Modern Compressible Flow Chapter 3 Notes

Chapter 3 is titled "One Dimensional Flow"
==Introduction==
===Bell XS-1===
Anderson, in Modern Compressible Flow, discusses the Bell XS-1, a bullet-shaped rocket-powered aircraft, piloted by Chuck Yeager, that broke the "sound barrier" on October 14, 1947. During the first flight of the Bell XS-1, the plane reached Mach 1.06, creating a bow shock that was detached from the front of the aircraft. In a later flight on March 26, 1948, the aircraft reached Mach 1.45 in a dive, and the shock wave attaced itself to the nose of the aircraft, creating a conical oblique shock wave.
===Compressible Flow Regimes===
The Bell XS-1 had to go from the subsonic regime to the transonic regime to the supersonic regime.
*In the subsonic regime, pressure disturbances propagate faster than the aircraft.
*In the transonic regime, a shockwave will start to form in front of the aircraft, and the aircraft is travelling at the pressure disturbance speed. There is a disturbance free zone ahead of the shock wave.
*In the supersonic regime, supersonic flow is everywhere. A shockwave is attached to the nose and the leading edges of the wings and horizontal stabilizer. Again, there is a disturbance free zone ahead of the shock wave. The aircraft is travelling faster than the pressure disturbance.
If the aircraft is treated as a disturbance travelling at different speeds,
*At subsonic speeds, pressure disturbances travel at speeds greater than the speed of the aircraft.
*At sonic speeds, the aircraft is travelling at the speed of the pressure disturbances and the disturbances coalesce in front of the aircraft into a shockwave.
*At supersonic speeds, the aircraft is travelling faster than the speed of the pressure disturbances. All of the disturbances happen behind the oblique or conical shockwave.
Another visual example of these three speeds (subsonic, sonic, and supersonic) is shown below at right.
===Normal Shock===
A normal shock is one that is perpendicular to the free stream flow. Shock waves in one-dimensional, or 1-D, flow are exclusively normal because in 1-D flows, flow-field properties vary only with a single coordinate direction. 
(see normal shock diagram) 
In the above case, the upstream flow is supersonic; however, conditions downstream impose the pressure p2 which is great enough that only a normal shock can produce the required pressure rise. Something very similar occurs in a supersonic nozzle if the nozzle back pressure is too high.
===Blunt Nose Normal Shock===
The bow shock wave for a blunt body in super sonic flow is "always" detached from the body. At locations close to the nose of the body, the shockwave is close to perpendicular to the free stream, and is, thus, a normal shock. 
The properties just behind the shock in the region near the centerline are essentially post normal shock properties. The centerline stream tube stagnates on the nose centerline, and the other stream tubes bend and start to flow parallel to the blunt base. As the flow turns around the blunt body it again becomes sonic and supersonic.
==One-Dimensional Flow Equations==
For a 1-D control volume with steady flow 
(see 1-D control volume Figure 3.6 in Modern Compressible Flow)
If the area is constant, the continuity equation becomes simply: 
The momentum equation reduces to: 
And the energy equation reduces to:
:\frac{\dot Q}{A} + p_1 u_1 + \rho_1\left(e_1+\frac{u_1^2}{2}\right)u_1 = p_2 u_2 + \rho_2\left(e_2+\frac{u_2^2}{2}\right)u_2 
If this is divided by mass flux (ρu)
:q + \frac{p_1}{\rho_1}+ e_1 +\frac{u_1^2}{2} = \frac{p_2}{\rho_2} + e_2 + \frac{u_2^2}{2}
Introducing the enthalpy and defining the heat per unit mass, the energy equation reduces to 
These 3 equations are the governing fundamental equations for steady 1-D compressible flow. They relate properties at two different stations, 1 and 2, along a 1-D, constant area path. These equations, although simplified from their original integral forms, still maintain the basic concepts of mass, momentum, and energy conservation.
==Speed of Sound and Mach Number==
Consider a small disturbance that is moving through a compressible gas, and assume that the conditions behind the wave after it has passed are incrementally disturbed from their initial values in the gas. 
