FLOW ON THE LEEWARD SIDE OF A SUPERSONIC SOURCE IN A

SUPERSONIC STREAM

International Space Science Institute

Team Meeting “Modeling Cometary Environments in the Context of the Heritage of the Giotto Mission to Comet

Halley”

19—24 November, 2012

M. G. LEBEDEV

• Unexpected and interesting phenomena can occur on the leeward (night) side of a comet in the solar wind flow. One of these phenomena can be the formation of return (circulatory) flow zones.

• The results of Dr. Benna’s simulation hint at this possibility

Calculation using the Babenko—Rusanov method (1980)

Calculation of supersonic flow past a supersonic source (windward side)

Formulation of the problem in the tail flow region

Streamlines General flow pattern

Velocity profile along the axis of symmetry

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Calculation of the flow on the leeward side using the Godunov method

Longitudinal velocity and pressure distributions in the X = 39 section

Calculated results

Density contours Pressure contours

Longitudinal velocity contoursVertical velocity contours

Another (time-dependent) formulation of the same problem with the formation of the return-flow zones

The similar situation can occur in subsonic flow past a supersonic source

Reflection of a shock wave from the axis of symmetry in uniform, wake, and source flows

11 – nozzle– nozzle; ; 2 2 – flame stabilizer– flame stabilizer3 3 – thermal wake – thermal wake from combustionfrom combustion4 4 – shock wave– shock wave5 5 –recirculation –recirculation zonezone

Hydrogen burns behind a cylindrical stabilizer, G.Winterfeld,1968.

Formation of return flow zones on reflection of an incident Formation of return flow zones on reflection of an incident shock from the axis of symmetry in a wake-type flowshock from the axis of symmetry in a wake-type flow

1 1 –– nozzle; nozzle; 22 – – jetlet; jetlet; 33 – – shock;shock;44 –– recirculation recirculation zonezone

G. F. Glotov, 1994

Low-pressure jetlet in a supersonic Low-pressure jetlet in a supersonic underexpanded jetunderexpanded jet

B.J. Gribben, K.J. Badcock, B.E. Richards. Numerical study of shock-reflection hysteresis in an underexpanded jet // AIAA Journal. 2000. V. 38. N. 2. P. 275—283.

M. Frey. Behandlung von Strömungsproblem in Racketendüsen bei Überexpansion // Inst. für Aerodynamik und Gasdynamik, Univ. Stuttgart. Dr.-Ing. Diss. 2001. (http://elib.uni-stuttgart.de/opus/volltexte/2001/800/pdf/diss_frey.pdf)

В.А. Горяйнов. О возможности реверса течения в свободных сверхзвуковых струях // Мат. моделирование. 2003. Т. 15. № 7. С. 86—92.

О.В. Бочарова, М.Г. Лебедев, А.В. Савин, Е.И. Соколов. Стационарные циркуляционные зоны в сверхзвуковых неравномерных потоках // XXI Школа-семинар ЦАГИ «Аэродинамика летательных аппаратов». Тезисы докладов М.: Изд. ЦАГИ. 2010. С. ??--??.

The existence of these experimentally observed structures was confirmed in numerical calculations.

So far, in the case of the source-type nonuniformity analogous structures were obtained only in numerical experiments

Shock reflection in imperfectly expanded jets

Numerical experiment by M. Frey

Our calculations of return flow zones in supersonic underexpanded jets (M = 3, n = 3.5)

To confirm these results, recently we calculated some flows with the formation of circulation, or return, or reverse, zones

The following numerical methods were employed

1. Godunov method (first order)

2. Method of adaptive artificial viscosity (second order of accuracy, on irregular, triangular grids)

developed by I.V. Popov and I.V. Fryazinov in Keldysh Institute of Applied Mathematics.

3. Babenko—Rusanov method (shock-fitting technique of the second order of accuracy).

For testing the technique the wake-type flow experimentally studied by Glotov was numerically modeled.

Numerical calculation (streamlines)Glotov’s experiment

Density c ontours for the above calculation

The problem of supersonic spherical source flow in a cylindrical channel

Calculated flow pattern at gamma = 1.4

Calculated flow pattern atgamma = 1.05

Calculations by the AAV method

Source flow with nonuniform angular velocity distribution (a maximum velocity is reached at the channel axis). The initial data correspond to the case of “uniform” source (gamma = 1.1). The return flow zone disappears.

In this case the velocity on the axis is minimum. As a result, the return flow zone enlarges.

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