Lecture #1: Stagnation Point Heating
1Lecture #1: Stagnation Point Heating
Background2 The kinetic energy of an entry vehicle is dissipated bytransformation into thermal energy (heat) as the entry systemdecelerates The magnitude of this thermal energy is so large that if all of thisenergy were transferred to the entry system it would be severelydamaged and likely vaporize– Harvey Allen - the blunt body concept Only a small fraction of this thermal energy is transferred to theentry system– The thermal transfer fraction is dependant on vehicle shape, size,aerodynamic regime and velocity– Near peak heating, 1% to 5% of the total thermal energy is transferred to theentry system– Example: at the peak heating point the freestream energy transfer forPathfinder was qÝ 12 V 3 4,000 W/cm2 but only about 110 W/cm2 (2.7%) wasactually transferred to the surface
Example3E V2 gohEnergy density:m2EntryMER V(km/s)E/m(MJ/kg)5.616Note that:Apollo11.466Water boils @ 2.3 MJ/kgCarbon vaporizes @ 60.5 MJ/kgMarsReturn14.098Galileo47.41130In each case goh is about 1% of total
Side Note: What Can We Test?Missionsof InterestLive here4
Blunt Body Rationale Why is a blunt body used forplanetary entry?– Slender body: low drag, highlymaneuverable– Blunt body: high drag, not verymaneuverable Blunt bodies generate strongshock waves– Efficient energy dissipation. Shockwaves convert kinetic energy tointernal energy. Result is: heatingof the gas, dissociation, ionization– Most of this energy is convected into the vehiclewake rather than transported to the surface– Intuitively, blunter is better (more bluntness equalsstronger shock). Hold that thought; we will come backto it 5
Blunt Body Rationale (2)Apollo Wake Flow Normal shock heats the gas tomany thousands of degrees Much of this heat is conductedinto the vehicle wake andpropogated downstream Can be tracked as a“velocity deficit” and persistslong downstream of thevehicle6
Definitions Heat Rate (q)– Instantaneous heat flux at a point on the vehicle (W/cm2) Heat Load (Q)– Integration of heat rate with time over a trajectory (J/cm2) Convective Heating– Heat flux to the vehicle from conduction ( gradT) Catalytic Heating– Heat flux to the vehicle due to surface facilitated chemical reactions– Commonly lumped with convective heating by convention Radiative Heating– Heat flux to the vehicle from radiation produced by excited atoms andmolecules in the shock layer7
What is Aerothermodynamics? Accurate and conservative prediction of the heatingenvironment encountered by an Earth or planetary entryvehicle Aerothermal modeling is coupled and entwined withThermal Protection System (TPS) design The TPS is designed to withstand the predicted environment with riskappropriate margin For ablative systems, the flowfield and TPS interact with each other innon-reversible manner; the physics themselves are coupled At its core, aerothermodynamics becomes the study ofan energy balance at the surface of the material Heat flux (with pressure & shear) used to select TPS material Heat load determines TPS thickness8
Principles of Aerothermal ModelsPlanetary AtmospheresMars&Venus: CO2/N2Titan: N2/CH4Giants: H2/HeEarth: N2/O2Thermal ProtectionSystem (TPS)Surface EnergyBalanceHot Shock Layer(up to 20000 K)Thermochemicalnonequilibrium,Ionization, RadiationqreradqradBoundary Layer(2–6000 K)Transport properties,Ablation productmixing, RadiationblockageV“Cool” Surface(2–3000 K)Surface kinetics,AblationqcondqcqmdotDesign Problem: Minimize conductioninto vehicle to minimize TPS mass/riskqcond qc qrad – qrerad – qmdotIncident AeroheatingMaterial Response9
Current State of the Art : CFD The current SOA involves the steady solution of the reactingNavier-Stokes equations via CFD or DSMC methods Full 3D simulations possible in hours to days Longer time required for the simulation of OML details (steps,gaps, seals, windows, etc.10
Pushing the Current State of the Art DES, DNS, LES Unsteady RANS (URANS) simulations of Supersonic RetroPropulsion flowfields; going on right now 11
NASA CFD Development StrategyLAURADPLRToday Structured, Finite Volume, mostly steady-state Also coupled to Radiation and Ablation codesUS3D-NASAFUN3D (LAURA-path)In 2-3 Years Unstructured, Finite Volume, low-dissipation schemes,DES/LES, DNS capability, well-balanced schemesDG (Discontinuous Galerkin)CESE (Conservation Element SolutionElement) Unstructured, higher order, unsteady, beyond finitevolumeIn 5-10 Years12
