Aerothermodynamics of high speed flows - AERO 0033–1 Lecture 2

Aerothermodynamics of high speed flowsAERO 0033–1

Lecture 2: Flow with discontinuities, normal shocks

Thierry Magin, Greg Dimitriadis, and Johan [email protected]

Aeronautics and Aerospace Departmentvon Karman Institute for Fluid Dynamics

Aerospace and Mechanical Engineering DepartmentFaculty of Applied Sciences, University of Liege

Room B52 +1/443Wednesday 9am – 12:00pm

February – May 2017

Magin (AERO 0033–1) Aerothermodynamics 2016-2017 1 / 34

Outline

1 Introduction: high-temperature gas dynamics

2 Flows with discontinuitiesExamplesRankine-Hugoniot jump relations

3 Normal shocks


von Karman Institute for Fluid Dynamics

“With the advent of jet propulsion, it became necessary to broaden the field of

aerodynamics to include problems which before were treated mostly by physical

chemists. . .”

Theodore Karman, 1958

“Aerothermochemistry” was coined by von Karman in the 1950s to denote this

multidisciplinary field of study shown to be pertinent to the then emerging aerospace era


AERO 0033–1 (30h Th +30h Pr – 5 ECTS)

) Course contents – This course introduces the students to theaerothermodynamic analysis of high speed flows. Two main subjectsare addressed:

Transonic and supersonic aerodynamicsHigh-temperature gas dynamics (today’s introduction)

The lecture notes are available online in the form of presentations

Supplementary textbooksJ. D. Anderson, Modern Compressible Flow: With HistoricalPerspective, McGraw-Hill, 2002P. A. Thompson, Compressible-fluid Dynamics, Advanced engineeringseries, 1988A. H. Shapiro, The dynamics and thermodynamics of fluid flow, TheRonald Press Company, New York, 1953


Some additional references

J. D. Anderson, Hypersonic and high-temperature gas dynamics,American Institute of Aeronautics and Astronautics, 2006

M. Mitchner, C. H. Kruger, Partially ionized gases, Wiley, 1973

G. Vincenti, C. H. Kruger, Introduction to physical gas dynamics,Wiley, 1965

L. C. Woods, The thermodynamics of fluid systems, OxfordUniversity Press, 1975

D. A. McQuarrie and J. D Simon, Physical chemistry, a molecularapproach, University Science Books, 1997

G. A. Bird, Molecular Gas Dynamics and the Direct Simulation ofGas Flows, Oxford University Press, 1994

T. E. Magin, Physical Gas Dynamics, von Karman Institute for FluidDynamics, 2012


Introduction: high-temperature gas dynamics

Outline


2 Flows with discontinuities

3 Normal shocks



Design of spacecraft heat shields requires the developmentof integrated codes for flow/radiation/material coupling



Motivation: new challenges for aerospace science

Design of spacecraft heat shieldsModeling of the convective and radiative heat fluxes for:

Robotic missions aiming at bringing back samples to EarthManned exploration program to the Moon and Mars

Intermediate eXperimental Vehicle of ESA Ballute aerocapture concept of NASA

Hypersonic cruise vehicles and ballutesModeling of flows from continuum to rarefied conditions for the nextgeneration of air breathing hypersonic vehicles and Entry Descent andLanding (EDL) technologies such as ballutes



Motivation: new challenges for aerospace science

Electric propulsion (EP)Today, 20% of active satellites operate with EP systems

STO-VKI Lecture Series Electric Propulsion Systems: from recent research developments to industrial

space applications

EP system for ESA’s gravity mission GOCE ⇠20,000 space debris > 10cm

Space debrisSpace debris, a threat for satellite and space systems and for mankindwhen undestroyed debris impact the Earth

STO-VKI Lecture Series Space Debris, In Orbit Demonstration, Debris Mitigation



Motivation: engineering design in hypersonics

Blast capsule flow simulation

COOLFluiD platform and Mutation library

Relevant quantities for engineers:Heat flux &Shear stress to the vehicle surface

) Complex multiphysics problemChemical nonequilibrium (gas)

Dissociation, ionization, . . .Internal energy excitation

Thermal nonequilibriumTranslational and internal energyrelaxation

Gas / surface interactionRadiationTurbulence (laminar-turbulenttransition)Rarefied gas e↵ects



