Mathematical Physics - Volume II - Numerical Methods

Chapter 4. Finite volume method

152

while the matrices A + , B + , C + , A − , B − , and C − are determined by expressions (4.20.2). The process of solving the first system of equations (4.129.1) takes place in the direction of increasing index ( i , j , k ) by replacing the known, previously calculated flow variable vector ∆ q ∗ n , while the second system of equations is solved in the direction of decreasing index ( i , j , k ) . On the body surface boundary conditions are defined by zero flow velocity vector with the adopted assumption of adiabatic flow and zero pressure gradient [1] and [38]

∂ H ∂ n

∂ p ∂ n

u = v = w = 0 ,

= 0 and

= 0 .

(4.129.4)

where H is the enthalpy of the fluid. Defining boundary conditions at the outer boundaries of the physical domain has already been explained in chapter 4.2.2.

1

0.75

0.5

0.25

-C p

0

-0.25

EULER (65 x 12 x 15) NAVIER-STOKES laminar (65 x 7 x 29) NAVIER-STOKES turbulent (65 x 7 x 29)

-0.5

-0.75

-1

0

0.2

0.4

0.6

0.8

1

x/l

Figure 4.19: Pressure distribution on a rectangular wing – plane of symmetry. Application of central difference scheme in calculation of the residual term R n in the equation (4.128.1) requires the introduction of additional terms in order to obtain stationary solutions of dynamic equations [45]. Additional dissipative terms are introduced in the same way as in the chapter 4.2.2. After the introduction of dissipative terms, the system the equations (4.129) receives the final form n I + β ∆ t h δ − ξ A + + δ − η B + + δ − ζ ( C + − C + v ) io n ∗ ∗ n I + β ∆ t h δ + ξ A − + δ + η B − + δ + ζ ( C − − C − v ) io n ∆ q n + ∆ t R n − R n 1 = 0 , (4.130) noting that the integration step has been determined previously by relations (4.12) and (4.12.1).