(see figure 3.7) 
Apply the conservation laws to this travelling disturbance wave.
===Continuity===
:\rho a=(\rho + d\rho)(a+da)
:\rho a=\rho a+\rho da+d\rho a+d\rho da
Keeping terms of only first order in differentials, because second order differentials are very, very small
:a = -\rho \frac{da}{d\rho}
===Momentum===
:p+\rho a^2 = (p+dp) + (\rho +d\rho)(a+da)^2
If you expand and cancel, and neglect all second order differential terms
:dp = -2\rho ada - a^2d\rho
Solving for da
:da = \frac{dp+a^2d\rho}{-2\rho a}
===Speed of Sound===
If this is substituted into the continuity equation
:a=-\rho \left(\frac{\frac{da}{d\rho} + a^2}{-2a\rho}\right)
Solving for a2
:a^2 = \frac{dp}{d\rho}
Because the disturbance wave has a very small strength, the change in pressure with respect to density is reversible, and no heat was generated or transfered. Hence (dp/dρ) is the isentropic derivative of pressure with respect to density. This equation is a fundamental expression for the speed of sound and is valid for all gases. It shows that the speed of sound is directly related to the compressibility of a gas. 
To show how the speed of sound relates to the compressibility of a gas:
:a^2 = \left(\frac{dp}{d\rho}\right)_{s=constant} = -\left(\frac{dp}{dv}\right)_s v^2 = -\frac{v}{\frac{1}{v}\left(\frac{dp}{dv}\right)_s}
Since the isentropic compressibility of a gas was defined in Chapter 1 as:
:\tau _s = \frac{1}{v}\left(\frac{dp}{dv}\right)_s
The speed of sound can be related to the compressibility of a gas as: 
 
Incompressible flow means that τs=0 or an infinite speed of sound in an incompressible medium (or M = ∞). 
For a thermally and calorically perfect gas, the isentropic relationship between pressure and density is
:p\rho^{-\gamma} = const
Differentiating this, and recalling that v = 1/ρ:
:\left(\frac{dp}{d\rho}\right)_s=\frac{\gamma p}{\rho}
Our expression for the speed of sound then becomes: 
Using the equation of state for a perfect gas p/ρ = RT, the speed of sound can be defined as: 
The two previous equations for the speed of sound hold for calorically and thermally perfect gases, but not for chemically reacting or real gases.
===Ratio of KE to Internal Energy in Flow===
For a flowing compressible fluid (assuming a calorically perfect gas), the ratio of kinetic energy to internal energy is:
:\frac{\frac{V^2}{2}}{e} = \frac{\frac{V^2}{2}}{c_v T} = \frac{\frac{V^2}{2}}{\frac{R}{(\gamma - 1)}T}=\frac{\frac{\gamma}{2}V^2}{a^2 \frac{1}{(\gamma - 1)}}=\frac{\gamma (\gamma -1)}{2}M^2
==Some Conveniently Defined Flow Parameters==
===Stagnation or Total Conditions===
Imagine a fluid element moving at some Mach number M, velocity V, with a static pressure p, and temperature T. Now imagine that same fluid element is isentropically brought to rest where the velocity and, consequently, Mach number are zero. The pressure and temperature increase from their starting state and are now at the stagnation, or total, state (the descriptors "stagnation" and "total" are synonymous). The total conditions are designated:
*p_o \text{ or } p_t \text{ — Total Pressure}
*T_o \text{ or } T_t \text{ — Total Temperature}
The total properties are associated with the fluid element while it is moving at velocity V. The actual pressure and temperature are known as the staticpressure and temperature. 