Example of aerospace mission killerInaccurate heat-flux predictions can be fatal for the crew / success ofa robotic missione.g. Lessons learned from Galileo’s entry into the Jovian atmosphere:

Stagnation point material recession ofheatshield was less than predicted

Ablation at frustrum and shoulder wasmuch higher than predicted

e.g. New carbon / resin composites require improved models



Gas-surface interaction: complex multiphysics problem

chemical(species((diffusion(

surface(recession(

virgin(material(

pyrolysis((zone(

porous(char(

hot(radia7ng(gas(

boundary(layer(

convec7ve((heat(flux(

radia7on(flux((

shock(heated(gas((T(~(10,000K)(

heat(transfer(

mechanical(erosion(

pyrolysis(gases(

reac7on(products(

OH( OH(

Radiative and convectiveheating

) Pyrolysis of phenolic resinC6H5 �OH (>200�C)

) Ablation of the charChemical mechanisms

oxidation (CO, CO2)nitridation (CN)

Phase changesmeltingsublimation (C, C2, C3)

Mechanical removalspallation, shear stress,melt removal



Three pillars for predictive engineering simulations

Computation cannot be truly predictive without the coupling tophysico-chemical models and experiments. . .

This coupling is precisely the verification and validation process!

Verification: Is the computational method implemented correctly?Validation : Are we solving the right equations?Uncertainty Quantification: Error bars in scientific predictions. . .



Heat-flux measurement in plasma jet obtained in VKIPlasmatron facility

Water-cooled calorimeter in a plasma jet stream, VKI Plasmatron,air, 8g/s mass flow, 3500 Pa pressure, 0.370 MHz frequency,

120 kW generator power, 0.16 cm torch diameter



Simulation of high-enthalpy flow in VKI Plasmatron facility

Inductively Coupled Plasma (ICP) wind-tunnelInjected gas

AirMass flow rate = 8 g/s, straight injectionTemperature = 300 K

Outlet pressure = 10 000 PaTorch characteristics (Plasmatron)

f = 0.37 MHzPower injected into the plasma = 60 kWConfiguration assumed to be axisymmetric

Dimensions [m]

0.16

0.15

0.486

0.05 0.05 0.05 0.05 0.050.127

0.21

8



Physico-chemical models for ICP wind-tunnel

Steady Navier-Stokes equations (including time-average Lorentzforces and Joule heating source terms)Helmholtz di↵usion equation for magnetic field (Maxwell equations)Example of high-temperature property: dynamic viscosity of air as afunction of temperature (at 1 atm pressure)) Viscosity cannot be directly measured at temperatures above 5000 K) Closure at microscopic level by means of the kinetic theory of gases

0 2500 5000 7500 10000 12500 15000Temperature [ K ]

0.0

0.5×10-4

1.0×10-4

1.5×10-4

2.0×10-4

2.5×10-4

η [

kg

m-1

K-1

]

Chapman-EnskogGupta-Yos’s mixture ruleWilke’s mixture rule

11-species air mixture at 1 atm pressure



Computed electromagnetic field

100

50

500

750 10001250

1500

150012501000

750500

250

250

100

12501500

1000750 25

0

15001250

750 1000

500

2000

2000

Induced electric field [V/m]



Computed temperature and flow fields

Temperature field [K]

Streamlines


Flows with discontinuities

Outline


2 Flows with discontinuitiesExamplesRankine-Hugoniot jump relations

3 Normal shocks


Flows with discontinuities Examples

Longshot Gun Tunnel at VKI

Schlieren visualisation of detached shocks in the Longshotfacility of the von Karman Institute for Fluid Dynamics

Nagamatsu probeStatic pressure

Reference probeStagnation point pressure andheat flux

FADS modelNose tip radius: 53 mmInstrumented with 5fast-response Kulite XCQ-093pressure sensors and 5 coaxialtype-E thermocouplesMeasurement locations: 1 atnosetip and 4 o↵-stagnationpoints at 45� equally spacedaround the probe



Electric Arc Shock Tube (EAST) facility at NASA ARC

Electric Arc Shock Tube facilityNASA Ames Research Center

Snapshot of radiating flow measured bymeans of a spectrometer

Wavelength resolved on x-axisSpatially resolved on y-axisRadiative intensity indicated by color



Shock tube operation

Wave propagation

Bursting of the diaphragm att = 0

Flow at rest and gases atambient temperatureDriver gas (region 4): highpressure p4Driven gas (region 1): lowpressure p1 ⌧ p4

