5.1 Shock Waves

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5.1 Shock Waves
CH.5 NORMAL SHOCK WAVES 5.1 Shock
Waves 1.Definition of Shock Wave and Normal
Shock Wave 1 Shock Wave The extremely thin
region in which the transition from the
supersonic velocity, relatively low pressure
state to the state that involves a relatively low
velocity and high pressure is termed a shock
wave. finite pressure disturbance, compressive
elastic wave of finite strength nonlinear
addition of the individual Mach waves emitted
from each point on the body of finite thickness
(or finite attack angle, not point projectile).
The shock wave is an abrupt disturbance that
causes discontinuous and irreversible changes in
such fluid properties as
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speed, which changes from supersonic to lower
speeds, pressure, temperature, and density.(18
p138) 2 Normal Shock Wave a shock wave which
is straight with the flow at right angles to the
wave. a plane surface of discontinuity normal
to flow direction stationary normal shock
waves and moving normal shock waves (T p80
Fig.5.7) 2. Case of Formation 1) Internal
Flow When a supersonic flow decelerates in
response to a sharp increase in pressure. (e.g.
flow in de Laval nozzle ) 2) External Flow
When a supersonic flow encounters a sudden
compressive change in flow direction. (e.g.
concave corner )
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A spectacular hydraulic jump at the end of a
spillway.
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3. Characteristics 1 irreversible, adiabatic
process As a result of the gradients in
temperature and velocity that are create by a
shock, heat is transferred and energy is
dissipated within the gas, and these processes
are the thermodynamically irreversible. (18
p138) 2 discontinuities in properties 3
thickness order of several
molecular mean free paths 4 shape A shock
wave is, in general, curved. However, many shock
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• waves that occur in practical situations are
  straight, being either at right angles to or at
  an angle to the upstream flow.
• 5 summary(18 p156)
• - remains constant and is equal
  to 1.
• decreases and approaches zero as
  approaches infinity.
• 
  increase with increasing
• 5. Difference between a shock wave and a sound
  wave
• 1 sound wave
• infinitesimal deflection of the stream due to
  the body
• (infinitesimal pressure disturbance)
• A Mach wave represents a surface across which
  some derivatives of the flow variables may be
  discontinuous while the

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• variables themselves are continuous. So the
  characteristic curves (or Mach lines) are
  patching lines for continuous flow.
• Governing eq. ? differential form
• 2 shock wave
• finite deflection of the stream due to the
  body finite pressure disturbance nonlinear
  addition of the sound wave
• A shock wave represents a surface across
  which the thermodynamic properties and the flow
  velocity as well as their derivatives are
  essentially discontinuous. So shock waves are
  patching lines for discontinuous flow
• 
  across shock waves.
• Governing eq. ? integral form
• Shock waves propagate faster than Mach waves
  do, and they

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show large gradients in pressure and in
density.(18 p140) 6. Formation of Shock (see
T p81 Fig.5.8)
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5.2 Stationary Normal Shock Waves
5.2 Stationary Normal Shock Waves 1. Basic
Assumptions one-dimensional steady flow
(stationary wave) frictionless flow (no
boundary layer) Shock process takes place at
constant area. shock wave ? streamlines
adiabatic flow without external work, negligible
body force 2. Governing Equations 1 control
volume (seeT p82 Fig. 5.9,1 p69 Fig. 4.8,
Fig. 4.9, 18 p144 Fig. 4.5) 2 unknown
variables
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in
x-direction 3 Rankine-Hugoniot Normal Shock Wave
Relation(18 p157-158 p158 Fig. 4.12) 1) The
set of equations which give
in terms of the shock strength
is often termed the Rankine-Hugoniot shock wave
relations.
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• 3) Rankine-Hugoniot Curve for
• At low pressure ratios, the Hugoniot curve and
  isentropic curve differ only slightly from each
  other, so that a weak shock appears like an
  isentropic process.

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- At the large pressure ratios which characterize
strong shocks, the density ratio reaches the
limiting value of in isentropic process,
however, the density ratio increases
constantly. 4) Remarks - Shock wave must
always be compressive, i.e. , that must
be greater than 1, i.e., the pressure must always
increase across the shock wave. And the density
always increases, the velocity always decreases,
and the temperature always increases across a
shock wave.
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Example 5.1 ltSol.gt
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5.3 Normal Shock Wave Relations in terms of Mach
Number

• 5.3 Normal Shock Wave Relations in terms of Mach
  Number
• 1. Eq. of state, speed of sound
• 2. Adiabatic process isoenergetic
• and also (18 p145-146)
• 3. Momentum Equation

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4. Continuity equation
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Since will be between 1 and 2, and since
is always greater than 1, it follows from this
equation that will always be less than 1,
i.e., the flow downstream of a normal shock wave
will always be subsonic. 5.( )-( )
(see 1 p71 Fig. 4.10, 18 p148, Fig. 4.7)
6. (see 1 p71, Fig. 4.11, 18 p150
Fig. 4.8)
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So expansion shock is
impossible. 2./ Other Important Normal Shock
Relation for a Calorically Perfect Gas in
terms of and 1. Properties ratio between
in front of and behind the shock wave
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often termed the strength of
the shock wave
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2. Shock Strength
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Example 5.2 ltSol.gt
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Example 5.3 ltSolgt
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• 3./ Limiting Cases of Normal Shock Wave
  Relations(see T p97 Fig. 5.17)
• 1.Very Strong Normal Shock a normal shock wave
  for which
• Thus, if tends to infinity,
  and tends to infinity but
  and tend to finite values. So the
  assumption that the gas remains thermally and
  calorically perfect will cease to be valid when
  the shock is very strong because very high
  temperatures will then usually exist behind the
  shock.
• 2.Very Weak Normal Shock Wave (ß1)
• The weak shock relations apply if .
• 1

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2 - if is not significantly larger
than unity ? - or criterion of
irreversibility of a weak normal shock wave
(1 p66 Fig.4.4) (1 p67 Fig.4.5)
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Example 5.4 ltSol.gt
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Normal shock wave at M1.5 A pattern of pairs of
weak oblique shock waves is produced by strips of
tape on the floor and ceiling of a supersonic
nozzel. They terminate at an almost straight and
normal shock wave, showing that the flow is
subsonic downstream. U.S. Air Force photograph,
courtesy of Arnold Engineering Development Center
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Cone-cylinder in supersonic free flight A
cone-cylinder of 12.5 deg. semi-vertex angle is
shot through air at M1.84. The boundary layer
becomes turbulent shortly behind the vertex, and
generates Mach waves that are visible in this
shadowgraph.Photographs by A.C. Charters
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Sphere at M4.01 This shadowgraph of a 1/2-inch
sphere in free flight through atmosphereic air
shows boundary layer separation just behind the
equator, accompanied by a weak shock wave, and
formation of the N-wave that is heard as a double
boom far away. The vertical line is a reference
cord. Photograph by A.C. Charters
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Model in wind tunnel
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