Notes #6MAE 533, Fluid Mechanics

S. H. Lam

lam@princeton.eduhttp://www.princeton.edu/∼lam

October 21, 1998

1 Different Ways of Representing T ◦

The speed of sound, a, is formally defined as√

(∂p/∂ρ)s. It is a function ofthat thermodynamic state variables of the gas.

The stagnation enthalphy of a steady flow streamtube is defined by”

h◦ ≡ h +1

2u2. (1)

It depends not only of the thermodynamic state of the gas, but also onthe magnitude of the flow velocity u. Using the perfect gas approximation(h = CpT , Cp = γR/(γ − 1), a2 = γRT ), we have:

h◦ = CpT (1 +γ − 1

2M2). (2)

where M = u/a is the Mach Number.We now define ao to be the stagnation speed of sound—i.e. the speed of

sound when the gas is hypothetically brought to rest adiabatically. For aperfect gas, we have h◦ = CpTo. Hence

a2o = γRT ◦ = a2(1 +

γ − 1

2M 2). (3)

In other words, ao is a measure of the stagnation temperature of the gas.

1

We next define a∗ to be the critical speed of sound—i.e. the speed ofsound when the gas is hypothetically bought to Mach one. Setting a = a∗and M = 1 in (3), we have:

a2o = γRT ◦ =

γ + 1

2a2∗. (4)

We have previously be introduced to the concept of ultimate velocity uultdefined by:

u2ult = 2h◦. (5)

It is the velocity that would be achieved in the steady flow stream tube whenthe gas temperature is zero. For a perfect gas, we have

u2ult =

2

γ − 1a2

o =γ + 1

γ − 1a2∗. (6)

Thus, ao, a∗ and uult are alternative measures of the stagnatin tempera-ture of the gas.

A new Mach Number, M∗, defined in terms of the critical speed of sound,is useful for oblique shock discussions. We have:

M∗ ≡u

a∗. (7)

Note that M∗ is dimensionless, is supersonic or subsonic when its value islarger or less than unity (just like M). Note that when M → ∞, we haveu → uult and the associated value of M∗ is finite!

2 Oblique Shocks

A normal shock is called a normal shock because the surface of discontinuityis normal (perpendicular) to the direction of the upstream velocity vector.When a normal shock is observed by an observer moving with velocity U‖along the surface of the shock wave, it would appear to the moving observeras an oblique shock. The surface of the shock is then oblique to the un-stream velocity vector, and the direction of the fluid velocity vector nowchanges abruptly across the shock surface. Obviously, the component of theupstream and downstream velocity vectors perpendicular to the shock sur-face obey the normal shock relation, the relevant Mach number being the

2

upstream “normal Mach Number”—the normal component of the upstreamMach Number. Equally obviously, the parallel components of the upstreamand downstream velocity vectors are identical (unchanged across the shocksurface).

It should be obvious that the relations across oblique shocks can be ob-tained from normal shock relations supplemented by some trigonometry. Inall fluid mechanics text books that deal with compressible flows, one canfind normal shock tables, giving the ratio of pressure, density, etc. across theshock as a function of the upstream Mach Number (for one value of γ, usually1.4). But no text book gives tables for oblique shocks—because the numberof tables needed to represent all interesting upstream Mach Number and theobliqueness is simply too large. Instead, the oblique shock relations are usu-ally presented graphically in a diagram commonly referred to as the shockpolar. A shock polar is a graphical representation of the downstream velocityvector for a fixed, given upstream velocity vector across a plane shock wave.

Figure 1: Oblique Shock Polar

Fig. 1 is a sketch of a shock polar. The coordinates are the velocity com-ponents in the x and y directions, normalized by a∗. The unit circle thereforeseparates subsonic flows (inside the unit circle) from supersonic flows down-stream of the oblique shock. The outer big circle with radius OC has lengthuult/a∗, and represents the maximum possible value of flow velocity from the

3

point of view of the energy equation (on this big circle, the fluid temperatureis zero). The upstream velocity vector, non-dimensionalized by a∗, is repre-sented by the line O-∞. The locus of all possible oblique shock solutions forthe downstream velocity vector is given by the tear-drop shape figure—theshock polar—with the pointed vertex of the tear-drop at the tip of the O-∞vector.

