BAIL 2006 Book of Abstracts - Institut für Numerische und ...

N.V. TARASOVA: Full asymptotic analysis of the Navier-Stokes equations in the
problems of gas flows over bodies with large Reynolds number
✬
✫
the application of the Euler equations for incompressible flow outside the boundary layer and the
classical equations of the compressible boundary layer near the body surface makes impossible
the agreement between both equation systems.
To consider the proposed problems from common positions another special dimensionless
variables for gas parameters in the Navier-Stokes equations are introduced. These new variables
are varied inside the studied flow area usually from 0 to 1 (in other words, they can vary to the
value of the order of unit). The main idea to use such special dimensionless variables is that in
this case all governing parameters that can be both small and not small are indicated explicitly
as the coefficients in the equation system. Among these parameters are M 2 ∞ and ∆T/T0 (∆T is
the temperature drop and T0 is the characteristic value of the temperature in the area).
It should be noticed that in the hyposonic flows the gravity force can affect significantly the
gas flow (in studies of free and induced convection). In term of asymptotic analysis it means
that it is necessary to take account of one more parameter, for example 1/Fr (Fr is the Frud
number) that can be both small and not small.
The full asymptotic analysis is carried out for all mentioned situations on the base of
comparison of all parameters mentioned above (M 2 ∞, ∆T/T0, 1/Fr) with the standard small
parameter ε in order of magnitude. As a result, the model for gas flow in both areas is formulated
and an attempt to construct the procedure of the agreement of the solutions in both areas is
made. It should be stressed that the equation systems derived under the assumptions 1) - 4)
differ from each other.
This model constructed for the cases 2) and 3) possibly can occupy the intermediate place
between two classical approaches when a gas is considered as incompressible or compressible one
over the whole flow field.
The equations describing hyposonic flows (M → 0) with the arbitrary values of the Reynolds
number (Re), ∆T/T0 and 1/Fr were derived in [2] from the Navier-Stokes equations on the base
of the asymptotic analysis with the Mach number (M) as a small parameter. In this work an
attempt to compare the model constructed for cases 2) and 3) with the results obtained in [2]
when considered flow with the large Reynolds number was made.
References
1. M. Van Dyke. 1962 In: Hypersonic Flow Research (Ed. F.R.Riddell). Academic Press.
2. A.I. Zhmakin, Yu.N. Makarov. 1985 Numerical modelling of hyposonic flows of viscous gas,
Dokl. AN SSSR, Mekh. Zh. i Gaza 280 (4), 827–830. [in Russian]
Speaker: TARASOVA, N.V. 144 BAIL 2006
2
✩
✪