Aerothermodynamics of high speed flows

Transcription

1 Aerothermodynamics of high speed flows AERO Lecture 2: Flow with discontinuities, normal shocks Thierry Magin, Greg Dimitriadis, and Johan Boutet Thierry.Magin@vki.ac.be Aeronautics and Aerospace Department von Karman Institute for Fluid Dynamics Aerospace and Mechanical Engineering Department Faculty of Applied Sciences, University of Liège Room B52 +1/443 Wednesday 9am 12:00pm February May 2017 Magin (AERO ) Aerothermodynamics / 34

2 Outline 1 Introduction: high-temperature gas dynamics 2 Flows with discontinuities Examples Rankine-Hugoniot jump relations 3 Normal shocks Magin (AERO ) Aerothermodynamics / 34

3 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 Kármán, 1958 Aerothermochemistry was coined by von Kármán in the 1950s to denote this multidisciplinary field of study shown to be pertinent to the then emerging aerospace era Magin (AERO ) Aerothermodynamics / 34

4 AERO (30h Th +30h Pr 5 ECTS) ) Course contents This course introduces the students to the aerothermodynamic analysis of high speed flows. Two main subjects are addressed: Transonic and supersonic aerodynamics High-temperature gas dynamics (today s introduction) The lecture notes are available online in the form of presentations Supplementary textbooks J. D. Anderson, Modern Compressible Flow: With Historical Perspective, McGraw-Hill, 2002 P. A. Thompson, Compressible-fluid Dynamics, Advanced engineering series, 1988 A. H. Shapiro, The dynamics and thermodynamics of fluid flow, The Ronald Press Company, New York, 1953 Magin (AERO ) Aerothermodynamics / 34

5 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, Oxford University Press, 1975 D. A. McQuarrie and J. D Simon, Physical chemistry, a molecular approach, University Science Books, 1997 G. A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Oxford University Press, 1994 T. E. Magin, Physical Gas Dynamics, von Karman Institute for Fluid Dynamics, 2012 Magin (AERO ) Aerothermodynamics / 34

6 Introduction: high-temperature gas dynamics Outline 1 Introduction: high-temperature gas dynamics 2 Flows with discontinuities 3 Normal shocks Magin (AERO ) Aerothermodynamics / 34

7 Introduction: high-temperature gas dynamics Design of spacecraft heat shields requires the development of integrated codes for flow/radiation/material coupling Magin (AERO ) Aerothermodynamics / 34

8 Introduction: high-temperature gas dynamics Motivation: new challenges for aerospace science Design of spacecraft heat shields Modeling of the convective and radiative heat fluxes for: Robotic missions aiming at bringing back samples to Earth Manned exploration program to the Moon and Mars Intermediate experimental Vehicle of ESA Ballute aerocapture concept of NASA Hypersonic cruise vehicles and ballutes Modeling of flows from continuum to rarefied conditions for the next generation of air breathing hypersonic vehicles and Entry Descent and Landing (EDL) technologies such as ballutes Magin (AERO ) Aerothermodynamics / 34

9 Introduction: high-temperature gas dynamics 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 debris Space debris, a threat for satellite and space systems and for mankind when undestroyed debris impact the Earth STO-VKI Lecture Series Space Debris, In Orbit Demonstration, Debris Mitigation Magin (AERO ) Aerothermodynamics / 34

10 Introduction: high-temperature gas dynamics Motivation: engineering design in hypersonics Relevant quantities for engineers: Heat flux & Shear stress to the vehicle surface Blast capsule flow simulation COOLFluiD platform and Mutation library ) Complex multiphysics problem Chemical nonequilibrium (gas) Dissociation, ionization,... Internal energy excitation Thermal nonequilibrium Translational and internal energy relaxation Gas / surface interaction Radiation Turbulence (laminar-turbulent transition) Rarefied gas e ects Magin (AERO ) Aerothermodynamics / 34

