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Behavior of shock trains in a diverging duct

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Abstract

A shock train inside a diverging duct is analyzed at different pressure levels and Mach numbers. Nonreactive pressurized cold gas is used as fluid. The structure and pressure recovery inside the shock train is analyzed by means of wall pressure measurements, Schlieren images and total pressure probes. During the course of the experiments, the total pressure of the flow, the back pressure level and the Mach number upstream of the compression region have been varied. It is shown that the Reynolds number has some small effect on the shock position and length of the shock train. However, more dominant is the effect of the confinement level and Mach number. The results are compared with analytical and empirical models from the literature. It was found that the empirical pseudo-shock model from Billig and the analytical mass averaging model from Matsuo are suitable to compute the pressure gradient along the shock train and total pressure loss, respectively.

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Abbreviations

a :

Speed of sound (m/s)

c :

Empirical constant 0.114

D :

Diameter or equivalent diameter of the duct (m)

H :

Duct height (m)

L p :

Length of the pseudo-shock (m)

Ma:

Mach number

Ma′:

Mean Mach number of the core flow

Ma′′:

Mean Mach number of subsonic outer region

\( {\bar{\text{M}}\text{a}}_{1} \) :

Mass averaged upstream Mach number

\( \dot{m} \) :

Mass flow (kg/s)

n :

Experimentally determined constant n = 2.2

p 0 :

Total pressure in the settling chamber (bar)

p/p1:

Ratio of local wall pressure to upstream static pressure

Rex :

Reynolds number based on the distance from the nozzle throat

Re*:

Reynolds number based on throat height

Reθ :

Reynolds number based on boundary layer momentum thickness

T o :

Total temperature in the settling chamber (K)

u :

Flow velocity (m/s)

w*:

Crocco number at sonic conditions \( w^{*} = \sqrt {{{\left( {\gamma - 1} \right)} \mathord{\left/ {\vphantom {{\left( {\gamma - 1} \right)} {\left( {\gamma + 1} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\gamma + 1} \right)}}} \)

w′ :

Crocco number in the isentropic core \( w' = {u \mathord{\left/ {\vphantom {u {\sqrt {2c_{p} T_{o} } }}} \right. \kern-\nulldelimiterspace} {\sqrt {2c_{p} T_{o} } }} \)

w 1,2 :

Crocco number upstream/downstream of pressure rise \( w_{1,2} = {{u_{1,2} } \mathord{\left/ {\vphantom {{u_{1,2} } {\sqrt {2c_{p} T_{o} } }}} \right. \kern-\nulldelimiterspace} {\sqrt {2c_{p} T_{o} } }} \)

x :

Distance downstream from the beginning of the pressure rise (m)

α1,2,3,4 :

Diverging half-angle of Laval nozzle (°)

δ:

Boundary layer thickness (mm)

δ*:

Boundary layer displacement thickness (mm)

θ:

Boundary layer momentum thickness for undisturbed flow (mm)

γ:

Isentropic exponent

σ:

Correction factor for the mass averaging pseudo-shock model

ξ:

Correction factor for the mass averaging pseudo-shock model

ρ:

Density (kg/m³)

μ:

Mass flow ratio between boundary layer and core flow

1:

Flow condition upstream of the shock system

2:

Flow conditions downstream of the shock system

e :

Free stream or edge of boundary layer

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Acknowledgments

The research reported in this paper was funded by the DFG (German research foundation) PAK 75/1 ‘A new gas dynamic process for the production of nanoparticles’. Thanks to Thomas Gawehn from the DLR Cologne and Nisar Al-Hasan from the Technical University Munich for many fruitful discussions.

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Authors and Affiliations

  1. Shock Wave Laboratory, RWTH Aachen University, 52056, Aachen, Germany

    A. Weiss, A. Grzona & H. Olivier

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  1. A. Weiss
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  2. A. Grzona
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  3. H. Olivier
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Correspondence to A. Weiss.

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Weiss, A., Grzona, A. & Olivier, H. Behavior of shock trains in a diverging duct. Exp Fluids 49, 355–365 (2010). https://doi.org/10.1007/s00348-009-0764-9

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