## Gas Dynamics & Supersonic FlowCompressible Flow Equations of Motion 1-D Isentropic Relations Wave Propagation Flow through Nozzles and Ducts 2-D Compressible Flow Prandtl-Meyer Expansion Shock Interactions Shock-Expansion Techniques for Aerofoils Method of Characteristics Unsteady Supersonic Flow Flow Tables/Software |
## Unsteady Gas Flow
Unsteady gas flows are among the most specialised topics in gasdynamics. Many aspects of compressible flow have already been studied in detail, including one dimensional flows, isentropic flows, shock waves and two dimensional supersonic flows. In addition there are many situations where the flow is unsteady such as in shock tubes, explosions and acoustics. To start with this section will cover :
- Moving normal shock waves
- reflection of a moving shock
- unsteady expansion waves
- shock tube flow.
Unsteady flow means that flow properties at points in the flow over time. These flows require additional considerations due to the rate of change of flow properties. In many cases a closed form analytical solution is not possible and numerical tools such as CFD will be required. In this section an understanding of the basic physics is covered so that flows have been restricted to variation in one dimension. Several simplifying assumptions are also required to arrive at a closed form solution. - The flow is generally inviscid. Shocks are thus lines of singularity where this assumption is broken.
- The flow is adiabatic. Heat transfer into or out of the system is not considered. Temperature changes will only occur due to internal state changes in the gas.
- The gas is perfect.
- The flow is
one dimensional and thus there are no area changes. I.e.
**dA=0**.
## Moving Normal Shock Waves.Compared to a stationary observer, there are cases such as for the shock tube and blast waves where the shock wave is moving. In many cases by a change of frame of reference to one that is moving with the shock wave, it is possible to analyse a moving shock by the same methods as has been shown previously for a stationary normal shock. ## Generation of a Moving Normal ShockIn a previous
section the formation of a shock in a piston-cylinder arrangement was
shown. The cylinder is filled with gas at rest. At t=0 the piston is
pushed at high speed to the right ( The speed of the
shock wave will be Figure
59. Unsteady Shock produced by Piston Motion.
The motion of shock
wave and piston can be represented by plotting position versus time
in an Figure
60. x-t Diagram for Piston induced Shock Motion.The
shock wave speed is typically expressed
in terms of a Shock Mach Number, ## Calculation of Flow behind a Moving Shock WaveIn a previous section the equations of motion governing the flow for a situation when the shock is stationary have been presented. A similar strategy can be taken in the case of a shock wave moving at constant speed. By applying a change of reference frame and applying an equal and opposite speed W everywhere to the flow, then the problem again becomes one of a stationary shock. Figure 61. Frame of Reference Change for Moving WaveThe equations governing this flow are, $ρ_1W=ρ_2(W-u^'_p)$ --- conservation of mass through shock (continuity) $P_1+ρ_1W^2=P_2+ρ_2(W-u^'_p)^2$ --- conservation of momentum $h_1+W^2/2=h_2+{(W-u^'_p)^2}/2$ --- conservation of energy The continuity equation can be arranged as follows, Substituting this into the momentum equation and simplifying gives, or Specific
enthalpy or in terms of specific volume This
result is called the If
it is now assumed that the gas is a perfect gas, ie. $$ρ_2/ρ_1={1+{γ+1}/{γ-1}(P_2/P_1)}/{{γ+1}/{γ-1}+ P_2/P_1}$$ A more convenient form of these equations is obtained by expressing them in terms of shock Mach number. The shock wave speed will be The particle speed for gas behind the shock will be, ## Shock Induced MotionFor a stationary shock , flow into the shock is supersonic and flow behind the shock is subsonic. For a moving shock due the change of reference, if the flow in-front of the shock is stationary, then the Mach number of the flow behind the shock can be calculated as follows, $$M_2=1/γ(P_2/P_1-1)({{2γ}/{γ+1}}/{P_2/P_1+{γ-1}/{γ+1}})^{1/2}({1+{γ+1}/{γ-1}P_2/P_1}/{{γ+1}/{γ-1}P_2/P_1+(P_2/P_1)^2})^{1/2}$$ as For
air with $γ=1.4$ the
maximum Mach
number of the induced flow is ## Distinction between Steady and Unsteady FlowsA change of frame of reference has been used to calculate the motion of the shock and values of the static properties of the gas. The ratio of these properties across the shock remain the same in both cases, steady flow – unsteady flow. There is however a difference when considering the stagnation conditions. For a steady flow normal shock stagnation temperature is unchanged through the shock wave, this is not the case for unsteady flow. ## Limits for Large Shock Mach NumberWhen the shock Mach number is large the equations can be simplified as follows ## Reflection of a Moving ShockThe moving shock impacts a stationary wall then it will be reflected back into the disturbed flow. The
sequence of events is shown in t=t
. It is then reflected and moves to the left at a speed of _{2}W.
The stationary wall will bring the flow to rest so the gas in state _{R}3
, between the reflected wave and the wall, is stationary. The
reflected shock has the opposite effect on the flow as compared to
the initial moving shock. The kinetic energy of the moving gas (2)
is converted to internal energy (3).
