Mesh Sizing
📐 Y+ Estimation
Estimates the first-cell wall distance (Δy) needed to achieve a target y+ value, given flow conditions and geometry type.
Wall Distance for Target Y+
Estimates Δy from flow conditions using skin friction coefficient correlation
$$y^+ = \frac{u_\tau \,\Delta y}{\nu}, \quad u_\tau = \sqrt{\frac{\tau_w}{\rho}}, \quad \tau_w = C_f \frac{\rho U^2}{2}, \quad \Delta y = \frac{y^+ \nu}{u_\tau}$$
Cƒ = 0.079·Re⁻⁰·²⁵ (Petukhov, turbulent pipe)
Cƒ = 0.0592·Re⁻⁰·² (Schlichting, turbulent flat plate)
Cƒ = 0.074·Re⁻⁰·² (approximate external body)
Free-stream velocity U
Reference length L
Density ρ
Dyn. viscosity μ
Target y⁺
Results
Y* — Pressure-Velocity Wall Unit
Wall-distance scaling when pressure gradient dominates over friction
$$y^* = \frac{y\,u_p}{\nu},\quad u_p = \left(\nu\,\frac{|dp/dx|}{\rho}\right)^{1/3}$$
Used in low-Reynolds-number models with significant streamwise pressure gradient. y* ≈ y+ in zero-pressure-gradient flow.
Wall distance y
Pressure gradient |dp/dx|
Density ρ
Dyn. viscosity μ
Results
Flow Classification
🌀 Dimensionless Numbers
Reynolds, Mach, Strouhal, Dean, and Péclet numbers with flow regime interpretation.
Reynolds Number
Re = ρ·U·L/μ — inertial vs viscous forces
$$\mathrm{Re} = \frac{\rho U L}{\mu} = \frac{U L}{\nu}, \qquad D_h = \frac{4A}{P_w}$$
Velocity U
Length/Diameter L
Density ρ
Dyn. viscosity μ
Results
Mach Number
Ma = U/a — compressibility effects
$$a = \sqrt{\gamma RT},\quad \mathrm{Ma}=U/a$$
Flow velocity U
Temperature T
Ratio γ (Cp/Cv)
Results
Strouhal Number
St = f·L/U — oscillation vs convection
$$\mathrm{St}=fL/U$$
Frequency f
Char. length L
Velocity U
Results
Dean Number
De = Re·√(r/R) — curved pipe secondary flow
$$\mathrm{De}=\mathrm{Re}\sqrt{r/R}$$
Reynolds number
Pipe radius r
Curvature radius R
Results
Péclet Number
Pe = U·L/D — advection vs diffusion
$$\mathrm{Pe}=UL/D=\mathrm{Re}\cdot\mathrm{Sc},\quad \mathrm{Sc}=\nu/D$$
Velocity U
Length L
Diffusivity D
Results
Turbulence Modeling
🌪 Turbulence Properties & Boundary Conditions
Compute k, ε, ω, and νₜ from turbulence intensity and length scale for k-ε and k-ω SST solvers.
Turbulence Inlet BCs — k, ε, ω, νₜ
For k-ε, k-ω SST, and Spalart-Allmaras solvers from I and ℓ
$$k=\tfrac{3}{2}(UI)^2,\quad \varepsilon=\frac{C_\mu^{3/4}k^{3/2}}{L_t},\quad \omega=\frac{\varepsilon}{C_\mu k}$$
Velocity U
Intensity I (%)
Length scale ℓ
ν (kin. visc.)
Results — Paste into solver inlet
Turbulence Intensity Estimate
From Reynolds number (pipe / free-stream)
$$I_\mathrm{pipe}=0.16\,Re^{-1/8},\quad I_\mathrm{ext}=0.16\,Re^{-1/8}$$
Reynolds number
Results
Turbulent Length Scale Estimate
From geometry type and characteristic dimension
$$\ell_\mathrm{pipe}=0.07D,\quad \ell_\mathrm{ext}=0.07D$$
Geometry type
Char. dimension
Results
Boundary Layer
📊 Boundary Layer Thickness
Flat-plate laminar and turbulent boundary layer properties, plus pipe hydrodynamic entry length.
