On Calculation of Flows of Heterogeneous Media in a Body-Force Field
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Journal of Engineering Physics and Thermophysics, Vol. 93, No. 4, July, 2020
ON CALCULATION OF FLOWS OF HETEROGENEOUS MEDIA IN A BODY-FORCE FIELD V. S. Surov
UDC 519.6:531.33
With the generalized equilibrium model of a gas-liquid mixture that was proposed by the authors earlier, the spreading of a liquid column in the presence of gravity forces is calculated using the nodal method of characteristics, which is generalized to a multidimensional case. Keywords: gravity, hyperbolic model of a medium, multidimensional nodal method of characteristics. Introduction. Many problems occurring in engineering physics are formulated with the use of balance laws of conservation with source terms [1]. Among the latter are, in particular, gravity forces. Importantly, the models described by balance relations must be hyperbolic, since the use of nonhyperbolic equations is known to lead to unstable solutions [2, 3]. In [4, 5], the influence of gravity forces on the flow of various media has been investigated numerically with finite-difference schemes. In the present work, in describing flows of heterogeneous media in a body-force field, use is made of a hyperbolic generalized equilibrium model from [6]. Numerical calculations were carried out with the nodal method of characteristics [7], which is generalized to a multidimensional case. Model of a Medium. Equations describing two-dimensional flow of a gas-liquid mixture with an incompressible disperse fraction ( ρs0 = const) in divergent form in the presence of gravity forces are of the form ∂α ∂αu ∂αv + + = 0, ∂t ∂x ∂y
∂αρs0 u ∂α ( p + ρs0 u 2 ) ∂αρs0 uv + + = − f x + αρs0 g x , ∂t ∂x ∂y
∂αρs0 v ∂αρs0 uv ∂α( p + ρs0 v 2 ) + + = − f y + αρs0 g y , ∂t ∂x ∂y ∂αρs0 es ∂α(ρs0 es + p )u ∂α(ρs0 es + p )v + + = − f x u − f y v + αρs0 ( g x u + g y v) , ∂t ∂x ∂y ∂ (1 − α)ρ 0g
+
∂t ∂ (1 − α)ρ 0g u ∂t ∂ (1 − α)ρ 0g v ∂t ∂ (1 − α)ρ 0g eg ∂t
+
∂x
∂ (1 − α)( p + ρ 0g u 2 )
+ +
∂ (1 − α)ρ 0g u
∂x ∂ (1 − α)ρ 0g vu ∂x
∂[(1 − α)(ρ 0g eg + p )u ] ∂x
+
+
+
+
∂ (1 − α )ρ 0g v ∂y
∂ (1 − α)ρ 0g uv ∂y
∂ (1 − α)( p + ρ 0g v 2 ) ∂y
∂[(1 − α)(ρ 0g eg + p )v] ∂y
= 0,
(1)
= f x + (1 − α)ρ 0g g x , = f y + (1 − α)ρ 0g g y ,
= f x u + f y v + (1 − α)ρ 0g ( g x u + g y v) ,
where gx and gy are the densities of body forces in the direction of the Ox and Oy axes, fx and fy are the densities of interfractional-interaction forces, which are a priori unknown and are determined in the process of integration of system (1) [5], ek = εk + 1/2(u2 + v2) is the specific total energy of the kth fraction (k = g, s), and εg = εg(p, ρ0g ) =
p ρ0g ( γ
− 1)
is the specific
South-Ural State University (National Research University), 76 Lenin Ave., Chelyabinsk, 454080, Russia; email: [email protected]. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 93, No. 4, pp. 911–917, July–August, 2020. Original article submitted March 4, 2019. 878
0062-0125/20/9304-0878 ©2020 Springer Science+Business Media, LLC
internal energy of the gas, where γ is the adiabatic exponent of the gas component. Notice
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