One-Dimensional Shock-Free Flow

For this and the following sections we assume steady, inviscid, iso-energetic, one-dimensional flow of a perfect gas.[57] Used is the nomenclature (x, u) given in Fig. 4.1, Section 4.1. The reader should note that we use the the same sym­bol “cpv for both the specific heat at constant pressure and for the pressure coefficient.

We treat in this section continuous shock-free flow as a prerequisite, and then in the Sub-Sections 6.3.1 and 6.3.2 normal and oblique shock waves.

The governing equations in Section 4.3 for mass, momentum and energy transport, eqs. (4.83), (4.27), (4.63) reduce for one-dimensional flow to (we omit dx)

d(pu) = 0,	(6.2)
pu du = – dp,	(6.3)
pudh = u dp.	(6.4)

The energy equation in differential form, eq. (6.4), can be rewritten by introducing the perfect-gas law, eq. (5.1), and some of the relations from eq. (5.10):

——— d — ] T и du = 0. (6-5)

Y- 1 pj

This, combined with eq. (6.3), yields the pressure-density relation for isentropic flow

— = constant = -^7. P7 Pt

The subscript ‘t indicates the reservoir or ‘total’ condition. pt is the total pressure. Eq. (6.6) can be generalized to

Here a is the speed of sound, the speed at which disturbances propagate through the fluid. Sound waves have so small amplitudes, that they can be considered as isentropic [4]: s = constant. Hence for perfect gas

In adiabatic flow the second law of thermodynamics demands:

s — sref ^ 0. (6.12)

Isentropic flow processes are defined by

s — sref = 0 : s = constant. (6.13)

The integrated energy equation eq. (6.5) in the familiar form for perfect gas reads, with cp the specific heat at constant pressure[58]

CpT + У«2 = cpTt. (6.14)

From this relation we see, that with a given total temperature Tt by expansion only a maximum speed Vm can be reached. This is given when the “static” temperature T reaches zero (expansion limit):

Vm = v72cpTt. (6.15)

The maximum speed can be expressed as function of at and y:

2 і 2 ci — U — Cv 1 f. 7-1 2 1 ’ and relate the critical speed of sound a* to the total temperature Tt:

і + ^м2′

Locally the “critical” or “sonic” condition is reached, when the speed u is equal to the speed of sound a

Eq. (6.14) combined with eq. (6.7) gives the equation of Bernoulli for compressible isentropic flow of a perfect gas:

12

-pur + p=pt.

The first term on the left-hand side is the dynamic pressure q (p is the “static” pressure)

Ч=ри2. (6.23)

The concept of dynamic pressure is used also for compressible flow. There, however, it is no more simply the difference of total and static pressure. Eq. (6.23) can be re-written for perfect gas as, for instance

4 = pu2 = lpM2. (6.24)

The free-stream dynamic pressure q= 0.5 pv^ is used also in aerother – modynamics to non-dimensionalize pressure as well as aerodynamic forces and moments.

For perfect gas we can relate temperature, density and pressure to their total values, and to the flow Mach number:

Tt=T, (6.25)

1

Pt=P (l + ^M2)7" , (6.26)

Pt=P (і + ^-М2У’ 1 . (6.27)

Similarly we can relate temperature, density, pressure and the speed u to their free-stream properties ‘to’:

with ^$TO UTO>/aTO •

The pressure coefficient Cp is

where qTO is the dynamic pressure of the free-stream. For perfect gas the pressure coefficient reads with eq. (6.24)

= (то1) (,U2)

The expansion limit (see above) is reached with p ^ 0. With this we get

the vacuum pressure coefficient

The pressure coefficient can be expressed in terms of the ratio local speed to free-stream value u, and the free-stream Mach number M„o

as well as in terms of the local Mach number M and its free-stream value

With the help of eq. (6.27) cp can be related to the total pressure pt

In subsonic compressible flow we get for the stagnation point (isentropic compression) with p = pt, u = (M =) 0

In the case of supersonic flow of course the total-pressure loss across the shock must be taken into account (see next section).

For incompressible flow we find the pressure coefficient with the help of eq. (6.22) and constant pt

u2

cP = 1——- —, (6.38)

uL

and note finally that at a stagnation point Cp for compressible flow is always larger than that for incompressible flow with cp = 1, see, e. g., [2].