58

11. TWO-DIMENSIONAL STEADY FLOW But now, we note a significant difference for M > 1
and M < 1 flow. For subsonic flow, M < ∞, (11.5)
We have studied transonic, one-dimensional flows, both is essentially Laplace’s equation, (3.3); so we can solve
time-steady and not. We now move to 2D, steady flows. this linear problem with our potential flow techniques
There are two mathematical approaches that can be use- of chapter 2. We also note that such flows are totally
ful: linearized potential flow, and characteristics. governed by their boundary conditions.1
A. The Nature of Steady, two-dimensional flow. However, if M > 1, the nature of equation (11.5)
changes. In particular, it becomes a wave equation; and
As we have noted already, subsonic and supersonic has as the general solution,
flows behave very differently. A formal way to demon-
strate this is through an extension of potential-flow φ(x, y) = F (x − βy) + G(x + βy) (11.6)
techniques to a general, compressible flow. Consider a
steady, irrotationa, inviscidl flow. Because ∇ × v = 0, where β 2 = M2 − 1, and F , G are any function. This
we choose a velocity potential, v = ∇φ. The continu- is getting back into the regime of characteristics – the
ity and Euler equations for steady flow are flow is determined only by those regions nearby from
1 which information can propagate.
∇ · (ρv) = 0 ; (v · ∇)v + ∇p = 0 (11.1)
ρ We can restate this using some results from formal
PDE theory. Equation (11.5) is a second order PDE; its

Now, we note the useful fact, still for steady flow nature depends depends on the sign of the 1 − M2
(whence?): term. If M < 1, the equation is elliptic – and does not
1 have any real characteristics. If M > 1, the equation is
v · ∇p = −c2s ∇ · v hyperbolic, and does have real characteristics. We can
ρ
exploit those characteristics in the solution of steady,
We can combine these results to get a second order two-dimensional problems; which is the point of this
equation in the velocity potential: chapter.
1 A final note: this breakdown into elliptic vs. hy-
∇φ · ∇(∇φ)2 = c2s ∇2 φ
 
(11.2) perbolic does not depend on the linearization; Schreier
2
demonstrates that the full potential system, (11.2) and
To be more explicit, we can pick Cartesian coordinates, (11.3) also changes type at M = ∞.
and write
∂2φ 2 2 ∂2φ 2 2 ∂φ ∂φ ∂ 2 φ B. Signal Propagation in Flows
2
(cs −φx )+ 2 (cs −φy ) = 2 (11.3)
∂x ∂y ∂x ∂y ∂x∂y If you don’t care for such a mathematical approach, we
These equations (11.2) and (11.3) are general, but not can try a more physical one. Consider the Mach con-
very useful as they are horribly nonlinear in φ. struction (due to Ernst Mach in an 1887 paper on super-
We can simplify it, however, by linearizing. In this sonic projectiles). Let some object emit a steady signal
section I describe one particular version of this: two- while moving at some speed v through a fluid. (This
dimensional Cartesian flow. That is, consider a simple, signal might, for instance, be the simple fact that the ob-
unperturbed flow – for instance, a uniform flow U in ject is moving and thus disturbing the local fluid). The
the x̂ direction. Add a thin body, such as an airfoil, signal propagates at cs relative to the fluid. If the object
nearly aligned with the x-axis. This will perturb the is moving subsonically, the signals “converge” ahead
flow slightly; we can specify the velocity potential to of the object, and “diverge” behind it: this leads to the
describe the perturbed flow. Going through the alge- familiar Doppler effect. Note, all of space eventually
bra (check Thompson, or Schreier, for the details), the receives the signals. The situation is different however,
linearized potential equation becomes if the object is moving supersonically. In this case the
spherical wavefronts emitted as the object moves, de-

∂2φ 2
2∂ φ fine an (upstream) cone of influence: fluid inside this
U 2 − c2s = cs (11.4) cone can receive information about the motion, while
∂x2 ∂y 2
Or, in terms of the Mach number,
1
Remember all those problems you solved in electrostatics, for

∂2φ ∂2φ the electric potential, where the boundary conditions determined
1 − M2 + 2 =0 (11.5)
∂x2 ∂y everything?
 59

fluid outside cannot. The opening angle of this cone is corner. Thus then tend to lean “forward”. Minus char-
acteristics come from ahead of a given point in the flow;
cs 1 at any point the intersection of plus and minus char-
sin µM = = (11.7)
v M acteristics determines θ and M (the latter through the
value of δ(M)). In particular, the flow angle and speed
which is called the Mach angle. The surface bounding
are constant along any given characteristic. Thus, the
the range of influence of the object is called the Mach
flow must follow the bending of the Mach lines. This
surface or characteristic surface; in two dimensions it
geometry forces the streamlines to diverge, so the flow
becomes the Mach lines, or simply characteristics.
must expand. Expanding a supersonic flow accelerates
it – just as in chapter 7.
By the way: where does the energy for the acceler-
ation come from?

