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One-Dimensional Flow

Modern Compressible Flow, Chap 39/30/02; 10/2/02; 10/7/02; 10/9/02

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One-Dimensional and Quasi-One-

Dimensional Flow By one-dimensional flow, we mean a flow

whose properties are functions of only one spatial

coordinate, typically the streamwise coordinate.Strictly, this requires constant streamtube area.

But when the variation of area A = A(x) is

gradual, we often assume the flow is one-

dimensional. We call it a quasi-one-dimensionalflow.

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One-Dimensional Flow Equations

For a steady 1-D flow with constant cross-

sectional area, no body force, and no shaft

work,

2211 uu

2

222

2

111

upup

22

2

2

2

2

1

1

uhq

uh

(3.2)

(3.5)

(3.9)

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Sound Wave

A sound wave, by definition, is a weak pressure wave with

negligible friction and thermal conduction effects. In other

words, the thermodynamic process inside the sound wave

is isentropic.

The sound speed, a, can be expressed as

So the sound speed is a direct measure of compressibility.

ss

vp

a

(3.18)

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Sound Speed and Mach Number

For thermally perfect gas and calorically perfect

gas,

At sea level, a = 340.0 m/s = 1117 ft/s

The Mach number is defined as M=V/a.

M 1, supersonic flow

RTp

a

(3.19-20)

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Total (Stagnation) Conditions - 1

For a steady, inviscid, adiabatic flow with

negligible body force, we can prove that

or

along the streamline.

0)2/( 2

Dt

VhD

.2/2 constVh

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Total (Stagnation) Conditions2

Total Temperature

Imagine a fluid element in a flow is brought to zero speed

adiabatically, the resulting temperature is defined as the

total temperature To, and the corresponding enthalpy as the

total enthalpy ho.

For a steady, inviscid, adiabatic flow with negligible body

force, ho = const. along a streamline. For a calorically

perfect gas, this implies To = const. along the streamline.

If all streamlines are from a common freestream, then both

To and ho are constant throughout the entire flow.

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Total (Stagnation) Conditions - 3

Stagnation speed of sound:

where the subscript o denotes the Mach

zero condition arrived via an isentropic

process.

Total (stagnation) density:

RTa

RTp

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Total (Stagnation) Conditions4

Total Pressure and Density

Imagine that a fluid element is brought to

zero speed isentropically, the resulting

pressure and density are defined as the totalpressure po and total density o,

respectively.

Since an isentropic process is adiabatic, theresulting total temperature is the same as

that defined before.

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Characteristic Conditions - 1

T* If a fluid element is brought to sonic speed

adiabatically, the resulting temperature is defined

as T*. The speed of sound at this hypothetical Mach 1 is

defined as a*, which for thermally perfect gas and

calorically perfect gas is

** RTa

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Characteristic Conditions - 2

Characteristic Mach number: V/a*,

where * denotes the reference Mach 1

condition arrived via an adiabatic

process.

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Energy Equations - 1

Without Heat Addition

For 1-D flow without heat addition, the energy

equation for calorically perfect gas can be

expressed in alternative forms as follows:

22

2

2

2

1 uTcu

Tc pp

2121

2

2

2

2

2

1

2

1uaua

2121

2

2

21

2

2

1

1

1upup

(3.22)

(3.24)

(3.25)

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From the previous page, we have

So for each point in the flow, there is anassociated a*. For an adiabatic flow, a* is

constant in the flow.

2*22

)1(2

1

21a

ua

(3.26)

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Similarly, for the stagnation condition, we

can prove

opp Tcu

Tc 2

2

2

211 M

TTo

(3.27)

(3.28)

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For isentropic, calorically perfect gas,

12)

2

11(

M

p

po

1

1

2)2

11(

Mo

(3.30)

(3.31)

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The earlier equations yields,

1

2*

2*

oo TT

aa

1*

1

2

o

p

p

11

*

12

o

112

2*

2

M

M

(3.34)

(3.36)

(3.35)

(3.37)

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1-D Flow Energy Equations

(Continued)

4.1833.012*

atTT

o

)4.1528.0()1

2(

1

*

atp

p

o

)4.1634.0()1

2( 1

1*

at

o

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M*

)1()1(

2

2*

2

M

M

From the above equation,

M* = 1 if M =1M* < 1 if M < 1

M* > 1 if M > 1

ifMM

1

1*

(3.37)

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Normal Shock

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Normal Shocks

The normal shock wave (and the shock waves ingeneral) is a sudden discontinuity for flow

properties.

By definition, a normal shock wave is a shockwave that isperpendicular to the flow. The flowvelocity decreases across the normal shock wave;the flow is supersonic ahead ofthe normal shockwave and subsonic afterthe shock wave.

