## Gas Dynamics & Supersonic FlowCompressible Flow Equations of Motion 1-D Isentropic Relations Wave Propagation Flow through Nozzles and Ducts 2-D Compressible Flow Prandtl-Meyer Expansion Shock Interactions Shock-Expansion Techniques for Aerofoils Method of Characteristics Unsteady Supersonic Flow Flow Tables/Software |
## 1-D Isentropic Flow Equations
For an isentropic flow, all the static properties such as
Eliminating where Multiplying throughout by $(γ-1)\/a^2$ yields,
This is the first relationship which connects temperature ratio with
Mach Number. Assuming isentropy and using the relation, $$ρ_0/ρ=(1+{γ-1}/2M^2)^{1/{γ-1}}$$ These relations prove very useful in calculating isentropic flows. Once Mach Number is known it is easy to calculate pressure, density and temperature as ratios of their stagnation values. These are tabulated as functions of Mach number in tables at the end of this section along with calculation scripts.
## Sonic Point as Reference
The preceding relations were arrived at with stagnation point as
the reference. It is also possible to choose the
a = a . Since ^{*}M = 1 , then u. As a consequence the energy equation, becomes
^{*} = a ^{*}Comparing with the temperature version of the energy equation then,
As a result for air with $$P^{\text"*"}/P_0=(2/{γ+1})^{γ/{γ-1}}=0.528$$ $$ρ^{\text"*"}/ρ_0=(2/{γ+1})^{1/{γ-1}}=0.634$$ It should be noted that a sonic point need not be present in the flow for the above equations to be applicable. ## Mass Flow Rate
An equation for mass flow rate can be determined in terms of Mach Number
of flow. The area in an isentropic flow where the Mach
Number becomes For an isentropic flow By noting that when
Substituting for terms such as This is a very useful result. It connects the
local area and local Mach Number. Tables at the end of this section list
this function, Figure 10 : Mach Number as a function of area.
## Equations of Motion in absence of Area ChangesThe equations that we have derived for mass, momentum and energy can be used for cases where there is no area change, $$h_1+1/2u_1^2=h_2+1/2u_2^2$$ |