Flat Plate BL Thickness
Laminar (Blasius) and turbulent Schlichting correlations
$$\delta_\mathrm{lam}=\frac{5x}{\sqrt{Re_x}},\quad \delta^*=\frac{1.72x}{\sqrt{Re_x}},\quad \theta=\frac{0.664x}{\sqrt{Re_x}}$$
Distance x
Velocity U
Kin. viscosity ν
Results
Pipe Entry / Development Length
Hydrodynamic and thermal entry lengths
$$L_{e,\mathrm{lam}}=0.06\,Re\,D,\quad L_{e,\mathrm{turb}}=4.4\,Re^{1/6}D,\quad L_t=0.05\,Re\,Pr\,D$$
Pipe diameter D
Reynolds number
Prandtl number Pr
Results
Internal Flow
🔵 Pipe & Channel Flow
Hagen-Poiseuille laminar flow, Darcy-Weisbach pressure drop with Moody friction factor.
Hagen-Poiseuille (Laminar)
Q = πR⁴ΔP / (8μL) — assumes Re < 2300
$$Q=\frac{\pi R^4\Delta P}{8\mu L},\quad \bar{U}=\frac{R^2\Delta P}{4\mu L},\quad U_\mathrm{max}=2\bar{U}$$
Pressure drop ΔP
Flow rate Q
Pipe radius R
Pipe length L
Dyn. viscosity μ
Results
Darcy-Weisbach Pressure Drop
ΔP = f·(L/D)·(ρU²/2) with Moody friction factor
$$\Delta P=f\frac{L}{D}\frac{\rho U^2}{2},\quad f_\mathrm{lam}=\frac{64}{Re},\quad \frac{1}{\sqrt{f}}=-2\log_{10}\!\left(\frac{\varepsilon}{3.7D}+\frac{2.51}{Re\sqrt{f}}\right)$$
Velocity U
Pipe diameter D
Pipe length L
Density ρ
Dyn. viscosity μ
Roughness ε
Results
Hydraulic Diameter
D_h = 4A/P for non-circular channels
$$D_h=4A/P_w$$
Width w
Height h
Outer D_o
Inner D_i
Base b
Height h_t
Results
Biomedical CFD
🩸 Non-Newtonian Viscosity Models
Effective viscosity from Power Law, Carreau, and Quemada models. Particularly relevant for blood flow simulation.
Power Law (Ostwald-de Waele)
μ_eff = K·γ̇ⁿ⁻¹
$$\tau=K\dot\gamma^n,\quad \mu_\mathrm{eff}=K\dot\gamma^{n-1}$$
Consistency K
Power index n
Shear rate γ̇
Results
Carreau Model (Blood Default)
μ_eff = μ_∞ + (μ₀−μ_∞)·[1+(λγ̇)²]^((n−1)/2)
$$\mu(\dot\gamma)=\mu_\infty+(\mu_0-\mu_\infty)\bigl[1+(\lambda\dot\gamma)^2\bigr]^{(n-1)/2}$$
μ₀ (zero-rate)
μ_∞ (inf-rate)
λ (time const.)
n (power index)
Shear rate γ̇
Results
Newtonian Validity Check
Identifies regions where non-Newtonian models are needed
$$\dot\gamma>100\ \mathrm{s}^{-1}:\ \text{Newtonian OK};\quad \dot\gamma<100\ \mathrm{s}^{-1}:\ \text{non-Newtonian}$$
Mean velocity U
Vessel radius R
Results
Biomedical CFD
💓 Pulsatile Flow & Wall Metrics
Womersley number, wall shear stress, and oscillatory shear index for cardiovascular flow simulations.
Womersley Number
α = R·√(2πf/ν) — pulsatile flow inertia
$$\alpha=R\sqrt{\omega/\nu},\quad \alpha<2:\ \text{quasi-steady},\quad \alpha\gg10:\ \text{inertia-dominated}$$
Vessel radius R
Heart rate / freq.
