Shock Waves (2018) 28:765–784
https://doi.org/10.1007/s00193-017-0772-2
ORIGINAL ARTICLE
Supersonic flow gradients at an overexpanded nozzle lip
M. V. Silnikov
1
· M. V. Chernyshov
2,3
Received: 30 March 2017 / Revised: 5 October 2017 / Accepted: 9 October 2017 / Published online: 13 November 2017
© Springer-Verlag GmbH Germany 2017
Abstract The flowfield of a planar, overexpanded jet flow
and an axisymmetric one are analyzed theoretically for a
wide range of governing flow parameters (such as the noz-
zle divergence angle, the initial flow Mach number, the jet
expansion ratio, and the ratio of specific heats). Significant
differences are discovered between these parameters of the
incident shock and the downstream flow for a planar jet and
for an axisymmetric overexpanded jet flow. Incident shock
curvature, shock strength variation, the geometrical curvature
of the jet boundary, gradients of total and static pressure and
Mach number, and flow vorticity parameters in post-shock
flow are studied theoretically for non-separated nozzle flows.
Flow parameters indicating zero and extrema values of these
gradients are reported. Some theoretical results (such as con-
cavities of incident shock and jet boundary, local decreases
in the incident shock strength, increases and decreases in the
static pressure, and the Mach number downstream of the inci-
dent shock) seem rather specific and non-evident at first sight.
The theoretical results, achieved while using an inviscid flow
model, are compared and confirmed with experimental data
obtained by other authors.
Communicated by F. Lu and A. Higgins.
B
M. V. Chernyshov
mvcher@mail.ru
1
Special Materials, Corp., 28A Bolshoy Sampsonievsky Ave.,
Saint Petersburg, Russia 194044
2
Department of Extreme Processes in Materials and Blast
Safety, Peter the Great Saint Petersburg Polytechnic
University, 29 Polytekhnicheskaya Str., Saint Petersburg,
Russia 195251
3
Department of Plasma and Gas Dynamics, Baltic State
Technical University “VOENMEH”, 1, 1st
Krasnoarmeyskaya Str., Saint Petersburg, Russia 190005
Keywords Supersonic overexpanded nozzle · Shock ·
Streamline · Curvature · Gradients
List of symbols
A Nozzle edge; the analysis is con-
ducted in its vicinity
AB The jet boundary
AT The incident shock wave
er Jet expansion (overexpansion) ratio
(relation of flow pressure at thenoz-
zle exit and the ambient one)
I Relation of stagnation pressures
downstream and upstream of the
shock
J Shock strength (staticpressureratio
across the shock wave)
J
c
Shock strength at the Crocco point
J
∗
Shock strength for sonic flow veloc-
ity downstream
J
Γ
Shock strength for the envelope line
of the family of shock polars
J
l
Shock strength for maximum flow
deflection (among all shocks at
given Mach number)
J
m
Shock strength corresponding to a
normal shock
J
p
Shock strength at constant pressure
(Thomas) point
K
σ
The geometrical curvature of the
shock
K
−
σ
= rK
σ
/sin θ,
K
σ
= rK
σ
Dimensionless geometrical curva-
tures of the incident shock intro-
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