3-2. Basic Equations

**Equations of motion (2D flow)**

*RHS = 0, d/dt = 0: Steady inviscid flow: Euler equations of Motion *

*For an ideal gas the equation of state *

(p the pressure,rho the density, e the total energy und gamma the ratio of specific heats of the gas).

These Euler equations include the modelling of rotational flow.

**Potential theory**:

Potential flow simplification: Irrotationality, isoenergetic flow

Potential Phi, with grad(Phi) = (u, v); Streamfunction Psi

Hodograph transformation to velocity components (Q, theta) as independent variables:

Physical plane (x, y), Hodograph plane (Q, theta)

Use of dimensionless variables: velocity Q, flow angle theta, density D, local Mach number M

Isoenergetic relations between density, D, Mach number M and velocity Q:

resulting in Eq. (3) to be linear because D = D(Q) and M = M(Q).

A solution Phi, Psi (Q, theta) of Eq. 3 allows for integration of the physical plane coordinates (x, y):

which allows to obtain flow models in the physical plane from finding solutions of Eq. (3) in the Hodograph plane

Reshaping the Hodograph variables yields a canonical form (Beltrami equations), which is the basis for inverse** methods of elliptic continuation** for the subsonic part, and **hyperbolic mathods of characteristics** for the supersonic part of transonic flow

There is an analogy to electrostatic potential distributions solved by an equivalent Beltrami system, representing a 2D Ohm's law with electric potential, current strength and conductivity in a 2D conductor

Compilation of gasdynamic variables suitable for modifications of the basic equations to become design tools:

Ratio of specific heats | |

Ideal gas equation of state | |

Isentropic conditions | |

Energy equation | |

Speed of sound a | |

Mach number | |

Critical conditions if M = 1 |

Parameters, made dimensionless by critical values:

**"Fictitious Gas"**

Parameter manipulation to change the speed of sound to become larger than the flow speed also for conditions q > a* (= Q > 1), resulting in subsonic flow phenomena in the supercritical domain:

With unchanged definition of the speed of sound

the following relations are postulated to hold within the domain Q > 1, describing the properties of a fictitious gas (F. G.) ,

with a free parameter 0 <= lambda <= 1:

within the limits of an **incompressible F. G. (D = 1)**, and a **sonic flow F. G. (A = Q)**

All cases suppress the phenomena of supersonic flow.

Interpretation of the F. G. as controlled energy addition and subtraction