Wave propagation at t = t⇤ > 0Expansion wave: region 3 ! 4Contact discontinuity wave:region 3 ! 2Shock wave: region 2 ! 1

Atmospheric entry shock layersimulated in region 2 behindshock wave


Flows with discontinuities Rankine-Hugoniot jump relations

Discontinuities in shock-tube flows

The interface between the regions 2 and 1 is called normal shockwaveIn the laboratory reference frame, the flow in region 1 is at rest(u1 = 0)As the normal shockwave propagates to the right with a velocity �, itincreases the pressure p2 of the gas behind it and induces a massmotion with velocity u2

In the shockwave reference frame (v = |u� �|), the normal shock waveappears stationary in space

The interface between the driver gas (region 3) and driven gase(region 2) is called contact surface

It propagates to the right with a velocity equal to the velocity of thegas in regions 2 and 3 (u2 = u3)The pressure is also preserved through a contact discontinuity (p2 = p3)



Shock wave at microscopic level

The density rises continuously in shockwavesThe shock thickness if of the order of the mean-free-path (averagedistance between two successive collisions of a molecule)The flow is rarefied ) failure of the fluid dynamical description

Normalized density distribution across shock wave in Argon at M=9

Alsmeyer experiment

Navier-Stokes eqs.simulation

Computational FluidDynamics (CFD)

Boltzmann eq.simulation

Direct SimulationMonte-Carlo (DSMC)



Euler eqs. as limit of the Navier-Stokes eqs.

Fluid dynamical description based on conservative equations for mass,momentum and total energy (in absence of external forces)

@t⇢+ @x

·(⇢u) = 0

@t(⇢u) + @x

·(⇢u⌦u + pI) + "@x

·⌧ = 0

@t(⇢E ) + @x

·(⇢uH) + "@x

·q + "@x

·(⌧·u) = 0

The Navier-Stokes equations are expressed by means of a closure. Forinstance, the viscous stress tensor is given by ⌧ = �2⌘S, with thevelocity gradient tensor S = 1

2

⇥@

x

u + (@x

u)T⇤� 1

3@x

·u IThe Navier-Stokes equations are written in a compact form as

@tU + @x

· F + "@x

· F d = 0

with U = (⇢, ⇢u, ⇢E )T and F = (⇢u, ⇢u⌦u + pI, ⇢uH)T

The Euler equations are obtained as the limit of the Navier-Stokesequations when the dissipative terms vanishes (" ! 0)



Rankine-Hugoniot jump relations

N-S and Euler solutions match in inviscid regions upstream (1) anddownstream (2) of a discontinuity where gradients vanishes (F d = 0)

Let us consider a point P on a discontinuity surface moving at speed�, with the normal n to the surfaceLet us assume the existence of a travelling wave solution U = U(y) tothe Navier-Stokes equations in the vicinity of P , with

Normalized density distribution across shock wave

Coordinate: y = x · n � �ntNormal component: �n = � · n

Travelling wave eq.��n

dUdy + d

dy (F ·n)+" ddy (F

d ·n) = 0

After integrationR y2y1(·) dy from

region 1 to region 2, the Rankine -Hugoniot jump relations for inviscidflows are derived

�n(U2 � U1) = (F · n)2 � (F · n)1Magin (AERO 0033–1) Aerothermodynamics 2016-2017 24 / 34

Normal shocks

Outline


2 Flows with discontinuities

3 Normal shocks


Normal shocks

Normal shock relations (inviscid flows)Rankine-Hugoniot relations for steady normal shocks� = 0, u = u ex , n = ex )

⇢2u2 = ⇢1u1

⇢2u22 + p2 = ⇢1u

21 + p1

⇢2u2H2 = ⇢1u1H1

These equations are valid for calorically perfect gases and mixtures ofgases in thermo-chemical equilibrium

For calorically perfect gases, further simplifications applyCalorically perfect gas law: p = ⇢RT with R = R

M and R = NAkB = 8.31 [J / (Kmole)]Specific energy: e = cvT , h = cpT , with cp � cv = R

Specific heat ratio: � =cpcv

Molecular gases: cv = 52R, � = 7

5 = 1.4

Atomic gases: cv = 32R, � = 5

3 = 1.66Entropy: s = cv ln

p⇢� = cp lnT � R ln p+ cp lnR

Speed of sound: a2 = (@p/@⇢)s = �p/⇢ = �RTMagin (AERO 0033–1) Aerothermodynamics 2016-2017 25 / 34

