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Aerospace 2020, 7, 123 4 of 15 2.2. Flow Field Computation The flow around the airfoil is assumed to be two-dimensional, compressible, and fully turbulent. The governing equations are the continuity, Navier–Stokes, energy, and transport equations of the turbulent kinetic energy k and its dissipation rate ε. The Kato–Launder k-ε turbulent model [32] with a wall function was employed to suppress the over-production of turbulent eddy viscosity around the leading-edge region. The governing equations are discretized using a second-order upwind TVD (total variation diminishing) scheme [33] for the inviscid terms, a second-order central difference scheme for the viscous terms, and an LU-ADI (lower upper-alternating direction implicit) scheme [34] for time integration. The first grid points from the airfoil surface are in the range of 50 to 300 in wall units. The L2 norm is employed as a convergence criterion. 2.3. Droplet Trajectory In the droplet trajectory computation, we assume that the size and concentration of droplets are sufficiently small. Additionally, the collision and splitting of the droplets are assumed to be negligible. Thus, the trajectory was calculated based on the Lagrangian approach using the one-way coupling method. In other words, the droplet motion is affected by the flow field, whereas the droplet does not affect the flow field. The motion equation of a droplet is a simplified Basset–Boussinesq–Oseen (BBO) equation and it is expressed, as follows: −→ dUp 3 ρg1−→−→ dt =4CDρ d UrUr. (1) dd −→ −→ where Up is the droplet velocity and Ur is the relative velocity between the droplet and the surrounding flow; ρg and ρd are the gas (air) and droplet density, respectively; dd is the droplet diameter, and CD is the drag coefficient. Because we assume that the droplet does not deform or rotate, the Schiller model [35] is used for the drag coefficient of a sphere; it is defined as follows: CD = 24 1 + 0.15Re0.687 , (2) Re d where Red is the droplet Reynolds number based on the droplet diameter dd, relative velocity Ur, fluid density ρ f , and fluid viscosity μ f . 2.4. Thermodynamics In the thermodynamic computation, which is, the calculation of ice growth, the weak coupling method is used because the time scales of the flow field and icing are significantly different. The EMM [29] is used as the icing model and is expressed, as follows: ∂Ti ∂t ∂Tw ∂t = ki ∂2Ti, (3) ρiCpi ∂y2i = kw ∂2Tw, (4) ρwCpw ∂y2w ρ∂Bi+ρ ∂Bw i∂tw∂t imines = m ̇ +m ̇ −m ̇ , (5) ρL∂Bi =k∂Ti−k∂Tw. (6) i f ∂t i ∂yi w ∂yw Here, T, k, L f , B, t, y, and ρ are the temperature, thermal conductivity, latent heat of solidification, thickness, time, wall normal, and density, respectively; subscripts i and w denote the ice and water, respectively. Additionally, m ̇ im, m ̇ in, and m ̇ es denote the mass flow rates of impingement, runback-in, and evaporation or sublimation, respectively. The EMM is based on the mass and energy conservationPDF Image | Anti-Icing Electric Heaters for Icing on the NACA 0012 Airfoil
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