Self-similarity parameter effect on critical characteristics of Hamel type compressible flow

Аuthors

Brutyan M. A.^1*, Ibragimov U. G.^2**

1. Central Aerohydrodynamic Institute named after N.E. Zhukovsky (TsAGI), 1, Zhukovsky str., Zhukovsky, Moscow Region, 140180, Russia
2. Moscow Institute of Physics and Technology (National Research University), 9, Institutskiy per., Dolgoprudny, Moscow region, 141701, Russia

*e-mail: m_brut@mail.ru
**e-mail: umar.ibragimov333@yandex.ru

Abstract

The article envisages the stationary flow of a viscous compressible gas flowing out the vertex of cone.

This flow is an analog of the famous Hamel flow between the two plane walls, inclined to each other at an angle. Unlike the classical case, axisymmetric compressible flow is considered. It is worth mentioning, that the studies on this theme were being carried out earlier by Williams (1967), Byrkin (1969) and others.

In this work the flow inside a cone is assumed radial, i.e. only one velocity component is not trivial: V^*=(u^*,0,0). Moreover, all flow gas-dynamic parameters are assumed dependent from the distance to the source r^* by the power law, while conductivity coefficients are temperature dependent by the Frost law η^*, κ^*∼T^*k, so:

It appeared that self-similarity solutions existed only at the certain relation between the self-similarity parameter m and the power law index k in the form of 2mk=1.

In this case, the system of Navier–Stokes equations written in spherical coordinates is being reduced to the system of nonlinear second-order ordinary differential equations (ODE). The problem key parameters are included to these equations and boundary conditions. They are α, Re₀, M₀, T_w, Q. It was established that the solution was determined by specifying only two parameters, while three others were being found automatically.

The nonlinear ODE system was being solved at various values of self-similarity parameters: m = 5·(k = 1), m≅0.66 · ( k = 0.76 ) and m = 1 · (k = 0.5 ).The first combination corresponds to Maxwell’s molecules model, second relates to the well-known Frost’s empirical model (η, κ~T^0.76 ), and the third corresponds to the Hard Spheres model.

It was established that the ODE system solution existed only at the cone semi-opening angle less than some critical value of α_*. With larger α, the medium continuity assumption becomes inapplicable (Kn=M₀/Re₀→1)  , so the gas temperature on the cone wall appears negative , which is physically unacceptable.

Thus, the presented work performed the analysis of viscous compressible gas flow in a cone at various values of self-similarity parameter m and set corresponding critical flow characteristics.

Keywords:

Navier–Stokes equations, exact solutions, viscous gas axisymmetric flow

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