Normal shocks

Normal shock relations for calorically perfect gases

The total enthalpy conservation can be expressed as

H2 = H1 ) �

� � 1

p2⇢2

+u222

=�

� � 1

p1⇢1

+u212

The non-linear algebraic system of 3 eqs. in 3 unknowns ⇢2, u2, andp2 has a a closed solution expressed as a function of the dimensionlessparameters: Mach number M1 = u1/a1 and �, with a1 =

p�RT1

For M1 > 1(derivation given further in this section)

⇢2⇢1

= u1u2

=(�+1)M2

12+(��1)M2

1> 1

p2p1

= 1 + 2��+1 (M

21 � 1) > 1

T2T1

= ( p2p1)/( ⇢2⇢1

) =h1 + 2�

�+1 (M21 � 1)

i 2+(��1)M2

1(�+1)M2

1

�> 1

M22 =

1+ ��12 M2

1

�M21�

��12

< 1


Normal shocks

Entropy across normal shocks

The entropy variation through a normal shock is

s2 � s1 = cp lnT2

T1� R ln

p2p1

= cv lnp2/p1

(⇢2/⇢1)�

This function is positive for M1 � 1 ) the 2nd law of thermodynamicsimposes that, for a calorically perfect gas, a shock wave may onlyhappen if M1 � 1

What is the origine of the entropy increase through a shock wave?Answer: the changes across the shockwave occur through a shortdistance of the order of the mean-free-path. Through the shockstructure, the gradients are very large. In turns, heat fluxes and viscousstresses are dissipative phenomena that generate entropy

Weak shock: M21 = 1 + " with 0 < " ⌧ 1

p2p1

= 1 + 2��+1 " and ( ⇢2⇢1

)�� = (1 + ")��⇣1 + ��1

�+1 "⌘�

p2p1( ⇢2⇢1

)�� ⇠ 1 + 23�(��1)(�+1)2

"3 +O("4)

) s2�s1cv

⇠ 23�(��1)(�+1)2

"3 +O("4): isentropic approximation valid for weak shocks


Normal shocks

Total and critical quantities

Consider a fluid element in an arbitrary flow travelling at velocity uwith static pressure p and temperature T

Total pressure p0 and total temperature T0

They are defined as quantities obtained by isentropically decelerating the

flow to rest

⇢H = cpT0 = cpT + u2/2s = cp lnT0 � R ln p0 = cp lnT � R ln p

Critical conditions

Let us imagine that a fluid element is adiabatically decelerated (if M > 1)or accelerated (if M < 0) until its velocity equals the speed of sound:u⇤ = a⇤ =

p�RT⇤. The velocity reached at critical condition is sonic


Normal shocks

Total and critical quantities

In steady inviscid flows, we will show in lecture 2 that the totalenthalpy is constant along a trajectory (pathline)

Therefore, in steady inviscid flows, one has,

H = cpT0 = cpT⇤ +

�RT⇤

2= 1

��1a⇤2 +

a⇤2

2= 1

2

� + 1

� � 1a⇤2

) The total temperature, critical temperature, and critical speed of soundare constant properties of a streamline

Through a normal shock) The total temperature is conserved T0,2 = T0,1

) The critical temperature and critical speed of sound are conserved:

T ⇤2 = T ⇤

1 and a⇤2 = a⇤1) The total pressure is a decreasing function: p0,2

p0,1= exp(� s2�s1

R )


Normal shocks

Total quantities (local quantities with definition still valid for viscous flows)

The total temperature is obtained from

T0

T= 1 + 1

2

u2

cpT= 1 + �R

2cp

u2

a2

= 1 + ��12 M2

The isentropic relations p0p =

⇣⇢0⇢

⌘�=

⇣T0T

⌘ ��1

yields the expressions

for the total pressure and total density

p0p

=⇣1 + ��1

2 M2⌘ �

��1

⇢0⇢

=⇣1 + ��1

2 M2⌘ 1

��1

For M ⌧ 1, using the Taylor expansion (1 + x)↵ = 1 + ↵x +O(x2),the Bernoulli relation for incompressible flows is retrieved

p0p

= 1 + �2M

2 +O(M4) ) p0 ⇠ p + 12⇢u

2


Normal shocks

Pitot probe measurement in the Plasmatron facilityHenri Pitot (1695-1771), French hydraulic engineer, is the inventor ofthe Pitot tubePitot probes are widely used in fluid dynamics (airspeed of aircraftand air and gas velocities in industrial applications)