Using the definitions for total pressure and temperature, the total density and total speed of sound can be defined as:
*\rho_o \text{ or } \rho_t = \frac{p_t}{RT_t} \text{ — Total Density}
*a_o \text{ or } a_t = \sqrt{\gamma RT_t} \text{ — Total Speed of Sound}
===Sonic (*) Conditions===
Similarly to the description of the total conditions, imagine the same fluid element moving at some Mach number M, velocity V, with a static pressure p, temperature T. Now instead of isentropically stagnating the element, imagine it is brought to a sonic state (M = 1) adiabatically; meaning the element is slowed down if it began supersonic, or sped up if it began subsonic. The new conditions where M = 1 are called the sonic conditions and are designated with a *. The sonic pressure and temperature are designated p* and T*, respectively. 
Again, similar to the total conditions description, additional properties can be described using these sonic conditions. Using the definitions for sonic pressure and temperature, the sonic density and speed of sound can be defined as:
*\rho^* = \frac{p^*}{RT^*} \text{ — Sonic Density}
*a^* = \sqrt{\gamma RT^*} \text{ — Sonic Speed of Sound}
Admittedly, sonic is a confusing descriptor to use with speed of sound, but it is used to describe the speed of sound in the fluid if that fluid has been adiabatically brought to a sonic condition.
==Alternate Forms of the Energy Equation==
===Steady, Single Stream Conservation of Energy Equation===
For no heat or work transfer: 
If we assume a calorically perfect gas: 
Since
:R= c_p-c_v \to c_p = \frac{\gamma R}{\gamma-1}
The energy equation becomes:
:\frac{\gamma R}{\gamma-1} T_1 + \frac{u_1^2}{2} = \frac{\gamma R}{\gamma-1} T_2 + \frac{u_2^2}{2}
or, because a2 = γRT: 
Note that:
:a^2 = \frac{\gamma p}{\rho}
so that yet another form is: 
If u2 = a*, i.e. T2 = T*, then:
:\frac{a_1^2}{\gamma-1} + \frac{u_1^2}{2} = \frac{a^{*2}}{\gamma-1} + \frac{a^{*2}}{2}
or: 
Thus, the speed of sound can be determined corresponding to the conditions at point 1, i.e. knowing a1 and u1 or a1 and M1 or u1 and M1. 
Note that so long as the flow is adiabatic, the initial energy of the flow is constant; a
* is constant for an adiabatic stream.
===The Total Temperature and Total Pressure of a Compressible Flow===
For an adiabatic, compressible, thermally and calorically perfect gas, the stagnation (or total) enthalpy is:
:h_t = h + \frac{u^2}{2}
:c_p T_t = c_p T + \frac{u^2}{2}
:T_t = T+\frac{u^2}{2c_p} = T\left(1+\frac{u^2}{2c_pT}\right)
Isentropic flow allows us to write:
:\frac{p_t}{p} = \left(\frac{\rho _t}{\rho}\right)^\gamma = \left(\frac{T_t}{T}\right)^{\frac{\gamma}{\gamma -1}}
The key point here is that these relations apply to each point in the flow, that is, at point i(x,y,z) there is an associated set of total conditions. For a flow that is everywhere adiabatic and reversible (isentropic), then:
:P_t = P_{ti} = \text{constant throughout the flow}
:T_t = T_{ti} = \text{constant throughout the flow}
:\rho_t = \rho_{ti} = \text{constant throughout the flow}
===Other Useful Relations===
Returning to the energy equation in the form:
:\frac{a_1^2}{\gamma-1} + \frac{u_1^2}{2} = \frac{a_2^2}{\gamma-1} + \frac{u_2^2}{2}
Let u_2 → 0, then a_2 = a_t, the stagnation sound speed. 