11 Introduction: high-temperature gas dynamics Example of aerospace mission killer Inaccurate heat-flux predictions can be fatal for the crew / success of a robotic mission e.g. Lessons learned from Galileo s entry into the Jovian atmosphere: Stagnation point material recession of heatshield was less than predicted Ablation at frustrum and shoulder was much higher than predicted e.g. New carbon / resin composites require improved models Magin (AERO ) Aerothermodynamics / 34

12 Introduction: high-temperature gas dynamics Gas-surface interaction: complex multiphysics problem convec7ve( heat(flux( surface( recession( radia7on( flux(( heat( transfer( hot( radia7ng( gas( OH( shock(heated(gas((t(~(10,000k)( reac7on( products( pyrolysis( gases( chemical( species(( diffusion( boundary( layer( mechanical( erosion( porous(char( pyrolysis((zone( OH( virgin(material( Radiative and convective heating ) Pyrolysis of phenolic resin C 6 H 5 OH (>200 C) ) Ablation of the char Chemical mechanisms oxidation (CO, CO 2) nitridation (CN) Phase changes melting sublimation (C, C 2, C 3) Mechanical removal spallation, shear stress, melt removal Magin (AERO ) Aerothermodynamics / 34

13 Introduction: high-temperature gas dynamics Three pillars for predictive engineering simulations Computation cannot be truly predictive without the coupling to physico-chemical models and experiments... This coupling is precisely the verification and validation process! Verification: Isthecomputationalmethodimplementedcorrectly? Validation :Arewesolvingtherightequations? Uncertainty Quantification: Errorbarsinscientificpredictions... Magin (AERO ) Aerothermodynamics / 34

14 Introduction: high-temperature gas dynamics Heat-flux measurement in plasma jet obtained in VKI Plasmatron facility Water-cooled calorimeter in a plasma jet stream, VKI Plasmatron, air, 8g/s mass flow, 3500 Pa pressure, MHz frequency, 120 kw generator power, 0.16 cm torch diameter Magin (AERO ) Aerothermodynamics / 34

15 Introduction: high-temperature gas dynamics Simulation of high-enthalpy flow in VKI Plasmatron facility Inductively Coupled Plasma (ICP) wind-tunnel Injected gas Air Mass flow rate = 8 g/s, straightinjection Temperature = 300 K Outlet pressure = Pa Torch characteristics (Plasmatron) f =0.37MHz Power injected into the plasma = 60 kw Configuration assumed to be axisymmetric Dimensions [m] Magin (AERO ) Aerothermodynamics / 34

16 Introduction: high-temperature gas dynamics Physico-chemical models for ICP wind-tunnel Steady Navier-Stokes equations (including time-average Lorentz forces and Joule heating source terms) Helmholtz di usion equation for magnetic field (Maxwell equations) Example of high-temperature property: dynamic viscosity of air as a function 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 species air mixture at 1 atm pressure η [ kg m -1 K -1 ] Chapman-Enskog Gupta-Yos s mixture rule Wilke s mixture rule Temperature [ K ] Magin (AERO ) Aerothermodynamics / 34

17 Introduction: high-temperature gas dynamics Computed electromagnetic field Induced electric field [V/m] Magin (AERO ) Aerothermodynamics / 34

18 Introduction: high-temperature gas dynamics Computed temperature and flow fields Temperature field [K] Streamlines Magin (AERO ) Aerothermodynamics / 34

19 Flows with discontinuities Outline 1 Introduction: high-temperature gas dynamics 2 Flows with discontinuities Examples Rankine-Hugoniot jump relations 3 Normal shocks Magin (AERO ) Aerothermodynamics / 34

20 Flows with discontinuities Examples Longshot Gun Tunnel at VKI Schlieren visualisation of detached shocks in the Longshot facility of the von Karman Institute for Fluid Dynamics Nagamatsu probe Static pressure Reference probe Stagnation point pressure and heat flux FADS model Nose tip radius: 53 mm Instrumented with 5 fast-response Kulite XCQ-093 pressure sensors and 5 coaxial type-e thermocouples Measurement locations: 1 at nosetip and 4 o -stagnation points at 45 equally spaced around the probe Magin (AERO ) Aerothermodynamics / 34

