The moment model for large mach number flows

Аuthors

Nikitchenko Y. A.

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

e-mail: nikitchenko7@yandex.ru

Abstract

The paper studies the physico-mathematical models of the flows, which are based on a system of moment equations. The main shortcoming of the moment equations systems is their short-wave instability. This flaw limits the use of the models in hypersonic regions of the flow.

In [1] the system of Grads moment equations is extended to polyatomic gases. The publication offers a method of constructing such system with a smaller number of additional assumptions. The obtained 24- moment system has the short-wave instability properties, which are similar to those of Grads 20-moment system.

Publication [2] showed that the main cause of the short-wave instability consists in the fact that the expressions for closure moments (4-th order moments), which were obtained by using the approximating distribution functions, do not satisfy the closure moments equations. This means that there is a discrepancy between the local and balance expressions for the closure moments. Thus the publication proposed the methods of the short-wave instability reduction.

One of the methods was connected with the expansion of the moment equations system. Additional members (matching additives) were introduced into the local expressions for closure moments. The resulting expressions were ins erted in to the closure moments equation. The differential equations of matching additives were obtained as a result. The original 24-moment system together with the matching additive equations created a 45-moment system of equations. This system allowed the research team to expand the region of solutions that are free from short-wave instability.

This work considers the method of [2], which is connected with the selection of the members with the highest order of magnitude from the matching additive equations. The paper considers two techniques of selecting the members with the highest order of magnitude.

The first method consists in excluding the non- equilibrium stresses even from the members with the highest order of magnitude. The moments of the 3rd order, which are included into the members with the highest order of magnitude, are left unaltered.

The second method implies excluding the deviatoric part from the moments of the 3rd order in addition to the above-mentioned alterations.

The algebraic expressions for matching additives are obtained as a result of this exclusion. The obtained algebraic expressions are included into the differential equations of the 3rd-order moments.

The usage of the proposed methods is illustrated on the example of the solution of the problem of the shock wave profile determination. This example shows that the obtained flow model allows the researcher to expand the region of solutions, which are free from the short-wave instability, close to the hypersonic Mach numbers ( M≈ 5 ). The second method of selection of members with the highest order of magnitude has significant advantages.

Keywords:

moment equations, closure moments, short-wave instability

References

1. Nikitchenko Yu.A. Polet, 2010, no.11, pp. 43-51.
2. Nikitchenko Yu.A. Modeli neravnovesnykh techenii (Models of nonequilibrium flows), Moscow, MAI, 2013, 160 p.