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10. ONE-DIMENSIONAL UNSTEADY
FLOW
In chapter 7 we introduced simple waves which carry
information about some disturbance in the fluid (such
as a piston moving or a sudden “hit”). In Chapter 9 we
studied the local structure of shock fronts. In this chap-
ter we return to the arguments of chapter 7, formalize
them and allow for waves that are not weak. That is,
we will formally introduce characteristics to solve un-
steady flow problems. One important application of this
is to shock formation and propagation.
A. Shock Formation: The Physical Picture
In Chapter 4, we treated sound waves as small-
amplitude perturbations, and found that they propagate
Figure 10.2. Schematic stages in the development of a
at a characteristic speed, cs , which depends only on wave which steepens into a shock. Note that all of the
the temperature of the fluid. In reality, however, any profiles contain the same area. From Faber figure 3.6.
such wave has higher density at its “center”, relative to
its surroundings, and has a smooth variation of density
with position. Figure 10.1 shows a discretized version SHOCK THICKNESS
of this. Physically, shocks cannot be infinitely thin surfaces,
although this is how they are treated mathematically.
In practice, the fluid variables cannot change instanta-
neously; there must be a smooth transition between up-
stream and downstream. This defines a shock thickness,
which is determined by viscosity, that is by interparticle
collisions.
We can find a scaling argument for this: consider
the balance between inertial and viscous forces, which
must apply at the shock:
dv d2 v
ρv ≃ν 2 (10.1)
dx dx
This suggests the shock thickness, ∆, is determined by
∆ ∼ ν/v. But, the viscosity can be found from ki-
Figure 10.1. Density profiles for layers of higher netic theory as ν ∼ λhvi, where λ is the mean free path
density, propagating to the right. From Faber figure 3.5. and hvi is the mean particle speed. Thus, noting that
hvi ∼ cs (check this out for an ideal gas if you don’t
believe it!), we would estimate ∆ ∼ λ. This makes
But the higher central densities lead to a higher cen- some physical sense: the information that the flow con-
tral pressure and temperature (assuming an adiabatic ditions have changed cannot be communicated over a
wave which is the most likely). Thus the central sig- distance shorter than a collision mean free path.
nal speed will be higher than that at the edges; there- This is not the entire picture, however. An ionized
fore the center will tend to overtake the leading edge. If gas – a plasma – can support shocks which are much
the wave amplitude is large enough, an initially smooth thinner than their collisional (Coulomb, cf.chapter 1)
waveform will be significantly distorted, and the wave mean free path. The details of how these shocks work
will “break”. That is, the wave profile will become are still being argued about. There seems to be agree-
triple valued at its leading edge; this leads to the reform- ment, however, that these shocks are supported by mi-
ing of the edge in a shock discontinuity (as sketched in croturbulence in the plasma: these small-scall fluctua-
Figure 10.2). tions transfer the energy and momentum from upstream
54
to downstream, and convert ordered to random energy, Thus, this function F is simply a constant times the lo-
just as interparticle collisions do in a shock in a neutral cal sound speed. Now, the pressure derivatives can be
gas. The width of a collisionless shock is typically a written in terms of F , as
fundamental plasma scale, such as the ion gyroradius or
the length derived from the plasma frequency, ∼ cs /ωp . ∂F ∂F ∂p 1 ∂p
= =
∂t ∂p ∂t ρcs ∂t
B. The Method of Characteristics ∂F ∂F ∂p 1 ∂p
= =
In chapter 7 we introduced weak waves – waves of such ∂x ∂p ∂x ρcs ∂x
small amplitude that the local conditions (density, pres-
so that (10.5) can be written as
sure) are not significantly disturbed. These waves prop-
agate at a constant speed, co , thus their loci in an (x, t)
∂ ∂
diagram are straight lines. ± (v ± cs ) (v ± F ) = 0 (10.8)
∂t ∂x
Our goal in this section is to extend this approach to
the general case, rather than limiting ourselves to small- At this point, each equation (the “+” and “-” ones) con-
amplitude or subsonic flows. To do this, we first find the tain only one derivative operator:
equations that describe the characteristics. Start with
the 1D momentum and continuity equations (in a uni- D+ ∂ ∂
= + (v + cs ) ;
form channel, no A terms, and with no external body Dt ∂t ∂x (10.9)
forces): D− ∂ ∂
= − (v − cs )
∂v ∂v 1 ∂p Dt ∂t ∂x
+v + =0 (10.2)
∂t ∂x ρ ∂x so that our two equations are, finally,
and
D+ D−
∂ρ ∂ρ ∂v (v + F ) = 0 ; (v − F ) = 0 (10.10)
+v +ρ =0 (10.3) Dt Dt
∂t ∂x ∂x
These equations turn out to be quite useful. We must
But this latter can be written, using ρ = ρ(p) and dp = interpret them physically, as follows. From (10.9) and
c2s dρ, as (10.10), we see that the quantities
1 ∂p v ∂p ∂v
+ + cs =0 (10.4) 2
ρcs ∂t ρcs ∂x ∂x J+ = v + F = v + cs
γ −1
(10.11)
Now, these last two can be added and subracted to form − 2
J =v−F =v− cs
two new equations, γ −1
∂v 1 ∂p ∂v 1 ∂p are constant along the lines described (and labelled) by
± + (v ± cs ) ± = 0 (10.5)
∂t ρcs ∂t ∂x ρcs ∂x
dx
As Thompson notes, it appears doubtful that this is any C+ : = v + cs ;
dt (10.12)
simplification, but it is nonetheless. Define a new funci- dx
−
ton F = F (p, s) (where s is the specific entropy), C : = v − cs
dt
which satisfies
Z p These lines are called characteristic lines. They de-
dp
F = (10.6) scribe the paths of signals (sound waves) travelling for-
po ρcs ward or backwards in a flow, at the local sound speed.
