In the previous lesson, we looked at normal shock waves.
In real life, these normal shocks are a limiting case and occur rather infrequently.
In fact, due to the shape of the body, most shocks that occur in supersonic flows over
objects are inclined at an angle relative to the incoming flow.
Such shocks are called oblique shock waves and are the focus of this lesson.
Oblique shocks generally occur when the supersonic flow is turned onto itself.
Let us understand this by looking at an illustration.
Consider a supersonic flow over a surface.
If a ramp is added to the surface at an angle theta such that the surface and the ramp make
a concave corner, the incoming supersonic flow gets deflected.
This deflection is towards the bulk flow i.e. the flow is turned onto itself.
To ensure that the supersonic flow is informed about the presence of the ramp, an oblique
shock wave is formed through which the flow gets deflected at the same angle as the deflecting
surface and becomes parallel to the wall.
Let us now mathematically describe the flow through an oblique shock wave.
Consider an oblique shock wave oriented at an angle beta to the incoming supersonic flow
which is at a mach number of M1.
The equivalent velocity is V1.
This velocity can be represented as a superposition of two components: 1) a velocity component
normal to the shock, u1, 2) and a velocity component parallel to the shock wave, Vt1.
The relationship between V1 and the velocity components is shown here.
Furthermore, the two velocity components are related to the oblique shock wave angle using
the given relationship.
It is easy to show that the continuity equation, assuming a steady flow, can be transformed
to the following form, where u2 is the downstream shock normal velocity.
The tangential component of the momentum equation, with a steady flow and no body force assumption,
can be transformed to the equation shown here.
Manipulating this equation using the continuity equation, we observe that Vt1=Vt2 i.e. the
shock parallel velocity component is the same upstream and downstream of an oblique shock.
The normal component of the momentum equation and the energy equation take the following
forms for an oblique shock wave.
If we observe closely, the governing equations of the flow through an oblique shock wave
are dependent only on the shock normal velocity i.e u1 and u2.
This means that all the theory developed for estimating the flow properties through a normal
shock wave in the previous lesson will hold as is for an oblique shock.
The only difference being that for the upstream mach number, we use the shock normal mach
number which is shown below.
For an oblique shock, the variation of flow properties across the shock is given by the
following relations.
In these relations, if you substitute for Beta as Pi/2, the exact normal shock wave
relations will be recovered.
This basically indicates that normal shock waves are just a unique case of oblique shocks
when the shock wave angle is equal to pi/2.
The mach number downstream of the oblique shock wave is given by the following equation.
From the velocity triangles upstream and downstream of the shock, we have the following relations.
Combining these relations, the continuity equation, and the density ratio relation across
the oblique shock wave, we can write an expression connecting the deflection angle, shock angle,
and mach numbers upstream of the shock wave as shown here.
This is the famous theta-beta-mach number relation.
The curves obtained from this relation are shown here.
There are quite a few observations that can be made from these curves.
Let’s look at them one by one.
For any given upstream mach number, there exists a unique maximum deflection angle i.e.
theta_max.
We will discuss what happens if theta is greater than theta_max a little bit later.
For any given upstream mach number and for all deflection angles less than theta_max,
there are always two solutions for the shock angle beta.
The larger value of beta is called the strong shock solution and the smaller value is called
the weak shock solution.
For the strong solution, the mach number behind the shock is always subsonic, on the other
hand, for the weak solution, which is more common in nature, the mach number behind the
shock is supersonic, except when theta is very close to theta_max.
A strong shock solution can be generated if the backpressure behind the shock is increased
by an external mechanism.
As discussed earlier, at theta=0, beta=pi/2 which is nothing but a normal shock.
Or beta can also take the value of mu resulting in a weak shock solution often referred to
as the mach wave.
For a fixed value of deflection angle theta, the weak solution shock angle is inversely
proportional to the upstream mach number.