The point N represents the normal shock, the point D represents the“detachment” solution, the point S represents the “sonic” solution, and thepoint ∞ represents the “nothing happened” solution. Also shown on thediagram is a sample solution point A (which happens to be supersonic asshown).

In general, we have:u‖1 = u‖2. (8)

The continuity equation across the shock wave says:

ρ2

ρ1=

u⊥1

u⊥2. (9)

The density ratio ρ2/ρ1 across the oblique shock is a function of M⊥∞ (fora given equation of state of the gas of interest) only and can be found fromthe normal shock relations.

To construct the oblique shock polar, one begins with a Cartesian coordi-nate system on a piece of graph paper, and locate the point ∞ to representthe upstream Mach Number M∞ of interest. This is the also the solutionpoint corresponding to the case of “nothing happened.”

Let the angle between the unstream velocity vector and the shock surfacebe denoted by σ. Hence, the point N corresponds to the normal shocksolution with σ = π/2. For the general case (with arbitrary σ), we first drawthe line OW on the diagram, the angle 6 (W-O-∞) is precisely σ. The relevantnormal Mach Number is simply M⊥∞ = M∞ sin σ, and is represented on thediagram by the line segment B-∞ which is perpendicular to the line O-W.One can easily find the density ratio ρ2/ρ1 for a normal shock with this M⊥∞,and the value of u⊥2/u⊥1 is then readily obtained by the use of (9). The pointA on the shock polar is obtained by (B-A)/(B-∞) = u⊥2/u⊥1 = ρ1/ρ2. Theangle 6 (A-O-∞) is the “turning angle” caused by the oblique shock, and isoften denoted by δ.

By changing the value of the shock wave angle σ from π/2 downward, thelocus of the solution points is the tear-drop shape oblique shock polar.

4

2.1 The Analytical Oblique Shock Polar

The analytical formula for the oblique shock polar for a perfect gas can befound in all the text books.

(v

a∗

)2

=(

U∞a∗

− u

a∗

)2 uU∞a2∗

− 1

2γ+1

(U∞a∗

)2 − uU∞a2∗

+ 1. (10)

2.2 Exercises

1. Show that for given M∞, the smallest value of σ (corresponds to the

weakest possible shock wave) is σ = arcsin(

1M∞

)= arctan

(1√

M2∞−1

).

This shock angle valid in the limit of very weak shock is called theMach Angle, frequently denoted by β.

2. Show that it is possible to obtain the value of p2− p1 across an obliqueshock from the shock polar by measureing the value of ∆u defined by:

∆u ≡ u2 − u1 (11)

which is shown on Fig. 1 as the line segment A-E. Hint: use x-momentumbalance as done in class.

3. When M∞ is asymptotically large, then M⊥∞ is also asymptoticallylarge except when the turning angle δ is asymptotically small. Hence,ρ2/ρ1 is approximately a constant (γ + 1)/(γ − 1). Convince yourselfthat in the large M∞ limit the shock polar is a circle (except whenδ << 1).

3 Weak Oblique Shocks

In the general case, the exact equations for the relations across oblique shocksare quite complicated. When δ is small, however, simple approximate equa-tions are available.

First of all, the shock is asymptotically weak in this limit, and is oftencalled a “Mach Wave” instead. The normal Mach Number M⊥∞ is approx-imately unity, and σ is approximately β. Using only trigonometry, we can

5

show that:

tanβ ≈ −∆u

U∞δ. (12)

Using the formula for β given previously, we have:

∆u ≈ − U∞δ√

M2∞ − 1

. (13)

3.1 Exercises

1. For weak oblique shock with δ << 1, show that ∆s ((s2 − s1)/R)is O(δ3), and is therefore negligible. In other words, in the leadingapproximation, weak shocks are isentropic.

2. Show that∆p

p∞≈ γM2

∞δ√M2∞ − 1

. (14)

where ∆p ≡ p2 − p1.

3. Take advantage of the fact that the flow is approximately isentropic,find ∆T/T where ∆T ≡ T2 − T1.

4. What happens when δ is small but negative?

6