21 Flows with discontinuities Examples Electric Arc Shock Tube (EAST) facility at NASA ARC Electric Arc Shock Tube facility NASA Ames Research Center Snapshot of radiating flow measured by means of a spectrometer Wavelength resolved on x-axis Spatially resolved on y-axis Radiative intensity indicated by color Magin (AERO ) Aerothermodynamics / 34

22 Flows with discontinuities Examples Shock tube operation Wave propagation Bursting of the diaphragm at t =0 Flow at rest and gases at ambient temperature Driver gas (region 4): high pressure p 4 Driven gas (region 1): low pressure p 1 p 4 Wave propagation at t = t > 0 Expansion wave: region 3! 4 Contact discontinuity wave: region 3! 2 Shock wave: region 2! 1 Atmospheric entry shock layer simulated in region 2 behind shock wave Magin (AERO ) Aerothermodynamics / 34

23 Flows with discontinuities Rankine-Hugoniot jump relations Discontinuities in shock-tube flows The interface between the regions 2 and 1 is called normal shockwave In the laboratory reference frame, the flow in region 1 is at rest (u 1 = 0) As the normal shockwave propagates to the right with a velocity,it increases the pressure p 2 of the gas behind it and induces a mass motion with velocity u 2 In the shockwave reference frame (v = u ), the normal shock wave appears 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 the gas in regions 2 and 3 (u 2 = u 3 ) The pressure is also preserved through a contact discontinuity (p 2 = p 3 ) Magin (AERO ) Aerothermodynamics / 34

24 Flows with discontinuities Rankine-Hugoniot jump relations Shock wave at microscopic level The density rises continuously in shockwaves The shock thickness if of the order of the mean-free-path (average distance between two successive collisions of a molecule) The flow is rarefied ) failure of the fluid dynamical description Alsmeyer experiment Navier-Stokes eqs. simulation Computational Fluid Dynamics (CFD) Boltzmann eq. simulation Direct Simulation Monte-Carlo (DSMC) Normalized density distribution across shock wave in Argon at M=9 Magin (AERO ) Aerothermodynamics / 34

25 Flows with discontinuities Rankine-Hugoniot jump relations 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 t x ( u) = t ( u)+@ x ( u u + pi)+"@ x = t ( E)+@ x ( uh)+"@ x q + "@ x ( u) = 0 The Navier-Stokes equations are expressed by means of a closure. For instance, the viscous stress tensor is given by = 2 S, withthe velocity gradient tensor S = 1 u +(@ x u) T 1 x u I The Navier-Stokes equations are written in a compact form t U 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-Stokes equations when the dissipative terms vanishes ("! 0) Magin (AERO ) Aerothermodynamics / 34

26 Flows with discontinuities Rankine-Hugoniot jump relations Rankine-Hugoniot jump relations N-S and Euler solutions match in inviscid regions upstream (1) and downstream (2) of a discontinuity where gradients vanishes (F d = 0) Let us consider a point P on a discontinuity surface moving at speed,withthenormaln to the surface Let us assume the existence of a travelling wave solution U = U(y) to the Navier-Stokes equations in the vicinity of P, with Coordinate: y = x n nt Normal component: n = n Travelling wave eq. n du dy + d dy (F n)+" d dy (F d n) =0 After integration R y 2 y 1 ( ) dy from region 1 to region 2, the Rankine - Hugoniot jump relations for inviscid flows are derived Normalized density distribution across shock wave n(u 2 U 1 )=(F n) 2 (F n) 1 Magin (AERO ) Aerothermodynamics / 34

27 Normal shocks Outline 1 Introduction: high-temperature gas dynamics 2 Flows with discontinuities 3 Normal shocks Magin (AERO ) Aerothermodynamics / 34