At upstream supersonic mach number below which no weak shock solutions are possible, the
deflection angle will correspond to the maximum possible deflection at this Mach number.
Going further below this mach number will result in a shock that is not attached to
the object anymore and is called a detached shock.
As mentioned earlier, when theta=0, the weak shock solution is that of wave referred to
as the mach wave.
It is basically an infinitely weak oblique shock wave through which the pressure jump
is 0.
Therefore, from the pressure ratio relation, it is easy to define the mach angle mu as
shown here.
At any point P in the flow field, there are two Mach lines, (+) and (-) intersecting a
streamline at the angle μ in 2D.
In 3D, Mach lines form a conical surface.
Let’s switch gears a little.
Till now, we have looked at the oblique shock wave formed at a sharp concave corner.
What happens if the corner is not sharp but smooth, as shown in this figure?
Well, for this type of geometry, we will notice the formation of weak mach waves at different
points along the curve.
These mach waves then coalesce at a point and form an oblique shock wave.
This is for the case where the radius of curvature of the curved corner is constant.
For real world geometries, it is possible for the radius to vary along the curve.
In such a situation, the flow can be extremely complex with the formation and interaction
of shock and expansion waves.
If you recall from the theta-beta-mach number chart, we saw that for a given mach number
there a maximum value of theta that is possible for the flow to turn.
What if the deflection angle is greater than this maximum?
In such a case, the shock wave will no longer be attached to the object but instead will
move upstream and form a detached curved shock, also known as a bow shock.
The flow behind the shock will be subsonic near the corner and slowly accelerate to supersonic
further down the surface.
One thing to remember here is that the value of theta-max is directly proportional to the
upstream mach number i.e. the higher the incoming mach number, the higher will be the possible
deflection angle required for creating an oblique shock.
Detached shocks are quite common since most supersonically traveling bodies have a blunt
nose cone.
Here is an illustration of a detached bow shock in front of a blunt body.
This bow shock is a blend of normal, oblique, and weak shocks.
Let’s now look at different regions of this bow shock.
We will also have the theta-beta-mach number chart handy for reference.
At point 0, the upstream flow is normal to the bow shock and therefore this point corresponds
to a normal shock wave.
Between points 0 and 2, a strong oblique shock wave exists and the flow behind the shock
is subsonic.
The angle of inclination of the wave in this region can be got from the T-B-M chart from
the strong wave solution section.
Point-2 corresponds to the separation point between the strong and weak solutions.
Point-3 corresponds to the location on the shock where the flow behind the shock is sonic.
Between points-3 and 5, a weak oblique shock wave exists, and the solution can be got from
the appropriate section of the TBM chart.
The flow behind the shock in this region is supersonic.
At point-5, the shock is infinitely weak and effectively acts as a mach wave.
This basically indicates that this region so far away from the body that its influence
is negligible.
Much of the discussion till now was regarding 2d planar shock waves and applies to geometries
like 2D wedges.
However, for 3D geometries like a cone, this discussion cannot be applied directly.
This is because the flow behind the shock is not uniform.
For a supersonic flow over a cone, the shock also takes the shape of a cone.
As shown here, a streamline representing the flow through the shock cone will start curving
behind the shock and eventually becomes parallel to the cone’s surface.
This is in contrast to what we saw with wedge where the streamlines and the flow are parallel
to the surface immediately behind the shock.
Furthermore, for the same flow conditions and deflection angle, the shock angle for
a cone is lesser than that of the wedge i.e. the shock is weaker for the cone.
This is because the flow is three dimensional and can move in any direction to avoid being
obstructed by the body.
This phenomenon is generally referred to as the three-dimensional relieving effect.
A cone can be considered as a ray originating from the vertex of the cone.
The flow properties can be assumed to be constant along each ray and vary only between rays.
Flows that satisfy this assumption are called conical flows.
The ordinary differential equation shown here is called the Taylor-Maccoll equation and
is used for finding the solution for conical flows.
With that, we will wrap up this lesson.