Why Engineering Methods?With present computational abilities, why use engineering methods? CFD is a powerful tool, but high-fidelity simulations remain time (andresource) consuming Some applications of simple relationships for calculating non-ablatingconvective and radiative heating– Negligible computation time– Included in most atmospheric trajectory codes-stag. pt. heating– Initial estimates of heating rates and loads for use duringconceptual design stage But most important: In this day of commodity supercomputers it is all too easy to run simulationswithout truly understanding the physics involved or the trends that areexpected. The fact that it “converged” doesn’t make it right. Engineeringmethods are based on sound approximations to theory and provide a valuablesanity check on CFD results13
Theory of Stag. Pt. Convective Heat Transfer Pioneering engineering theories were developed in the1950’s (missile technology)Lees, L. “Laminar Heat Transfer Over Blunt-Nosed Bodies at HypersonicSpeeds,” Jet Propulsion, pp. 256-269, Apr. 1956Fay, J.A. and Riddell, F.R., “Theory of Stagnation Point Heat Transfer inDissociated Air,” Journal of Aeronautical Sciences, Feb. 1958 Extensions to higher velocities were required to accountfor chemistry and ionization Many extensions and simplifications followed for specificapplications, non-Earth atmospheres14
Theory of Stag. Pt. Convective Heat Transfer (2)15 Early correlations for convective heating have the form:12 qÝs V Rn 3 Why? At first cut, one might expect heat flux to the3 surface to beproportional to freestream energy flux ( 12 V ) From previous discussion one would expect convectiveheat flux to decrease as bluntness (Rn) increases, but with what functionality? (insert brief derivation here)
Fay-Riddell Method16 Convective: derived from boundarylayer and stagnation point theoriesFay & Riddell (1958):w walle edgeBoundary layer eqns, similarity transformationVelocity gradient from mod. Newtonian theory (1/Rn)due 1 2 pe p dx R eSignificant advance, but still requires many quantities that arenot readily available to designerAllows for chemistry effects, non-unity Pr, Le (Prandtl, Lewisnumbers)
Simplified Methods17Chapman Equation (Earth):1 2 3 hw 4 q s 1.63 10 V 1 Rn h Th C pTdt 12 V 20 Sutton Graves:“hot wall correction” canfrequently be neglected inhypersonic flow (hw h )1 2 3q s k V Rn k 1.7415e-4 (Earth)k 1.9027e-4 (Mars)(SI units) Calculatedfor specific atmosphere (Earth or Mars), accounting for thermodynamics. Above assume a fully catalytic surface; equivalentexpressions for non catalytic wall are available.
Hot Wall Correction Term18 Negligible above about 100 W/cm2 assuming radiative equilibrium Actual effect is smaller than this for ablative TPS0.7Enthalpy Ratio0.6 0.5HWC 1 0.4hwh 0.3 0.20.1Radiative EquilibriumApproximateAblative Correction0110100q (W/cm2) – log scale1000
Generalized Chapman Method19 hw Cmnqc,0 ( ) (V ) 1 ;Rn h Earth :Mars:m 0.5,m 0.5,n 3n 3.04C is derived for problem of interestPowerful design tool - can be used to approximate heating froma small number of CFD “anchor points” even away from thestagnation point by letting C, m, and n be curve fit coefficients
Comparison of Data to Correlations20
Nuance – Effective Nose Radius21 Prior correlations are straightforward and require onlyreadily available quantities However, there is a nuance. All are dependent on theeffective nose radius of the vehicle under investigation For a hemisphere, Reff Rn, but corrections are required forother vehicle shapes. For example, Apollo was a truncated sphere, with aneffective radius almost twice the base radius of thecapsule. MER/MSL use sphere-cones, where the conicalflank increases the effective radius of the nose For bodies with a rounded corner, Zoby and Sullivan havecomputed tables of effective radius as a function of Rb/Rnand Rc/Rb:Zoby, E. and Sullivan E, “Effects of Corner Radius on Stagnation Point Velocity Gradients onBlunt Axisymmetric Bodies,” Journal of Spacecraft and Rockets, Vol. 3, No. 10, 1966.