The static pressure, temperature and density allincrease across the normal shock wave.

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Shock Waves

If the flow is supersonic (relative to a moving

vehicle), then Vinf> ainfso the sound waves can no

longer propagate upstream ahead of the vehicle.Instead, they coalesce ahead of the vehicle,

forming a thin shock wave. See Fig. 4.5.

The shock wave is usually a few mean free path

thick, say, 10-5 cm for air at standard conditions.

The flow is adiabatic across the shock waves.

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Normal Shock Relations - 1

Prandtl Relation

For calorically perfect gas,

21

2* uua

1*

2

*

1MM

So the Mach number behind the normal shock

wave is always subsonic.

or(3.47)

(3.48)

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For calorically perfect gas with a given ,

M2, 2/1,p2/p1, and T2/T1 are functions of M1 only:

2/)1(

]2/)1[(12

1

2

12

2

M

MM

)1(1

21

2

1

1

2

Mp

p

2

1

2

1

1

2

)1(2

)1(

M

M

(3.51) from (3.37) and (3.48)

(3.57)

(3.53)

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Furthermore,

2

1

2

12

1

1

2

1

2

)1(

)1(21

1

21

M

MM

h

h

T

T

(3.59)

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Substituting eqs. (3.57) and (3.59) into

we have

1

2

1

2

12 lnln p

p

RT

T

css p

11

21ln

)1(

)1(211

21ln

2

1

2

1

2

12

112

MR

M

MMcss p

(3.60)

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From the 2nd law, s2>s1 only if M1>1. So the onlyphysically possible process is .

For ,

At M1 =1, M2 =2. Then we have Mach wave with no finiteproperty changes across the wave. Also,

Note that for thermally perfect gas, the changes across thenormal shock wave depend on both M1 and T1. Forchemically reacting gases, they depend on M1, T1 and p1.

11 M

1,,,11

2

1

2

1

2

2

T

T

p

pM

1

1M

6lim,378.0lim 12

211

MM M

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For a stationary normal shock, the total

enthalpy is constant across the shock wave.

For calorically perfect gas, since h = cpT,the total temperature is constant across the

normal shock wave:

021TT

o (3.61)

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For calorically perfect gas, the total pressure

decreases across the normal shock:

If the shock wave is not stationary, neither thetotal enthalpy nor the total temperatures are

constant across the shock wave.

R

ss

o

oe

p

p/12

1

2

(3.64)

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Hugoniot Equation The normal shock can be viewed as thermodynamic device

that compresses gas following the Hugoniot equation:

Since e1 is a function p1 and v1 and e2 a function of p2 and

v2, for given p1 and v1 upstream of a normal shock, each

point on the Hugoniot curve (the p2v2 curve represented bythe Hugoniot equation) represents a different shock with a

different upstream M1. See Figure 3.11.

)(2

2121

12 vvppee

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1-D Flow with Heat Addition - 1

Forsupersonic flow in region 1 when heat isadded

a. Mach number decreases, M2p1d. Total pressure decreases, po2T1

f. Total temperature increases, To2>To1For cooling, all trends are opposite.

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1-D Flow With Heat Addition - 2

Forsubsonic flow in region 1 when heat isadded

a. Mach number increases, M2>M1b. Velocity increases, u2>u1c. Pressure decreases, p2

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1-D Flow With Heat Addition - 3Rayleigh Curve (Fig. 3.13)

The Rayleigh curve is a curve for a give set of upstream

condition on the Mollier (h-s) diagram for 1-D heat

addition flow. Each point on the curve corresponds to a

different value of q added or taken away.

For the 1-D flow, the effect of heat addition is always to

drive the Mach number towards 1.

The maximum entropy location corresponds to the sonic

location. The flow is said to be choked at this conditionbecause of any further increase in q is not possible without

a drastic revision of the upstream condition in region 1.

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1-D Flow With Friction - 1

Forsupersonic inlet flow, the effect of friction on

the downstream flow is

a. Mach number decreases, M2

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1-D Flow With Friction - 2

Forsubsonic flow, the effect of friction on

downstream flow is

a. Mach number increases, M2>M1b. Velocity increases, u2>u1

c. Pressure decreases, p2

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1-D Flow With Friction - 3Fanno Curve (3.15)

The Fanno curve is a curve on the Mollier (h-s) diagramfor a given upstream condition for different amount offriction (different length of pipe).

The maximum entropy condition corresponds to the soniccondition at which the flow is choked. Friction alwaysdrive the Mach number towards 1.

Once the sonic condition is reached at the exit, anyincrease in pipe length is not possible without drastic

revision of the inlet condition.

Within the framework of 1-D theory, it is not possible tofirst slow a supersonic flow to the sonic condition and thento further slow it to subsonic speeds also by friction.

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