Kin. viscosity ν
Results
Wall Shear Stress (Pipe)
Laminar: τ_w = 4μQ/(πR³) or from ΔP
$$\tau_w=\frac{4\mu Q}{\pi R^3},\quad \mathrm{TAWSS}=\frac{1}{T}\int_0^T|\tau_w|\,dt$$
Flow rate Q
Pressure drop ΔP
Pipe length L
Radius R
Dyn. viscosity μ
Results
Oscillatory Shear Index (OSI)
Quantifies WSS directionality over a cardiac cycle
$$\mathrm{OSI}=\frac{1}{2}\!\left(1-\frac{|\int\tau_w\,dt|}{\int|\tau_w|\,dt}\right)$$
Enter time-series of WSS values (space or comma separated):
Results
Structural FEA
🔩 Elastic Constants Conversion
Convert between Young's modulus E, Poisson's ratio ν, shear modulus G, bulk modulus K, and Lamé constants.
Isotropic Elastic Constants
Given any two independent constants, compute all others
$$E=2G(1+\nu)=3K(1-2\nu),\quad G=\frac{E}{2(1+\nu)},\quad K=\frac{E}{3(1-2\nu)}$$
Provide two of these
Young's E
Poisson's ν
Shear G
Bulk K
Lamé λ
Results
Structural FEA
⚡ Stress Analysis & Failure Criteria
Von Mises equivalent stress, principal stresses, Tresca criterion, and safety factors.
Von Mises Stress (2D / 3D)
Distortion energy failure criterion
$$\sigma_\mathrm{VM}=\sqrt{\sigma_x^2-\sigma_x\sigma_y+\sigma_y^2+3\tau_{xy}^2}\quad(2\mathrm{D})$$
σ_x
σ_y
τ_xy
σ₁
σ₂
σ₃
Yield strength σ_y
Results
Pressure Vessel / Cylinder
Lamé equations for thick/thin-walled vessels
$$\sigma_h=pr/t\ (\text{thin}),\quad \sigma_h(r)=\frac{pr_i^2}{r_o^2-r_i^2}\!\left(1+\frac{r_o^2}{r^2}\right)\ (\text{Lam\'{e}})$$
Internal pressure P
Inner radius r_i
Outer radius r_o
Yield strength σ_y
Results
Structural FEA
📏 Beam Deflection & Second Moment of Area
Classical beam deflection formulas and moment of inertia for common cross-sections.
Beam Deflection
Cantilever, simply supported, UDL cases
$$\delta_\mathrm{cant}=\frac{PL^3}{3EI},\quad \sigma_\mathrm{max}=\frac{PLc}{I}$$
$$\delta_\mathrm{SS}=\frac{PL^3}{48EI},\quad \sigma_\mathrm{max}=\frac{PLc}{4I}$$
$$\delta_\mathrm{UDL}=\frac{5wL^4}{384EI}$$
Load F (or w)
Length L
Young's E
2nd moment I
Neutral axis c
Results
Second Moment of Area
I for rectangle, circle, hollow circle, I-beam
$$I=\frac{bh^3}{12},\quad A=bh$$
Width b
Height h
$$I=\frac{\pi D^4}{64},\quad A=\frac{\pi D^2}{4}$$
Diameter D
$$I=\frac{\pi(D_o^4-D_i^4)}{64}$$
Outer D_o
Inner D_i
Results
All Engineers
📐 Geometry
Areas, volumes, moments of inertia, centroids, surface areas — circles, rectangles, triangles, ellipses, spheres, cylinders, cones, and more.
2D Shape Properties
Area, perimeter, centroid, second moment of area
$$A_\mathrm{circle}=\pi r^2,\quad I_x=\frac{\pi r^4}{4},\quad A_\mathrm{rect}=bh,\quad I_x=\frac{bh^3}{12}$$
Radius r
Results
Width b
Height h
Results
Base b
Height h
Side c (optional)
Results
Semi-major a
Semi-minor b
Results
Outer radius R
Inner radius r
Results
3D Shape Properties
Volume, surface area, centroid height
$$V_\mathrm{sphere}=\tfrac{4}{3}\pi r^3,\quad V_\mathrm{cyl}=\pi r^2 h,\quad V_\mathrm{cone}=\tfrac{1}{3}\pi r^2 h$$
Radius r
Results
Radius r
Height h
Results
Base radius r
Height h
Results
Length a
Width b
Height c
Results
Major radius R
Tube radius r
Results
Triangle Solver Auto-solve — Law of Cosines / Sines
Enter any 3 known values (side or angle) — all others computed
$$c^2=a^2+b^2-2ab\cos C,\qquad \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$$
Leave one field blank. Angles in degrees. At least one side must be known.