The effect is to produce a high temperature and pressure in region
(3).The
x-t diagram for the reflected shock is shown in $u^'_p$. This motion is brought to rest when the reflected shock impacts the
particles downstream location.
Figure 62. Reflection of a
Moving Shock
## Calculation of Reflected Shock Speed and PropertiesFor the reflected shock,the governing equations are applied to a frame of reference where the reflected shock is stationary.
For this case, $$P_2+ρ_2(W_R+u^'_p)^2=P_3+ρ_3W_R^2$$ $$h_2+{(W_R+u^'_p)^2}/2=h_3+ W_R^2/2$$ By definition, By substitution and rearranging the above equations with the Mach number definitions, Thus
the reflected shock Mach number , γ).
In
the limiting case of When M_{s}$$T_3/T_1≈{2(γ-1)(3γ-1)}/{(γ+1)^2}M_s^2$$ Pressure and Temperature reach very high values behind a reflected shock. ## Unsteady Expansion Waves
Expansion
waves are the ones through which the gas expands. Similar to moving
shock waves, there are also moving or unsteady expansion waves. In
the piston-cylinder arrangement, it is possible to have the piston
moving away from the gas instead of into the gas as shown in As the piston moves to the right the gas in the cylinder expands, its pressure decreases. This occurs through a sequence of expansion waves. Unlike the shock compression situation, each expansion wave is generated in a medium that has a lower temperature and pressure than the previous instant and hence the wave speed is smaller. Consequently every subsequent wave travels slower than the one before. There is no coalescing of waves, instead they move apart.
## Shock Tube FlowThe above theory for unsteady flows is used in a typical application call a Shock Tube. This is a short duration facility creating moving shocks along a tube into a test region. It has many uses. It is used by Physicists and Chemists to study high temperature and high enthalpy gas effects and in the creation of chemical lasers. Aerodynamicists use it to study flow about objects places in a test section for supersonic or hypersonic flow. ## Workings of a Shock Tube
A
shock tube essentially consists of a high pressure gas region
initially separated form a low pressure gas region by a thin
diaphragm. The chamber containing the high pressure gas (region 4) is
called a Driver Gas and the one containing the low pressure gas
( The flow is initiated by rupturing the diaphragm separating the two regions. This can be done up over pressure or more accurately by a hydraulically actuated cutting needle.
The
rupture of the diaphragm causes the high pressure gas to rush into
the low pressure driven tube. This gas motion produces a piston
effect and thus creates a shock wave moving at speed
At
a small time after initiation, there will be four states, the
additional ones will be ( There are a number of variations in terms of hardware that can be used to create a shock tube. The diaphragm may be a plastic sheet or aluminium plate depending on the initial pressure ratio. Instead of a diaphragm a fast action valve may be used. The sheet rupture has the advantage of being faster in creating a shock wave but the disadvantage of sending pieces of plastic or aluminium down the tube with the flow. In many cases the diaphragm is scribed with a 'petal' pattern so that its rupture is better controlled and extra debris in the flow is avoided.
## Governing Equations for Shock Tube FlowEquations developed for moving shock and expansion waves can be applied to solve the shock tube flow.
Across the contact surface pressure and velocity does not change, so For the moving shock wave, Assuming isentropic flow through the expansion waves, which can be rearranged to by equating the speeds and pressures at the contact surface and rearranging, gives,
This
is an expression relating the pressure ration of the shock produced
in comparison to the initial diaphragm pressure ratio. The Mach
number of the shock produced can be calculated form this Figs 69 and 70 show shock
pressure ratio and shock Mach number as a function of diaphragm
pressure ratio for the case where both driven and driver gas are
ideal air, γ= 1.4.
Based
on the above equations, it is clear that a stronger diaphragm
pressure ratio produces a stronger moving shock. However, to get
extremely fast shocks requires enormous pressure ratios
The
ratio of acoustic speeds,
It
is usually difficult to maintain significant temperature differences
between the two initial regions but it is quite easy to use different
driver and driven gases to maximise
In
the case where a Helium driver gas, R J/kg K, is used with air, _{4} = 2077γ, _{1}
= 1.4R J/kg K, as the driven gas then _{4} = 287a. In this case, the required diaphram pressure ratio, _{41}
= 0.34P,
is only about 10% of that required for the air-air case in order to
produce a Mach 3 shock wave._{41}## X-t Diagram for a Shock Tube.
The
X-t diagram for a shock tube is shown in X.
The flow behind the shock is uniform but the shock will reflect from
the end wall and “clean” flow at _{T}X is lost once the
reflected shock reaches the test location. The length of the test
time is “_{T}tt”. Due to the high velocity of the wave the test times
will be in the order of millisecs or less. If the test point is
moved further away from the end of the tube then this location will
be quickly impacted by the contact discontinuity and again only a
very short test duration is possible.
## PROGRAM for Simulation of Shock Tube Flow.
The
following program allows the simulation of shock tube flow. Data on
the driver and driven gas properties can be entered along with the
intial pressures in regions
Shock-Tube Simulation ( |