Are solutions of the Mach-Area-Relation unique?
Depends, can be one fixed area but two differnent solutions for Ma.
Then one sub and one supersonic.
Why is there a kink in the state distribution for a certain back pressure?
Split up of two possible solutions: one subsonic (Ma about 1) and one isentropic (ideal adapted flow solution).
-> Solution depends on pe.
What properties do hyperbolic processes usually have?
Kind of direction and kind of speed.
How is the characteristic velocity defined?
How is the characteristic curve defined?
Why are characteristic quantities constant along characteristic curves?
If y is characteritsic quantity, then it does not change, because dy=0, as long as inside of braket is 0 and this is when f(y) is built over dx/dt
Name all differences between linear and non linear transport processes.
Shape in linear processes does not change, whereas in non-linear case it changes dramatically.
What causes convergent and divergent characteristics?
Depending on characteristic distribution of velocity.
If velocity is not constant and therefore increasing or decreasing, the slope changes and we have either convergent or divergent types.
How does a shock form?
If we have convergent characteristocs, there will be a certain limit time where these characteristics intersect.
-> This is how shocks form.
Information collides into one point and thereby a loss of information happens (equvalent to the increase of entropy).
How can the shock speed Vs be computed?
Apply Rankine-Hugoniot conditons and properly find out shock speed.
How are characteristic velocities defined for systems of PDEs?
(Partial differential equation)
Same way as for single PDEs, but contribute eigenvalues and if they are real numbers, there are the characteristic velocities of the underlying transport processes.
-> Compatibility conditions
Give the compatibility conditions for 1-D time dependent Euler equations.
Linear
Non-linear
Linear:
Non-linear:
Linearize the euler equations in their characteristic form.
Which types of linear time dependend waves do exist?
Compression wave
Expansion wave
Entropy wave
What happens when M=1 with the wave speed for Y1?
then this wavespeed will be zero, because difference of speed of sound and velocity is zero.
Then Y1 does not lead to any information transport, neither upstream nor downstream, because resulting characteristic velocity would be zero.
Are linear compression waves equal to shock waves?
No, shockwaves tend to have the same tendency in changing states as compression waves. BUT: Characteristics can notbintersect and shockwaves are as per definition something where characteristics intersect.
Linear compression waves are somehow like a limiting approximation of shock waves when shock intensity approaches zero.
-> Highly simplified version of shockwave. BUT there are never shockwaves in linear theory.
How does a leftward running compression affect the flow velocity?
Ich hätte gesagt:
Decreases flow velocity by delta u.
How does a leftward running expansion wave affect flow velocity?
Increases velocity by delta u.
How does a rightward running expansion affect flow velocity?
Decreases velocity by delta u.
How does a rightward running compression wave affect flow velocity?
Which types of nonlinear time dependent waves do exist?
non smooth:
shock waves
entropy waves (jumps in entropy and desnity)
smooth:
expansion wave (can never be non smooth, no jumps possible)
entropy wave
Why are non linear characteristic curves no longer parallel?
Slope of curve depends on characteristic velocity which is not longer constant.
In some sense yes: Both generate pressure increase and change in velocity.
BUT: There are still not the same, because there is no shock in the linear wave. Shocks require intersection of characteristics and in linear case characteristics can never intersect.
Why are rarefraction (=expansion) fans develop?
We have different states at head and tail and these two differ from each other. Very different characteristic velocities and inbetween all states have to be possible
-> Because it is a continous process (no jumps)
-> Expansion fan possible
Can the nonlinear compatibility relations used to compute shocks?
No, because they require that the solution is smooth but they can be used to compute isentropic compression (smooth compression) along waves which have not jet formed a shock.
How does velocity change across rarefaction fan?
Same as in linear case:
When running in one direction, velocity increases in opposite direction.
Is there a difference in maximum velocity between steady and unsteady flow?
Yes!
State definition of Knudsen number and explain its relevance.
lambda = mean free path
L = characteristic length of a problem
We assume that continuum hypothesis holds -> Knudsen number
determines flow regime of gas
How is speed of sound defined in the framework of thermodynamics?
-> for ideal gas
Which effects have to be neglected when Euler Equations are applied?
