Irrotational Fan Blade Flow and Thermal effects :

SummaryHere 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 \(\psi\).

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);


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


[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).