Page 107 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

92 BASICS OF FLUID MECHANICS AND INTRODUCTION TO CFD

for i1=1:2*n for j=1:2*n
if j “=i
=~(P(i,5)-P(j,1))*cos(P(j,7))-...

(P(i,6)-P(j,2)) *sin(P(j,7));
B=(P(i,5)-P(j,1))72+(P(i,6)-P(j,2))°2;
E=(P(i,5)-P(j,1))*sin(P(j,7))-...
(P(i,6)-P(j,2))*cos(P(j,7));
I(i,j)=1/2*sin(P(i,7)-P(j,7))*...
log (1+(P(j,8)72+2*A*P(j,8))/B)-...

cos(P(i,7)-P(j,7))*(atan((P(j /E)-atan(A/E));
,8)+A)
Ip(i,j)#-1/2*cos(P(i,7)-P(j,7))*...
log(1+(P(j ,8)~2+2*A*P(j,8))/B)-...
sin(P(i,7)-P(j,7))*(atan((P(j,8)+A)/E)-atan(A/E)) ;

else 1(i,i)=pi;Ip(i,i)=0;
end;
end; end;
Lp=I\ sin(P(:,7));
VPU=cos(P(: ,7))+Ip*Lp; V=VPU*U;
cp=1-VPU.°2;
for i=1:2*n disp(Li cp(i) V(i)]);end;

for 1=1:2*n
plot3((P(i,5) ,P(i,5)+eps], [(P(i,6) ,P(i,6)+eps] , [0,cp(i)]);
set (gca,’view’, [95,20]);
xlabel (‘x’) ;ylabel (‘y’) ;zlabel (/cp’) ;hold on;

end;
,6);P(1,6)],zeros(2*n+1,1),’.’);
(PC:
plot3([P(:,5);P(1,5)],
plot3({P(:,5);P(1,5)], (P(:,6);P(1,6)],...
-10*ones(2*n+1,1),’.’);
grid;
hold off;

We remark the low-pressure region between the two cylinders.

7. Almost Potential Fluid Flow

By almost (slightly) potential flows, we understand the flows in which
the vorticity is concentrated in some thin layers of fluid, being zero out-
side these thin layers, and there is a mechanism for producing vorticities
near boundaries.
For such models the Kutta—Joukovski theorem does not apply and
the drag may be different from zero, which means one can avoid the
D’ Alembert paradox.

There are many situations in nature or in engineering where the
viscous flows can be considered, in an acceptable approximation, as