A graphical version of this slide is available. In the text only version presented here, * denotes multiplication, / denotes division, ^ denotes exponentiation, ^2 means quantity squared, sqrt means square root. r is the density, p is the pressure, T is the temperature and g is the gas constant. u is the velocity, s is the speed of sound, A is the area, M is the Mach number, q is the dynamic pressure, and gam is the ratio of specific heats. pt, Tt, and rt are total values of the variable. astar is the area at choked conditions.

Equation 1. M = u / s

Equation 2. s = sqrt(gam * p /r) = sqrt(gam * g * T)

Equation 3. p / r ^gam = constant = pt / rt ^ gam

Equation 4. p / pt = (r / rt) ^ gam = (T / Tt) ^ (gam / (gam-1))

Equation 5. q = .5 * r * u ^2 = .5 * gam * p * M ^2

Equation 6. p / pt = (1 + .5 * (gam - 1) * M ^2) ^ -(gam / (gam-1))

Equation 7. T / Tt = (1 + .5 * (gam - 1) * M ^2) ^ -1

Equation 8. r / rt = (1 + .5 * (gam - 1) * M ^2) ^ -1 / (gam -1)

Equation 9. a / astar = ((.5 * (gam + 1))^ -((gam + 1) / (2 * (gam -1))) * ((1 + .5 * (gam - 1) * M ^2) ^ ((gam + 1) / (2 * (gam -1)))) / M

As a gas is forced through a tube, the gas molecules are deflected
by the walls of the tube. If the speed of the gas is much less than
the speed of sound of the gas, the density of the gas remains
constant. However, if the speed of the flow is nearer the speed
of sound we must consider compressibility
effects on the gas. The density of the gas will vary locally as
the gas is compressed by the walls of the tube. Regardless of the
speed, the flow conditions can be determined by conserving mass,
momentum, and energy
in the flow. For compressible flows with a smooth, gradual flow
turning, the change in flow properties are given by the **isentropic
flow relations** which are presented on this slide. Isentropic
means constant entropy which implies a
**reversible** process from the second
law of thermodynamics.

On this slide we have collected many of the important equations
which describe an isentropic flow. The derivation of the equations is
given on a separate slide. Beginning at the
top left, equation (1), the Mach number is
the ratio of the speed of the flow (V) to the speed of sound (s). The
speed of sound, in turn, depends on the
properties of the gas, as shown in equation (2). As shown on the
derivation slide, if we begin with the
entropy equations for a gas, it can be
shown that the pressure and density of an isentropic flow are related
as shown in the first part of equation (3). We can determine the
value of the constant by defining **total conditions** to be the
pressure and density when the flow is brought to rest isentropically.
The "t" subscript used in many of these equations stands for "total
conditions". (You probably already have some idea of total conditions
from experience with Bernoulli's equation).
Using the equation of state, we can easily
derive equations (4) and (5) from equation (3). Equations (6), (7),
and (8) are derived by considering the
total enthalpy, of the flow. While
equation (9) can be derived from the compressible
mass flow equation and defining the starred conditions to occur
when the flow is **choked** and the Mach number is equal to one.
Notice the important role that the Mach number plays in all the
equations on the right side of this slide. If the Mach number of the
flow is determined, all of the other flow relations can be
determined. Similarly, determining any flow relation (pressure ratio
for example) will fix the Mach number and set all the other flow
conditions.

The isentropic flow relations apply to gradual changes in the flow and the
same equations can be used to describe the flow through a tube or flow past
an object. If, however, the turning of the flow is abrupt and decreases the
flow area, shock waves are generated and the isentropic
relations are no longer valid because the flow is **irreversible **. The
flow is then governed by the oblique or normal
shock relations depending on the free stream Mach number
and the amount of flow turning.

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*byTom
Benson
Please send suggestions/corrections to: benson@grc.nasa.gov *