Vorlesungsfragen

mit

• λ: mittlere freie Weglänge der Moleküle

• L: charakteristische Länge (z. B. Rohrdurchmesser)

Relevanz:

• Sie beschreibt das Verhältnis zwischen molekularer Bewegung und makroskopischer Länge.

• Für Kn < 0.01: Kontinuumsmechanik ist gültig → Navier-Stokes-Gleichungen anwendbar.

• Für Kn > 0.1: Molekulare Effekte werden wichtig → Gasdynamik erfordert kinetische Modelle (z. B. Boltzmann-Gleichung).

• Wichtig bei sehr kleinen Längen (z. B. Mikrokanälen) oder sehr dünner Atmosphäre (z. B. Höhenflug).

Die Schallgeschwindigkeit aaa ist in der Thermodynamik definiert als:

mit

• p: Druck

• ρ: Dichte

• Ableitung bei konstanter Entropie (isentropisch)

Für ideale Gase:

mit

• γ: isentroper Exponent (cp/cv​)

• R: spezifische Gaskonstante

• T: Temperatur

1. Viscosity → No shear stresses (no internal friction)

2. Heat conduction → No heat transfer due to temperature gradients

3. Diffusion → No mass transfer due to concentration gradients

H = U + p*V or H = E + p/rho

shocks (changes that are not smooth)

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1. Viscous dissipation → No friction or shear → no entropy production

2. Heat transfer → No heat exchange with surroundings

3. Shock waves → Shocks are irreversible and produce entropy

4. Mixing or diffusion → These cause entropy increase

The material derivative describes the rate of change of a quantity (e.g. temperature, velocity) as experienced by a moving fluid particle.

It combines local and convective changes:

Where:

• ϕ: any scalar or vector field (e.g. temperature, velocity)

• ∂ϕ/∂t: local (Eulerian) time change at a fixed point

• v*∇ϕ: change due to movement through a spatial gradient

Meaning: It tracks how a property ϕ\phiϕ changes along the path of a fluid element — essential for describing unsteady and spatially varying flows.

The momentum and total energy equations are called balance laws, not strict conservation laws, because:

• They include source terms:

  • Momentum balance includes body forces (e.g. gravity)

  • Energy balance includes work (e.g. pressure forces) and heat transfer

• A conservation law requires that a quantity is conserved without external input or loss. → Only mass is conserved in a strict sense (no source or sink).

Conclusion: Momentum and energy can be added to or removed from the system by external actions → they obey balance, not strict conservation.

Total enthalpy h0h_0h0​ is constant when the flow is:

• Adiabatic (no heat transfer)

• No work is done on or by the fluid (no shaft work, no electrical work)

• No viscous dissipation (frictionless flow)

• Steady flow (no time dependence)

Summary: In an isentropic, steady, inviscid, adiabatic flow without body forces or external work, total enthalpy remains constant along a streamline.

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Only at the smallest cross-section

no, shocks are always a jump in entropy (not const. entropy and no smooth change)

Static Temperature when M = 1 is reached

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The relative mass flow density ​

is the mass flow rate per unit area. For a perfect gas in isentropic flow, it is often expressed in terms of Mach number:

In dimensionless (normalized) form, relative to stagnation conditions:

This shows how the mass flow per area changes with Mach number under isentropic conditions.

Note: This expression reaches a maximum at M=1M = 1M=1 → used in choked flow and nozzle design.

not necessarily; if the flow is supersonic in the div. part of the nozzle, the ma differs

Bc. of the split up of two possible solutions, either we stay subsonic or reach supers.

1. last subsonic solution (critical subsonic solution)

2. isentropic solution/ ideal adapted flow solution

the kink is at A_min and is the point where it is decided

Normal shock solutions stem from the non-linear nature of the 1D Euler equations, which can mathematically admit multiple solutions.

The key idea:

• When analyzing compressible flows, we must check if multiple solutions exist and whether they are physically admissible.

In the case of a normal shock:

• A jump from supersonic (M > 1) to subsonic (M < 1) is physically possible, but

• A jump from subsonic to supersonic would require a decrease in entropy, which violates the second law of thermodynamics → not physically allowed

Conclusion: Only solutions that satisfy the conservation laws and result in increasing entropy (or at least no decrease) are physically valid. This ensures the shock wave proceeds from M > 1 to M < 1.