Hence:
:\frac{a_t^2}{\gamma-1} = \frac{a^2}{\gamma-1} + \frac{u^2}{2}
But since:
:\frac{a^2}{\gamma-1} + \frac{u^2}{2} = \frac{\gamma +1}{2(\gamma -1)}a^{*2}
Then for isentropic flow:
===Ratios of Sonic to Stagnation Quantities for Dry Air===
For dry air, where γ=1.4
:\frac{T^*}{T_t} = 0.833
:\frac{p^*}{p_t} = 0.528
:\frac{\rho^*}{\rho_t} = 0.634
===An "Unfortunate" Definition===
If we return to the energy equation once more, written as
:\frac{a^2}{\gamma-1} + \frac{u^2}{2} = \frac{\gamma +1}{2\left(\gamma-1\right)}a^{*2}
Now, divide by u2 (the velocity on the left-hand side), to get:
:\frac{\left(\frac{a}{u}\right)^2}{\gamma -1} + \frac{1}{2}=\frac{\gamma +1}{2\left(\gamma-1\right)}\frac{a^{*2}}{u^2}
Define M
* as the ratio of the actual velocity to the sonic speed of sound:
:M^* = \frac{u}{a^*}
Then the previous equation becomes:
:\frac{\left(\frac{1}{M}\right)^2}{\gamma -1} + \frac{1}{2}=\frac{\gamma +1}{2\left(\gamma-1\right)}\frac{1}{M^{*2}}
Solving for M2: 
This M
* is not the sonic Mach number, but rather a reference Mach number. Anderson notes that M and M
* are correlated in that if,
*M
* = 1, M = 1
*M
* > 1, M > 1
*M
* < 1, M < 1
:\lim_{M \to \infty} M^* = \sqrt{\frac{\gamma +1}{\gamma -1}}
==Normal Shock Relations==
A normal shock is a thin (a few molecular mean free paths or ~10-7m for air a standard conditions) discontinuity in the flow; all properties but total enthalpy, or total temperature, change. Total enthalpy remains constant because a shock is assumed to be adiabatic.
===Algebraic solution of the equations of motion across a normal shock===
For the purposes of our analysis, the normal shock is assumed to be a discontinuity in a control volume, across which the flow properties suddenly change. The standard convention for normal shock relations is to designate the region upstream of the normal shock as station 1 and the region downstream as station 2. The one-dimensional flow equations for this control volume analysis are the same as always:
*Continuity
::\rho_2 u_2 = \rho_1 u_1
*Momentum
::p_2 + \rho_2 u_2^2 = p_1 + \rho_1 u_1^2
*Energy
::h_2 + \frac{u_2^2}{2} = h_1 + \frac{u_1^2}{2}
If we assume a calorically perfect gas, we can also introduce the following thermodynamic relations:
:p = \rho RT ~\text{ (Thermally Perfect)}
:h = c_p T ~~\text{ (Calorically Perfect)}
With these 5 equations, and knowledge of all upstream conditions (ρ1, u1, p1, h1, T1), we can solve for the downstream conditions (ρ2, u2, p2, h2, T2). 
To solve for these 5 unknowns, first we divide the momentum equation by the continuity equation:
:\frac{p_1}{\rho_1 u_1} - \frac{p_2}{\rho_2 u_2} = u_2 - u_1
Since a2 = γp/ρ:
:\frac{a_1^2}{\gamma u_1} - \frac{a_2^2}{\gamma u_2} = u_2 - u_1
Using one of our previously defined alternate forms of the energy equation, we can write:
:a_1^2 = \frac{\gamma + 1}{2}a^{*2}-\frac{\gamma -1}{2}u_1^2
:a_2^2 = \frac{\gamma + 1}{2}a^{*2}-\frac{\gamma -1}{2}u_2^2
But since the shock is adiabatic, a1* = a2*, and these two equations can be combined yielding:
:\frac{\gamma + 1}{2\gamma u_1 u_2}\left(u_2 - u_1\right) a^{*2} + \frac{\gamma-1}{2\gamma}\left(u_2 - u_1\right) = u_2 - u_1
Dividing by (u2 - u1) yields:
:\frac{\gamma + 1}{2\gamma u_1 u_2}a^{*2} + \frac{\gamma-1}{2\gamma} = 1
When solved for a
*2, we get the Prandtl Relation: 
The Prandtl Relation is a useful intermediate relation for normal shocks. It relates the upstream and downstream velocities across a normal shock to the speed of sound at M= 1.