where po is some (useful) reference state, and the inte- This, the quantity v+F is constant for an observer trav-
gration is carried out at constant entropy. For an adia- elling at velocity v + cs , or on any plane perpendicular
batic perfect gas, the F integral can be done straight- to the x axis which moves at v + cs (that is, with a for-
forwardly: ward, or positive, sound wave in a flow with velocity v.
this is the plus characteristic, or the C + wave). Simi-
dp dρ 2dcs 2
Z Z Z
larly, the quantity v − F is is constant for an observer
F = = cs = = cs
ρcs ρ γ−1 γ−1 travelling at velocity v − cs (that is, a backward or neg-
(10.7) ative sound wave; the minus characteristic or the C −
55
regions reached by no characteristic.
1. PISTON MOTION IN A CHANNEL
This may be made more concrete by the example of
piston withdrawal, or advance, in a 1D channel. Let the
piston position be described by X(t); withdrawal has
dX/dt < 0 and advance has the opposite sign.
Figure 10.3. Characteristic lines, along which
J + = v + F and J − = v − F are constant. The solution at
point d is uniquely determined by the initial data at points a
and b. From Thompson figure 8.3
wave). These quantities, J ± = v ± F , are called the
Riemann invariants.
But these characteristics allow the simple solution
of an initial value problem (conceptually; in practice
numerical integration may be required). Refer back to Figure 10.5. The geometry and characteristics
Figure 10.4, which assumes we have a time-dependent associated with smooth piston withdrawal. PP labels the
problem in which initial values are specified over the path of a particle starting at a particular point on the x axis.
range a < x < b. Consider point d in that figure. It is From Thompson figure 8.17.
connected by a plus (forward direction) characteristic to
point a, and a minus (backward direction) characteristic We first notice that C − characteristics come in from
to point b. That means the quantity Ja+ = v +F at point ahead, from the undisturbed flow region (which has v =
d is the same at point a (since C+ connects the two 0, cs = cs,o . On these lines,
points), and the quantity Jb− = v − F is the same at
point d and point b (since C− connects them). Thus, 2 2
we v(x, t) − cs = cs,o (10.13)
know the solution uniquely: v = 12 Ja+ + Jb− ; cs = γ −1 γ −1
γ−1
Ja+ − Jb− .
4 This relation allows us to find v from cs,o , or vice versa.
Now, the C + characteristics connect the “influenced
flow region” to the piston motion. On these lines,
dX γ − 1 dX
v= ; cs = cs,o + (10.14)
dt 2 dt
In this latter, we have noted that a typical straight C + ,
which propagates into a uniform region, characteristic
carries nearly constant cs and v values. Thus, on C + we
can identify v with dX/dt at the piston (assuming slow
enough withdrawal that the fluid doesn’t separate); and
Figure 10.4. Regions of influence for the same we have used (10.13) as well. We note that the inverse
initial-data problem as in Figure 10.4. From Thompson slope of the C + characteristics is v+cs = cs,o +dX/dt.
figure 8.4 As this quantity becomes smaller, the characteristics
become more and more vertical. We also note that the
Going further, referring to Figure 10.4, we can see second expression in (10.14) shows that cs < cc,o for
the extent of the solution to this problem. The solu- a withdrawing piston; the expansion of the gas cools
tion v(x, t) is completely determined by the initial val- it.These relations (10.13) and (10.14) effectively solve
ues in the region enclosed by the triangle abd. Outside the problem; they specify the values along each char-
this region, it is partly influenced in regions where one acteristic, and thus all regions of (x, t) space are ad-
characteristic reaches; and it is totally undertermined in dressed.
