Page 383 - Computational Fluid Dynamics for Engineers

374 12. Compressible Navier-Stokes Equations

Left-top corner i ? i j , close to inflow and the line of symmetry, i = 1,
J = J

9Ql,J _ E 1 + 1/2,J - El-1/2,J F l,J+l/2 - F l,J-l/2
dt Ax Ay

(Ey)i+i/2,j ~ (Ey)i-i/2,J (Fy)i,j+i/2 C ^ ) I , J - I / 2
+ Re Ax Ay

Note that 1 - 1/2 =^> 0 and J + 1/2 ^> J + 1

E\ + \,j+Ei j rp rp Fl,J+Fl,J-l
9Qi,j " "~2—~ ~~ ^o.J ^ i , J + i
d£ Z\x ^2/
F
E
1 (-Bv)i+i/2,J _ ( v)o,J , (-^)l,J+l ~ ( v)l,J-l /2
+ +
Re Ax ^
Right-top corner i ? / j , close to the line of symmetry and outflow,
i = I, j = J

dQl,J _ E I+l/2,J - F I-1/2,J F I,J+l/2 ~ F I,J-l/2
dt Ax Ay
F
F
E
(E v) I+l/ 2ij ~ ( v)l-l/2,J ( v)l,J+l/2 - ( v)l,J-l/2
+ Re Ax Ay
Note that / + 1/2 => I + 1 and J + 1/2 =^> J + 1

°Ql,J ^/+1,J 2 ^/,J+1 2
9^ Ax Ay
- (E v) I_ 1^ 2ij -
(E V) I+1J (E v) IJ+l (E v) IJ_ 1/ 2
Ax Ay

References

[1] Cebeci, T. and Cousteix, J.: Modeling and Computation of Boundary-Layer Flows,
Horizons Pub., Long Beach, Calif, and Springer, Heidelberg, 1998.
[2] Hirsh, C : Numerical Computation of Internal and External Flows, Volume 2, John
Wiley and Sons, N.Y., 1988.
[3] Beam, R. M. and Warming, R. F.: An implicit Factor Scheme for the compressible
Navier-Stokes Equations, AIAA Journal, Vol. 16, No. 4, 1978, pp. 393-402.
[4] Jameson, A., Schmidt, W. and Tukel, E.: Numerical Solutions of the Euler Equations
by Finite Volume Methods Using Runge-Kutta Time-Stepping Schemes, AIAA 14 th
Fluid and Plasma Dynamics Conference, June 23-25, 1981.