Side a
Side b
Side c
Angle A°
Angle B°
Angle C°
Results
Coordinate Geometry
Distance, midpoint, line equations, angle between lines, circle equation
$$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2},\quad \text{slope}=\frac{y_2-y_1}{x_2-x_1}$$
x₁
y₁
x₂
y₂
Results
Enter slope and one point, or two points.
Slope m
Point (x₀, y₀)
Results
Centre (h, k)
Radius r
Results
All Engineers
∫ Calculus Tools
Numerical differentiation, numerical integration (Simpson / Gauss), Taylor series, gradient / divergence / curl in Cartesian coordinates, and common engineering transforms.
Numerical Differentiation
Forward, backward, central difference — enter f(x±h) values
$$f'(x)\approx\frac{f(x+h)-f(x-h)}{2h},\quad f''(x)\approx\frac{f(x+h)-2f(x)+f(x-h)}{h^2}$$
f(x − h)
f(x)
f(x + h)
Step size h
Results
Numerical Integration
Trapezoidal rule, Simpson's 1/3, Simpson's 3/8 — up to 9 equally-spaced points
$$\int_a^b f\,dx \approx \frac{h}{3}(f_0+4f_1+2f_2+\cdots+4f_{n-1}+f_n)\quad \text{(Simpson)}$$
Enter equally-spaced y-values (at least 3). For Simpson's: use 3, 5, 7, or 9 points (n+1 odd).
x start (a)
x end (b)
y values (comma separated)
Results
Taylor Series Approximation
sin, cos, eˣ, ln(1+x), (1+x)ⁿ — value and error estimate
$$f(x)=\sum_{n=0}^N \frac{f^{(n)}(a)}{n!}(x-a)^n$$
Function
x (radians for trig)
Terms to use (1–12)
Results
Vector Calculus — Gradient, Divergence, Curl
Numerical evaluation at a point from finite-difference stencils
$$\nabla f=\left(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z}\right),\quad \nabla\cdot\mathbf{F}=\frac{\partial F_x}{\partial x}+\frac{\partial F_y}{\partial y}+\frac{\partial F_z}{\partial z}$$
Enter scalar f at 6 neighbouring points (central difference). All h = step size.
f(x+h,y,z)
f(x-h,y,z)
f(x,y+h,z)
f(x,y-h,z)
f(x,y,z+h)
f(x,y,z-h)
Step size h
Results
Enter vector field components F=(Fx,Fy,Fz) at neighbouring points.
Fx(x+h)
Fx(x-h)
Fy(y+h)
Fy(y-h)
Fz(z+h)
Fz(z-h)
Step size h
Results
∇×F = (∂Fz/∂y−∂Fy/∂z, ∂Fx/∂z−∂Fz/∂x, ∂Fy/∂x−∂Fx/∂y)
Fz(y+h)
Fz(y-h)
Fy(z+h)
Fy(z-h)
Fx(z+h)
Fx(z-h)
Fz(x+h)
Fz(x-h)
Fy(x+h)
Fy(x-h)
Fx(y+h)
Fx(y-h)
Step size h
Results
Unit Converters
🔄 All Unit Converters
Live unit conversion — type a value to see all equivalents instantly.
MME / ChE / EE
🌡️ Heat Transfer
Conduction, convection, radiation, thermal resistance, and Fourier / Newton / Stefan-Boltzmann laws.
Fourier Conduction / Thermal Resistance
Flat wall, cylindrical, and composite wall
$$q=\frac{kA\,\Delta T}{L},\quad R_{th}=\frac{L}{kA},\quad q_\mathrm{cyl}=\frac{2\pi kL\,\Delta T}{\ln(r_2/r_1)}$$
Conductivity k
Area A
Thickness L
ΔT
Results
Conductivity k
Length L
Inner radius r₁
Outer radius r₂
ΔT
Results
Forced Convection — Nusselt / h
Dittus-Boelter (pipe), Churchill-Bernstein (cylinder), flat plate
$$\mathrm{Nu}=hL/k,\quad \mathrm{Nu}_\mathrm{pipe}=0.023\,Re^{0.8}Pr^n,\quad \mathrm{Nu}_\mathrm{plate}=0.664\,Re^{0.5}Pr^{1/3}$$
Dittus-Boelter: n=0.4 heating, n=0.3 cooling. Valid Re>10,000, 0.6<Pr<160.
