基于simple算法的流场模拟计算文档格式.docx
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和
是边界上的值,为己知。
上式中任一方程都可写成:
除第一及最后一个方程外,其余方程可写为:
这些方程可通过消元和回代两个过程求解。
现引入记号:
则,
即:
在边界点,j=1与j=n+1,
为了求解方程组,首先要对方程组按的形式编排,并明确其中的系数
。
从j=2起,计算出
,直到j=n。
由于在边界位置(n+1)
的数值是已知的,因此,可连续计算出
3、计算结果对比
本文基于simple算法用编写MATLAB程序对整个流场进行计算,另外借助fluent计算流体力学软件数值模拟,通过对比分析计算结果,得出整个流场的流速矢量图和速度云图。
上图为两种计算方法的计算结果,发现两者速度分布趋势一致,编程计算结果数值偏大,原因在于计算采用一阶迎风格式,节点数值趋近于其上游节点值。
下图为速度云图,入口段效应的影响导致流动未达到充分发展。
MATLAB程序:
%%%%%%%%%%%%%%%%%%%%%%%%
dx=5e-4;
dy=5e-4;
den=998;
dyna=1001.6e-6;
a=0.01/dy+3;
%x节点%
b=0.15/dx+1;
%y节点%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%边界条件的设置、初始值的设置%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
T=0.2*ones(a,b);
U1_old=zeros(a,b);
P1_old=zeros(a,b);
%初值%
U1_old(3:
a-2,:
)=T(3:
);
%加速收敛,边界条件%
uw_sum(1,b)=0;
ue_sum(1,b)=0;
de1_sum(1,b)=0;
m=1;
A(1:
a-4,2:
b-1)=0.2;
%初值为1%
e22=1;
whilee22>
0.01%整个流场迭代%
if m>4
break
end
m=m+1;
forj=2:
1:
b-1;
ifj>2
break
end
n=1;
%迭代次数%
e=1;
%初次迭代%
B=[0;
0;
0.0823455;
0.149211;
0.196652;
0.224194;
0.23674;
0.241087;
0.242138;
0.242214;
0.242125;
0.242083;
0.242125;
0.242214;
0.242138;
0.241087;
0.23674;
0.224195;
0.196652;
0.149212;
0.0823459;
0;
0];
U1_old(3:
a-2,1)=B(3:
a-2,1);
U1_old(3:
a-2,2)=B(3:
a-2,1);
whilee>
0.001%迭代,算两列,其他为已知%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%对(i,j)点求uxin%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
fori=3:
1:
a-2
Fe1(i,j)=den*(U1_old(i,j)+U1_old(i,j+1))*dy/2;
%对流强度%
Fw1(i,j)=den*(U1_old(i,j-1)+U1_old(i,j))*dy/2;
ap_u1(i,j)=max(Fe1(i,j),0)+max(-Fw1(i,j),0)+4*dyna;
%离散方程对应系数%
ae_u1(i,j)=dyna+max(-Fe1(i,j),0);
aw_u1(i,j)=dyna+max(Fw1(i,j),0);
an_u1(i,j)=dyna;
as_u1(i,j)=dyna;
A1_u(i,i-1)=-an_u1(i,j);
A1_u(i,i)=ap_u1(i,j);
A1_u(i,i+1)=-as_u1(i,j);
b1_u(i,j)=aw_u1(i,j)*U1_old(i,j-1)+ae_u1(i,j)*U1_old(i,j+1)...
+(P1_old(i,j)-P1_old(i,j+1))*dy/2;
%(i,j+1)点的源项%
end
A1_u(1,1)=1;
%附加点%
A1_u(2,2)=1;
%边界点%
A1_u(a-1,a-1)=1;
A1_u(a,a)=1;
b1_u(1,j)=0;
b1_u(2,j)=0;
%边界点%
b1_u(a-1,j)=0;
b1_u(a,j)=0;
U1_new(:
j)=inv(A1_u)*b1_u(:
j);
%得到(i,j+1)点的速度计算值%
%对(i,j)点求p%
uw_sum(1,j)=0;
ue_sum(1,j)=0;
de1_sum(1,j)=dy/ap_u1(2,j);
for i=3:
a-2
uw_sum(1,j)=uw_sum(1,j)+U1_old(i,j-1);
ue_sum(1,j)=ue_sum(1,j)+U1_new(i,j);
de1_sum(1,j)=de1_sum(1,j)+dy/ap_u1(i,j);
end
P1_fix(1,j)=(uw_sum(1,j)-ue_sum(1,j))/de1_sum(1,j);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%对(i,j)点进行速度修正%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
fori=3:
U1_fix(i,j)=(dy/ap_u1(i,j))*P1_fix(1,j);
%速度修正%
end
U1_old(3:
a-2,j)=U1_new(3:
a-2,j)+U1_fix(3:
a-2,j);
P1_old(2:
a-1,j)=P1_old(2:
a-1,j)+0.4*P1_fix(1,j);
ee=uw_sum(1,j)-ue_sum(1,j);
e=max(max(abs(ee)));
%判断的是否合理?
%
n=n+1;
N(1,j)=n;
ifn>
1000
break
end
A((a-4)*m+1:
(a-4)*(m+1),2:
b-1)=U1_old(3:
a-2,2:
b-1);
%赋值,即m从1开始%
end
end
E(1:
a-4,2:
j)=A((a-4)*m+1:
(a-4)*(m+1),2:
j)-A((a-4)*(m-1)+1:
(a-4)*m,2:
j);
e22=max(max(abs(E)));
end%整个流场迭代结束%
y=0:
10;
%结果的输出%
u=U1_old(2:
2:
a-1);
u
plot(y,u)