Viscous effects
Heat transfer / heat sources
Gravity
How is total enthalpy related to total energy?
Which effect can not take place in isentropic flow?
Entropy remains constant
Shock waves
Explain concept of “material derivative”.
describes rate of change of a physical quantity as it moves along with a fluid particle
combines local and convective changes
Adiabatic exponent?
Why are the balance law for momentum and total energy balance laws but not conservation laws (in a strict sence)?
State the thermal and the caloric EoS for a perfect gas.
Under which conditions is total enthalpy constant?
Does the total conditions relation for the temperature require s=const?
No, don’t need do be assumed but for ptot and roh,tot!
Hm unsicher, ChatGPT sagt doch s muss konstant sein damit andere Relationen angewendet werden können.
Was ist hier richtig?
At which position in a Nozzle can M=1 be reached?
Samllest cross section
Can flows featuring shocks be isentropic?
No, because shocks increase entropy. Its not a smooth change, but its a jump.
Give the definition of critical Ma number.
Critical Ma number is a dimensionless velocity.
What does “critical temperature” mean in a fluid mechanics sense?
Static temperature when Ma=1 is reached.
Sketch relative mass flow density against Ma number.
Why is total temperature constant across a steady normal shock?
Because total enthalpy is constant:
H = h0 = cp * T0 and cp stays constant -> Therefore T has to be constant as well.
Hbefore = Hafter
Prandtls shock relation:
Normal shocks
With M > 1 and M^* < 1.
How is relative mass flow density defined?
Qualitative state changes across steady normal shock:
Sketch total pressure loss across steady normal shock.
Sketch the pressure distribution for subsonic and ideally adapted supersonic flow.
purple part can’t be described with our theory
Explain idea of normal shock solutions.
If something is non linear, equations might have multible solutions.
-> Check is these solutions are physically possible.
Jumps from sub to supersonic, therefore from Ma<1 to Ma>1, would result in a decrease in entropy and this is for us not possible.
-> No possible solution.
-> In this case: 1-D Euler equations
Why do all transsonic nozzle flows feature the same mass flow (for given p0, T0)?
If Ma=1, mass flow density is at ist maximum and can’t reach any higher values. In this case everything is in the convergent part of the nozzle and stays the same even for different exit pressures.
Change only possible with change in A or total pressure.
Why is our theory not sufficient for explanazion of all possible back pressures?
Some are not presentable with 1D theory, might have 2D or more, some also after nozzle…
What does homentropic mean and is it identical with isentropic?
Homentropic: entropy is constant in the whole flow field
Isentropic: entropy is constant along streamlines
-> Not identical!
What does isoenergetic mean?
Total enthalpy is constant in the whole flow field.
How is Mach angle defined?
Give formulas for linearized characteristic quantities and curves?
Can we apply this theory in subsonic flow reagions?
No! Because many quantities are only defined if MA > 1.
-> Not for subsonic or transsonic regions
How do compression and expansion waves turn the flow?
How are changes in velocity related to changes in pressure coefficient?
Multiply by -2.
State D’Alembert’s paradox.
2D steady, invisicid, subsonic flow
Lift is possible, but Drag is never zero
Can this “wave drag” be negative?
No
What is “wave drag”?
Drag caused by waves, typically due to losses in shock wave.
In linear theory there are no losses, but still have resulitng pressure forces in flow direction due to waves -> wave drag
Does a fat, cambered airfoil make sense for supersonic flight?
No, thickness is causing drag, camber is cuasing drag and none of them is causing Lift.
-> Wings are normally very slendered
How can Lift be generated in supersonic flight?
Angulation or angle of attack
Is M>1 lift without drag possible?
No, because only angulation causes lft, but this also causes drag.
Sketch the Ma cone as an evelope of unsteady waves.
How can normal shocks be related to oblique shocks?
If tangent component of a shock is zero, then its a normal shock.
-> Normal and oblique shocks are related over the tangent component wether it is zero or not.
Can oblique shocks lead to Ma = 1 after the shock?
Yes, supersonic, sonic and subsonic possible.
Is it important to use the red or the blue curve in epicycloid diagram?
No, doesn’t matter.
State Prandtls shock relation.
