11. Introduction to Compressible Fluid Mechanics

Governing Equations

• From compressible Navier-Stokes equations, to Euler equations for compressible fluids with negligible viscosity and heat conduction effects. Under these assumptions, the diffusive parts of the equations disappear, and discontinuities (e.g. normal shocks, oblique shocks, or contact discontinuities) may appear in the field: as fields are not differentiable and not even continuous across a discontinuity, differential equations fail and jump conditions from integral equations are required to connect 2 regions of smooth fields.

• The definition of ideal flow requires the additional assumption of absence of shocks, beside negligible viscosity and heat conduction. Without shocks, differential equations hold in the whole domain and entropy equation shows that the flow is homoentropic, \(s(\mathbf{r},t) = \overline{s}\), if the inflow is uniform.

• If the flow is also irrotational, the governing equations can be written as a PDE for kinetic potential. Without shocks, differential equations hold in the whole domain and vorticity equation shows that the flow is irrotational in the whole domain if the inflow is irrotational.

• Acoustics …linearized equations…

Integral and differential form of the governing equations are provided, along with jump conditions: jump conditions play a crucial role in compressible inviscid flows governed by Euler equations where discontinuities (shocks and contact discontinuities) appear as a result of vanishing diffusion terms, in the limit of negligible viscous stress and heat conduction.

Vorticity, entropy and Bernoulli theorems

Vorticity equation and entropy equation are discussed in detail in order to use them and their conclusions in different models of a flow.

• Vorticity equation

• Entropy equation

• Bernoulli equation

Mathematical nature of equations of physics

The (local) mathematical nature of the problem depends on the (local) value of the Mach number. As an example, for Euler equations1 and derived models

• in subsonic regions:

  • in steady condition, the governing equations are elliptic

  • in unsteady condition, the governing equations are hyperbolic

• in supersonic regions:

  • in steady contditions, the governing equations are hyperbolic

  • in unsteady conditions, the governing equations are hyperbolic

As an example, the mathematical nature of governing equations of steady flows naturally apperas as a function of the (local) Mach number \(M\):

• in the potential equation for irrotational homo-entropic flows. The (global) nature of the linearized potential equation depends on the Mach number of the reference flow \(M_{\infty}\);

• in the characteristic method for steady compressible flows (todo collect all the material, now here and there, e.g. in Math:PDE:hyperbolic equations): in a d-dimensional domain, \(d+2\) families of characteristic surfaces exist in supersonic regions, while only \(d\) families exist in subsonic flows. Thus, it’s not possible to use the method of characteristics: the solution in a point \(P\) is not function of the values of \(d+2\) variables on the \(d+2\) characteristic lines through \(P\), but depends on the solution in its neighborhood (typical condition for elliptic equations).

Introduction to hyperbolic problems:

• Link to Math jbook: hyperbolic problems, method of characteristics, integral balance, jumps,…

• include script avaiable on GDrive in \(\texttt{fvm}\) folder

Canonical flows

• 1-dimensional ideal flow

• quasi 1-dimensional flows

• (quasi) 1-dimensional non-ideal flows: with friction (Fanno), and with heat flux (Rayleigh)

• 2-dimensional flows:

  • shocks: normal shocks and oblique shocks (\(\theta - \beta - M \))

  • rarefaction waves: Prandtl-Meyer exapnsion

  • discontituities

Acoustics

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1

Dissipative terms make Navier-Stokes equations elliptic in steady conditions and parabolic in unsteady conditions.