Nuance – Effective Nose Radius (2)When does it matter?Can the flow “tell” that the nose is finite?45 Sphere-ConeSupersonic Oblique ShockReff Rn60 Sphere-ConeSubsonic ShockReff Rn22
Theory of Stag. Pt. Radiative Heat Transfer Theory is less intuitive, more involved Atoms or molecules are excited bycollisions. Excited species can emit aphoton that carries energy with it2h Upper Level (U)h h Photons are emitted isotropically, andtravel effectively instantaneously Radiative heating is the integration ofthose photons that hit the surface timesthe energy they carry; intuitively shouldbe proportional to the size of theradiating volume Partition functions for excited statesimply a near exponential dependence ontemperature Radiation is coupled to the fluidmechanics for two reasons: Emitted photons carry energy out of controlvolume (adiabatic cooling) Photons can be absorbed in the boundarylayer and heat the gas23StimulatedEmissionSpontaneousEmissionh DE EU-ELAbsorptionLower Level (L)Ni Ngi e h EikT g e EjkTgi e QjjLTE-PlasmaEikT
Relative Importance of Radiation vs ConvectionRadiative (neglecting coupling effects)Radiative (including coupling effects)ConvectiveNose Radius 4.5mAltitude 60 kmAdapted from Anderson, Hypersonic and High Temperature Gas Dynamics, Fig. 18.1024
Theory of Stag. Pt. Radiative Heat Transfer25Martin:1.0 1.6 8.5Ýq r rn V EarthDirect dependence on Rn agrees withintuitive argument about radiating volumeTauber-Sutton: a mÝqr Ci rn f i V Earth : a 1,Mars: a 0.526,m 1.2m 1.2 based on tabulated data,equilibrium shock theoryfi are tabulated, near exponentialat moderate velocity Theory is less intuitive, more involved. Typically relies on tablelookups and has limited range of validityFortunately, radiation is not a major issue for many problems ofinterest: Mars (moderate velocity), LEO return, Titan
Importance of Radiative Cooling The shock layer is cooled by the emission of photons. Clearly this effect willbecome more important as a larger fraction of the total shock layer energy isconverted to photons Tabular or engineering expressions for stagnation point radiation typicallyinclude the radiative cooling effect However it is very important to recognize this phenomenon when computingradiation from CFD data (inherently uncoupled operation) Goulard proposed a non-dimensional parameter that is essentially the ratio oftotal energy flux to that lost to radiation:2q R, unc 1 V 32 The net radiative heating can then be computed from (Tauber-Wakefield): q R, uncq R, coup 1 0.7 Where is an atmosphere-specific constant 2 for Titan 3.45 for Earth 3 for Mars/Venus 26
Example - Galileo ProbeRadiative (no coupling)Radiative (including coupling)Convective (no blowing)Convective (blowing)Adapted from Anderson, Hypersonic and High Temperature Gas Dynamics, Fig. 18.1627
Wall Temperature Estimation28 How hot does the TPS surface get? A body radiates heat at a rate proportional to the 4th power of its temperature Stefan-Boltzmann Law: q rerad T4 where is the emissivity of the TPS ( 1 for a blackbody), is the Stefan-Boltzmannconstant ( 5.67e-8 W/m2/K4), and T is the wall temperature (assumes the ambienttemperature is much lower) The wall heat flux balance is in general given by the sum of heat into the materialminus reradiation, conduction, and material response. A primary function of TPSis to minimize conduction (good insulator), and thus, neglecting materialresponse we can assume that:q rerad q conv q Rwhich can readily be solved for Tw. Examples: (Tw 1600 K) Orbiter peak heating MER peak heating (Tw 1725 K) Orion peak heating (Tw 3360 K)– by this point we are overpredicting by 20% due to material response effects