Oblique shocks.
How does the shock angle relate to the Ma angle for very weak shocks?
For every weak shock, the Ma number after thne shock will be nearly the same as before shock. This will lead to Ma angles which are nearly the same.
-> If we assume shock intensity near zero, then Ma- and shock angle will be the same.
To shich direction do we have to sketsch the nonlinear wave?
Always to local flow velocity.
Galilean invariance:
The laws of motion are the same in all inertial frames.
Can normal shocks lead to post-shock Ma number > 1?
No, Prandtl-shock relation.
-> Not possible
Refractive index?
n=c/c0
c = local speed of ligth
c0 = general speed of light
n= 1 + k * density
k = gladstone-dale constant
Give three options for visulization of compressible flow features.
Schlieren technique
Visulization of density variations
1st derivative
Shadowgraphy
Visulization of denisty variations
2nd derivative
Mach-Zehner interferometry
Sketch and explain schlieren images.
Schlieren pictures also can be taken using a prims. Then resulting pictures are coloured in blue or orange.
Sketch and explain shadow picture.
You know its shadowgraph, if shockwaves are composed of black AND white light
How are schlieren and shadow pictures related to each other?
What does schlieren knife do?
Cuts out portion of light under specific angle which can be selected by schlieren knife.
-> Cut out portion leads to a darker portion in schlieren image.
How can a boundary layer in a nozzle be visualized?
Use schlieren knife and position in normal direction to light.
-> Get density gradient in normal direction.
Typical example of parabolic PDE?
Heat equation
Types of PDEs and characteristics.
Parabolic PDE
try to reduce gradients by smoothing conditions over time
speed of process is just related to gradients
there is no process direction
Elliptic PDE
LaPlace equation
there is no time involved in process
gives optimal conditions for prescribed voundary data
Hyperbolic PDE
wave equation
have speed and direction
we need initial conditions, but under the expection thaht the domain is infinitely large, we don’t have to take any boundary conditions into account
phi is constant along curves
Linearized compatibility conditions.
Ideal gas law.
pV=nRT
Linearized cimpatibility conditions for 2-D subsonic flows.
Linearized wave pattern.
-> Draw!
-> Think about entropy wave.
Non-linearized wave pattern.
Non-liniear
expansion alsways ends in an expansion fan.
compression ends in a shot.
Relative overspeed and pressure coefficient.
Where does flow separation occure and why?
Blie circles:
At this position in reality, qou would never have flow separation, because p decreases. Therefore boundary condition is also reduced -> stabalized flow
Green dots:
Separation usually happens, when flow has to overcome a positive pressure gradient.
-> If pressure increases in flow direction (compression wave), there is a risk that flow separation occurs.
What happens if a channel would continue at its current end for an unlimited period of time without any geometry changes? Would the behavior of waves eventuelly subside in the sense of this theory?
No, in linear theory there is no damping mechanism present.
-> Waves would remain forever
But in reality:
Compression waves are shocks, and because of shock losses -> reduction of Ma number
Wave would stop when e.g. Ma=1 is reached
shocks cause entropy changes and therefore pressure losses
also viscid effects would damp these effects
Which general behavior must the shock have when interacting with expansion fan?
The more the expansion interacts with the shock, the weaker the shock will be.
-> Shock will be turned to alpha,infinity
Why is M-angle after the wedge smaller than at inlet flow?
M after expansion smaller than M,inlet
Smaller Ma number causes smaller angle
shock cuases loss in total pressure
Increases entropy
increases speed of sound
-> Decrease of M
-> Decrease of M angle
How would flow problem change qualitatively, if instead of the corner a continous redirection would exist?
post shock M number would not change
shock angle would not change
no sharp corner expansion, more ‘leisurely’ continous expansion
shock curvature occures earlier
State two typical applications for experiments in shock tubes. For each application, state the advantages of a shock tube in comparison to a continous wind tunnel.
Hypersonic aerodynamics
( Wind tunnels cannot produce M>5 flows)
Measurement of combustion delay time via shocks
(Wind tunnels cannot generate well-defined moving shocks)
When is linearized consideration justified?
When delta,u << c,ref
-> All changes suffieciently small
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