Mathematical Physics - Volume II - Numerical Methods

Chapter 4. Finite volume method

138

Characteristic form of Euler equations in a one-dimensional flow field, written in vari ables W , implies an uncoupled system of equations ∂ W ∂ t + Λ ∂ W ∂ x = L − 1 e Q , (4.93) bearing in mind that variations of characteristic variables δ w 1 = δρ − 1 c 2 δ p , δ w 2 = δ u + 1 ρ c δ p and δ w 3 = δ u − 1 ρ c δ p (4.94) extend along the corresponding characteristics at velocities u , u + c and u − c , respectively. In the case of a three-dimensional flow variations of characteristic variables become: δ w 1 = δρ − 1 c 2 δ p , δ w 2 = ˆ k x δ w − ˆ k z δ u , δ w 3 = ˆ k y δ u − ˆ k x δ v , δ w 4 = l k δ v + 1 ρ c δ p , δ w 5 = − l k δ v + 1 ρ c δ p , (4.95) where u , v and w are projections of the velocity vector v on the axes of the Cartesian coordinate system. Quantities ˆ k x , ˆ k y and ˆ k z in expressions (4.95) denote the projections of the unit vector l k on corresponding coordinate axes, where the vector l k determines the direction of wave propagation. Boundary conditions The considerations in the chapter 4.2.3 have a direct influence on the number of boundary conditions imposed on the physical boundaries in the one-dimensional non-viscous flow of the ideal fluid. The number of boundary conditions directly depends on the direction of wave propagation in a fluid along the characteristics at the boundaries of the flow domain. The problem will be well posed if complete information on characteristics directed towards the flow domain or oppositely oriented characteristics can be obtained by the appropriate combination of known conservative or basic ( nonconservative ) variables .

t

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1 P

P 0

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( ) b Subsonic flow

( ) a Supersonic flow

Figure 4.9: Boundary conditions in one-dimensional flow.

Number of physical boundary conditions, ie. variables, to be imposed at the boundaries of the flow domain depends on direction of propagating waves, ie. on the way the