From this relation, we can see that: 
Which relates the Mach number ahead of the shock to the Mach number behind it. Since the flow ahead of a shock must always be supersonic, M1 > 1, and thus M
*1 > 1, M
*2 must be subsonic, and thus M2is also subsonic. 
Previously,Anderson had shown that there was a relation between M and M
*''
:M^2 = \frac{2}{\left(\frac{\gamma +1}{M^{*2}}\right) - (\gamma - 1)}
When this is solved for M
*2''
:M^{*2} = \frac{(\gamma +1)M^2}{2+(\gamma -1)M^2}
If this is then substituted into our previous relation between M
*2 at station 1 and 2:
:\frac{(\gamma +1)M_1^2}{2+(\gamma -1)M_1^2} = \left [ \frac{(\gamma +1)M_2^2}{2+(\gamma -1)M_2^2} \right ]^{-1}
Solving for M22to get the relation between the upstream and downstream Mach number: 
Note that M2 only depends on M1 for a given ratio of specific heats.
Also, for M1 = 1, M2 = 1 thus the normal shock is "infinitely weak" when M1= 1 (so nothing changes across this "Mach wave"). 
As the upstream Mach number approaches infinity, the downstream Mach number reduces to:
:M_2 = \sqrt{\frac{\gamma -1}{2\gamma}}
For air with γ = 1.4, as M1 approaches infinity, M2= 0.378. 
All other properties downstream of the normal shock depend only on their upstream values and M1 (for a given γ). 
From continuity, for example:
:\frac{\rho _2}{\rho _1} = \frac{u_1}{u_2} = \frac{u_1^2}{u_1 u_2} = \frac{u_1^2}{a^{*2}} = M_1^{*2}
Since:
:M_1^{*2} = \frac{(\gamma + 1)M_1^2}{2+(\gamma - 1)M_1^2}
Then: 
===Solving for the Pressure and Temperature Ratios across the Shock===
Starting with the momentum equation, put is into the form:
:p_2-p_1 = \rho_1 u_1^2 - \rho_2 u_2^2 = \rho_1 u_1(u_1-u_2) = \rho_1 u_1^2 \left(1-\frac{u_2}{u_1}\right)
Divide by p1 and recognize that:
:\frac{\rho_1 u_1^1}{p_1} = \frac{\gamma u_1^2}{a_1^2} = \gamma M_1^2
The previous equation becomes:
:\frac{p_2 - p_1}{p_1} = \gamma M_1^2 \left(1-\frac{u_2}{u_1}\right)
However since:
:\frac{u_2}{u_1} = \frac{\rho_1}{\rho_2} = \frac{2+(\gamma - 1)M_1^2}{(\gamma + 1)M_1^2}
The pressure ratio across a shock can be expressed as: 
The temperature ratio follows from the ideal gas equation of state, p = ρRT:
:\frac{T_2}{T_1} = \frac{p_2}{p_1}\frac{\rho_1}{\rho_2}
Substituting the density and pressure ratios, the temperature ratio becomes: 
For a gas with constant specific heats, all of the conditions downstream of a normal shock depend only on the upstream Mach number and ratio of specific heats. At higher speeds where the temperature increase across the shock causes molecular dissociation, the downstream properties depend on M1, γ1, and T1.
===Limiting Property Ratios for Calorically and Thermally Perfect Gas===
Assuming the gas is calorically and thermally perfect, what are the limiting values of velocity, density, pressure, and temperature ratios as M1approaches infinity. 