28 Normal shocks Normal shock relations (inviscid flows) Rankine-Hugoniot relations for steady normal shocks =0, u = u e x, n = e x ) 2 u 2 = 1 u 1 2 u p 2 = 1 u p 1 2 u 2 H 2 = 1 u 1 H 1 These equations are valid for calorically perfect gases and mixtures of gases in thermo-chemical equilibrium For calorically perfect gases, further simplifications apply Calorically perfect gas law: p = RT with R = R M and R = N Ak B =8.31 [J / (K mole)] Specific energy: e = c v T, h = c pt,withc p Specific heat ratio: = cp c v Molecular gases: c v = 5 2 R, = 7 5 =1.4 Atomic gases: c v = 3 2 R, = 5 3 =1.66 Entropy: s = c v ln p = c p ln T R ln p + c p ln R Speed of sound: a 2 =(@p/@ ) s = p/ = RT c v = R Magin (AERO ) Aerothermodynamics / 34

29 Normal shocks Normal shock relations for calorically perfect gases The total enthalpy conservation can be expressed as H 2 = H 1 ) p 2 + u = p u2 1 2 The non-linear algebraic system of 3 eqs. in 3 unknowns 2, u 2,and p 2 has a a closed solution expressed as a function of the dimensionless parameters: Mach number M 1 = u 1 /a 1 and,witha 1 = p RT 1 For M 1 > 1 (derivation given further in this section) 2 1 = u 1 u 2 = ( +1)M2 1 2+( 1)M 2 1 > 1 p 2 p 1 = (M2 1 1) > 1 T 2 =( p 2 )/( 2 )= T 1 p 1 h (M2 1 1)i apple 2+( 1)M1 2 ( +1)M1 2 M2 2 = M 2 1 M < 1 > 1 Magin (AERO ) Aerothermodynamics / 34

30 Normal shocks Entropy across normal shocks The entropy variation through a normal shock is s 2 s 1 = c p ln T 2 T 1 R ln p 2 p 1 = c v ln p 2/p 1 ( 2 / 1 ) This function is positive for M 1 1 ) the 2nd law of thermodynamics imposes that, for a calorically perfect gas, a shock wave may only happen if M 1 1 What is the origine of the entropy increase through a shock wave? Answer: the changes across the shockwave occur through a short distance of the order of the mean-free-path. Through the shock structure, the gradients are very large. In turns, heat fluxes and viscous stresses are dissipative phenomena that generate entropy Weak shock: M1 2 =1+" with 0 <" 1 p 2 =1+ 2 p 1 +1 " and ( 2 ) =(1+") 1 p 2 ( 2 ) 1+ 2 p ( 1) ( +1) 2 " 3 + O(" 4 ) " ) s 2 s 1 2 ( 1) c v 3 ( +1) 2 " 3 + O(" 4 ): isentropic approximation valid for weak shocks Magin (AERO ) Aerothermodynamics / 34

31 Normal shocks Total and critical quantities Consider a fluid element in an arbitrary flow travelling at velocity u with static pressure p and temperature T Total pressure p 0 and total temperature T 0 They are defined as quantities obtained by isentropically decelerating the H = cp T flow to rest 0 = c p T + u 2 /2 s = c p ln T 0 R ln p 0 = c p ln T 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 Magin (AERO ) Aerothermodynamics / 34

32 Normal shocks Total and critical quantities In steady inviscid flows, we will show in lecture 2 that the total enthalpy is constant along a trajectory (pathline) Therefore, in steady inviscid flows, one has, H = c p T 0 = c p T + RT 2 = 1 1 a 2 + a 2 2 = a 2 ) The total temperature, critical temperature, and critical speed of sound are constant properties of a streamline Through a normal shock ) The total temperature is conserved T 0,2 = T 0,1 ) The critical temperature and critical speed of sound are conserved: T2 = T 1 and a 2 = a 1 p ) The total pressure is a decreasing function: 0,2 p 0,1 =exp( s 2 s 1 R ) Magin (AERO ) Aerothermodynamics / 34

33 Normal shocks Total quantities (local quantities with definition still valid for viscous flows) The total temperature is obtained from T 0 T = 1+ 1 u 2 2 c p T =1+ R u 2 2c p a 2 = M2 p = 0 The isentropic relations p 0 for the total pressure and total density p 0 p 0 = = = T0 T M M yields the expressions For M 1, using the Taylor expansion (1 + x) =1+ x + O(x 2 ), the Bernoulli relation for incompressible flows is retrieved p 0 p =1+ 2 M2 + O(M 4 ) ) p 0 p u2 Magin (AERO ) Aerothermodynamics / 34

