在數值計算過程中，對於計算結果的准確性和效率有很高的要求，但是這兩者之間往往互相矛盾；而使用柯朗數可用於平衡兩者。
1、柯朗數的定義：
C = sqrt(gh)*t/s
其中，t是時間步長，s是網格在水平方向的間距。
柯朗數的意義在於表示了在單位時間步長中，有多少個網格的信息發生了移動。經過正確的調整，可以更好地加速收斂和增強解的穩定性。
2、C語言實現柯朗數計算：
依據上述方程，在實際計算中采用C語言實現計算固液界面上的柯朗數，結果如下：
1 void localCourantNumber() 2 { 3 4 5 double rhoe,rhon,rhot; 6 7 for(i=2;i<=nxm-1;i++) //Calculation of local Courant number only at internal faces 8 { 9 ieast = i + 1; 10 dxpe = xc[ieast] - xc[i]; 11 fxe = (xf[i]-xc[i])/dxpe; 12 fxp = 1.0 - fxe; 13 14 for(j=2;j<=nym;j++) 15 { 16 jnorth = j + 1; 17 dypn = yc[jnorth] - yc[j]; 18 fyn = (yf[j] - yc[j])/dypn; 19 fyp = 1.0 - fyn; 20 21 for(k=2;k<=nzm;k++) 22 { 23 ktop = k + 1; 24 dzpt = zc[ktop]-zc[k]; 25 fzt = (zf[k] - zc[k])/dzpt; 26 fzp = 1.0 - fzt; 27 28 29 30 //Calculating density at cell interface 31 rhoe = fxp * rho[i][j][k] + fxe * rho[ieast][j][k]; 32 33 /* rhoe = 2.0 * rho[i][j][k] * rho[ieast][j][k]/( rho[i][j][k] + rho[ieast][j][k]);*/ 34 35 s = (yf[j]-yf[j-1])*(zf[k]-zf[k-1]); 36 vole = dxpe * s; 37 38 39 //Sum of courant numbers of outflow faces of donor cell 40 Ce[i][j][k] = fabs(Fe[i][j][k]/(rhoe*vole))*dt; 41 42 /* printf("Ce=%e\n",Ce[i][j][k]);*/ 43 } 44 } 45 } 46 47 48 for(i=2;i<=nxm;i++) //Calculation of local Courant number only at internal faces 49 { 50 ieast = i + 1; 51 dxpe = xc[ieast] - xc[i]; 52 fxe = (xf[i]-xc[i])/dxpe; 53 fxp = 1.0 - fxe; 54 55 for(j=2;j<=nym-1;j++) 56 { 57 jnorth = j + 1; 58 dypn = yc[jnorth] - yc[j]; 59 fyn = (yf[j] - yc[j])/dypn; 60 fyp = 1.0 - fyn; 61 62 for(k=2;k<=nzm;k++) 63 { 64 ktop = k + 1; 65 dzpt = zc[ktop]-zc[k]; 66 fzt = (zf[k] - zc[k])/dzpt; 67 fzp = 1.0 - fzt; 68 69 70 71 //Calculating density at cell interface 72 rhon = fyp * rho[i][j][k] + fyn * rho[i][jnorth][k]; 73 74 /* rhon = 2.0 * rho[i][j][k] * rho[i][jnorth][k]/( rho[i][j][k] + rho[i][jnorth][k]);*/ 75 76 77 s = (xf[i]-xf[i-1])*(zf[k]-zf[k-1]); 78 voln = s * dypn; 79 80 81 //Sum of courant numbers of outflow faces of donor cell 82 Cn[i][j][k] = fabs(Fn[i][j][k]/(rhon*voln))*dt; 83 84 /* printf("Ce=%e\n",Ce[i][j][k]);*/ 85 } 86 } 87 } 88 89 for(i=2;i<=nxm;i++) //Calculation of local Courant number only at internal faces 90 { 91 ieast = i + 1; 92 dxpe = xc[ieast] - xc[i]; 93 fxe = (xf[i]-xc[i])/dxpe; 94 fxp = 1.0 - fxe; 95 96 for(j=2;j<=nym;j++) 97 { 98 jnorth = j + 1; 99 dypn = yc[jnorth] - yc[j]; 100 fyn = (yf[j] - yc[j])/dypn; 101 fyp = 1.0 - fyn; 102 103 for(k=2;k<=nzm-1;k++) 104 { 105 ktop = k + 1; 106 dzpt = zc[ktop]-zc[k]; 107 fzt = (zf[k] - zc[k])/dzpt; 108 fzp = 1.0 - fzt; 109 110 111 112 //Calculating density at cell interface 113 rhot = fzp * rho[i][j][k] + fzt * rho[i][j][ktop]; 114 115 /* rhot = 2.0 * rho[i][j][k] * rho[i][j][ktop]/( rho[i][j][k] + rho[i][j][ktop]);*/ 116 117 118 s = (xf[i]-xf[i-1])*(yf[j]-yf[j-1]); 119 volt = s * dzpt; 120 121 122 //Sum of courant numbers of outflow faces of donor cell 123 Ct[i][j][k] = fabs(Ft[i][j][k]/(rhot*volt))*dt; 124 125 126 127 /* printf("Ce=%e\n",Ce[i][j][k]);*/ 128 } 129 } 130 } 131 132 133 for(i=2;i<=nxm;i++) //Calculation of local Courant number only at internal faces 134 { 135 136 for(j=2;j<=nym;j++) 137 { 138 139 for(k=2;k<=nzm;k++) 140 { 141 142 COutD[i][j][k] = Ce[i][j][k] + Cn[i][j][k] + Ct[i][j][k]; 143 /* printf("COutD=%lf\n",COutD[i][j][k]);*/ 144 /* printf("Ce=%e\n",Ce[i][j][k]);*/ 145 /* printf("Cn=%e\n",Cn[i][j][k]);*/ 146 /* printf("Ct=%e\n",Ct[i][j][k]);*/ 147 } 148 } 149 } 150 151 152 }
3、柯朗數使用的注意事項：
在fluent中，用courant number 來調節計算的穩定性與收斂性。一般來說，隨着courantnumber 的從小到大的變化，收斂速度逐漸加快，但是穩定性逐漸降低。所以具體的問題，在計算的過程中，最好是把Courant number 從小開始設置，看看迭代殘差的收斂情況，如果收斂速度較慢而且比較穩定的話，可以適當的增加courant number 的大小，根據自己具體的問題，找出一個比較合適的courant number，讓收斂速度能夠足夠的快，而且能夠保持它的穩定性。
Generally, in the explicit schemes of differential method, Courant number is an important number which should be less than 1 in order to assure the stability. However, if the Courant number is too small, much computation time will be lost. So the Courant number could be one of those important parameters affecting computation cost and stability. we could use Courant number to control the time step in the transient simulation in CFD codes. Here is some configuration parameters which could be used in simulation with OpenFOAM。