Anderson (p.91) gives these ratios (along with their values in air) as:
:\lim_{M_1 \to \infty} M_2 = \sqrt{\frac{\gamma-1}{2\gamma}} = 0.378
:\lim_{M_1 \to \infty} \frac{\rho _2}{\rho _1} = \frac{\gamma + 1}{\gamma - 1} = 6
:\lim_{M_1 \to \infty} \frac{u_2}{u_1} = \frac{\gamma - 1}{\gamma + 1} = \frac{1}{6}
:\lim_{M_1 \to \infty} \frac{p_2}{p_1} = \infty
:\lim_{M_1 \to \infty} \frac{T_2}{T_1} = \infty
===Entropy Change Across a Normal Shock===
The entropy change in the flow passing through a shock wave is:
:\Delta s_{shock} = s_2 - s_1 = c_p \ln \frac{T_{t2}}{T_{t1}} - R \ln \frac{p_{t2}}{p_{t1}}
However, shocks are considered to be adiabatic, so Tt1 = Tt2, so:
:\Delta s_{shock} = - R \ln \frac{p_{t2}}{p_{t1}}
Note that we can also address the change in entropy across the shock as:
:\Delta s_{shock} = c_p \ln \frac{T_{2}}{T_{1}} - R \ln \frac{p_{2}}{p_{1}}
or, 
This equation shows that the change in entropy across a shock is only a function of γ, cp, R, M1.
:
If M1 < 1	\Delta s_{shock} < 0(Physically impossible)
If M1 = 1	\Delta s_{shock} = 0(Isentropic Mach wave)
If M1 > 1	\Delta s_{shock} > 0
Only conditions where Δsshock ≥ 0 are permitted by the second law of thermodynamics (in cases where no work or heat is transfered); therefore, shock waves cause an entropy increase in the flow and can only exist if M1 > 1.
The physical mechanism by which the shock waves create entropy is attributed to viscous friction due to large gradients of velocity over a very short distance (shock wave thickness). 
The result of the friction is a large temperature gradient within the shock wave, also creating entropy.
The shock strength is defined as the rise in entropy across the shock.
===Total Pressure Ratio across a Shock Wave===
Since M2 is known in terms of M1 (for a given γ), pt2/pt1 can be found in terms of M1 and γ. 
The relation is: 
Anderson provides a figure showing these ratios as a function of M1 for γ = 1.4 (Fig. 3.10, p. 94). Anderson's text provides normal shock tables for γ = 1.4 as an appendix.
==Hugoniot Equation==
The algebraic solution of the conservation equations across a normal shock were obtained in terms of γ and M1. Alternate forms of the solution exist in terms of primitive thermal and fluid flow properties. Returning to the continuity equation:
:u_2 = u_1 \frac{\rho_1}{\rho_2}
Substitute this into the momentum equation:
:p_1 + \rho_1 u_1^2 = p_2 + \rho_2 \left(u_1 \frac{\rho_1}{\rho_2} \right) ^2
Solve for u12:
:u_1^2 = \frac{p_2-p_1}{\rho_2-\rho_1}\left(\frac{\rho_2}{\rho_1}\right)
Similarly it can be shown that:
:u_2^2 = \frac{p_2-p_1}{\rho_2-\rho_1}\left(\frac{\rho_1}{\rho_2}\right)
Next the energy equation is:
:h_1 + \frac{u_1^2}{2} = h_2 + \frac{u_2^2}{2}
or:
:e_1 + \frac{p_1}{\rho_1} + \frac{u_1^2}{2} = e_2 + \frac{p_2}{\rho_2} + \frac{u_2^2}{2}
Substituting then for u12 and u22:
:e_1 + \frac{p_1}{\rho_1} + \frac{1}{2} \left [ \frac{p_2-p_1}{\rho_2-\rho_1}\left(\frac{\rho_2}{\rho_1}\right) \right ] = e_2 + \frac{p_2}{\rho_2} + \frac{1}{2} \left [ \frac{p_2-p_1}{\rho_2-\rho_1}\left(\frac{\rho_1}{\rho_2}\right) \right ]
This simplifies algebraically to:
:e_2-e_1 = \frac{p_2-p_1}{2}\left(\frac{1}{\rho_1} - \frac{1}{\rho_2}\right)
or: 
This last equation is called the Hugoniot equation. It holds for ideal gas, real gas, or reacting gas, as long as the gas is in thermodynamic equilibrium (i.e. have definite e, h, p, T, etc).
