Mathematical Physics - Volume II - Numerical Methods

4.2 Solution of Euler equations

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surface of the control volume Ω . The following applies to the continuity equation: W = ρ and F = ( ρ u , ρ v , ρ w ) T .

(4.2)

In the momentum equation W and F become: W =   ρ u ρ v ρ w and while in the energy equation

F =  

( ρ u 2 + p , ρ uv , ρ uw ) T , ( ρ uv , ρ v 2 + p , ρ vw ) T , ( ρ uw , ρ vw , ρ w 2 + p ) T ,

(4.3)

W = ρ e t (4.4) In the expressions (4.2)-(4.4) variables p , ρ , u , v , w , e t and H represent pressure, density, velocity vector projections on three Cartesian coordinate axes, energy and enthalpy, respectively. For the perfect gas system of equations (4.2)-(4.4) is extended by defining the energy of the fluid and F = ( ρ Hu , ρ Hv , ρ Hw ) T .

1 γ − 1

p ρ +

1 2

( u 2 + v 2 + w 2 ) ,

e t =

(4.5)

having in mind the relation between enthalpy H and energy e t

p ρ ,

H = e t +

(4.6)

whereby the system of Euler equations is finally closed. In the numerical approach in solving Euler equations by the finite volume method the com putational domain is discretized to the appropriate number of hexahedron-shaped cells, and for each mentioned cell, a system of equations that approximates system (4.2)-(4.6) is set up. If inside each cell ( i , j , k ) value of the dependent variable W is known, an approximate form of system of equations (4.1) is obtained: ∂ ∂ t ( h i , j , k W i , j , k )+ Q ( W ) i , j , k = 0 , (4.7) where Q ( W ) i , j , k represents the flux of flow quantities through the boundaries of observed cell. Cell volume, denoted by h i , j , k in the expression (4.7), can be considered constant in time. Flux of flow quantities through the boundaries of the observed cell is calculated as follows: Q ( W ) i , j , k = F ( W ) · S | i + 1 / 2 , j , k − F ( W ) · S | i − 1 / 2 , j , k + F ( W ) · S | i , j + 1 / 2 , k − − F ( W ) · S | i , j − 1 / 2 , k + F ( W ) · S | i , j , k + 1 / 2 − F ( W ) · S | i , j , k − 1 / 2 . (4.8) The quantity S i + 1 / 2 , j , k in the relation (4.8) represents a surface vector of the computational cell between the grid points ( i , j , k ) and ( i + 1 , j , k ) , at whereby the flux through the observed cell surface is calculated by averaging the quantity F ( W ) , calculated in points ( i , j , k ) and ( i + 1 , j , k ) . Fluxes through the remaining five surfaces of observed hexahedron are determined by an analogous procedure. The use of a central difference scheme requires introduction of additional terms in the approxi mate equation (4.7) in order to obtain stationary solutions of dynamic equations [45]. Supplementary dissipative terms, known in the literature as artificial viscosity terms, have a role to prevent even-odd decoupling of the numerical solution in equations (4.7), on the one hand, and the occurrence of high-frequency oscillations of the solution in the shock wave zone, on the other hand. Presence of dissipative terms also provides the uniqueness of the numerical solution, which is shown in the