4.8.7 The Shock Tube
The previous examples lead us naturally to apply the numerical procedure to the
shock tube [9–11] which is an important device for the study of shock waves and
shock wave interactions in the laboratory.
An exact analytical solution to the shock tube problem can be obtained using the
method of characteristics [12]. As a result, the shock tube problem has become an
important test for the accuracy of computational fluid dynamic (CFD) codes.
Accordingly, the numerical solution can be compared with the analytical solution
which allows one to ascertain how accurately the code resolves shock discontinuities
and reproduces the correct density, pressure and velocity profiles.
A shock tube consists of a long tube that is divided into two chambers by a
diaphragm. One side contains gas at high pressure, p 4 , and the other side contains gas
at a lower pressure, p 1 . Both chambers can contain different gases and, hence, have
different gas constants, R, and different specific heat ratios, γ, although in the
example to be discussed here we will assume that both chambers contain air with
γ ¼ 1.4. At t ¼ 0 the diaphragm is ruptured and the high-pressure gas, called the
driver gas, rushes into the low-pressure chamber called the driven section. The
interface or contact surface between the two gases that were initially separated by
the diaphragm, behaves like a piston which drives a shock wave into the
low-pressure gas to increase the pressure. During the time interval that the shock
wave moves down the tube, an expansion wave propagates into the high-pressure
chamber to reduce the pressure. Figure 4.30 illustrates the flow in the different
regions following the rupture of the diaphragm. The gases in regions 2 and 3 are
at the same pressure and they move with the same velocity [9], whilst the strength of
the shock wave, p 2 /p 1 , and the velocities of the gases are dependent on the initial
pressure ratio, p 4 /p 1 , across the diaphragm: the required expressions are derived
below.
By using Eqs. (3.27) and (3.41) we have the following relationships for the
velocity U s of the shock wave and the particle velocity u p behind the shock in
terms of the pressure ratio across the shock;
U S ¼ c 1
γ À 1
ð
Þþ γ þ 1
ð
Þ
p 2
p 1
2γ
"
# 1=2
ð4:53Þ
and
u 2 ¼
c 1
γ
p 2
p 1
À 1
2γ
γþ1
γÀ1
γþ1 þ
p 2
p 1
! 1=2
,
ð4:54Þ
174
4 Numerical Treatment of Plane Shocks
The previous examples lead us naturally to apply the numerical procedure to the
shock tube [9–11] which is an important device for the study of shock waves and
shock wave interactions in the laboratory.
An exact analytical solution to the shock tube problem can be obtained using the
method of characteristics [12]. As a result, the shock tube problem has become an
important test for the accuracy of computational fluid dynamic (CFD) codes.
Accordingly, the numerical solution can be compared with the analytical solution
which allows one to ascertain how accurately the code resolves shock discontinuities
and reproduces the correct density, pressure and velocity profiles.
A shock tube consists of a long tube that is divided into two chambers by a
diaphragm. One side contains gas at high pressure, p 4 , and the other side contains gas
at a lower pressure, p 1 . Both chambers can contain different gases and, hence, have
different gas constants, R, and different specific heat ratios, γ, although in the
example to be discussed here we will assume that both chambers contain air with
γ ¼ 1.4. At t ¼ 0 the diaphragm is ruptured and the high-pressure gas, called the
driver gas, rushes into the low-pressure chamber called the driven section. The
interface or contact surface between the two gases that were initially separated by
the diaphragm, behaves like a piston which drives a shock wave into the
low-pressure gas to increase the pressure. During the time interval that the shock
wave moves down the tube, an expansion wave propagates into the high-pressure
chamber to reduce the pressure. Figure 4.30 illustrates the flow in the different
regions following the rupture of the diaphragm. The gases in regions 2 and 3 are
at the same pressure and they move with the same velocity [9], whilst the strength of
the shock wave, p 2 /p 1 , and the velocities of the gases are dependent on the initial
pressure ratio, p 4 /p 1 , across the diaphragm: the required expressions are derived
below.
By using Eqs. (3.27) and (3.41) we have the following relationships for the
velocity U s of the shock wave and the particle velocity u p behind the shock in
terms of the pressure ratio across the shock;
U S ¼ c 1
γ À 1
ð
Þþ γ þ 1
ð
Þ
p 2
p 1
2γ
"
# 1=2
ð4:53Þ
and
u 2 ¼
c 1
γ
p 2
p 1
À 1
2γ
γþ1
γÀ1
γþ1 þ
p 2
p 1
! 1=2
,
ð4:54Þ
174
4 Numerical Treatment of Plane Shocks