The Hugoniot equation can be written as:
:\Delta e_{shock} = -p_{mean}\Delta v
This is analogous to the differential for reversible work in open systems:
:d W_{rev} = -p d v
===Hugoniot Curve===
For a simple compressible gas, any thermodynamic state variable can be written as a function of two other state variables, e.g. for a thermally and calorically perfect gas,
:e = e(p,v) = c_vT = c_v\frac{p}{\rho R} = c_v\frac{pv}{R}
Thus the Hugoniot equation can be written as:
:c_v\frac{p_2v_2}{R}-c_v\frac{p_1v_1}{R} = \frac{p_2-p_1}{2}\left(v_1 - v_2\right)
Since:
:\frac{c_v}{R} = \frac{1}{\gamma-1}
The previous equation can be reduced to:
:\frac{1}{\gamma-1}(p_2v_2-p_1v_1) = \frac{p_2-p_1}{2}\left(v_1 - v_2\right)
It can be shown that:
:\frac{p_2}{p_1} = \frac{\frac{\gamma+1}{\gamma-1}-\frac{v_2}{v_1}}{\left(\frac{\gamma+1}{\gamma-1}\right)\frac{v_2}{v_1} - 1}
Anderson gives this equation on page 101 as:
:\frac{p_2}{p_1} = \frac{\left(\frac{\gamma+1}{\gamma-1}\right)\frac{v_1}{v_2}-1}{\frac{\gamma+1}{\gamma-1} - \frac{v_1}{v_2}}
In any case, the Hugoniot curve is a curve of pressure versus specific volume. The flow upstream of the shock wave defines one point on this curve.
See figure 3.11 in the book for a depiction of the Hugoniot curve.
How can we determine point 2 on the Hugoniot curve (the state downstream of the shock)?
by numeric iteration of the Hugoniot equation
take equation 3.67 and write as:
:u_1^2 = \frac{p_2 - p_1}{\rho_2 - \rho_1}\frac{\rho_2}{\rho_1} = \frac{p_2 - p_1}{\frac{1}{v_2} - \frac{1}{v_1}}\frac{v_1}{v_2}This can be rearranged to: This is the slope of a straight line passing through both points (p1,v1) and (p2,v2).
==One-Dimensional Flow with Heat Addition==
The system of governing equations that applied to the shock wave region, which is infinitesimal in thickness can also apply to a finite region or length of flow.
Consider a length of constant area pipe or duct or streamtube. The flow is assumed to be non-adiabatic, i.e. heat is added or subtracted, per unit mass, of magnitude q.
The governing equations are:
Continuity
Momentum
Energy 
Also:
:h=c_pT
:p=\rho RT
The energy equation implies that,
:q = \left(h_2+\frac{u_2^2}{2}\right) - \left(h_1+\frac{u_1^2}{2}\right) = \left(c_pT_2+\frac{u_2^2}{2}\right) - \left(c_pT_1+\frac{u_1^2}{2}\right)
Heat addition changes the total enthalpy of the flow, and for constant cp (calorically perfect gas), the total temperature as well.
===Ratios of Properties Across the Control Volume===
Since:
:\rho u^2 = \rho a^2 M^2 = \rho \frac{\gamma p}{\rho}M^2 = \gamma p M^2
Momentum conservation is then:
:p_1\left(1+\gamma M_1^2\right) = p_2\left(1+\gamma M_2^2\right)
From the perfect gas equation of state, p = ρRT:
:\frac{T_2}{T_1} = \frac{p_2}{p_1} \frac{\rho_1}{\rho_2} = \frac{p_2}{p_1} \frac{u_2}{u_1}
Using the calorically and thermally perfect gas definition of Mach number the ratio of u2/u1 can be defined as:
:\frac{u_2}{u_1}=\frac{M_2}{M_1}\sqrt{\frac{T_2}{T_1}}
From this equation, the temperature ratio can be derived as: 
Since:
:\frac{\rho_2}{\rho_1} = \frac{p_2}{p_1}\frac{T_1}{T_2}
The density ratio is: 
The ratio to total pressures and temperature can be derived from the isentropic relationships between static and total conditions and the ratios of static conditions that were previously derived.