The stagnation pressure is measured at the probe nose pPitotFor low Mach number flows, pPitot = p0 ⇠ p + 1

2⇢u2

In low Reynolds number flows (e.g . subsonic plasma flows): pPitotdeparts from p0 due to viscous e↵ects [Barker, 1922]

10144

10145

10147

10146

1014410147

10146 10145

10146

10147

10148

Pressure field around a Pitot probe in a plasma jet at Mach = 0.1(�p = 1 Pa)

p=10,145 Pa, ⇢ = 6.6 kg/m3, u=137.2 m/s ) p0=10,207 PaThe Reynolds number based on the probe radius (3 mm) is Re = 19 and pPitot=10,225 Pa


Normal shocks

Prandtl’s relation for normal shocks

Prandtl’s relation, a⇤2 = u1u2, is derived as followsTotal enthalpy conservation: �

��1p2⇢2

+u222 = �

��1p1⇢1

+u212 = 1

2�+1��1a

⇤2

Alternative form: p2⇢2

=� � 1

2�

✓� + 1

� � 1a⇤2 � u22

◆

p1⇢1

=� � 1

2�

✓� + 1

� � 1a⇤2 � u21

◆

Introducing the previous relations in the ratio of momentum to massshock relations

⇢2u22 + p2⇢2u2

=⇢1u21 + p1

⇢1u1) u2 +

p2⇢2u2

= u1 +p1

⇢1u1

yields Prandtl’s relation after some algebra

The critical Mach number M⇤ = u/a⇤, can be obtained from

1

� � 1a2 +

u2

2= 1

2

� + 1

� � 1a⇤2 ) 1

� � 1

1

M2+

1

2= 1

2

� + 1

� � 1

1

M⇤2


Normal shocks

Application of Prandtl relation and critical Mach number

The critical Mach number behaves as the local Mach number butremains finite at high speeds

M⇤2 =(� + 1)M2

2 + (� � 1)M2)

M⇤ < 1 if M < 1M⇤ = 1 if M = 1M⇤ > 1 if M > 1

M⇤ ! �+1��1 if M ! 1

Alternative form of Prandtl relation for normal shocks: M⇤1M

⇤2 = 1

For a normal shock M1 > 1 ) M⇤1 > 1 ) M⇤

2 < 1 ) M2 < 1

The normal shock relations are then easily derived as follows⇢2⇢1

= u1u2

=u21

u1u2=

u21a⇤2

= M⇤21 =

(�+1)M21

2+(��1)M21

M⇤22 = 1

M⇤21

) (�+1)M22

2+(��1)M22=

2+(��1)M21

(�+1)M21

) M22 =

1+ ��12 M2

1

�M21�

��12

p2 � p1 = ⇢1u21 � ⇢2u22 = ⇢1u1(u1 � u2) = ⇢1u21(1� u2u1)

) p2�p1p1

= �M21 (1� u2

u1) = �M2

1

✓1� (�+1)M2

12+(��1)M2

1

◆= 2�

�+1 (M21 � 1)

T2T1

= ( p2p1)/( ⇢2⇢1

) =h1 + 2�

�+1 (M21 � 1)

i 2+(��1)M2

1(�+1)M2

1

�


Normal shocks

Exercise

Consider a shock-tube facility closed at both ends with a diaphragmseparating a region of high-pressure gas on the left (region 4) from aregion of low-pressure gas on the right (region 1). When the diaphragm isbroken at t = 0 s (for instance by mechanical means), a shock wavepropagates into section 1 and an expansion wave propagate into section 4.As the normal shock-wave propagate to the right with a constant velocity�, it increases the pressure of the gas behind it (region 2), and induces amass motion with velocity u2.

1 Derive the Rankine-Hugoniot jump relations between the regions 1 and 2 with the gasvelocities u1 and u2 in the laboratory frame.

2 Using the change of variables v = u � � for the velocity in the shockwave frame, showthat the normal shock relations are satisfied in this reference frame.

⇢2v2 = ⇢1v1

⇢2v22 + p2 = ⇢1v

21 + p1

h2 +12 v

22 = h1 +

12 v

21


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Aerothermodynamics of high speed flows - AERO 0033–1 Lecture 2