Numerical Simulation Tools

Atmospheric Parameters & Total Conditions

Altitude

Mach Number

Formulas:
T0 = T * (1 + (gamma-1)/2 * M^2)
P0 = P * (1 + (gamma-1)/2 * M^2)^(gamma/(gamma-1))
q = 0.5 * rho * V^2
a = sqrt(gamma * R * T)

y+ Calculator (Boundary Layer Mesh Design)

Altitude

Mach

Chord Length (m)

Turbulence Model y+ Targets

Model	Target y+	Description
k-omega SST	~ 1	Requires fine near-wall mesh
Standard k-epsilon	~ 30	Standard wall function
LES/DNS	~ 1	Direct simulation

Formulas:
y+ = rho * u* * y / mu = u* * y / nu
u* = sqrt(tauw/rho), tauw = 0.5 * Cf * rho * V^2
First layer: Dy = y+ * nu / u*
Cf = 0.074 * Re^(-0.2) (Schlichting)

Boundary Layer Calculator

Altitude (km)

Mach

Position x (m)

Note: Shape factor H = delta*/theta:
H ~ 2.0-2.5 = Laminar
H ~ 1.2-1.5 = Turbulent

Formulas:
Turbulent: delta = 0.37 * x / Re_x^(1/5)
delta* = 0.046 * x / Re_x^(1/5)
theta = 0.036 * x / Re_x^(1/5)
Re_x = rho * V * x / mu

Drag & Lift Calculator

Altitude (km)

Mach

Reference Area (m2)

Angle of Attack (deg)

Drag Coefficient Reference

Body Type	Cd	Application
Sphere	0.47	Reference
Cylinder	0.99	Bluff body
Airfoil 0deg	0.006	Flat plate
Airfoil 10deg	0.08	With AoA
Car	0.30	Ground vehicle
Truck	0.70	Commercial

Formulas:
Drag D = Cd * q * A
Lift L = Cl * q * A
Cl = 2pi * alpha (rad), thin airfoil
q = 0.5 * rho * V^2

Reynolds Number Calculator

Altitude (km)

Mach

Characteristic Length (m)

Flow Regime:
Flat plate: Re_crit = 5x10^5
Sphere/Cylinder: Re_crit = 2x10^5
Re > Re_crit = Turbulent

Formula:
Re = rho * V * L / mu = V * L / nu