One possible solution procedure
#Specify p1, T1, ρ1, u1, M1, γ, Tt1, pt1, etc for all upstream conditions
#Specify q
#Calculate Tt2 = Tt1 + q/cp
#Iteratively, trial and error, solve for M2 from equation 3.84
#Solve for p2, T2, ρ21, pt2 from equations 3.78, 3.81, 3.82, 3.83
===Sonic Reference Conditions===
Consider a flow at a given Mach number, M1, with properties p1, T1, γ, etc. Assume that a quantity of heat has been added, enough to drive M2 to a sonic value, i.e. M2 = 1. Then p2 = p
*, T2 = T
*, etc.
The ratios of parameters under this case are:
Note that the starred quantities are constants for each flow; therefore, they can be tabulated as functions of γ and Mach number, where M is the Mach number upstream, before heat addition.
How can we use the tabulated data to solve a Rayleigh flow problem? Assume that Rayleigh tables have been given,
#Specify p1, T1, ρ1, u1, M1, etc for all upstream conditions
===Summary of Physical Changes with Heat Addition===
When heat is added to a:	When heat is added to a:
Supersonic Flow 	Subsonic Flow
M2 < M1 	M2 > M1
p2 > p1 	p2 < p1
T2 > T1 	T2 > T1 for M1 < γ-1/2 else T2 < T1
Tt2 > Tt1 	Tt2 > tt1
pt2 < pt1 	pt2 < pt1
u2 < u1 	u2 > u1
==One-Dimensional Flow with Friction (Fanno Flow)==
Although this course pertains to inviscid, or frictionless, flow, constant area duct flow with wall friction can be easily treated. This section treats one-dimensional flow without heat transfer, but with friction.
(See figure 3.14)
(start on 10/9)
The equation of state can be used to get a density ratio:
Finally, the total pressure ratio:
===Sonic Conditions===
Use of the equations for frictional, adiabatic flow are again expedited by defining sonic reference conditions. Let M2 = M
* = 1, where p2 = p
*, T2 = T
*, Pt2 = Pt
*, and ρ2 = ρ
*.
Then the property ratios are:
:\frac{T}{T^*} = \frac{\gamma +1}{2+(\gamma-1)M^2}
:\frac{p}{p^*} = \frac{1}{M}\left(\frac{\gamma +1}{2+(\gamma-1)M^2}\right)^{0.5}
:\frac{\rho}{\rho^*} = \frac{1}{M}\left(\frac{2+(\gamma-1)M^2}{\gamma +1}\right)^{0.5}
:\frac{p_t}{p_t^*} = \frac{1}{M}\left(\frac{2+(\gamma-1)M^2}{\gamma +1}\right)^{\frac{\gamma +1}{2(\gamma-1)}}
Denoting L = L* as the length at which the friction coefficient causes the flow to accelerate (subsonic) or decelerate (supersonic) to M = M
* = 1, then:
:\int_0^{L^*} \frac{4fdx}{D} = \left [ -\frac{1}{\gamma M^2} - \frac{\gamma +1}{2\gamma}\ln\left(\frac{M^2}{1+\frac{\gamma -1}{2}M^2}\right) \right ] _M ^1
Introduce a mean friction coefficient,
:\bar f = \frac{1}{L^*}\int_0^{L^*}f dx
and the previous equation can be written:
:\frac{4 \bar f L^*}{D} = \frac{1-M^2}{\gamma M^2} - \frac{\gamma +1}{2\gamma}\ln\left(\frac{M^2}{1+\frac{\gamma -1}{2}M^2}\right)
For each value of M on the right hand side, a unique value for 4fL
*/D is generated on the left hand side. That is,
:\frac{4 \bar f L^*}{D} = f(\gamma ,M)
where M is the inlet Mach number. We therefore create a table, Table A.4, of 4fL
*/D as a function of M (for γ = 1.4).
==Historical Note: Sound Waves and Shock Waves==
==Summary==