Example: Shuttle Orbiter For the Shuttle-Like entry previously studied, what is the stagnation pointheating rate and the wall temperature at 60 km altitude? Assume a 1m noseradius and a TPS emissivity of 0.8– 3.1459e-4 kg/m3– V 3.535 km/s– qw 1.7415e-4*(3.1459e-4/1)0.5*(3535)3 13.6 W/cm2 (Sutton-Graves)– qR 0 (Tauber-Sutton)– Tw [(13.6*1e4)/(0.8*5.67e-8)]0.25 1316 K29
30Further Reading
Engineering Methods Tauber, M., “A Review of High-Speed, Convective Heat Transfer Computation Methods,” NASATP-2914, Jul. 1989.Tauber, M., Bowles, J., and Yang, L., “Use of Atmospheric Braking During Mars Missions,” Journalof Spacecraft and Rockets, Sept.-Oct. 1990, pp. 514-521.Tauber, M., Yang, L. and Paterson, J., “Flat Surface Heat-Transfer Correlations for Martian Entry,”Journal of Spacecraft and Rockets, March-April 1993, pp.164-169.Compton, D. L. and Cooper, D. M., “Free-Flight Measurements of Stagnation Point Convective HeatTransfer at Velocities to 41,000 ft/sec,” NASA TN D-2871, Jun. 1965.Marvin, J. G. and Deiwert, G. S., ”Convective Heat Transfer in Planetary Atmospheres,” NASA TRR-224, Jul. 1965.Kaattari, G. E., “Effects of Mass Addition on Blunt Body Boundary Layer Transition and HeatTransfer”, NASA TP-1139, 1978.Tauber, M. E. and Sutton, K., “Stagnation Point Radiative Heating Relations for Earth and MarsEntries”, Journal of Spacecraft and Rockets, Jan.-Feb. 1991, pp. 40-42.Page, W. A. and Woodward, H. T., “Radiative and Convective Heating during Venus Entry”, AIAAJournal, Oct. 1972, pp.1379-1381.Tauber, M. E., “Some Simple Scaling Relations for Heating of Ballistic Entry Bodies”, Journal ofSpacecraft and Rockets, July 1970, pp. 885-886.Chapman, G.T., “Theoretical Laminar Convective Heat Transfer & Boundary Layer Characteristicson Cones at Speeds to 24 km/s,” NASA TN D-2463, 1964Sutton, K. and Graves, R.A., “A General Stagnation Point Convective Heating Equation for ArbitraryGas Mixtures,” NASA TR- R-376, 1971Fay, J.A, and Riddell, F.R, “Theory of Stagnation Point Heat Transfer in Dissociated Air,” J.Aeronautical Sciences, 25, 1958, pp. 73-85,121.31
32Lecture #2: Distributed Heating andTrajectory Effects
33Distributed Heating - Sphere It can be shown that the heat transfer rate along the bodyvaries according toqq stag cos for angles as large as 45 (in theory) and 70 (in practice) This expression permits us to integrate the total heat flux into a sphericalnose as qdA q stag cos dAdA 2 yRn d 2 Rn2 sin d /2 qdA 2 Rn2q stag sin cos d Rn2q stag 0 For a laminar boundary layer, the heat input to a hemisphere is equalto the product of stag. point heating times the projected area
34Distributed Heating - Sphere (2)
35Local Similarity - Flat Faced Cylinder Local similarity methods (see e.g. Anderson) can beextended to other geometries Take for example a flat-faced cylinder with a roundedcorner For this case, local similarity theory (and moresophisticated methods) show that the stagnation point isnot the highest heating location; rather heating is higheron the corner– Physically, the large favorable pressure gradient causes the boundarylayer to thin. This increases the magnitude of h, which increases heattransfer per previous arguments. The magnitude of increase is inverselyrelated to the radius of curvature.