Re
Pr
Fluid k
Diameter D
Mode
Results
Laminar: Nu=0.664·Re^0.5·Pr^(1/3). Turbulent: Nu=0.037·Re^0.8·Pr^(1/3).
Re_L
Pr
Fluid k
Length L
Results
Thermal Radiation — Stefan-Boltzmann
Blackbody, greybody, and net exchange between surfaces
$$q=\varepsilon\sigma A(T_1^4-T_2^4),\quad \sigma=5.67\times10^{-8}\ \mathrm{W/(m^2 K^4)}$$
Emissivity ε
Surface area A
Surface temp T₁
Surroundings T₂
Results
Extended Surface (Fin) Efficiency
Rectangular, spine, and annular fins
$$m=\sqrt{hP/(kA_c)},\quad \eta=\tanh(mL)/(mL),\quad q=\eta hPL\,\Delta T$$
Conv. coeff h
Conductivity k
Fin width w
Fin thickness t
Fin length L
T_base − T_∞
Results
MME / Civil / Aerospace
🔧 Material & Fracture Mechanics
Stress intensity factors, fatigue life (S-N), torsion, thick-wall pressure vessels, and buckling.
Stress Intensity Factor — Mode I
Through-crack in infinite plate; Irwin K_I = σ√(πa)
$$K_\mathrm{I}=Y\sigma\sqrt{\pi a},\quad a_\mathrm{cr}=\frac{1}{\pi}\!\left(\frac{K_\mathrm{IC}}{Y\sigma}\right)^2$$
Remote stress σ
Half crack length a
Geometry factor Y
Fracture toughness K_IC
Results
Fatigue Life — Basquin S-N
Cycles to failure from stress amplitude and material constants
$$\sigma_a=\sigma_{f}^{\prime}(2N_f)^b,\quad N_f=\tfrac{1}{2}\!\left(\sigma_a/\sigma_{f}^{\prime}\right)^{1/b},\quad \sigma_a/S_e+\sigma_m/S_u=1$$
Stress amplitude σ_a
Fatigue strength coeff σ_f′
Fatigue strength exponent b
Mean stress σ_m
Ultimate strength S_u
Results
Torsion of Circular Shaft
Shear stress, angle of twist, polar moment of inertia
$$\tau_\mathrm{max}=Tr/J,\quad \phi=TL/(GJ),\quad J_\mathrm{solid}=\pi d^4/32$$
Torque T
Diameter d
Length L
Shear modulus G
Results
Torque T
Outer diam d_o
Inner diam d_i
Length L
Shear modulus G
Results
Thick-Wall Pressure Vessel & Column Buckling
Lamé hoop/radial stress; Euler critical load
$$\sigma_\theta=\frac{pr_i^2(r_o^2+r^2)}{r^2(r_o^2-r_i^2)},\qquad P_\mathrm{cr}=\frac{\pi^2 EI}{(KL)^2}$$
Internal pressure p
Inner radius r_i
Outer radius r_o
Results
K: 0.5=fixed-fixed, 0.7=fixed-pinned, 1.0=pinned-pinned, 2.0=fixed-free
Elastic modulus E
2nd moment I
Column length L
End condition K
Results
EE / BME
⚡ Electrical Engineering
RC/RL/RLC circuits, filter design, Ohm's law, power, dB, and signal encoding.
Ohm's Law & DC Power
Solve any two of V, I, R, P
$$V=IR,\quad P=VI=I^2R=V^2/R$$
Enter any two fields — the others are computed.
Voltage V
Current I
Resistance R
Power P
Results
RC / RL / RLC Circuits
Time constants, cutoff frequency, resonance, Q-factor
$$\tau_{RC}=RC,\;\; f_c=\frac{1}{2\pi RC},\;\; f_0=\frac{1}{2\pi\sqrt{LC}},\;\; Q=\frac{\omega_0 L}{R}$$
⚡ Auto-solve: leave R or C blank — enter τ or fc below to back-calculate.