Figure 11.1. Repeating Figure 7.3, showing Mach’s

construction for the Mach cone and Mach angle. Note in
this figure, the flow is moving to the right relative to the
object; alternatively one can envision the object as moving
to the left. From Thompson figure 5.2.

C. How Does Supersonic Flow Turn a Corner? Figure 11.2. Flow turning corners, with streamlines and
plus (forward) characteristics (Mach lines) shown. The left
Now, we illustrate the use of characteristics in steady, diagram shows flow into a bending wall. The streamlines
supersonic flow with one example: flow around a cor- come closer together as they follow the wall around, so the
ner. flow must compress and decelerate. A shock forms where
the characteristics intersect. The right picture shows the
Before we hit the math, however, let’s think about opposite case, flow around a corner. The characteristics
the physics. Consider supersonic flow along a wall, and diverge, as does the flow; this expansion results in an
let the wall have a corner. Remember that information acceleration. From Kundu figure 15.18
about the corner cannot propagate very effectively up-
stream, so the incoming fluid cannot “know” very easily
about the corner. Just how, then, does the flow turn the
corner? Now, consider the converse case: flow “into” a wall
To be specific, think about flow around a wall which which bends, as in Figure 11.1a. The geometry forces
turns a corner away from the flow, as in Figure 11.2b. the flow to be compressive, and thus (still being super-
Since we’re working with supersonic flow, we can think sonic) it must slow down. Here, however, the corner-
back to the channel flow of Chapter 7, in particular flow turning takes place through a set of shocks; it cannot
into an expanding channel. The flow must expand, and be smooth. We can see this by drawing in characteris-
being supersonic it must also speed up. This can be tics again. The nature and angle of the characteristics
seen in terms of characteristics (also called Mach lines follows the same rules as above; the angle of the wall
in this context). In the next section we will show that sets the angles of the Mach lines which arise from it.
the Riemann invariants in this situation are given in But now, it’s apparent that these lines must intersect: a
terms of two angles: θ, the angle of the flow relative shock will form. (Formally: think about characteristics
to some axis (say the wall before it turns), and δ(M, intersecting. If “adjacent” lines carry different values
the Prandtl-Meyer angle defined in chapter 9. The in- of θ ± δ, the values of θ and/or δ will not be uniquely
variants are θ ± δ(M). Our boundary (“initial”) condi- defined at the intersection. That is not good. The fluid
tion must be along the wall: the flow (streamline) must will respond by setting up a shock – an infinitely thin
parallel the wall, at the wall. Thus, the characteristics (well, nearly) jump between “front” and “back” prop-
must be straight lines which originate from the wall. erties.
Now, the plus (forward) characteristics have values of By the way: when the flow decelerates, where does
θ which go more and more negative, moving along the the energy go?
 60

D. One example: Prandtl-Meyer flow

Now, let’s do this mathematically. A specific exam-
ple of flow around corners is the case of flow around
a sharp corner (as in Figure 3.14 of Faber). The flow
expands through a “fan” of Mach lines centered at the
corner, called the Prandtl-Meyer expansion fan. The
mach number increases along a streamine through the
fan, while the pressure falls along that streamine. Each
mach line is inclined at µM to the local flow direction.
Thus, supersonic flow turns via a sequence of standing
oblique shocks.
This can be described in detail with characteristics. Figure 11.3. (a) Streamline coordinates for the
We could formally start with the second order PDE, characteristic solution. The flow velocity is u, at angle θ to
(11.5), and find its characteristics. However, Thompson the x-axis; the distance along a streamline is s; n is the
vector normal to the streamline. (b) Two Mach waves, m+
(his chapter 9) has an alternative approach which might and m− , intersecting a streamline, (c) A geometrical way to
be more illuminating. Following him, we search for remember the Mach angle. From Thompson figure 9.6
characteristics by starting with the basic, 2D, steady-
flow equations:
the two equations are
√
∇ · (ρv) = 0 ; ∇ × v = 0 M2 − 1 ∂v ∂θ
 2 (11.8) tan µM + =0
v √ v ∂n ∂s
ρ∇ + ∇p = 0
2 M2 − 1 ∂v ∂θ tan µM sin θ
+ tan µM =−
v ∂s ∂n r
We write these in streamline coordinates: s is the dis-
tance along the streamline, at local angle v̂; n̂ is the (11.13)
local normal, and θ is the flow angle (relative to some
reference direction). Figure 11.2 illustrates the geome- But the first term in this second equation is ∂δ/∂s, if
try. The zero-curl equation becomes δ(M) is the Prandtl-Meyer function, defined in (9.27)
above. Putting δ(M) into (11.13) where we can, and
∂v ∂θ adding or subtracting the two, gives us our characteris-
+v =0 (11.9)
∂n ∂s tic equations:
which is the first equation of motion. For the moment, 
∂ ∂