36Distributed Heating - FF Cylinder (2)
37Distributed Heating - Approximate Methods Many other approximate methods have been developed forthe calculation of heating on other geometries, e.g. wings,attachment lines. Detailed assessment is beyond the scope of theselectures, but the interested student can read further in:Tauber, M.E., “A Review of High Speed Convective Heat TransferComputation Methods,” NASA TP 2914, 1989which is included as a handout for this course.
Real World Examples Laminar FlowMSL Shape in T5Predicted Stardust Heating38
39Trajectory Effects The discussion up to now has focused on the calculationof an instantaneous heat flux (primarily at the stagnationpoint). However, the heating on the vehicle is obviously coupledto the trajectory flown, and thus it is important to developexpressions that quantify the relationship between heatingand trajectory. You have already learned two basic trajectory equations(Allen-Eggers and Equilibrium Glide); lets start with AllenEggers For simplicity, lets use the simplest of convective heatingrelationships:1q s 2 V 3
40Intuition (1) Two identical ballistic vehicles enter the atmosphere. Oneis on a steep entry trajectory and one is on a shallow entrytrajectory. Which has the higher peak heat flux? Load?shallow steep
41Intuition (2) Two ballistic vehicles enter the atmosphere on an identicalflight path angle. One has a higher ballistic coefficient.Which has the higher peak heat flux? Load?high low
42Allan-Eggers Trajectory Equation V Vatm exp Ce h / H Vatm exp C oVatm Velocity at atmospheric interface m/CDAExponential atmosphere assumed Ballistic entryC 02 sin Substitute above for V into approximate heating equation:12 3q s Vatmexp 3C o Differentiate w.r.t density:12 V 123atm exp 3C o 3C 3 Vatm exp 3C o o 12 dq s d
43Allan-Eggers Trajectory Equation (2) Looking for a maximum of qs, which should occur when dqs/q 0: 1 6C 0 o So the density of maximum convective heating is: *q max o6C sin 3H For a given atmospheric scale height, the density (altitude) of peakheating increases with ballistic coefficient and flight path angle
44Allan-Eggers Trajectory Equation (3) So, in the exponential atmospheric model sin 3H oe h*/ H sin h* ln H 3H o The altitude and velocity of peak heating are given by: *q maxV *q maxh sin H ln 3H o C o Vatm exp Vatme 1 / 6 0.846Vatm o 6C
45Allan-Eggers Trajectory Equation (4) As in the case of the previously derived expression for the velocity atpeak deceleration, the velocity at peak heating is a function only of theentry velocity. Recall that Vgmax 0.606Vatm. Therefore, peak heating occurs earlier inthe entry than peak deceleration. In fact, it can be shown thathq* max 1.1h*g max We are now in the position of being able to calculate the peakstagnation point convective heat rate for a ballistic entry vehicle Substitute the evaluated expressions for Vand into theqmaxqmaxSutton-Graves Equation:12 1 sin 23q s, max k .6055Vatm Rn 3H 1 In addition to the nose radius dependence shown earlier, we now seethat peak heating rate increases with increasing ballistic coefficientand flight path angle
46Heat Load Stagnation point heat load is just the time integration of the heat fluxkQs Rn 12V 3dt How do we convert this to an integral that we now how to evaluate(redefine dt through change of variables)? Lets borrow some logic of Motion:from the Equationssin dh;dsV dsdtdsdhdt VV sin Using the exponential atmosphere model we can write this in terms of d
47Heat Load (2) Exponential atmosphere model oe h / H Differentiate: Substitute into dt:d o h / H e dhHHHd dt V sin Now we can substitute into the heat load integral:2 2C k VatmH o 12Qs q sdt exp d
Mar 28, 2009 · Mars: a 0.526, m 1.2 f i are tabulated, near exponential at moderate velocity Tauber-Sutton: Theory is less intuitive, more involved. Typically relies on table lookups and has limited range of validity Fortunately, radiation is not a major issue for many problems of interest:
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