Live Heat Transfer Simulation

Hover over the canvas to see the temperature

Parameters

Thermal diffusivity (α) 0.50

Airspeed (m/s) 100

Surface temperature (K) 500

Angle of attack (°) 5

\underbrace{\frac{\partial T}{\partial t}}_{\text{Rate of temperature change}} = \underbrace{\alpha}_{\text{Thermal diffusivity}} \left( \underbrace{\frac{\partial^2 T}{\partial x^2}}_{\text{Diffusion in } x} + \underbrace{\frac{\partial^2 T}{\partial y^2}}_{\text{Diffusion in } y} \right) + \underbrace{\dot{Q}}_{\text{Heat source}}

T	Temperature field [K]
t	Time [s]
α = k / (ρ · c_p)	Thermal diffusivity [m²/s]
k	Thermal conductivity [W/(m·K)]
ρ	Density [kg/m³]
c_p	Specific heat capacity [J/(kg·K)]
∂T/∂t	Partial derivative of temperature with respect to time (how fast the temperature changes at a fixed point)
∂²T/∂x²	Second partial derivative in x (curvature of the temperature profile in the horizontal direction)
∂²T/∂y²	Second partial derivative in y (curvature of the temperature profile in the vertical direction)
Q̇	Volumetric heat generation rate [W/m³]

\underbrace{\frac{\partial T}{\partial t}}_{\text{Local rate of change}} + \underbrace{u \frac{\partial T}{\partial x}}_{\text{Convection in } x} + \underbrace{v \frac{\partial T}{\partial y}}_{\text{Convection in } y} = \underbrace{\alpha \left( \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} \right)}_{\text{Thermal diffusion } (\alpha \nabla^2 T)}

u	Velocity component in the x-direction (horizontal) [m/s]
v	Velocity component in the y-direction (vertical) [m/s]
u · ∂T/∂x	Convective heat transport in x (rate at which the flow carries heat horizontally)
v · ∂T/∂y	Convective heat transport in y (rate at which the flow carries heat vertically)
α · ∇²T	Thermal diffusion (Laplacian), the spread of heat from hot to cold regions through molecular conduction

\underbrace{T_{i,j}^{\,n+1}}_{\text{New temperature}} = \underbrace{T_{i,j}^{\,n}}_{\text{Current temperature}} + \Delta t \left\{ \underbrace{\alpha \left[ \frac{T_{i+1,j}^n - 2T_{i,j}^n + T_{i-1,j}^n}{\Delta x^2} \right]}_{\text{Diffusion in } x \text{ (central difference)}} + \underbrace{\alpha \left[ \frac{T_{i,j+1}^n - 2T_{i,j}^n + T_{i,j-1}^n}{\Delta y^2} \right]}_{\text{Diffusion in } y \text{ (central difference)}} - \underbrace{u \frac{\Delta_x T}{\Delta x}}_{\text{Upwind convection } x} - \underbrace{v \frac{\Delta_y T}{\Delta y}}_{\text{Upwind convection } y} \right\}

Tⁿ⁺¹_i,j	Temperature at grid point (i, j) at the next time step
Tⁿ_i,j	Temperature at grid point (i, j) at the current time step
Δt	Time step size [s]
Δx, Δy	Grid spacing in the x and y directions [m]
T_i+1,j, T_i−1,j	Temperatures at neighboring grid points in x (right, left)
T_i,j+1, T_i,j−1	Temperatures at neighboring grid points in y (above, below)
Central difference	Second-order approximation of ∂²T/∂x² using three grid points
Upwind scheme	First-order approximation of ∂T/∂x that uses the upstream direction based on flow velocity sign

\underbrace{Re = \frac{\rho \, U_\infty \, L}{\mu}}_{\text{Reynolds number}} \qquad \underbrace{Pr = \frac{\mu \, c_p}{k}}_{\text{Prandtl number}} \qquad \underbrace{Nu = \frac{h \, L}{k}}_{\text{Nusselt number}}

Re	Reynolds number (ratio of inertial forces to viscous forces), which indicates laminar or turbulent flow
U_∞	Freestream airspeed [m/s]
L	Characteristic length (chord length) [m]
μ	Dynamic viscosity of air [Pa·s]
Pr	Prandtl number (ratio of momentum diffusivity to thermal diffusivity)
Nu	Nusselt number (ratio of convective to conductive heat transfer at the surface)
h	Convective heat transfer coefficient [W/(m²·K)]

Re < 5×10⁵	Laminar flow regime (smooth, ordered flow)
Re ≥ 5×10⁵	Turbulent flow regime (chaotic flow with more mixing)
Re^1/2	Square-root dependence on Reynolds number (laminar boundary layer scaling)
Re^4/5	Near-linear dependence on Reynolds number (turbulent boundary layer scaling)
Pr^1/3	Cube-root dependence on Prandtl number (thermal boundary layer ratio)

\underbrace{y_t}_{\text{Half-thickness}} = \underbrace{5t}_{\text{Scaling factor}} \left[ \underbrace{0.2969\sqrt{x}}_{\text{Leading edge}} - \underbrace{0.1260\,x}_{\text{Linear term}} - \underbrace{0.3516\,x^2}_{\text{Quadratic}} + \underbrace{0.2843\,x^3}_{\text{Cubic}} - \underbrace{0.1015\,x^4}_{\text{Trailing edge}} \right]

y_t	Half-thickness of the airfoil at chordwise position x [m]
t	Maximum thickness ratio = 0.12 (12% of chord for NACA 0012)
x	Normalized chordwise coordinate (0 = leading edge, 1 = trailing edge)
0.2969√x	Dominant term at the leading edge, which gives the airfoil its rounded nose
−0.1015x⁴	Dominant term at the trailing edge, which tapers the profile to a point