Airflow model#
A small box of air is a useful starting point. Air enters or leaves the box, and pressure or friction changes its motion. We need to account for all the mass and all the forces that change motion.
Momentum is mass times velocity, in kg m/s. Conservation means not creating or destroying a quantity in the accounting. Momentum changes when an external force acts; its rate of change must equal that force.
Physical assumptions#
A continuum model treats air as smoothly distributed material, not individual molecules. A field assigns a value at every location and time, such as velocity throughout a room. Density is mass per volume: kg/m³ means kilograms in each cubic metre. Incompressible here means taking air density as constant: a moving parcel does not appreciably change volume. This is a low-speed approximation, not a claim that air cannot be compressed.
Viscosity resists neighboring layers sliding at different speeds. Newtonian means the resisting stress is proportional to the rate of deformation. Stress is force per area; pressure is the part pushing equally in all directions, measured in pascals (Pa = N/m²). Unsteady flow changes with time at a fixed location; steady flow does not. We do not assume that air stays attached: separation means it peels away from the cloth.
Symbols and operations#
Let \(x,y,z\) be position coordinates (m) and \(t\) time (s). Bold \(\mathbf{u}\) is local air velocity, not necessarily incoming wind.
Symbol |
Meaning and example choice |
SI unit |
|---|---|---|
\(\mathbf{u}\) |
Local velocity; solved, not fixed |
m/s |
\(\rho\) |
Constant density; choose 1.225 to match the demo |
kg/m³ |
\(p\) |
Local pressure; solved, not fixed |
Pa |
\(\eta\) |
Dynamic viscosity; illustrative choice \(1.8\times10^{-5}\) |
Pa s |
\(\mathbf{g}\) |
Gravity; choose (0, −9.81, 0) |
m/s² |
Actual air properties depend on conditions and should be established for the experiment. Viscosity is absent from the demo’s airflow solve, not real air.
A derivative measures a rate of change. \(\partial/\partial t\) measures time change at a fixed position; \(\partial\) means other variables are held fixed. The symbol \(\nabla\) organizes spatial derivatives, each introducing a factor of 1/m:
\(\nabla p\) is pressure’s direction and rate of increase (Pa/m).
\(\nabla\cdot\mathbf{u}\) is divergence: net outward volume-flow rate per unit volume (1/s). The dot combines matching directional derivatives.
\((\mathbf{u}\cdot\nabla)\mathbf{u}\) measures velocity change encountered by moving through spatially varying flow (m/s²).
\(\nabla^2\mathbf{u}\) adds second spatial derivatives of each velocity component, measuring variation relative to neighbors (1/(m s)).
The conservation equations#
Constant-density mass conservation is
The signed total volume outflow from an arbitrarily small fixed box is zero: whatever leaves must be replaced. Outflow need not be equal through each face.
Momentum conservation gives the incompressible Navier–Stokes equation:
Read this as density times acceleration equals pressure, viscous, and gravity forces per volume. Every term is in N/m³. The negative pressure gradient drives air toward lower pressure.
For an illustrative 1 Pa drop over 0.10 m, the gradient magnitude is 10 Pa/m. Alone it would cause acceleration \(10/1.225\approx8.16\) m/s²; other terms generally act too. The derivation from momentum and Newtonian stress is in MIT’s Navier–Stokes notes, equations (346)–(347). Our equation writes constant density and dynamic viscosity explicitly and adds gravity as a force per volume. It does not choose a numerical mesh or turbulence model.
Hydrostatic pressure supports still air’s weight. Subtracting that background can consistently remove the corresponding fluid gravity term. It does not remove gravity from the cloth.
Air at the moving sail#
A boundary condition specifies behavior at a surface of the calculation. Impermeable means air cannot pass through the cloth. No slip means adjacent air shares the cloth’s velocity along the surface. Normal means perpendicular to the surface; tangential means along it.
Let \(\mathbf{r}\) be a particular cloth point’s position (m) and \(\mathbf{v}\) its velocity (m/s). The derivative \(d/dt\) follows that same point; \(d\mathbf{r}/dt\) is its velocity. On either face,
Cloth moving sideways at 0.10 m/s requires air touching it to share that velocity there. The normal part prevents penetration, and the tangential part expresses no slip. The no-slip condition is stated in the MIT notes, Section 13.2.
Velocity changes from its surface value to the outer flow over a thin region called the boundary layer. The disturbed air downstream forms the wake. Viscosity affects both and whether separation occurs: omitting it does not just omit a small friction force.
Far away, prescribe \(\mathbf{W}\), incoming velocity in m/s from the introduction. Initial conditions specify the air and cloth state at the starting time. Ground, mast, and boom add boundaries if their obstruction is included. Cloth permeability, its ability to pass air, needs measurements and a different interface law if that effect is wanted.