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Piston advance can be treated in the same way, tak- ary region ahead of the wave. On these characteristics,
ing dX/dt > 0. The difference is that a shock forms J − = Jo− = constant everywhere, so that
ahead of the piston. The analysis above carries through 2 2
just the same – we again note that v+cs = cs,o +dX/dt J− = v − cs = − cs,o ; (10.15)
is the inverse slope of the C + lines. But now, with γ−1 γ−1
dX/dt > 0, the slope decreases moving along the path and thus
of the accelerating piston: the characteristic lines must 2
cross. As we will see next, a shock forms at the point v= (cs − cs,o ) (10.16)
γ−1
where they first intersect.
Thus, fluid within compression regions (cs > cs,o )
moves in the +x direction, the direction of travel of
the wave. In addition, any portion of the C + wave must
move with speed
γ+1 2
v + cs = cs − cs,o (10.17)
γ−1 γ−1
Both of these connections verify the monotonic in-
crease of wave speed with wave amplitude (cs − cs,o )
which gives rise to the distortion of the wave into a
shock. We can visualize this from characteristics also.
The slope of each characteristic, in a (t, x) diagram,
is the inverse of the local sound speed – so regions
Figure 10.6. The geometry and characteristics of higher overdensity have higher cs , and thus lower
associated with smooth piston advance. From Thompson slopes. Thus, characteristics from some region will in-
figure 8.21. tersect at some later time – shown as point s in the fig-
ure.
2. CONNECTION TO SHOCK FORMATION
We argued in §10.1 that simple waves can steepen into
shocks; this is based on the local propagation speed is
an increasing function of the local density; so overdense
regions move faster, and overtake the less dense regions
ahead.
We can treat this more formally using characteris-
tics. Let the initial wave be traveling in the +x di-
rection (a forward, C + , wave), into stationary gas with
po , ρo , cs,o values far ahead of our disturbance.
Figure 10.8. A characteristic diagram (wave diagram),
showing characteristics intersecting at what becomes a
shock. From Thompson figure 8.12
When is this time? In general, two nearby pieces
of fluid have similar values of the quantity u = v +
cs ; say they vary by δu (take this to be positive, just
to pick a specific case). Referring to Figure 10.7, we
see that the shock first forms, then, at the point where
this derivative dx/du is a minimum – which occurs at
Figure 10.7. The geometry, and progressive distortion,
of the steepening waveform described by characteristics in
the steepest point in the wave (that is along the leading
Figure 10.9. From Thompson Figure 8.10. edge, as in Figure 10.2).
Note, after the shock forms, our method of charac-
Now, the region around the disturbance includes C − teristics must be extended to include jumps at the shock
characteristics which come in from the uniform station- itself.
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3. TRAFFIC SHOCKS describes the car density at t = 0. But each point in the
(x, t) plane can be mapped back, along its local char-
Finally, let’s consider a different approach: characteris- acteristic, to a starting point xo . Thus, if things are
tics and shocks in traffic flow. single-valued, the solution can be found directly from
ρ(x, t) = f [xo (x, t)]. (If things are multivalued, that
is characteristics from more than one xo pass through a
given (x, t), then we find that a shock develops. Exam-
ples of both of these are given in the homework.
Figure 10.9. A traffic shock; from Thompson figure 7.3.
References
That is: consider the flow of cars along a one lane I’m mostly following Thompson here, also Faber on
road (no passing!). We can take ρ as the density of cars shock development. Traffic shocks can also be found as
(per length), v as their speed, and q = ρv as the traffic examples in PDE books.
flow (what are the units here?). Conservation of cars is
expressed by
∂ρ ∂q
+ =0 (10.18)
∂t ∂x
But this is, of course, just the continuity equation. Now,
traffic-flow people simplify this by assuming the flow
can be taken as a function only of the density: q =
q(ρ).1 This allows the basic equation to be written
∂ρ ∂ρ
+ c(ρ) =0 (10.19)
∂t ∂x
if c(ρ) = ∂q/∂ρ.
Now, we know how to treat the equation (10.19):
the solution ρ is constant along the characteristic lines
dx
= c(ρ) (10.20)
dt
Why? Consider the case when the “total derivative” is
zero:
Dρ ∂ρ dx ∂ρ
= + =0
Dt ∂t dt dx
But this is just equation (10.18) if the dx/dt term is
given by (10.20). Thus, (10.19) shows that ρ is constant
along the “path” in (x, t) space given by (10.20). That
is, the lines on which ρ is constant, are simple waves,
straight lines in the (x, t) plane.
Typically, the problem is set up in terms of initial
conditions: ρ(x, 0) = f (x), that is some function f (x)
1
A typical choice assumes the velocity is v(ρ) = V (1 − ρ/ρo ),
ρ < ρo . That is, a linear relation between density and speed, up
to some maximum density ρo at which gridlock sets in and no
further motion occurs.