Because at Mach 1 (choked flow), the mass flow rate reaches its maximum for a given total pressure and total temperature:

• When M=1M = 1M=1 at the throat of the nozzle, → the flow is choked → the mass flow becomes independent of the exit pressure

• Any further decrease in back pressure does not increase the mass flow → only leads to changes in the diverging section (e.g. shocks or expansion)

• The converging section of the nozzle stays the same for all choked conditions

Therefore: All transonic flows with M=1M = 1M=1 at the throat have the same mass flow, and higher mass flow is only possible by increasing either:

• the throat area A∗A^*A∗

• the total pressure p0p_0p0​

• or the total temperature T0T_0T0​

This is the essence of maximum mass flow condition in compressible nozzles.

The quasi-1D theory (steady, isentropic, area-varying flow) cannot explain all possible back pressure scenarios because:

• It neglects non-isentropic effects like shocks and viscous losses

• It assumes 1D flow, but some phenomena (e.g. oblique shocks) are inherently 2D or 3D

• It cannot model shock-induced flow separation, boundary layers, or shock reflections

• For strongly over-expanded nozzles, the flow may become unstable or asymmetric, which 1D theory cannot capture

Example:

• Oblique shocks and shock-boundary layer interactions require multi-dimensional models

Conclusion: The basic theory is a good approximation for many cases, but realistic nozzle flows under all back pressures require more advanced models, including shock theory, 2D/3D flow and viscous effects

1) Schlieren technique

2) Shadowgraphy

3) Mach-Zehnder interferometry

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1. Schlieren photography – Visualizes density gradients using light refraction – Commonly used to observe shock waves and expansion fans

2. Shadowgraphy – Also shows density changes, but based on second derivative of refractive index – Simpler setup than Schlieren, but less sensitive

3. Interferometry – Measures precise density fields by observing interference patterns in light – Allows quantitative flow field analysis

The Schlieren knife (or knife edge) cuts out a portion of the light in the focal plane of the Schlieren setup.

• It blocks light rays that are deflected due to changes in fluid density, which affect the refractive index.

• The angle and position of the knife determine how strongly these density variations appear (e.g., as brighter or darker regions).

• By adjusting the knife’s angle, different aspects of the flow (e.g., shocks, shear layers) can be emphasized.

This selective blocking enhances the visibility of compressible flow phenomena like shock waves or expansion fans.

Schlieren: 1st Derivative of Density

Schadow: 2nd Derivative of Density

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Both Schlieren and Shadowgraph techniques visualize effects caused by variations in fluid density — but they differ in what quantity they highlight:

• Schlieren imaging visualizes the first spatial derivative of density (∂ρ/∂x or ∂ρ/∂y), i.e. density gradients. → It is sensitive to the direction of the gradient and is adjustable via the knife edge orientation.

• Shadowgraph imaging visualizes the second spatial derivative of density (∂²ρ/∂x² or ∂²ρ/∂y²), i.e. changes in the density gradient (density curvature). → It is more sensitive to sharp changes, like shock waves.

Schlieren Technique with the knife in normal direction so not in flow direction ?????

Some kind of direction and some kind of speed

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Hyperbolic processes typically have the following properties:

• Directional wave propagation: Information travels along specific paths called characteristics, which define directions in space-time.

• Finite propagation speed: Disturbances or signals propagate at finite speeds, so effects do not appear instantaneously at a distance.

• Possibility of discontinuities: Solutions can develop shocks or sharp fronts even from smooth initial data due to nonlinear effects.

• Well-posedness of initial value problems: Given initial conditions, solutions evolve uniquely and continuously over time.

These properties are fundamental in systems describing waves and compressible flows (e.g., Euler equations).

A characteristic curve is a path in space-time along which information or disturbances travel. For 1D flow, its slope is given by:

where uuu is flow velocity and ccc is speed of sound.

d_phy/d_t = 0 for the characteristic curve

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Characteristic quantities are constant along characteristic curves because:

• Along these curves, the governing PDEs reduce to ordinary differential equations (ODEs).

• The ODEs describe invariants that do not change as information propagates.

• Physically, these represent wave properties or Riemann invariants conserved along the path of wave propagation.

Thus, characteristic quantities remain unchanged along their respective characteristic curves.