Summary*Here we will learn how to deal with a multi-physics system of PDEs on a Complex geometry, with multiple meshes within one problem. We also learn how to manipulate the region indicator and see how smooth is the projection operator from one mesh to another.*

Incompressible flow

Without viscosity and vorticity incompressible flows have a velocity given by: $$ u=\left(\begin{matrix}{\partial \psi \over \partial x_{2} }\\ -{\partial \psi \over \partial x_{1}} \end{matrix}\right), \quad ~where~\psi~~is~solution~of~\quad \Delta \psi =0 $$ This equation expresses both incompressibility (\(\nabla\cdot u=0\)) and absence of vortex (\(\nabla\times u =0\)).

As the fluid slips along the walls, normal velocity is zero, which
means that \(\psi\) satisfies:
$$ \psi ~constant~on~the~walls. $$
One can also prescribe the normal velocity at an artificial boundary, and this translates into
non constant Dirichlet data for \(\psi\).

Airfoil

Let us consider a wing profile \(S\) in a uniform flow. Infinity will be represented by a large circle \(C\) where the flow is assumed to be of uniform velocity; one way to model this problem is to write

\(\) |

\( \Delta \psi =0 in \Omega, \qquad \psi |_{S}=0, \quad \psi|_{C}= {u_\infty}y, \) |

where \(\partial\Omega=C\cup S \)

The NACA0012 Airfoil An equation for the upper surface of a NACA0012 (this is a classical wing profile in aerodynamics) is: $$ y = 0.17735\sqrt{x}-0.075597x- 0.212836x^2+0.17363x^3-0.06254x^4.$$ Example[potential.edp]

// file potential.edp real S=99; border C(t=0,2*pi) { x=5*cos(t); y=5*sin(t);} border Splus(t=0,1){ x = t; y = 0.17735*sqrt(t)-0.075597*t - 0.212836*(t^2)+0.17363*(t^3)-0.06254*(t^4); label=S;} border Sminus(t=1,0){ x =t; y= -(0.17735*sqrt(t)-0.075597*t -0.212836*(t^2)+0.17363*(t^3)-0.06254*(t^4)); label=S;} mesh Th= buildmesh(C(50)+Splus(70)+Sminus(70)); fespace Vh(Th,P2); Vh psi,w; solve potential(psi,w)=int2d(Th)(dx(psi)*dx(w)+dy(psi)*dy(w))+ on(C,psi = y) + on(S,psi=0); plot(psi,wait=1);

A zoom of the streamlines are shown on Figure [figpotential].

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[figpotential] Zoom around the NACA0012 airfoil showing the streamlines (curve \(\psi=\) constant). To obtain such a plot use the interactive graphic command: "+" and p. Right: temperature distribution at time T=25 (now the maximum is at 90 instead of 120). Note that an incidence angle has been added here (see Chapter 9).