34 10146 Normal shocks Pitot probe measurement in the Plasmatron facility Henri Pitot ( ), French hydraulic engineer, is the inventor of the Pitot tube Pitot probes are widely used in fluid dynamics (airspeed of aircraft and air and gas velocities in industrial applications) The stagnation pressure is measured at the probe nose p Pitot For low Mach number flows, p Pitot = p 0 p u2 In low Reynolds number flows (e.g. subsonic plasma flows): p Pitot departs from p 0 due to viscous e ects [Barker, 1922] Pressure field around a Pitot probe in a plasma jet at Mach = 0.1( p =1Pa) p=10,145 Pa, =6.6 kg/m 3, u=137.2 m/s ) p 0 =10,207 Pa The Reynolds number based on the probe radius (3 mm) is Re = 19 and p Pitot =10,225 Pa Magin (AERO ) Aerothermodynamics / 34

35 Normal shocks Prandtl s relation for normal shocks Prandtl s relation, a 2 = u 1 u 2, is derived as follows p Total enthalpy conservation: 2 + u2 2 p = 1 1 Alternative form: p = a 2 u2 2 p 1 1 = a 2 u u2 1 2 = a 2 Introducing the previous relations in the ratio of momentum to mass shock relations 2 u p 2 2 u 2 = 1u p 1 1 u 1 ) u 2 + p 2 2 u 2 = u 1 + p 1 1 u 1 yields Prandtl s relation after some algebra The critical Mach number M = u/a, can be obtained from 1 1 a2 + u2 2 = a 2 ) M = M 2 Magin (AERO ) Aerothermodynamics / 34

36 Normal shocks Application of Prandtl relation and critical Mach number The critical Mach number behaves as the local Mach number but remains finite at high speeds M 2 = ( +1)M2 2+( 1)M 2 ) M < 1 if M < 1 M =1 if M =1 M > 1 if M > 1 M! +1 if M!1 1 Alternative form of Prandtl relation for normal shocks: M 1 M 2 =1 For a normal shock M 1 > 1 ) M 1 > 1 ) M 2 < 1 ) M 2 < 1 The normal shock relations are then easily derived as follows 2 = u 1 = u2 1 = u2 1 1 u 2 u 1 u 2 a 2 = M1 2 = ( +1)M2 1 2+( 1)M1 2 M2 2 = 1 M1 2 ) ( +1)M2 2 2+( 1)M2 2 ) M2 2 = M 2 1 M ) p 2 p 1 p 1 = M 2 1 (1 u 2 u 1 )= M 2 1 = 2+( 1)M2 1 ( +1)M 2 1 p 2 p 1 = 1 u1 2 2 u2 2 = 1u 1 (u 1 u 2 )= 1 u1 2(1 u 2 ( +1)M1 1 2 T 2 T 1 =( p 2 p 1 )/( 2 1 )= 2+( 1)M 2 1 h (M2 1 1)i apple 2+( 1)M 2 1 ( +1)M 2 1 u 1 ) = 2 +1 (M2 1 1) Magin (AERO ) Aerothermodynamics / 34

37 Normal shocks Exercise Consider a shock-tube facility closed at both ends with a diaphragm separating a region of high-pressure gas on the left (region 4) from a region of low-pressure gas on the right (region 1). When the diaphragm is broken at t =0s(forinstancebymechanicalmeans),ashockwave propagates 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 a mass motion with velocity u 2. 1 Derive the Rankine-Hugoniot jump relations between the regions 1 and 2 with the gas velocities u 1 and u 2 in the laboratory frame. 2 Using the change of variables v = u for the velocity in the shock- wave frame, show that the normal shock relations are satisfied in this reference frame. 2 v 2 = 1 v 1 2 v p 2 = 1 v p 1 h v 2 2 = h v 2 1 Magin (AERO ) Aerothermodynamics / 34