tan µM sin θ
let’s work in cylindrical coordinates. The continuity + tan µM (θ + δ(M)) = +
∂s ∂n r
equation becomes, in these coordinates,  
∂ ∂ tan µM sin θ
1 ∂ρ 1 ∂v ∂θ sin θ − tan µM (θ − δ(M)) = −
+ − =− (11.10) ∂s ∂n r
ρ ∂s v ∂s ∂n r
(11.14)
This is the first equation of motion we want. Next, the
momentum equation (in the s direction) becomes These can be interpreted as follows. (Compare equa-
tions (10.10) in our earlier discussion of characteris-
∂v 1 ∂p tics.) At any point along the streamline, two charac-
v + =0 (11.11)
∂x ρ ds teristics (a.k.a. Mach waves) can be found. They head
out at angles ±µM relative to v, which defines the lo-
These last two combine, using dp = c2s dρ, as
cal streamline. (These are the analogs of C ± in chapter

1 ∂v ∂θ sin θ 10).
M2 − 1 + = (11.12)
v ∂s ∂n r But the left hand side of equations (11.14) are just
which is the second equation of motion. the derivatives of θ ± δ(M) along the lines m± . To
see this, recall that (by the chain rule) the derivative of
Now we use (11.9) and (11.11) to look for charac-
some function F along m+ is
teristics. In terms of the mach angle,
dF ∂F ds ∂F dn
= +
p
tan µM = 1/ M2 − 1 dm+ ∂s dm+ ∂n dm+
 61

But, ds/dm+ = cos µM and dn/∂m+ = sin µM ; so everywhere. The other characteristics have
that
  θ − δ(M) − 2θ − δ1 on m−
dF ∂F ∂F
= cos µM + tan µM
dm+ ∂s ∂n where δ1 = δ(M1 ). From this, we get
A similar result obtains for derivatives along m− .
Thus, the two characteristic equations for steady, two- on m− : θ = constant and
dimensional flow can be written, (11.17)
δ(M) = δ(M)1 = θ = constant
d sin µM sin θ
(θ + δ(M)) = +
dm+ r (11.15) Thus, each m− characteristic is a straight line, with
d sin µM sin θ
(θ − δ(M)) = − constant δ(M) and θ values everywhere along it. There
dm− r will then be a first intersting m− characteristic, on
Now, there are more general ways to treat characteristic which θ = 0◦ and δ(M) = δ1 = δ(M1 ); and a last
problms, if the derivative of some function along a char- intersting one, on which θ2 = −∆, and δ2 = δ1 + ∆;
acteristic is not zero.2 For now, let’s switch back from the final M value can be found from this, through the
cylindrical to plane geometry: that means take r → ∞ Prandtl-Meyer function.
in (11.15). This gives

θ + δ(M) = constant on m+
(11.16)
θ − δ(M) = constant on m − References
Mostly from Schreier and Thompson, here.
and these equations are good analogs of (10.10). Thus,
we have our characteristics (m± ) and our invariants
[θ ± δ(M)].

Figure 11.4. The geometry used in the example of a

Prandtl-meyer solution. The wall makes a sudden turn by
angle ∆ away from the initial flow direction. m−
characteristics starting from the sudden turn, are illustrated.
From Thompson figure 9.13

Finally, then, we can use this formalism to describe

Prandtl-Meyer flow around a corner. To pick an exam-
ple, consider upstream flow at M1 , meeting a corner
that turns by some angle ∆. The flow angle θ has a dis-
continuity at the corner; the interesting characteristics,
m− , diverge from here.
Now, the entire flow field is covered by characteris-
tics m+ which originiate in the uniform, upstream re-
gion 1 (where θ → 0): they have
θ + δ(M) = δ1

2
We didn’t bother with this in chapter 10, feel free to bug a math-
ematician if you’re interested.