Mathematical Physics - Volume II - Numerical Methods

Chapter 4. Finite volume method

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calculated for very coarse grid ( 65 × 11 × 15 ) and shown on Figure 4.3 together with Swanson and Turkel results [45], calculateded using an explicit numeric scheme. As can be seen from Figure 4.3, good agreement of the values of the pressure coefficients is present near the leading edge of the airfoil, with minor deviations in in the immediate vicinity of the shock wave. The reason for the mentioned disagreement lies in the application of a very coarse grid. Using a finer grid would certainly lead to a qualitative improvement in the zone of very strong pressure variations.

0.12

0.1

0.08

LU STEGER

0.06

0.04

0.02

0

C L

-0.02

-0.04

-0.06

-0.08

-0.1

-0.12

0

100

200

300

400

Phase angle (°)

Figure 4.4: Dependence of lift coefficient on phase angle.

Example of non-stationary flow, shown in Figure 4.4, refers to the oscillatory translational wing motion with airfoil NACA 65A010 along the span. Change of the lift coefficient C L of the airfoil in the plane of symmetry of the wing, depending on the phase angle of the given oscillatory wing motion, is compared with the corresponding Steger results [44]. A good match of the lift coefficient followed by small phase angle deviation is quite obvious. Numerical stability and fast convergence of system of differential equations solution represent a significant improvement in relation to the application of the explicit method. LU implicit factorization by applying flux “decomposition” allow the use of very large integration steps thanks to its stability, even for Courant number CFL ≥ 20. Increased scheme stability made the exposed procedure acceptable in the analysis of non-stationary flows in three-dimensional space, when the integration step has crucial role. On the other hand, the presence of only two factors in the three-dimensional flow case reduces the factorization error. Relative to application of the classical ADI scheme, the presented procedure also showed advantage in terms of CPU time savings. The way of solving systems of equations does not require a large memory space and it consists of the inversion of fifth-order square matrices. Correct program coding allows easy vectorization and application on supercomputers. The implemented modification of the scheme at the boundaries of the physical domain provided precise definition of boundary conditions, and thus high accuracy of the solution. Results obtained by applying this numerical procedure in the analysis of three-dimensional fluid flow [31] showed a good agreement with the available literature data, based on different