• Thickness: t (max as fraction of c)
• Camber: mean camber line
• Angle of attack: α between chord line and flow

Thin Airfoil Theory

Lift slope:

dC_L/dα = 2π (per radian)

Zero-lift angle α₀ (depends on camber).

Wing Planforms

• Rectangular: Simple, low efficiency
• Tapered: Better, optimized
• Delta: High speed, low supersonic drag
• Swept: Delay wave drag

Induced Drag

C_Di = C_L²/(π AR e)

Compressibility Effects

Critical Mach number: First sonic point on airfoil. Drag divergence Mach: Significant drag rise begins.

Prandtl-Glauert correction:

C_L = C_L,incompressible/√(1 - M²)

Dimensional Analysis

Buckingham Pi Theorem

If equation involves n variables with k fundamental dimensions:

• Write dimension matrix
• Find rank
• Form n - k dimensionless groups

Common Dimensionless Numbers

Dynamic Similarity

Two flows similar if Re, M, Fr (as relevant) match.

Experimental Techniques

Wind Tunnels

• Subsonic: Continuous or intermittent
• Transonic: Porous walls, slotted throat
• Supersonic: Converging-diverging nozzle
• Hypersonic: High M, low density

Measurement Techniques

Similarity Scaling

• Complete similarity: all relevant Pi groups equal
• Partial similarity: critical groups matched

Common Errors to Avoid

• Confusing static, dynamic, and stagnation pressures
• Applying Bernoulli incorrectly (not along streamline, including friction)
• Using laminar formulas for turbulent flow
• Ignoring compressibility at high Mach numbers
• Incorrectly applying continuity (mass balance)
• Confusing viscosity types (dynamic vs kinematic)
• Using wrong friction factor correlation
• Forgetting to convert to consistent units

Key References

• Fluid Mechanics by Kundu, Cohen, and Dowling — Comprehensive textbook
• Boundary-Layer Theory by Schlichting — Definitive BL text
• Fundamentals of Aerodynamics by Anderson — Aerodynamics intro
• Theory of Wing Sections by Abbott & von Doenhoff — Airfoil data