Irrotational Movements

Contents

Coordinate System and Hypothesis

The coordinate system used has the xx component in the direction of the flight velocity1, and zz to the up. Force in xx is drag, and in zz is lift.

Considerations:

  • Negligible viscosity, except in zones such as boundary layer and wake
    • Re=ρ∞U∞c/μ≈4×106\mathrm{Re}=\rho_\infty U_\infty c/\mu\approx 4\times10^6 for conventional flight. The
  • Adiabatic movement
    • The heat conduction negligible with RePr≫1\mathrm{RePr}\gg1, given Re≫1\mathrm{Re}\gg1
  • Negligible static forces such as gravity given gL≪V2gL\ll V^2

Equations of Movements

Differential equations in partial derivatives for the movement of the fluid, given negligible viscosity and heat conduction

Continuity:

∂ρ∂t+∇⋅(ρV)= ⁣Dρ ⁣Dt+ρ∇⋅V=0\frac{\partial \rho}{\partial t}+\nabla\cdot(\rho\mathbf V)=\frac{\DD \rho}{\DD t}+\rho\nabla\cdot\mathbf V=0

Conservation of momentum:

 ⁣DV ⁣Dt=∂V∂t+V⋅∇V=∂V∂t+∇(V22)−V×(∇×V)=−1ρ∇p\frac{\DD\mathbf V}{\DD t}=\frac{\partial\mathbf V}{\partial t}+\mathbf V\cdot\nabla\mathbf V=\frac{\partial\mathbf V}{\partial t}+\nabla\left( \frac{V^2}{2} \right)-\mathbf V\times(\nabla\times\mathbf V)=-\frac{1}{\rho}\nabla p

with V=∣V∣V=|\mathbf V|.

Evolution of entropy:

 ⁣Ds ⁣Dt=∂s∂t+V⋅∇s=0\frac{\DD s}{\DD t}=\frac{\partial s}{\partial t}+\mathbf V\cdot\nabla s=0

in perfect gas model, this equation can be substituted with p/ργ=const.p/\rho^\gamma=\text{const.}

Euler-Bernoulli Equation

If the vorticity conserves and zero ∇×V=0\nabla\times\mathbf V=0 the velocity can be represented as the potential of velocities Φ=Φ(x,y,z,t)\Phi=\Phi(x,y,z,t), such that V=∇Φ\mathbf V=\nabla\Phi.

Applying this to the equation of conservation of momentum, we obtain the equation of Euler-Bernoulli:

∂Φ∂t+∣∇Φ∣22+∫ ⁣dpρ=F(t)\boxed{\frac{\partial \Phi}{\partial t}+\frac{|\nabla\Phi|^2}{2}+\int\frac{\dd p}{\rho}=F(t)}

For liquids:

ρ∂Φ∂t+ρV22+p=G(t)\rho\frac{\partial \Phi}{\partial t}+\rho\frac{V^2}{2}+p=G(t)

For perfect gas, with the velocity of the sound a2=γRTa^2=\gamma RT:

∂Φ∂t+V22+a2γ−1=G(t)\frac{\partial \Phi}{\partial t}+\frac{V^2}{2}+\frac{a^2}{\gamma-1}=G(t)

Footnotes

  1. Pointing towards the tail of the aircraft. Also knows as the wind’s system. ↩