Resistance R
Capacitance C
Results
Resistance R
Inductance L
Results
Resistance R
Inductance L
Capacitance C
Results
Decibels & Signal Levels
Voltage gain, power gain, dBm, SNR, and noise figure
$$G_\mathrm{dB}=20\log_{10}(V_2/V_1)\ \text{(voltage)},\quad G_\mathrm{dB}=10\log_{10}(P_2/P_1)\ \text{(power)}$$
Type
Input value
Output value
Results
Power (mW)
Power (dBm)
Results
Op-Amp Configurations
Inverting, non-inverting, difference, integrator, differentiator
$$A_v^\mathrm{inv}=-R_f/R_{in},\quad A_v^\mathrm{non-inv}=1+R_f/R_1,\quad V_o=\frac{R_f}{R_{in}}(V_+-V_-)$$
R_in
R_f
V_in
Results
R₁ (to ground)
R_f
V_in
Results
R_in
R_f
V+ (non-inv input)
V− (inv input)
Results
BME / Clinical
🫀 Biomedical Engineering
Cardiac output, vascular resistance, Poiseuille flow, Windkessel model, Nernst potential, and electrode impedance.
Cardiovascular Hemodynamics
Cardiac output, stroke volume, vascular resistance, Poiseuille
$$\mathrm{CO}=\mathrm{SV}\times\mathrm{HR},\quad \mathrm{SVR}=\mathrm{MAP}/\mathrm{CO},\quad Q=\pi R^4\Delta P/(8\mu L)$$
Stroke volume SV
Heart rate HR
MAP
Results
Hagen-Poiseuille: Q = πΔP·r⁴/(8μL) — valid laminar, Newtonian, fully developed flow.
Pressure drop ΔP
Vessel radius r
Length L
Viscosity μ
Results
Nernst & Goldman Equations
Equilibrium potential, Goldman membrane voltage
$$E_X=\frac{RT}{zF}\ln\frac{[X]_o}{[X]_i},\quad V_m=\frac{RT}{F}\ln\frac{P_K[K]_o+P_{Na}[Na]_o+P_{Cl}[Cl]_i}{P_K[K]_i+P_{Na}[Na]_i+P_{Cl}[Cl]_o}$$
Ion charge z
Outside conc [X]_o
Inside conc [X]_i
Temperature T
Results
Goldman-Hodgkin-Katz for K⁺, Na⁺, Cl⁻. Permeabilities relative to P_K.
[K⁺]_out / [K⁺]_in (mM)
[Na⁺]_out / [Na⁺]_in
[Cl⁻]_out / [Cl⁻]_in
P_K : P_Na : P_Cl
Results
MME / EE / Aerospace
🎛️ Dynamics & Control
Mass-spring-damper, natural frequency, damping ratio, PID tuning, and projectile motion.
Mass-Spring-Damper System
Natural frequency, damping ratio, response classification
$$\omega_n=\sqrt{k/m},\quad \zeta=c/(2\sqrt{km}),\quad \omega_d=\omega_n\sqrt{1-\zeta^2}$$
Mass m
Spring constant k
Damping coefficient c
Results
Projectile Motion
Range, max height, time of flight — vacuum and with drag
$$R=v_0^2\sin2\theta/g,\quad H=v_0^2\sin^2\!\theta/(2g),\quad t_f=2v_0\sin\theta/g$$
Initial speed v₀
Launch angle θ
Initial height h₀
Gravity g
Results
Civil / Environmental
🏗️ Civil & Geotechnical
Manning's equation, Darcy seepage, soil bearing capacity, hydraulic jump, and retaining wall stability.
Manning's Open-Channel Flow
Discharge, velocity, and channel design for uniform flow
$$Q=\frac{1}{n}A\,R_h^{2/3}S^{1/2},\quad R_h=A/P_w$$
Manning's n
Channel width b
Flow depth y
Slope S
Results
Full circular pipe (D = diameter, R_h = D/4).
Manning's n
Diameter D
Slope S
Results
Darcy's Law — Groundwater Seepage
Flow through porous media and soil permeability
$$Q=kiA,\quad i=\Delta h/L,\quad v=ki$$
Hydraulic conductivity k
Head difference Δh
Flow path length L
Cross-section area A
Results